Mathematical modelling as a research tool in the ... · Tag der mündlichen Prüfung: 01.06.2017 Vorsitzender des Promotionsorgans: Prof. Dr.-Ing. Reinhard Lerch Gutachter: Prof

Mathematical modelling as a research tool in the cyanobacteria

cultivation

Mathematische Modellierung im Mikroalgenanbau

Der Technischen Fakultät

der Friedrich-Alexander-Universität

Erlangen-Nürnberg

zur

Erlangung des Doktorgrades Dr.-Ing

vorgelegt von

Hugo Fabian Lobaton Garcia

aus Palmira, Kolumbien

II

Als Dissertation genehmigt

von der Technischen Fakultät

der Friedrich-Alexander-Universität Erlangen-Nürnberg

Tag der mündlichen Prüfung: 01.06.2017

Vorsitzender des Promotionsorgans: Prof. Dr.-Ing. Reinhard Lerch

Gutachter: Prof. Dr. rer. nat. Rainer Buchholz

Prof. Dr.-Ing. Clemens Posten

III

Acknowledgments

The present work was part of my research activities at the institute of bioprocess engineer in

the University of Erlangen-Nuremberg, headed by Prof. Rainer Buchholz. Special thanks are

dedicated to Prof. Rainer Buchholz for the opportunity to enter in the microalgae world as

well as the helpful discussions about of the practice in microalgae cultivation. Dr.-Ing. Bar-

bara Klein and Dr.-Ing Stephanie Stute are gratefully acknowledged for the initial support

in the laboratory duties. I also thank to Dr.-Ing Holger Hübner and Philipp Schwerma, M.Sc.

for her critical comments on the dissertation.

Thanks to all members of the group of bioprocesses for their valuable contributions through-

out the course of graduate study. Special thanks are also dedicated to the person who helps

me to improve the language in this dissertation and the biggest recognitions go but surely to

my parents who always supported me in this process.

The laws of Nature are but the mathematical thoughts of God. (Euclid)

IV

Abstract

This study will present the development of mathematical model that will be used in a multi-

product strategy for the A. platensis cultivation. Exopolysaccharide with promissory biolog-

ical activities and phycocyanin with interesting properties for the cosmetic and food indus-

tries are the target products. The possible simultaneous production of both high value prod-

ucts with reasonable productivities will be examined. In order to achieve this goal, a mathe-

matical model was developed in three steps. The first one was to model the growth curve for

A. platensis in different culture condition and to test the model fitting performance. Sec-

ondly, the model was used to interpret the results concerning the product formation (phy-

cocyanin and exopolysaccharides). Finally, the model was used to calculate the time and

nitrate concentration additions in order to enhance the co-production of exopolysaccharides

and phycocyanin. The model was tested compared to several culture conditions and it was

able to predict accurately the growth curve of A. platensis after the variations in the flow rate

from 1 to 5 vvm, initial carbon dioxide (0.035 %-3 %) and an incident light intensity on PBR

surface (60-600 µmol m-2 s-1).

Concnerning to product formation, the experimental results show that phycocyanin mass

fraction is degraded as results of the complete nitrate depletion and nitrate additions during

the cultivation help to keep constant this molecule until new macro-element limitation ap-

pear. According to the model, bicarbonate is this limitation. Therefore, a kinetic law for

phycocyanin formation that include this phenome was proposed and linked to the core

model. Regarding the exopolysaccharides formation, this work shows that not only nitrate

depletion is necessary to trigger its formation, as the experiments with nitrate additions

shows better exopolysaccharides production. The exopolysaccharides formation is en-

hanced perhaps as a result of nitrate and other macro-element limitation i.e. phosphate. Fi-

nally, the current work has demonstrated that by controlling nutrient additions such as ni-

trate, reasonable productivities in both products phycocyanin (38 mg l-1 d-1) and exopolysac-

charides (32 mg l-1 d-1) could be obtained.

V

Zusammenfassung

Diese Arbeit stellt die Entwicklung eines mathematischen Modells vor, welches in einer

Multiprozessstrategie für die Kultivierung von A. platensis verwendet wird. Exopolysaccha-

ride, die vielversprechenden biologischen Aktivitäten aufweisen, sowie Phycocyanin, wel-

ches interessanten Eigenschaften für die Kosmetik- und Lebensmittelindustrie besitzt, stehen

dabei als Produkte aus dem Cyanobakterium im Fokus. Dabei wird die gleichzeitige Her-

stellung von den hochwertigen Produkten unter hohen Produktivitätsansprüchen untersucht.

Um dieses Ziel zu erreichen, wurde ein mathematisches Modell in drei Schritten entwickelt.

Zunächst wurde das Wachstums für A. platensis in unterschiedlichen Kulturbedingungen

modelliert und die Modellbefestigungsleistung zu testen. Als nächstes wurde das Modell

verwendet, um die Ergebnisse der Produktkinetiken für Phycocyanin und Exopolysaccharide

zu interpretieren. Schließlich wurde das Modell zur Berechnung der zeitabhängigen Nitrat-

zugaben verwendet, um die Koproduktion von Exopolysacchariden und Phycocyanin zu ver-

bessern.

Das Modell wurde anhand mehrerer Kulturbedingungen getestet und es war in der Lage, die

Wachstumskurve von A. platensis bei Variationen der Durchflussrate von 1 bis 5 vvm, bei

einer CO2-Beimischung von 0,035 % bis 3 % und Lichtintensitäten zwischen 60 und 600

μmol m-2 s-1 genau vorherzusagen.

Die experimentellen Ergebnisse zeigen, dass einerseits die intrazelluläre Phycocyankonzent-

ration sinkt, wenn Nitrat limitierend vorliegt und andererseits dass Nitratzugaben während

der Kultivierung dazu beitragen, die Ausbeute von Phycocyanin konstant zu halten, bis wei-

tere Limitationen auftreten. Gemß dem Modell-Bicarbonat ist diese Beschränkung Bicarbo-

nat. Daher wurde ein kinetisches Gesetz für die Phycocyaninbildung vorgeschlagen und mit

dem Kernmodell verknüpft. Hinsichtlich der Bildung von Exopolysacchariden zeigt diese

Arbeit, dass nicht nur eine Nitratlimitierung notwendig ist, um ihre Bildung auszulösen, da

die Experimente mit Nitratzusätzen eine bessere Produktion von Exopolysacchariden zeigen

Schließlich hat die vorliegende Arbeit gezeigt, dass durch Kontrolle von Nährstoffzusätzen

wie Nitrat Produktivitäten bei beiden Produkten Phycocyanin (38 mg l-1 d-1) und Exopoly-

saccharide (32 mg l-1 d-1) erhalten werden konnten.

Table of contents

Acknowledgments ...................................................................................................... III

Abstract ....................................................................................................................... IV

Zusammenfassung........................................................................................................ V

List of figures ........................................................................................................... VIII

List of tables............................................................................................................... XII

List of abbreviations ............................................................................................... XIII

List of symbols .......................................................................................................... XIV

1 Introduction .................................................................................................. - 7 -

2 State of art ................................................................................................... - 10 -

2.1 The cyanobacteria Arthrospira platensis and its relevance .......................... - 10 -

2.1.1 Phycocyanin: a high value product ............................................................... - 11 -

2.1.2 Phycocyanobilin biosynthetic metabolic pathway ....................................... - 12 -

2.1.3 Exopolysaccharide properties and biosynthesis ........................................... - 16 -

2.2 Mathematical modelling in biotechnology ................................................... - 19 -

2.3 Biomass kinetics models in cyanobacteria ................................................... - 20 -

2.3.1 Genome-scale metabolic models in cyanobacteria ....................................... - 20 -

2.3.2 Macroscale models – Monod type kinetics .................................................. - 21 -

2.3.3 Numerical solutions of the differential equations ......................................... - 22 -

2.3.4 Formation kinetics in biomass and nutrient consumptions .......................... - 23 -

2.3.5 Formation kinetics in product ....................................................................... - 24 -

2.3.6 State of art – Kinetics models in Arthrospira platensis................................ - 25 -

2.4 Modelling photobioreactors features ............................................................ - 26 -

2.4.1 Modelling light supply in photobioreactors.................................................. - 26 -

2.4.2 Mass transfer in photobioreactors ................................................................. - 28 -

2.4.3 Modelling turbulence and light/dark cycles ................................................. - 30 -

3 Project relevance ......................................................................................... - 33 -

4 Proposed mathematical model .................................................................. - 34 -

4.1 Photobioreactor characterization .................................................................. - 35 -

4.1.1 Energy dissipation rate estimation by using CFD ........................................ - 35 -

4.1.2 Light absorption ............................................................................................ - 36 -

4.1.3 Mass transfer coefficient estimation ............................................................. - 36 -

4.2 Macroscale mathematical model .................................................................. - 36 -

5 Materials and methods ............................................................................... - 40 -

VII

5.1 Microalga and media composition ............................................................... - 40 -

5.2 Preculture ...................................................................................................... - 40 -

5.3 Photobioreactor setup ................................................................................... - 40 -

5.4 Analytic determinations ................................................................................ - 42 -

5.4.1 Biomass quantifications and pH measurements ........................................... - 42 -

5.4.2 Nitrate determinations .................................................................................. - 42 -

5.4.3 Phycocyanin determinations ......................................................................... - 42 -

5.4.4 Exopolysaccharides (EPS) quantifications ................................................... - 43 -

5.5 Experimental plan ........................................................................................ - 44 -

6 Results .......................................................................................................... - 45 -

6.1 Photobioreactor computational fluid dynamics characterization: Energy

dissipation rates and liquid velocities ........................................................... - 45 -

6.2 Biomass cultivation results ........................................................................... - 48 -

6.2.1 Gas flow rate experiments and model validation ......................................... - 48 -

6.2.2 Carbon dioxide experiments and model validation ...................................... - 50 -

6.2.3 Light intensity experiments and model validation ....................................... - 53 -

6.3 Product formation results.............................................................................. - 55 -

6.4 Nitrate model validation ............................................................................... - 58 -

6.5 Effects of nitrate on phycocyanin and exopolysaccharides production ....... - 59 -

6.6 Phycocyanin kinetic model ........................................................................... - 62 -

6.7 Literature data simulations ........................................................................... - 63 -

7 Discussion .................................................................................................... - 67 -

7.1 Influence of flow rate on biomass ................................................................ - 67 -

7.2 Influence of carbon dioxide on biomass ....................................................... - 69 -

7.3 Influence of light intensity on biomass and nitrate consumption ................. - 70 -

7.4 Model fitting performance for biomass ........................................................ - 71 -

7.5 Phycocyanin production: biosynthesis, steady state and degradation .......... - 72 -

7.6 Phycocyanin kinetic model ........................................................................... - 75 -

7.7 Exopolysaccharide production ..................................................................... - 76 -

8 Possible experimental errors ..................................................................... - 79 -

9 Model limitations ........................................................................................ - 80 -

10 Final remarks and further study ............................................................... - 81 -

11 Annex ........................................................................................................... - 83 -

12 References .................................................................................................... - 95 -

List of figures

Figure 2-1 A microscope picture from Arthrospira platensis - Cyanobacteria size

around 200 µm with multicellular helical-shaped (Kamata et al. 2014). ..... - 10 -

Figure 2-2 Representation of the light harvesting complex – phycoerythrin (PE),

phycocyanin (PC) and allophycocyanin (APC) together with the Photosystem I

(PSI) and Photosystem II (PSII) (Guan et al. 2007)..................................... - 11 -

Figure 2-3 Molecular structure Phycocyanobilin - Covalently linked to the L-cystein

amino acid to assemble the protein (Brown et al. 1990). ............................. - 13 -

Figure 2-4 Phycocyanobilin and Chlorophyll a pathways beginning from 5-ALA

with relevant Co-factors (Brown et al. 1990)............................................... - 14 -

Figure 2-5 Effect of light intensity on phycocyanin content after 72 hours of

cultivation in the cells of Synechococcus sp. (Takano et al. 1995). ............. - 15 -

Figure 2-6 Insolated Exopolysaccharide from A. platensis (A) and purified EPS –

withe EPS powder isolated by tangential flow filtration followed by freeze

drying (Reichert 2016). ................................................................................ - 17 -

Figure 2-7 A schematic view of key pathways of central metabolism in Arthrospira

platensis - It is supposed that in stress conditions the metabolic pathway is

redirected to glucose-1-phosphate and then to Glycogen or exopolysaccharides

(Cogne et al. 2003). ...................................................................................... - 19 -

Figure 2-8 A schematic representation of the modelling process - one equation may

include a description of how the rate of growth of the biomass depends on the

substrate (Nitrate, light, carbon, etc.) quantities in the photobioreactor, whereas

another equation may include a description of how the substrates are consumed.

- 22 -

Figure 2-9 Particle track at 0.5 m s−1 (top) superposed with a radial light distribution

for two different biomass concentrations of 0.5 and 2 g l−1 (bottom); the light

absorption in the biosuspension was calculated by means of the hyperbolic

model with a light intensity of IO = 150 μE m−2 s−1 at the reactor surface (Perner-

Nochta and Posten 2007). ............................................................................ - 28 -

Figure 2-10 Typical particle trajectories in the draft tube and the split columns - Only

one recirculation is shown for each reactor, while both the front and the top view

of the trajectories are shown respectively in the r–z plane and the cross-sectional

plane. Solid lines inside the figures represent the walls and internals (Luo et al.

2003). - 28 -

Figure 2-11 Gas mass transfer (left) and carbon dioxide in water equilibrium and

carbon uptake at different pH and mechanisms (right) (Markou et al. 2013).- 29

-

Figure 2-12 Schematic of eddies – The higher the energy dissipation rate, the smaller

the eddies size (Sokolichin and Eigenberger 1999). .................................... - 30 -

Figure 4-1 Graphical structure of the mathematical model- The model was divided

in photobioreactor characterization block and cyanobacteria kinetic model

block. - 34 -

Figure 4-2 Bubble column mesh to perform the CFD simulations - 0.05 m diameter

and 0.675 m height. ...................................................................................... - 35 -

Figure 5-1 Photobioreactor screening module (PSM) (Walter et al. 2003). ........ - 41 -

IX

Figure 5-2 Cultivation set up with a gas humidifier in the inlet and cooling system in

the outlet. ...................................................................................................... - 42 -

Figure 5-3 96-well plate with reference and culture supernatant in the different

experiments. ................................................................................................. - 43 -

Figure 6-1 Average energy dissipation rates and average fluid velocities for different

gas flow rates in the photobioreactor. Each data point is an average from all of

the local values in the mesh. ........................................................................ - 45 -

Figure 6-2 Local energy dissipation rates in a front view of the in the photobioreactor

for different gas flow rates (1 vvm and 5 vvm) after 40 s of simulation time. .... -

46 -

Figure 6-3 Local liquid velocities in 1 vvm and 5 vvm in a front view of the in the

photobioreactor for different gas flow rates (1 vvm and 5 vvm) after 40 s of

simulation time. ............................................................................................ - 47 -

Figure 6-4 Simulated and experimental growth of A. platensis in different gas flow

rates. The lines show the results from the simulation of the following conditions:

60 µmol m-2 s-1, 0.035 % carbon dioxide, 30°C, which corresponds to the same

experimental conditions implement in the PSM cultivation. ....................... - 49 -

Figure 6-5 Light attenuation after 100 hours of cultivation in different gas flow rates.

The lines show the results from the simulation of the following conditions: 60

µmol m-2 s-1, 0.035 % carbon dioxide, 30°C, which corresponds to the same


Figure 6-6 Simulated and experimental pH in different gas flow rates. . The lines

show the results from the simulation of the following conditions: 60 µmol m-2 s-

1, 0.035 % carbon dioxide, 30°C, which corresponds to the same experimental

conditions implement in the PSM cultivation. ............................................. - 50 -

Figure 6-7 Simulated and experimental biomass in 3 % and 0.035 % carbon dioxide

(Top) Simulated and experimental pH at 3 % and 0.035 % carbon dioxide

(Bottom). The lines show the results from the simulation of the following

conditions: 60 µmol m-2 s-1, 1 vvm, 30°C, which corresponds to the same


Figure 6-8 Simulated and experimental biomass in 3 % and 0.8 % of carbon dioxide

(Top) Simulated and experimental pH at 3 % and 0.8 % carbon dioxide

(Bottom). The lines show the results from the simulation of the following

conditions: 60 µmol m-2 s-1, 1 vvm, 30°C, which corresponds to the same

experimental conditions implement in the PSM cultivation ........................ - 52 -

Figure 6-9 Simulated dissolved carbon dioxide in 3 % and 0.8 % of carbon dioxide.

The lines show the results from the simulation of the following conditions: 60

µmol m-2 s-1, 1 vvm, 30°C, which corresponds to the same experimental


Figure 6-10 Simulated and experimental growth at different incident light intensity

on PSM surface. The lines show the results from the simulation of the following

conditions: 1.4 % carbon dioxide, 1 vvm, 30°C, which corresponds to the same


Figure 6-11 Internal light intensity at different incident light intensity on PSM

surface. The lines show the results from the simulation of the following

conditions: 1.4 % carbon dioxide, 1 vvm, 30°C, which corresponds to the same

experimental conditions implemented in the PSM cultivation. ................... - 54 -

Figure 6-12 Simulated nitrate at different incident light intensity on PSM surface. The

lines show the results from the simulation of the following conditions: 1.4 %

X

carbon dioxide, 1 vvm, 30°C, which corresponds to the same experimental


Figure 6-13 Experimental phycocyanin mass fractions at different carbon dioxide

concentrations. Experimental conditions implemented in the PSM cultivation: 1

vvm, 30°C. ................................................................................................... - 56 -

Figure 6-14 Experimental phycocyanin mass fractions at different incident light

intensity on PSM surface. Experimental conditions implemented in the PSM

cultivation: 1.4 % carbon dioxide, 1 vvm, 30°C. ......................................... - 57 -

Figure 6-15 Experimental biomass at different incident light intensity on PSM

surface. (Top) Simulated nitrate at different incident light intensity on PSM

surface. (Bottom). The lines show the results from the simulation of the

following conditions: 1.4 % carbon dioxide, 1 vvm, 30°C, which corresponds to

the same experimental conditions implement in the PSM cultivation ......... - 58 -

Figure 6-16 Experimental and simulated biomass in 600 µmol m-2 s-1 (Top)

Experimental and Simulated nitrate in 600 µmol m-2 s-1 (Bottom). The lines

show the results from the simulation of the following conditions: 1.4 % carbon

dioxide, 1 vvm, 30°C, 0.9 g l-1 initial NO3-, which corresponds to the same


Figure 6-17 Experimental biomass in control and nitrate Fed batch experiment (Top)

Experimental and simulated nitrate in control and nitrate Fed batch experiment

(Bottom) The lines show the results from the simulation of the following

conditions: 600 µmol m-2 s-1, 1.4 % carbon dioxide, 1 vvm, 30°C, 0.9 g l-1 initial

NO3-, which corresponds to the same experimental conditions implement in the

PSM cultivation. ........................................................................................... - 60 -

Figure 6-18 Phycocyanin mass fractions in control and nitrate fed-batch experiment.

Experimental conditions implemented in the PSM cultivation: 600 µmol m-2 s-

1,1.4 % carbon dioxide, 1 vvm, 30°C, 0.9 g l-1 initial NO3-. ........................ - 61 -

Figure 6-19 Exopolysaccharides mass fractions in control and nitrate fed-batch

experiment. Experimental conditions implemented in the PSM cultivation: 600

µmol m-2 s-1,1.4 % carbon dioxide, 1 vvm, 30°C, 0.9 g l-1 initial NO3- ....... - 62 -

Figure 6-20 Experimental and simulated phycocyanin mass fractions in different

experimental conditions. The lines show the results from the simulation of the

following conditions:,1.4 % carbon dioxide, 1 vvm, 30°C, which corresponds to

the same experimental conditions implement in the PSM cultivation. ........ - 63 -

Figure 6-21 Simulated and experimental biomass from A. platensis (data from

Reichert 2016). The experimental points correspond to cultivation in an open

pond with a diameter of 5 m. The lines show the results from the simulation of

the same experimental conditions applied by Reichert (2016). ................... - 64 -

Figure 6-22 Simulated and experimental biomass from A. platensis (Experimental

data from Jing 2015) (Top) Simulated and experimental nitrate concentrations

(Experimental data from Jing 2015) (Bottom). The lines show the results from

the simulation of the same experimental conditions applied by Jing (2015).- 65

-

Figure 6-23 Simulated and experimental phycocyanin concentration from A.

platensis (Experimental data from Jing 2015). The lines show the results from

the simulation of the same experimental conditions applied by Jing (2015).- 66

-

Figure 7-1 Graphical representation of the medium alkalization mechanism. ..... - 68 -

XI

Figure 7-2 Simulated bicarbonate concentrations in the nitrate fed batch experiment.

The line shows the results from the simulation of the following conditions: 1.4 %


conditions implement in the PSM cultivation .............................................. - 76 -

Figure 7-3 Simulated phosphate concentrations in different culture conditions The

lines show the results from the simulation of the following conditions: 1.4 %


conditions implement in the PSM cultivation .............................................. - 78 -

XII

List of tables

Table 2-1 Exopolysaccharide production in different light intensities for A. platensis

(Cogne et al. 2003) ....................................................................................... - 18 -

Table 4-1 Summary of parameters and conditions used in the simulation of Arthrospira

platensis ....................................................................................................... - 39 -

Table 5-1 Experimental plan ................................................................................... - 44 -

Table 7-1 Experimental growth rates for different pH in Arthrospira platensis. ... - 70 -

Table 7-2 Final remarks on model fitting performance. ......................................... - 72 -

XIII

List of abbreviations

ALA δ-aminolaevulinic acid

APC Allophycocyanin

CARPT Computer-automated radioactive particle tracking

Ca-Sp Calcium Spirulan

CFD Computational fluid dynamics

DW Biomass dried weight

EPS Exopolysaccharides

GEM Genome scale metabolic

PBPs Phycobilinproteins

PC Phycocyanin

PCB Phycocyanobilin

PE Phycoerythrin

PSM Photobioreactor screening module

PSU Photosynthetic units

vvm Gas volume flow per unit of liquid volume per minute

(volume per volume per minute)

L/D Light and Dark cycles

EPS Exopolysaccharides

XIV

List of symbols

HCO2 Henry’s constant of CO2 (bar l mol-1)

𝐴𝑐𝑠 Cross-sectional area (m2)

𝑀𝐸̅̅̅̅̅ Maintenance without shear effects (h-1)

𝑐𝐶𝑂2 CO2 concentration in gas phase (%)

𝜏𝑐 Critical shear stress (Pascal)

∅ Time fraction in light zone (s)

µmax Maximum specify growth rate (h -1)

B Bicarbonate concentration (g l -1)

bo Initial bicarbonate ion concentration (g l -1)

C Dissolved carbon dioxide (g l -1)

Cp Compensation point (µmol m-2 s-1)

Dp Cyanobacteria diameter (µm)

Ea Scattering coefficient (m2 kg-1of antenna)

Es Absorption coefficient (m2 kg-1 biomass)

I Light intensity (µmol m-2 s-1)

Io Incident light intensity on PBR surface (µmol m-2 s-1)

Kb Monod-half saturation constant of carbon (g l -1)

Ki Monod-half saturation constant of light intensity for bio-

mass (µmol m-2 s-1)

Kip Light inhibition constant (µmol m-2 s-1)

Kla Carbon dioxide volumetric Mass transfer coefficient (h -1)

Kli Monod-half saturation constant of light intensity for phy-

cocyanin kinetic (µmol m-2 s-1)

km Extinction coefficient for shear stress (Pascal-1)

Kn Monod-half saturation constant of nitrate (g l -1)

Kpc Monod-half saturation constant of phycocyanin (g gbio-

mass-1)

N Nitrate concentration (g l -1)

no Initial nitrate concentration (g l -1)

P Gas inlet absolute pressure (bar)

P Phosphate (g l -1)

Pc Phycocyanin concentration (g l -1)

Pk Acid dissociation constant

Po Initial phosphate concentration (g l -1)

Q Volumetric flow rate (l h-1)

R Bubble column radius (m)

Rpc Phycocyanin formation rate (h -1)

V Dissolved carbon dioxide concentration (g l -1)

Vs Superficial gas velocity (m s-2)

X Total biomass concentration (g l -1)

Xa Active biomass concentration (g l -1)

xao Initial biomass (g l -1)

yb/x Bicarbonate consumption yield (g gbiomass-1)

yn/x Nitrate consumption yield (g gbiomass-1)

yp/x Phosphate consumption yield (g gbiomass-1)

Zpc Mass fraction of phycocyanin (gphycocyanin gbiomass-1)

𝑓 Light/dark frequencies (Hz)

𝜀 Energy dissipation rate (m2 s-3)

VIII

𝜆 Eddy length (µm)

𝜇𝑙 Fluid viscosity (Pascal×s)

𝜌𝑙 Fluid density (kg m-3)

𝜏 Shear stress (Pascal)

- 7 -

1 Introduction

The cyanobacterium Arthrospira platensis is a prokaryotic photoautotrophic microorganism

that is successfully cultivated for the commercialization as whole biomass due to its high pro-

tein content and promising valuable substance. For instance, phycocyanin ─ a light harvesting

protein present in A. platensis ─ has recently drawn the interest of the food and cosmetic indus-

tries due to its bright blue colour and its strong antioxidant capacities. Additionally, other po-

tential compounds, which are present in A. platensis are gama-linoleic acid ─ an essential un-

saturated fatty-acid and Calcium-Spirulan ─ a sulphated exopolysaccharide with promissory

biological activities (Borowitzka 2013; König 2007; Pulz and Gross 2004). Although A. platen-

sis is successfully cultivated in open raceways, the low productivities of biomass dried weight

(DW) reached in these cultivations (0.04 g DW l-1 d-1) (Jiménez et al. 2003) and the unreliable

product quality (i.e phycocyanin) seem to require the cultivation of this cyanobacteria in pho-

tobioreactors.

Biomass productivities up to 20 times higher than those reached in raceway cultivations

(Bezerra et al. 2011; Chen et al. 2013) have been found by using closed photobioreactor sys-

tems. Although growth is relevant in cyanobacteria production, as in many other biotechnolog-

ical processes, the product formation is the main goal. Therefore, to become economically fea-

sible, the cultivation of cyanobacteria in photobioreactors requires optimal product yields cou-

pled with low plant investment and operating costs (Bertucco et al. 2014). Optimal strategies

for nutrient delivery, as well as the accurate use of light, are the key parameters that could make

this technology promising in order to achieve higher efficiencies. In addition, the efficiency of

the process can be enhanced with a multiproduct strategy already proposed for cyanobacteria,

which implies the recovery of more than one useful compound.

A. platensis use phycocyanin as a light-harvesting protein and consequently, light intensity and

nitrate are crucial factors in the accumulation of this phycobiliprotein (Chen et al. 2013). How-

ever, the values of optimal nitrate and light intensity for phycocyanin production found in re-

search papers are inconsistent, possibly due to different bioreactor configurations and culture

conditions (Xie et al. 2015). For instance, (Chen et al. 2013) reported that the maximal phy-

cocyanin productivity (125 mg l-1 d-1) of A. platensis was obtained at 700 µmol m-2 s-1. In con-

trast to this, Xie et al stated inhibition in growth by using the same light intensity (Xie et al.

2015). The discrepancy in these results can be explained by dissimilar light paths, diverse light

- 8 -

qualities or different initial cell densities used in both cases (0.5 g l-1 (Chen et al. 2013) In

addition, in batch cultivation, nutrients like nitrate, carbon and light decrease with the time and

therefore, the cyanobacteria composition (phycocyanin, exopolysaccharides protein and lipids)

also suffered quantitative and qualitative changes. Therefore, a mathematical model can facili-

tate the investigation of the optimal conditions according to particular cultivation settings.

A. platensis – when cultivated under photosynthetic growth conditions − produces different

quantities of an associated exopolysaccharide (EPS) depending on experimental parameters

(Borowitzka 2013; Pulz and Gross 2004). According to Filali et al, the exopolysaccharide sugar

composition consists of 6 neutral sugars − xylose, rhamose, fructose, galactose, mannose, glu-

cose; 2 uronic acids − glucuronic and galacturonic acid; 2 unidentified sugars, as well as some

sulphate groups, whose localization is still not completely clear (Filali Mouhim et al. 1993).

The possibility of stimulating polysaccharide release by means of an optimization of the culture

conditions has been poorly considered.

The primary source of inorganic carbon for A. platensis is the bicarbonate ion (HCO3-) (Cornet

et al. 1998). The bicarbonate present in the medium (around 117 mM) is consumed by the cya-

nobacteria to support its growth. With the purpose of avoiding carbon limitations and taking

advantage of the carbon dioxide captures, A. platensis cultures may be provided with additional

CO2. In addition, in microalgal culture technics, carbon dioxide is also used to control the pH

(Pawlowski et al. 2014). A carbon dioxide line is opened or closed automatically according to

an established pH set point. This control technique has provided promising performance results

with an adequate reduction of the CO2 losses (Pawlowski et al. 2014). However, this implies

the necessity of sensors in the reactor to measure these variables, which can increase investment

and operating costs. A predictive CO2 supply based on a mathematical model can ease the re-

search activities and overtake the already mentioned challenge.

Mathematical models are valuable tools to investigate and optimize bioprocess (Fleck-

Schneider et al. 2007). Therefore, several attempts have been conducted to simulate A. platensis

growth (Cornet et al. 1992; Levert and Xia 2001). Nonetheless, all of them have been applied

just for low cell densities (< 1g l-1) and they have neglected carbon limitations in culture. In

addition, phycocyanin and exopolysaccharide formation laws have not been deeply docu-

mented.

- 9 -

The aim of this study is to extend these models to higher biomass conditions of A. platensis in

bubble columns. The model will be used also to investigate the phycocyanin and exopolysac-

charide formation under different scenarios and the proof of concept of controlling nutrients for

optimal product formation.

- 10 -

2 State of art

The theoretical part of this thesis will firstly describe the cyanobacteria Arthrospira platensis

together with its compounds phycocyanin and exopolysaccharides, as well as their relevance

within various industries. Furthermore, the importance of mathematical models and simulations

in biotechnological processes will be explained and then applied on this particular group of

cyanobacteria. Not last, various aspects like the light supply, mass transfer and turbulence will

be considered within this chapter.

2.1 The cyanobacteria Arthrospira platensis and its relevance

Arthrospira sp. are filamentous prokaryote cyanobacteria with at least 38 species already de-

scribed in the literature. The most representative strains in the industry are Arthrospira platensis

and Arthrospira maxima. A. platensis is a multicellular helical-shaped alga and the morpholog-

ical features are highly dependent on the parameters of cultivation. The literature describes a

big range of lengths and spirals: from 200 µm with 6 spirals to 1150 µm with 12 spirals (Figure

2-1). This size range depends on the culture environments. For example, nitrogen limited cul-

tures seem to have a long shape due to the glycogen accumulation. However, even if the cells

are extending their shape, there is no evidence of cell division (Cornet et al. 1998)

Figure 2-1 A microscope picture from Arthrospira platensis - Cyanobacteria size around 200 µm

with multicellular helical-shaped (Kamata et al. 2014).

The quantity of carbohydrates in dried A. platensis powder represents around 10 to 20 % of the

total algae mass. These carbohydrates can be found in both intracellular and extracellular envi-

ronment, while the percentage of lipids varies between 6 to 10 %. Gamma-linolenic acid, a fatty

- 11 -

acid sold as a dietary supplement, represents almost 30 % of A. platensis lipids (Cornet et al.

1998). The high concentration of proteins in A. platensis is an advantage and it is one of the

reasons why the World Health Organization indicates this alga as a healthy food. The percent-

age of proteins varies between 59 to 71 % in dry weight depending on the culture conditions.

Instead of chloroplast, A. platensis have special structures called phycobilisomes (Cornet et al.

1998) which are localized in the thylakoid membrane (Figure 2-2). These granules contain phy-

coerythrin (PE), phycocyanin (PC) and allophycocyanin (APC), which are important phyco-

bilinproteins (PBPs) in the photosynthesis process together with chlorophyll A.

Figure 2-2 Representation of the light harvesting complex – phycoerythrin (PE), phycocyanin (PC)

and allophycocyanin (APC) together with the Photosystem I (PSI) and Photosystem II

(PSII) (Guan et al. 2007).

2.1.1 Phycocyanin: a high value product

The phycocyanin (PC) is the main protein-pigment with the highest quantity in A. platensis

(4 % to 20 %) (Chen et al. 2013; Xie et al. 2015). The major industrial applications of phycocy-

anin protein are related to its antioxidant capacity and to its bright blue which makes it a poten-

tial pigment for food and cosmetic industry. The purity of phycocyanin is measured by the

absorbance ratio A620/A280 with the purity values of 0.7 and 4.58 depending on food grade and

analytic grade. The blue chromophore group in phycocyanin is the Phycocyanobilin molecule

(Figure 2-3). It is covalently linked to the L-cystein amino acid. Phycocyanobilin biosynthesis

will be described afterwards in more detail.

It is important to highlight that the maximum phycocyanin productivity (840 mg l-1 d-1) was

achieved with the cultivation of Galdieria sulphuraria in heterotrophic conditions. To achieve

this, a high rate of biomass production at high biomass concentration (83.3 DW g L-1) together

with a strain that has relatively a high specific phycocyanin concentration (15.6 mg g-1)

- 12 -

(Querques et al. 2015) were produced in continuous culture mode. This serious drawback of

photosynthetic microorganisms compared to heterotrophic cultivations could be partially dis-

regarded in view of some advantages, of both environmental and economic impact, that photo-

autotrophic cultivations could achieve : (i) they are capable of utilising energy from the sunlight

instead of glucose; (ii) many strains can grow in brackish or in waste waters ; (iii) it is possible

to utilise as carbon source the CO2 emitted by industrial plants ; (iv) the economy of the process

could be enhanced by recovering more than one useful compound, with a multiproduct strategy

as already proposed for cyanobacteria

The phycobilinproteins (PBPs), in particular phycocyanin, have been widely used as nutritional

ingredients, natural dyes and florescent markers for pharmaceuticals such as antioxidants and

anti-inflammatory reagents (Querques et al. 2015). The phycobilinproteins serve as valuable

fluorescent tags with numerous applications in flow cytometry, fluorescence and activated cell

sorting. These applications exploit the unique physical and spectroscopic properties of phyco-

bilinproteins. In addition, because of the high molecular absorptivity of these proteins at visible

wavelengths, they are convenient markers in applications like gel electrophoresis, isoelectric

focusing and gel exclusion chromatography (Querques et al. 2015).

Phycocyanin is used also as colorant in food (chewing gums, dairy products, jelly, etc.) and in

cosmetics such as lipstick and eye liners in Japan, Thailand and China. Phycocyanin is one of

the most important natural blue pigment used in food and biotechnology because of its color,

fluorescence and antioxidant properties with estimated marker is around 5 to 10 million of dol-

lar yearly and growing (Querques et al. 2015).

The extraction of phycobilinproteins involves cell rupture and release of these proteins from

the inside of the cell. Several methods have been developed for the separation and purification

of PC, such as density gradient centrifugation, ammonium sulfate precipitation, chromatog-

raphy method and aqueous two phase extraction. One of the most difficult steps in the extraction

of phycobilinproteins is the wall cell break. Thus, the use of variations in the osmotic pressure,

abrasive conditions, chemical treatment, freezing-thawing, sonification and tissuelyser are nec-

essary (Kumar et al. 2014). Mechanically cell disintegration methods are currently preferred

for large-scale operations.

2.1.2 Phycocyanobilin biosynthetic metabolic pathway

The Phycocyanobilin (Figure 2-3) is the blue chromophore in the phycocyanin molecule and it

shares common biosynthetic pathways with heme and chlorophyll to the level of protoporphyrin

- 13 -

IX. Both are assembled from δ-aminolaevulinic acid (ALA), which is synthesized from gluta-

mate via the C5 pathway, whereas the pathways diverge upon metalation with either iron or

magnesium (Figure 2-4). In the later stages of phycocyanobilin synthesis, the protoheme is bro-

ken down to bilieverdin. And finally, cyanobacteria possess ferredoxin-dependent bilin reduc-

tases primarily for the synthesis of Phycocyanobilin from bilieverdin (Brown et al. 1990)

Figure 2-3 Molecular structure Phycocyanobilin - Covalently linked to the L-cystein amino acid to

assemble the protein (Brown et al. 1990).

Next, the effect of light conditions and nitrate concentrations in the biosynthesis of this mole-

cule will be addressed.

2.1.2.1 Light

The light harvesting protein that belongs to the photosynthetic units, together with the mass

fractions phycocyanin and its chromophore group are regulated by light. According to Rubio

cells are known to adapt their number and size of photosynthetic units (PSUs) or phycobili-

somes to the available irradiance when this is constant for a prolonged period (Rubio et al.

2003). This phenomenon is also known as photoadaptation. Moreover, the cells that are accli-

mated to high irradiance have fewer PSUs and contain less chlorophyll and phycocyanin than

the same type of cells that are growing under low irradiance (Rubio et al. 2003). In contrast to

this, the concentration of PSUs in the cell increases under low irradiance. It is important to

highlight that all PSUs reach their steady-state value in hours, either from a culture transfer

from low light to high light or vice versa (Ritz et al. 2000).

5- ALA

Light induction

- 14 -

PORPHOBILINOGEN

UROPORPHYRINOGEN III

COPROPORPHYRINOGEN III

PROTOPORPHYRINOGEN IX

PROTOPRPHYRIN IX

CHLOROPHYLL a PROTOHEME IX α

PHYCOCYANOBILIN

Figure 2-4 Phycocyanobilin and Chlorophyll a pathways beginning from 5-ALA with relevant Co-

factors (Brown et al. 1990).

This photoadaptation process has been found not only in all light harvesting chlorophylls, phy-

coerythrin and phycocyanin, but also within all light harvesting carotenoids like fucoxantin and

peridinin (Dubinsky and Stambler 2009). For example, the effect of light intensity on phycocy-

anin content in the cyanobacterium Synechococcus sp. was addressed by (Takano et al. 1995).,

who found a maximum of phycocyanin production at 25 µmol m-2 s-1 (Figure 2-5), whereas

light intensities above or below this value decrease the phycocyanin content. Additional studies

have shown that a light increment above the optimal level leads to a reduction in the expression

of phycocyanobilin structural genes (Lau et al. 1977), fact that can be related to the photoadap-

tation process.

- 15 -

Figure 2-5 Effect of light intensity on phycocyanin content after 72 hours of cultivation in the cells

of Synechococcus sp. (Takano et al. 1995).

2.1.2.2 Nitrate

The main nitrogen source of A. platensis is the nitrate (NO3-) ion which is 29 mM in the standard

spirulina medium. Nitrate is carried through membrane by active transport and then reduced to

ammonium (NH4+) (equations below), after which it is incorporated into glutamic acid in the

amino acid metabolism. The proportion of the reducing power that needs to be used for nitrogen

assimilation can be reduced to 10 % of the total by assimilating ammonium. It is reasonable for

cyanobacteria to show a strong preference for ammonium (NH4+) over nitrate (NO3

-), whereas

the use of ammonia salts, mainly NH4Cl and (NH4)2SO4, has been reported. However, they

cannot replace the total amount of nitrate because high quantities of ammonia seem to be toxic

(Converti et al. 2006).

NO3- + 2 reduced-ferredoxin + 2 H+ => NO2

- + 2 oxidized-ferredoxin + H2O

NO2- + 6 reduced-ferredoxin + 7 H+ => NH4

+ + 6 oxidized-ferredoxin + 2 H2O

Furthermore, it has been found that under complete nitrate deprivations, phycocyanin is de-

graded in some cyanobacterium species (Gilbert et al. 1996; Lau et al. 1977). A study made in

Synechococcus sp shows the loss of spectrophotometrically measurable phycocyanin after the

resuspension in nitrate-free medium. Once the nitrate was restored, phycocyanin was resynthe-

sized and its content began to increase (Takano et al. 1995). However, low but non-limiting

- 16 -

nitrate conditions have been also found positive for the biosynthesis of phycocyanin. Moreover,

(Xie et al. 2013) found a better lutein formation by maintaining the nitrate concentration below

2.2 mM without reaching complete depletion.

2.1.3 Exopolysaccharide properties and biosynthesis

A. platensis is cultivated under photosynthetic growth conditions and produces different quan-

tities of an associated exopolysaccharide (Figure 2-6) depending on the culture conditions

(König 2007; Pulz and Gross 2004). According to the early literature the exopolysaccharide

sugars composition consists of 6 neutral sugars, xylose, rhamnose, fructose, galactose, man-

nose, glucose, 2 uronic acids: glucuronic and galacturonic acid and 2 unidentified sugars and

the presence of sulphated groups, whose localization is still not completely clear. Uronic acids

were found to contribute to 40 %. (Filali Mouhim et al. 1993). Another study it has been re-

ported that cyanobacterial EPS are not only composed of carbohydrates but also of other mac-

romolecules such as polypeptides enriched with glycine, alanine, valine, leucine, isoleucine and

phenylalanine have been reported in the EPS. Recently, a partial EPS characterization indicated

by using elemental analysis and a bicinchoninic acid (BCA) reaction that the EPS were hetero-

polysaccharides that contain carbohydrate (13%) and protein (55 %) moieties. In addition, gas

chromatography analysis of the carbohydrate portion of the EPS indicated that it was composed

of seven neutral sugars: galactose (14.9 %), xylose (14.3 %), glucose (13.2 %), fructose

(13.2 %), rhamnose (3.7%), arabinose (1%), mannose (0.3%) and two uronic acids: galac-

turonic acid (13.5 %) and glucuronic acid (0.9 %)(Trabelsi et al. 2009).

However, an additional study shows 57 % protein and 43 % carbohydrates with the following

sugar composition: galactose (4.3 %), xylose (6.2 %), glucose (17.3 %), frucose (15.2 %), rham-

nose (49 %), mety-rhamose (17 %), mannose (2.3 %) and glucuronic acid (3.95 %) (König

2007).

- 17 -

Figure 2-6 Insolated Exopolysaccharide from A. platensis (A) and purified EPS – withe EPS powder

isolated by tangential flow filtration followed by freeze drying (Reichert 2016).

The differences found in the sugar composition can be explained by the fraction of the extract

where the monosaccharides were analysed. For example, an study analysed the sugar composi-

tion from the EPS in the supernatant (culture medium) and also in an extract that contains the

EPS from the external layers of cells and they found a difference in the molar rations of the

monosaccharide composition between the both fractions (König 2007; Nie et al. 2002; Xia et

al. 2001).

Many studies have reported that the exopolysaccharides from A. platensis have a particular

variety of biological applications including antiviral, inhibitory effects on corneal neovascular-

ization and booster of the immune system (Chaiklahan et al. 2013). Calcium Spirulan (Ca-Sp)

(Hayashi et al. 1996) is a sulphated exopolysaccharide (17.7 mg g-1 exopolysaccharides) that

was found to inhibit the replication of several enveloped viruses, including Herpes simplex

virus type, human cytomegalovirus, measles virus, influenza A virus and HIV-1 virus. It was

also revealed that Ca-Sp selectively inhibited the penetration of the virus into host cells with a

high relevance for the antiviral activity of the sulphated groups and for the molecular confor-

mation by chelation calcium ion with sulfated groups). Ca-Sp is also particularly interesting for

its use in cosmeceuticals. It stimulates the metabolic activity of human skin fibroblast cell lines,

which are responsible for collagen synthesis and for the firmness of skin (Pulz and Sandau

2009). With increasing age, the collagen synthesis drops significantly, so a main target in the

cosmetic research is the development of anti-aging products that are capable of enhancing the

metabolism of fibroblast. A 36% enhancement of collagen synthesis was found by applying

CA-Sp at 10 µg ml -1. It was also found that UV exposed fibroblasts showed a higher vitality,

if Ca-Sp had been added prior to or even after radiation (Pulz and Sandau 2009).

- 18 -

In addition, the exopolysaccharides show non-Newtonian behavior and strong pseudo-plastic

characteristics at a concentration of 0.02 g EPS l-1 and therefore, due to its rheological charac-

teristics, EPS from A. platensis has been proposed as possible substitute for agar-agar study

shows an increment in the medium relative viscosity with the increment of the EPS in medium

(Filali Mouhim et al. 1993). This increment can also generate additional challenges in A. platen-

sis production and scale-up.

Under unfavorable growing conditions many algae shift their metabolic pathways toward the

biosynthesis of exopolysaccharides. In batch cultivations, this can occur during the course of

the cultivation as a result of the changes in substrates like light, nitrate, phosphate and carbon.

However, the possibility of stimulating polysaccharide release by means of an optimization of

the culture conditions has been poorly considered. A research studied the light influence in the

exopolysaccharides formation. Table 2-1 shows the values of exopolysaccharides in different

light conditions. Incident light intensity on PBR surface up to 1000 µmol m-2 s-1 favor the pro-

duction of exopolysaccharides by reaching a maximum of 0.54 g polysaccharides g-1 biomass.

It has been identified that exopolysaccharides can be delivered as result of an overflow in the

metabolism (Staats et al. 2000), which generates a drain the excess of ATP (Cogne et al. 2003)

by them. Similar to phycocyanin production, low, but not growth limiting, concentrations of

nitrate have demonstrated to enhance the accumulation of exopolysaccharide (Staats et al.

2000). For example, (Abd El Baky et al. 2014) found the best exopolysaccharide production at

nitrate concentrations between 0.2 g L-1 to 0.5 g L-1 nitrate, whereas the highest values of growth

were found in concentrations of 1.8 g L-1 of nitrate. Analyzing the general metabolism for A.

platensis (Figure 2-7) proposed by (Cogne et al. 2003). It is supposed that in stress conditions

the metabolic pathway is redirected to glucose-1-phosphate and then to Glycogen or exopoly-

saccharides. Excess of energy in form of ATP may lead the exopolysaccharide production.

Table 2-1 Exopolysaccharide production in different light intensities for A. platensis (Cogne et al.

2003)

Irradiance

(µmol m-2 s-1)

Exopolysaccharide

(g polysaccharides g-1 biomass)

92 0.1

228 0.33

731 0.50

1000 0.54

- 19 -

Figure 2-7 A schematic view of key pathways of central metabolism in Arthrospira platensis - It is

supposed that in stress conditions the metabolic pathway is redirected to glucose-1-phos-

phate and then to Glycogen or exopolysaccharides (Cogne et al. 2003).

2.2 Mathematical modelling in biotechnology

Mathematical models are valuable tools to predict and understand the behavior of biological

systems. In addition, they can be used to describe the past performance and predict the future

performance of biotechnological processes. They can be applied to processes operating at many

different levels, from the action of an enzyme within a cell, to the growth of that cell within a

commercial scale bioreactor (Nielsen et al. 1991).

The majority of the goals of modeling cell factories are related to understanding and predicting

their behavior when they are perturbed either internally through genetic modifications, or ex-

ternally by changing various environmental factors. The model purpose defines the complexity

of the modeling problem and will influence all subsequent steps of the modeling procedure

(Nielsen et al. 1991).

- 20 -

Mathematical models can be powerful tools in both fundamental research and applied research

and development. Some models contribute to the understanding of how cells function, while

other models allow us to use laboratory and pilot-scale data to make predictions about how a

commercial scale bioreactor must be designed and operated in order to give optimal perfor-

mance. For example, mathematical models have been widely used within classical fermenta-

tion, glucose fed-batch strategies to enhance ethanol production or the microbiological L-leu-

cine production (Georgiev et al. 1997). Other biotechnological products which have benefited

from mathematical modelling and simulation of fermentations are: citric acid (Zlateva et al.

1993) and penicillin-G (Menezes et al. 1994).

Furthermore, many different types of biotechnological systems and processes, such as the op-

eration of metabolic pathways within a cell, the expression of genes within a cell, the death of

cells during a sterilization process, the growth of cells in a bioreactor and the action of enzymes

can be modeled (Nielsen et al. 1991). Nonetheless, in the field of cyanobacteria biotechnology,

few mathematical models have been developed to simulate cyanobacteria growth with less or

more details depending on the purpose for which the model was built. The level of accuracy in

the kinetic equations varies from the classical Monod approaches to more sophisticated genome

scale models and they will be described in the next section

2.3 Biomass kinetics models in cyanobacteria

In the following theoretical part, the advantages and the current limitations of the modern ge-

nome scale models are described, as well as the classical macroscale modelling base on differ-

ential equations couple with production or/and consumption kinetics for cyanobacteria cultiva-

tions. In addition, a short description of the numerical approaches to solve the models is given.

Finally, the general approaches for biomass and product formation in microalgal culture are

addressed.

2.3.1 Genome-scale metabolic models in cyanobacteria

In modern, system-level microbial metabolic engineering, genome scale metabolic reconstruc-

tions (GEMs) are used to integrate and analyze large omics datasets as well as to evaluate de-

signs in silico. A GEM maps annotated metabolic genes and proteins to reactions based on the

current best understanding of a given organism. A growing number of metabolic engineering

studies in classical microbial strains have demonstrated the use of well-curated GEMs to gen-

erate strain designs that are neither intuitive nor obvious (Baart and Martens 2012).

- 21 -

However, currently there are a few genome scale reconstructions available for cyanobacteria.

For example, the metabolic network of the cyanobacterium Synechocystis sp. has been deeply

addressed by various research groups in the world. More recently quasi-steady state simulations

of the A. platensis metabolic network have been done. A simpler metabolic network was built

up with 121 reactions and 134 metabolites including biomass synthesis, production of a growth

associated polysaccharide and energy aspects (Cogne et al. 2003).. Another more sophisticated

metabolic network of A. platensis was generated by (Klanchui et al. 2012), which accounts for

771 metabolic genes, 712 metabolites and 868 reactions More than 85 % of the total reactions

were associated with genes. The simulated results were validated by experimental evidence and

showed satisfactory agreement under three different growth conditions: autotrophic, hetero-

trophic, and mixotrophic.

Genome-scale models are able to estimate the flux distribution of the entire metabolism. How-

ever, they are still immature due to the lack of information in most of microalgal species over

all cellular biochemical reactions. In addition, most of the studies have focused only on the

quasi-steady state under constant light and nutrient regimes. Static metabolic studies can give

a first insight in the behavior of continuous cultivations, but until now they couldn’t be success-

fully implemented in batch or fed-batch cultivations due to changes in the culture conditions

over the time, which probably implies higher computational resources (Baart and Martens 2012;

Baroukh et al. 2015)

2.3.2 Macroscale models – Monod type kinetics

Macroscale models are based on differential equations and simple consumption or formation

kinetic terms. As it was already mentioned they have been widely used in classical fermenta-

tions. Differential equations describe, in a simplified manner, how the key physical and bio-

logical phenomena operate. Figure 2-8 shows a simplified illustration of how a model that con-

sists of differential equations might be applied to a cyanobacteria process.

As Figure 2-8 displays, one equation may include a description of how the rate of growth of the

biomass depends on the substrate (Nitrate, light, carbon, etc.) quantities in the photobioreactor,

whereas another equation may include a description of how the substrates are consumed. These

equations describe the rate of change in biomass, substrates and products but not the actual

value of these variables. Therefore, these equations need to be solved and under some condi-

tions these differential equations can be either analytically integrated or simplified in order to

give algebraic equations, but this is very often not the case and a numerical solution is required.

- 22 -

Figure 2-8 A schematic representation of the modelling process - one equation may include a de-

scription of how the rate of growth of the biomass depends on the substrate (Nitrate, light,

carbon, etc.) quantities in the photobioreactor, whereas another equation may include a

description of how the substrates are consumed.

2.3.3 Numerical solutions of the differential equations

Runge-Kutta methods are iterative techniques, which are used in temporal discretization for the

approximation of solutions for ordinary differential equations. The Runge-Kutta algorithm lets

us solve a differential equation numerically (i.e. approximately) and it is known to be very

accurate and well-behaved for a wide range of problems.

Consider the single variable problem: dx

dt= f(t, x) with initial condition x(0) = x0. Suppose that

xn is the value of the variable at time tn. The Runge-Kutta formula takes xn and tn and calculates

an approximation for xn+1 at a brief time later, tn+h. It uses a weighted average of approximated

values of f (t, x) at several times within the interval (tn, tn+h). The formula is given as it follows:

Equation 1: xn+1 = xn +h

6× (a + 2b + 2c + d)

Equation 2: a = f(tn,, xn)

Equation 3: b = f (tn +h

2, xn +

h

2× a)

Equation 4: c = f (tn +h

2, xn +

h

2× b)

Equation 5: d = f(tn + h, xn + h × c)

- 23 -

To run the simulation, we start with x0 and find x1 using the formula above. Then we plug in

x1 to find x2 and so on. The Runge-Kutta method is included in Matlab thought a routine called

ODE45, which was used in this work to solve the differential equations set.

2.3.4 Formation kinetics in biomass and nutrient consumptions

The right side in the biomass differential equation, i.e (Equation 6); can be set with simple mac-

roscopic kinetic laws that relates growth rate to extracellular concentration in limiting substrates

(i.e. light, nitrate, etc.) These laws are established mainly with Monod model, where Ki is the

substrate concentration that reduces the maximum growth rate to the half.

Equation 6: 𝑑𝑥

𝑑𝑡= µ𝑚𝑎𝑥 ×

𝐼

𝐼+𝐾𝑖×

𝑓1

𝑓1+𝐾𝑓1×

𝑓1+𝑛

𝑓1+𝑛+𝐾𝑓1+𝑛

Equation 7: 𝑑𝑓1

𝑑𝑡= −𝑌𝑓1

𝑥

×𝑑𝑥

𝑑𝑡

Although several limiting factors can be added to the Equation 6, the main drawback in formu-

lating general kinetic laws in limiting conditions is to find simple macroscopic laws that relate

growth rate to extracellular concentration in limiting substrates (Bungay 1994). This requires

not only a complete understanding of the mechanisms involved in the limiting process, but also

to study them at different levels (metabolism, physiology, etc) (Lee et al. 2015) In the case of

photosynthetic micro-organisms, these mechanisms become more complex because two or

more limiting factors interact, since these micro-organisms are always cultivated in limited light

conditions (Lee et al. 2015).

Currently, the main variation in the model presented above consists in how the effect of light

on the growth rate is modeled 𝐼

𝐼+𝐾𝑖. Simple models do not include the photo inhibition and photo

adaptation processes in cells (Merchuk and Wu 2003) and they are directly related to the max-

imum specific growth rate as the Equation 6 states. More sophisticated kinetic models include

photo inhibition and photo adaptation processes in cells such as the one proposed by Eilers and

Peeters and advance by Wu and Merchuk (Eilers and Peeters 1988; Wu and Merchuk 2002)

who describe the relationship between growth and biomass with four equations instead of one.

This approach has been successfully used by Al-Dahhan (Luo and Al-Dahhan 2004) and other

authors in their simulations of the red marine cyanobacteria Porphyridium sp.

The Eilers and Peeters model assumes that the PSU (photosynthetic units) has three stages: the

resting stages x1, the activated state x2 and the inhibited state x3. The probabilities of the state

transitions following a photon capture are supposed to be proportional to the light intensity, or

these reactions are assumed to be first- order reactions with reaction constants of 𝛼I for x1→x2

- 24 -

and βI for x2→x3. Since the other-state transitions can happen in the dark, the reaction constants

for these reactions are assumed to be constant, i.e. δ for x2→x1 and γ for x3→x2. Nonetheless,

the model needs a lot of empirical parameters and its use with other cyanobacteria are very

scarce in the literature, which limits the application on other species. (Eilers and Peeters 1988).

All nutrient consumptions can be correlated with theoretical consumptions yields (Equation 7),

which are calculated from Stoichiometric equations and the biomass formation rate dx

dt. In ad-

dition, light also has to be supplied in each time step. However, this “nutrient” is more chal-

lenging to model because light varies with time and also in the space as consequence of the

biomass growth. Further in this work, it will be explained how light (I) is model by taking into

account the light absorption and scattering properties by the cyanobacteria.

2.3.5 Formation kinetics in product

The biomass composition in the cells changes with the time. However, most of the available

mathematical models for bioprocesses are unstructured. This means that the biomass is consid-

ered as one entity and is described only by its concentration, whereas product formation is

linked to the biomass formation by linear relations (Equation 8) (Nielsen et al. 1991). In other

words the product mass fraction Zp is constant during the cultivation; however this assumption

is not completely correct.

Equation 8: 𝑑𝑝

𝑑𝑡= 𝑍𝑝 ×

𝑑𝑥

𝑑𝑡

Advanced structured models consider that the biomass can change its composition during the

time of cultivation as a result of the changes in substrates. Especially the Zp is not constant and

it should change as consequence of the metabolic pathway shifting. Structured models describe

not only the biomass kinetics, but also in particular the product formation kinetics for transient

operation, using a small set of parameters which often have a biological meaning. Therefore,

product mass fractions have to be modeled by introducing a simple macroscopic law that links

the biological phenomena in the cell with the biosynthesis and degradation of product during

the batch cultivation. Consequently, a new differential equation for phycocyanin mass fraction

dZp

dt that describes its changes during the time is necessary:

Equation 9: 𝑑𝑝

𝑑𝑡= 𝑍𝑝 ×

𝑑𝑥

𝑑𝑡

Equation 10: 𝑑𝑍𝑝

𝑑𝑡=?

- 25 -

The challenge is to describe in a suitable manner the right term in the Equation 10 because of

the lack of information about the mechanisms that trigger biosynthesis and degradation of the

cyanobacteria components, particularly phycocyanin and exopolysaccharides. At first glance,

microorganisms appear to be very different, but a closer study reveals that they have a number

of basic functions in common. For example, they all need catabolism of substrate to create

energy for growth and maintenance of cell functionality

In microalgae modelling, the number of studies that have included the kinetic of product for-

mation is very scared and limited to low cell densities. A study includes in their model a mech-

anism to describe the phycoerythrin formation. Their mathematical model was well suited to

understand the complex variations in the biomass composition for low cell densities cultivations

of Porphyridium purpureum (Csőgör et al. 2001; Fleck-Schneider et al. 2007). In the case of A.

platensis, a model proposed by (Levert and Xia 2001) incorporates the exopolysaccharide for-

mation mechanism (Levert and Xia 2001). However, the model was validated just in low light

conditions that produce low cell densities (1g l-1). It limits the model extrapolation to other

conditions, such as in the case of using high light intensity (>100 µmol m-2 s-1).

To sum up, the product formation is the goal in many biotechnological processes, so that the

kinetics of product formation can be more relevant than the kinetics of growth. Therefore, it is

necessary to understand the interrelations between the cultivation conditions and the product

formation and to implement these in mathematical models. Yet, the kinetic description for phy-

cocyanin and EPS formation has been barely addressed and restricted just for low cell densities

(1 g l-1)

2.3.6 State of art – Kinetics models in Arthrospira platensis

In the follow part, it will show the most relevant scientific contributions in the mathematical

modelling of A. platensis. The first approach to mathematical modelling in A. plantesis was

done by Huang and Chen in 1986. They described the A. platensis growth by using a simple

Monod-type model which include just light as limiting factor (Huang and Chen 1986). The

Monod-half saturation constant was 124 µmol m-1 s-1 and µmax of 0.083 h-1. In 1987, a model

which includes photoinhibition was proposed by (Lee and Erickson 1986). A new photoinhibi-

tion constant was incorporated into the Monod mechanism, which has extended the model ca-

pabilities to high illuminations. It was not just until 1994, when Cornet proposed a model that

considered not just light- limitation but also other limiting-factors like nitrate and phosphate

- 26 -

(Cornet et al. 1998). This model was validated until a biomass concentration of 1 g l-1. How-

ever, nowadays higher biomass concentrations are reached. Therefore, these models have to be

extending to high cell densities. A recent research from 2015 developed a model for cell con-

centrations higher that 1 g l-1(Del Rio-Chanona et al. 2015). This model was validated with

experiments from the literature (Xie et al. 2015). The model proposed in this work was also

tested using this experimental points and it will discuss in section 6.7

In almost of the models already mentioned, the biomass composition does not change with the

time, which means that the products of interest, phycocyanin or exopolysaccharides are con-

sider constant during the cultivation. However, as it was already mentioned in section 2.1.2 and

2.13, it knows that biomass compounds variated with the time as the nutrients started to be

consumed. Such variations in biomass activity and composition require a complex description

of the cellular metabolism and a structured approach to the modeling of cell kinetics. However,

in general it is very difficult experimentally to obtain sufficient mechanistic knowledge about

the cell physiology and metabolism for the development of a realistic structured model. More-

over, the mechanism of cell growth is complex and not yet completely understood.

2.4 Modelling photobioreactors features

The site (photobioreactor), where the reactions mentioned above take place, should be described

in detail because some parameters like light, turbulence and mass transfer deepens specifically

of the photobioreactor geometry and operational settings. Furthermore, the current state of art

regarding different methods to simulate light distribution and to estimate the mass transfer co-

efficients in photobioreactors will be presented in the next part. In addition, the influence of

turbulence in microalga culture and the ways to estimate values of turbulence generates in pho-

tobioreactors will be approached.

2.4.1 Modelling light supply in photobioreactors

As we already mentioned, light has to be supplied in every time step to the Equation 6 and it is

known that one of the most important factors that control cell growth in a photobioreactor is

light availability. Perhaps the most unique substrate to model is light since its availability does

not depend just on time, but also on space. The attenuation of light in culture media that contains

cells creates a heterogonous radiation field, which is responsible for the local kinetics (Equation

6). Therefore, it is necessary to consider the effects of the light intensity available in each point

of the reactor on the growth rate.

- 27 -

For low cell densities (<0.1 g l-1) the Beer Lamber law describes the light attenuation by the

cells along the light path. However, for values higher than 0.1 g l-1, the Beer Lamber equation

overestimated the light attenuation by the cyanobacteria because it does not take into account

the scattering coefficient in the algae.

Light attenuation inside of liquid medium depends on two independent phenomena: absorption

by pigments and scattering by the whole cell. The scattering of radiant light energy makes the

mathematical description of light transfer extremely complex, since the available energy in any

point of the reactors derives from the main source of light and from all the directions, because

the light is scattered by the suspension.

One of the most used approaches to simulate light is the two-flux model proposed by Cornet.

This equation is the one-dimensional analytical solution for the radiative transfer model pro-

posed by (Cornet et al. 1998). The most accurate way to simulate the profile in the reactors is

the solution referring to the radiative transfer model in three dimensions, although it consumes

high computer resources.

Once the space light intensities are estimated, the average light intensity to every time step has

to be calculated. There are two methods to estimate this value: the first requires a simple calcu-

lation of the average light value in the light path, whereas the second method requires the con-

crete position of the cyanobacteria in the reactor for each time step. This position can be

matched with the light available in any point of the reactors (Figure 2-9). Although, the last

approach is the more suitable, the lagrangian trajectory information of the cells movement in

the reactor cannot be easily established. The movement of algal cells through light gradients is

very complex, but two recent approaches target this problem theoretically and experimentally.

(Perner et al. 2003) used (CFD) computational fluid dynamic modelling to predict particle tra-

jectories in a tubular photobioreactor equipped with a helical mixer (Zhang et al. 2012) used

(CARPT) computer-automated radioactive particle tracking to actually measure the trajectories

of a small radioactive particle in bubble column and airlift bioreactors (Figure 2-10)

- 28 -

Figure 2-9 Particle track at 0.5 m s−1 (top) superposed with a radial light distribution for two different

biomass concentrations of 0.5 and 2 g l−1 (bottom); the light absorption in the biosuspen-

sion was calculated by means of the hyperbolic model with a light intensity of

IO = 150 μE m−2 s−1 at the reactor surface (Perner-Nochta and Posten 2007).

Figure 2-10 Typical particle trajectories in the draft tube and the split columns - Only one recirculation

is shown for each reactor, while both the front and the top view of the trajectories are

shown respectively in the r–z plane and the cross-sectional plane. Solid lines inside the

figures represent the walls and internals (Luo et al. 2003).

2.4.2 Mass transfer in photobioreactors


et al. 1998) enters to cell by an active transport mechanist (Figure 2-11). The intracellular is

the dehydrated via the carbonic anhydrase to CO2, which is incorporate into the Calvin cycle

- 29 -

via the rubisco. This carbon is dispatched for the synthesis of all macromolecules in the cells

from the 3-phosphoglycerate as a key intermediate metabolite.

The bicarbonate present in the standard Zarrouk medium (around 117 mM) is consumed by the

cyanobacteria in order to support its growth. A theoretical value of mass conversion yields of

bicarbonates (2.57 g HCO3- g-1 biomass) was calculated by Cornet et al. it was calculated

based on stoichiometric equations for the biomass formation (Cornet et al. 1998) This yield can

be used to calculate the bicarbonate consumption during the batch cultivation.

With the aim of avoiding carbon limitations and taking advantage of the carbon dioxide cap-

tures, A. platensis cultures may be provided with additional carbon via carbon dioxide injection.

The carbon dioxide should be transferred from the gas phase to the liquid phase. Figure 2-11

shows the steps in the gas mass transfer processes. All these processes from gas phase to liquid

phase can be resumed in one empirical parameter, namely the volumetric mass transfer coeffi-

cient (kla). It is mainly controlled via the flow rate in bubble columns, air-lift and flat plate

photobioreactors (Kantarci et al. 2005). Several studies have demonstrated that the volumetric

mass transfer coefficient increases with gas velocity, gas density and pressure and decreases

with increasing solid concentration and liquid viscosity. Many empirical correlations to calcu-

late the kla are available in the literature and are summarized in the work of (Kantarci et al.

2005).

Figure 2-11 Gas mass transfer (left) and carbon dioxide in water equilibrium and carbon uptake at

different pH and mechanisms (right) (Markou et al. 2013).

Once the dissolved carbon dioxide reacts with the water, it forms other two main species, CO3-

2 and HCO3-. The equilibrium between them could be observed in Figure 2-11. Therefore, de-

pending on the pH, new bicarbonate can be generated from the injected CO2 to support the A.

platensis growth.

- 30 -

2.4.3 Modelling turbulence and light/dark cycles

The culture of cyanobacteria in a photobioreactor should be carried out at a relative high cell

density because this maximizes the biomass productivity and improves the economy in the re-

covering process. However, high cell density requires a level of mixing that ensures a proper

use of the available light, through the light/dark effect, and nutrients (Molina Grima et al. 2000).

In bubble columns and air-lift reactors, similarly to the mass transfer, mixing is controlled via

the flow rate (superficial aeration) and the rheological properties of the fluid.(Rodríguez et al.

2009) (Doshi and Pandit 2005; Pandit and Doshi 2005)

High level of turbulence is characterized by small eddies Figure 2-12. However, smaller eddies

can affect cyanobacteria size growth rate and morphology. The dissipation of the kinetic energy

of turbulence (the energy associated with turbulent eddies in a fluid flow) is the rate at which

the turbulence energy is absorbed by breaking the eddies down into smaller and smaller eddies

until it is ultimately converted into heat by viscous forces. It is expressed as the kinetic energy

per unit mass per second, with units of velocity squared per second (m2 s-3). The damage on the

algae is caused by the shear stress (𝜏), which expresses the parallel force on the surface of the

cell. This force originates from the movement transfer (turbulence) and its value can be calcu-

lated with the following formula that relates energy dissipation rate 𝜀, cyanobacteria diameter

𝑑𝑝 and micro eddy length 𝜆. Therefore, the estimation of energy dissipation rate by experi-

mental technics or simulations is necessary.

There is an evidence that the hydrodynamic shear stress (τ), is the cause of cell damage and

consequently, it mediates the production of reactive oxygen species (ROS) and lipid oxidation

within cells (Rodríguez et al. 2009). For example, the shear sensitivity dinoflagellate Protocer-

atium reticulatum shows agitation-associated shear stress damage threshold lower than 0.16

mPa (Rodríguez et al. 2009). Cyanobacterium are more resistant that dinoflagelletes. In A.

platensis different values that ranged from 0.2 to 1 Pa have been reports as critical shear.

(Mitsuhashi et al. 1995)

Figure 2-12 Schematic of eddies – The higher the energy dissipation rate, the smaller the eddies size

(Sokolichin and Eigenberger 1999).

- 31 -

However, as it was already mentioned, high turbulence is necessary in order to achieve a better

light usage. It is also important to highlight that cyanobacteria, which grow in a photobioreactor

are exposed to natural dark and light cycles as a consequence of the biomass increase in the

photobioreactor. Due to the self-shading effect among algal cells, the light regime inside an

outdoor photobioreactor is characterized by a light gradient with full light at the light-exposed

surface (photic zone) and total darkness in the interior of the photobioreactor (aphotic zone).

The scale-up of photobioreactors cannot be properly done because the light distribution within

the culture is non-homogenous as consequence of this self-shading effects (Camacho et al.

2011). Therefore, as it was already mentioned, light availability is influenced by the position of

the algae in the reactor. By increasing the level of agitation in the reactor, the light availability

per cell seems to increase as result of faster movement between the dark and light zones.

Some studies have found an increase in the biomass productivity in over-saturating illumination

with the increase of the frequency of the light-dark cycles (Luo and Al-Dahhan 2004; Molina

Grima et al. 2000). The magnitude of photo inhibition under continuous strong light is always

greater than under intermittent light. This explains the observation that intermittent light can

provide a higher productivity than the equivalent continuous illumination when the cyanobac-

teria are exposed to light intensities over the saturation point.

Phototrophic microorganism growth can only be sustained if the sufficient amount of reducing

equivalents and energy is produced in the light reaction of the photosynthesis. Under continuous

saturated light intensity, the electrons are produced in a higher rate that the consumption rate in

the dark reaction. The rests of them have to be stored in electron pool and an overflow of elec-

trons will cause photoinhbition, while in a proper intermittent light the electrons generated in

the flash time will be available during the light/dark (L/D) cycle in synchronized way (Rubio

et al. 2003; Wu and Merchuk 2002). However, the size of the electron pool seems only suffi-

cient to achieve full light integration at flash times around 1ms, whereas during longer flash

times the pool will overflow during the light phase, resulting in a loss of photosynthetic effi-

ciency and a decrease in the biomass yield on light (Luo and Al-Dahhan 2004)

As it was already mentioned, depending on the mixing characteristics of the culture suspension,

algal cells will be exposed to different series of light/dark (L/D) cycles. A highly defined mixing

pattern that produces light–dark cycles at a given frequency is required for enhancing produc-

tivity through the flashing-light effect. In contrast, it was concluded that chaotic mixing is not

as effective in enhancing productivity as is organized mixing (Degen et al. 2001; Zhang et al.

2012). Therefore, the uses of photobioreactors with baffles or helical flow promoters have been

- 32 -

proposed in order to create and organized mixing. To sum up, mixing seems to have positive

and negative effects in cyanobacteria culture and they have to be included and linked with the

cyanobacteria kinetics in order to generate a robust model that helps in the cyanobacteria pre-

dictions.

- 33 -

3 Project relevance

Arthrospira (Spirulina) platensis is a filamentous cyanobacterium that has become important

as a source for commercially produced nutraceutical compounds such as phycocyanin, exopol-

ysaccharides (Calcium-Spirulan) and gama-linoleic acid. To understand biological mechanisms

and to optimize production processes, rational design guided by experience is the most common

method currently used. However, experiments are time consuming and expensive and gener-

ally, they generate noisy data. Therefore, predicting the behavior of cyanobacteria during the

culture processes under different culture conditions is highly desirable for both commercial and

scientific reasons and can be an aid in the design of experiments

In batch, the rate of overproduction of metabolites by cyanobacteria is limited or activated by

the depletion of required substrates or by the accumulation of metabolic products and inhibitors.

Therefore, it becomes imperative to identify the parameters that have a significant impact on

product production and to create a mathematical model that can assist the investigation, opti-

mization and scale up of A. platensis growth and phycocyanin and exopolysaccharide produc-

tion in batch cultivation. Although, several mathematical models have been built to simulate

the growth of A. platensis (Cornet et al. 1992; Levert and Xia 2001), most of these studies have

been validated just for low cell densities (<1g l-1) and limitations of carbon or nitrate has been

neglected. In addition, the product formation kinetic and in particular the one for phycocyanin

and exopolysaccharides formation have been barley addressed.

The aim of this study is create a model for higher biomass conditions of A. platensis in bubble

column and then use the model to predict the A. platensis in different cultivation scenarios.

Once the biomass growth is validated, the model will be used to investigate phycocyanin and

exopolysaccharide formation during the batch cultivation under different culture conditions. In

addition, the model will be used to explore a possible controlling nutrient strategy to enhance

the phycocyanin and exopolysaccharide production.

- 34 -

4 Proposed mathematical model

In order to simulate the growth of A. platensis, a macro-scale mathematical model was devel-

oped in two stages (Figure 4-1). The first block incorporates the characterization of the photo-

bioreactor features (i.e. light attenuation, energy dissipation rate and mass transfer coefficient)

by using CFD or/and empirical correlations. This information was then used in the second stage

as input for the kinetic model, which was composed with different equations such as the nitrate

supply, biomass formation and the product formation.

Figure 4-1 Graphical structure of the mathematical model- The model was divided in photobioreac-

tor characterization block and cyanobacteria kinetic model block.

Photobioreactor features Energy dissipation rate

Mass transfer coefficient

Nutrients supply

Biomass formation

Product formation

Light supply

Kinetic model

- 35 -

4.1 Photobioreactor characterization

4.1.1 Energy dissipation rate estimation by using CFD

The energy dissipations rates (𝜀) in the photobioreactor were found by using CFD. In a previous

work, a bubble column with 0.10 m diameter and 0.45 m height and clear liquid height of 0.30

m was simulated using an Eulerian model and the k-𝜀 model for turbulence. The commercial

software Fluent was used to solve the model. All simulation conditions were validated against

the experimental mixing time and gas hold-up (Lobatón et al. 2011). For this work, the same

model and simulation conditions were used for a bubble column with different dimensions:

0.05 m diameter and 0.675 m height. The clear liquid height was 0.45 m and the bubble column

was aerated by air through a sparger (Figure 4-2). The volumetric flow rate was varied in the

range of 1 to 5 vvm. Once the simulations reached the steady state, the energy dissipation rate

ε was obtained as the average of all grid points.

Figure 4-2 Bubble column mesh to perform the CFD simulations - 0.05 m diameter and 0.675 m

height.

Kolmogorov’s theory describes how energy is transferred from larger to smaller eddy, how

much energy is contained by eddies of a given size and how much energy is dissipated by eddies

of each size. Furthermore, the smaller the eddy, the more damage on the cell occurs. The dam-

age on the algae is caused by the shear stress, which expresses the parallel force on the surface

- 36 -

of the cell. This force originates from the movement transfer (turbulence) and its value can be

calculated with the following formula that relates energy dissipation rate 𝜀, cyanobacteria di-

ameter 𝑑𝑝 and micro eddy length 𝜆.

Equation 11: 𝜆 = (𝜇𝑙

𝜌𝑙)

3

4. 𝜀−

1

4

Equation 12: 𝜏 = 0.0676. (𝑑𝑝

𝜆) . (𝜌𝑙. 𝜇𝑙. 𝜀)0.5

4.1.2 Light absorption

Light was modeled with the radiative transfer model (one-dimensional two-flux model)

(Cornet et al. 1998) for a bubble column with a radius of 0.025 m:

Equation 13: I(r, t) = Is ×1r

R

×2∗cos h(δ×

r

R)

cos h(δ)+α×sin h(δ)

The coefficients α and δ show the dependence of light attenuation from the biomass

concentrations and from the phycocyanin mass fractions:

Equation 14: α = [Ea∗(Zpc+0.009)

(Ea∗(Zpc+0.009)×+Es)]

1/2

Equation 15: δ = [𝐸𝑎 ∗ (Zpc + 0.009) + 𝐸𝑠] × xt × α × R

4.1.3 Mass transfer coefficient estimation

The mass transfer coefficient was calculated by using an empirical correlation for bubble col-

umns proposed by (Kantarci et al. 2005).

Equation 16: 𝑘𝑙𝑎 = 0.467 × 𝑣𝑔0.88

Equation 17: 𝑣𝑔 =𝑄

𝐴𝑐𝑠

4.2 Macroscale mathematical model

A mathematical model for A. platensis growth and phycocyanin formation was developed under

the following assumptions:

1. A well-mixed homogeneous model for bubble columns operated in batch mode for the

liquid phase and continually for the gas phase. The depletion of carbon dioxide in the

gas phase was neglected due to short gas residence time in the short-sized bubble col-

umn.

- 37 -

2. The relationship between the CO2 partial pressure and its equilibrium in the liquid phase

was simplified with Henry´s law. Consequently, the changes within the dissolved car-

bon dioxide concentration were modeled using the following equation:

Equation 18: 𝑑𝑐

𝑑𝑡= 𝑘𝑙𝑎 × (

𝑃×𝐶𝐶𝑂2

𝐻𝐶𝑂2

− 𝑐)

3. The changes in the bicarbonate depend on the rate of microalga consumption:

Equation 19: 𝑑𝑏

𝑑𝑡=

𝑑𝑐

𝑑𝑡− 𝑦𝑏

𝑥

×𝑑𝑥

𝑑𝑡

4. In cyanobacteria cultures, the changes in pH occur mainly as a result of carbon con-

sumption. However, the pH variation derived from other nutrients or degradation of the

excreted metabolites can be neglected (Camacho Rubio et al. 1999). The pH calcula-

tions were based on the Henderson-Hasselbalch equation:

Equation 20: 𝑝𝐻 = 𝑝𝑘 + 𝑙𝑜𝑔 ([𝑏]

[𝑐])

5. The biomass was modeled using a kinetic Monod model with light, bicarbonate and

nitrate as limitations:

Equation 21: 𝑑𝑥𝑎

𝑑𝑡= µmax (𝑝ℎ) ×

𝐼

𝐼+𝐾𝑖+𝐼2

𝐾𝑖𝑝

×𝑏

𝑏+𝐾𝑏×

𝑛

𝑛+𝐾𝑛× 𝑥𝑎 − 𝑀𝑒 × 𝑥𝑎

6. 𝑀𝑒 is a maintenance constant, which accounts for cellular damage due to adverse envi-

ronments, i.e. high shear stress. Wu and Merchuk found that the shear stress below crit-

ical level does not have any effect, whereas beyond the critical shear stress, the mainte-

nance term increases exponentially (Wu and Merchuk 2002). The CFD results in this

work shows by using 1 vvm, the maximum shear stress calculated was 0.3 Pa, which is

below the critical shear stress (1 Pa) reported in A. platensis. Therefore, the 𝑀𝑒 term

was neglected.

Equation 22: 𝑀𝑒 = 𝑀𝐸̅̅̅̅̅ × 𝑒𝑘𝑚(𝜏−𝜏𝑐)

7. Under nitrate limitation conditions, the continuous increase in the biomass concentra-

tion results from the accumulation of intracellular glycogen and exopolysaccharides.

Therefore, Equation 21 was modified to include this effect and the new equation was

described as follow:

Equation 23: 𝑑𝑥

𝑑𝑡= µmax (𝑝ℎ) × (

𝐼

𝐼+𝐾𝑖+𝐼2

𝐾𝑖𝑝

×𝑏

𝑏+𝐾𝑏×

𝑛

𝑛+𝐾𝑛+

𝑍𝑝𝑐

𝑍𝑝𝑐+𝐾𝑝𝑐×

𝐾𝑛

𝑛+𝐾𝑛) × 𝑥

8. According to Cornet et al, the mass fraction of phycocyanin (Zpc) remains constant in

the exponential or linear phase before the appearance of nutrient limitations (Cornet et

al. 1998). However, the experimental results in this work show that the mass fraction of

- 38 -

phycocyanin varies also in the exponential phase. Therefore, an equation that describes

the variation of the phycocyanin content in the cell was set up:

Equation 24: 𝑑𝑧𝑝𝑐

𝑑𝑡= 𝑅𝑝𝑐 × (

𝐾𝑙𝑖

𝐼+𝐾𝑙𝑖 − (

𝐼

𝐼+𝐾𝑙𝑖+

𝐾𝑛

𝑛+𝐾𝑛+

𝐾𝑏

𝑏+𝐾𝑏) ) × 𝑧𝑝𝑐

9. The nitrate and phosphate consumption was described using the following equation:

Equation 25: 𝑑𝑛

𝑑𝑡= −𝑌𝑛

𝑥×

𝑑𝑥

𝑑𝑡 Equation 26:

𝑑𝑝

𝑑𝑡= −𝑌𝑝

𝑥×

𝑑𝑥

𝑑𝑡

The model was simulated in Matlab and the parameters and conditions used for simulation are

shown in Table 4-1. The program code is included in the annex 1.

- 39 -

Table 4-1 Summary of parameters and conditions used in the simulation of Arthrospira platensis

Parame-

ter

Value

Range

Reference/

Comments

Parame-

ter

Value

range

Reference/

Comments

Consumption yields Io 60-600 experimental

condition

yb/x 2.57 (Cornet et al.

1998) No 0.91.82

experimental

condition

yn/x 0.45-

0.56

(Cornet et al.

1998) Xo 0.3-0.6

experimental

condition

yp/x 0.024 (Cornet et al.

1998) 𝑪𝑪𝑶𝟐

0.035-3 experimental

condition

Empirical parameters Bo 9.8 experimental

condition

Ki 72-120

(Chen et al. 2013;

Cornet et al.

1998) P 1.0

experimental

condition

Kip 400 (Del Rio-Chanona

et al. 2015) po 0.2

experimental

condition

Kil 28 estimated from

experimental data R 0.025

experimental

condition

Kb 3×10-6 (He et al. 2012) Bicarbonate/carbonate system con-

stants

Kn 1×10-2

5.3×10-3

(Cornet et al.

1998; Levert and

Xia 2001) pK 6.4

(Keymer et al.

2014)

kla 30

estimated from

empirical correla-

tion Cp 4.5

(Cornet et al.

1998)

µmax 0.073-

0.09

(Chen et al. 2013;

Cornet et al.

1998)

𝐇𝐂𝐎𝟐 30.04 (He et al. 2012)

Ea 4300

(Cornet et al.

1998)- estimated

from experi-

mental data

kca 8.9×10-

3 (Kern 1960)

Es 730

(Cornet et al.

1998)- estimated

from experi-

mental data

Empirical parameters for turbulence

Kpc 2×10-4 estimated from

experimental data 𝝉𝒄 0.3

(Wu and

Merchuk 2002)

Rpc 0.02-

0.038

estimated from

experimental data km

1.6×10-

3

(Wu and

Merchuk 2002)

Initial conditions 𝑴𝑬̅̅ ̅̅ ̅ 0.0-

0.059

(Wu and

Merchuk 2002)

- 40 -

5 Materials and methods

5.1 Microalga and media composition

Arthrospira platensis N-39 was obtained from NIES, Japan and pre-cultured in Zarrouk Me-

dium. The medium is composed of (per litre): 13.61g NaHCO3, 4.03g Na2CO3, 2.5g NaNO3,

1g K2SO4, 1g NaCl, 0.5g K2HPO4, 0.2g MgSO4.7H2O, 0.04g CaCl2.2H2O and 5ml of trace

metal solution, which consists in (per litre): 0.7g FeSO4.7H2O, 0.8g EDTA, 0.01g H3BO3,

0.002g MnSO4.4H2O, 0.001g ZnSO4.7H2O, 0.001g Co(NO3)2.6H2O, 0.001g Na2MoO4.2H2O.

5.2 Preculture

The preculture was grown in Erlenmeyer and constantly agitated by mixer plate at 110 rpm,

25°C and 40 µmol m-2 s-1. Cells in exponential growth were used for the following experiments.

5.3 Photobioreactor setup

A. platensis was cultivated in 1 L bubble column photobioreactor (Figure 5-1) (Walter et al.

2003) with a gas flow rate of 48 l h-1(1vvm) and 240 l h-1 (5vvm). The working volume was 0.8

L. The gas supplied was just air (0.035% (v/v) CO2) or mixtures of air and CO2, depending on

the experimental trial. All the trials were carried out at 30°C, form 60 to 700 µmol m-2 s-1 of

light intensity (illuminated by fluorescent lamps surrounding the reactor), initial biomass from

0.3 to 0.5g l-1 and gauge pressure of 1 bar. The inlet and outlet flows of the reactors were con-

nected with a micro filter (Midisart 2000 0.2µm PTFE, Sartorius Stedim) during the whole

cultivation time in order to keep the process in sterile conditions.

In preliminary trials, it was observed, that the liquid phase in PSM decreased in high flow rates.

Therefore, in order to reduce the liquid losses in the experiment with high flow rates, a humid-

ifier in the in inlet and a cooling system in the outlet were used.

- 41 -

Figure 5-1 Photobioreactor screening module (PSM) (Walter et al. 2003).

The function of the humidifier was to saturate the inlet air of water in order to decrease the

aqueous mass transfer inside the reactors, avoiding the reactor`s level variation. A heat plate

was used in addition to the display in order to optimize its efficiency. The mass of water in

saturated air increases by heating, therefore, its usage can ensure the maintenance of the reac-

tor's level. On the other hand, the overheating could increase the aqueous mass transfer from

the gas to liquid phase, enhancing the culture level.

A cooling system was set up after the outlet in order to condense the water in the gas phase,

avoiding the exit of water (Figure 5-2). The both inlet and outlet flows of the PSM were con-

nected with a micro filter (Midisart 2000 0,2µm PTFE, Sartorius Stedim) during the whole

cultivation time in order to keep the process in sterile conditions.

- 42 -

Figure 5-2 Cultivation set up with a gas humidifier in the inlet and cooling system in the outlet.

5.4 Analytic determinations

5.4.1 Biomass quantifications and pH measurements

Culture samples were daily collected from the bubble columns in sterile conditions. The bio-

mass (g l-1) was measured directly after a biomass lyophilisation process. In addition, the optical

density was determined by measuring the absorbance at a wavelength of 750 nm using a spec-

trophotometer (Specord 210-A, Shimadzu). The pH was also measured for every sample with

the pH-Electrode MP 220 (Mettler Toledo).

5.4.2 Nitrate determinations

The nitrate concentration in the medium was determined by a colorimetric method. The cali-

bration between the absorbance and nitrogen concentration was established using sodium ni-

trate as the standard and they were determinate by the absorbance at 210 nm.

5.4.3 Phycocyanin determinations

Phycocyanin measurements were done as following: samples were centrifuged at 4000 rpm, 25

°C and 10 minutes and the medium was discarded and the biomass was freeze-dried. Further-

more, the dried biomass was weighed and used to extract the phycocyanin content. 0.01g or

- 43 -

0.6g dried weight of the total dried biomass was disrupted by Tissuelyser at 50Hz for 3 minutes

and then suspended into 4 ml of 0.15 M phosphate buffer and 0.4ml of streptomycin. This

suspension was maintained at 4 °C for one hour and agitated every 30 minutes. After this pro-

cedure, it was centrifuged at 4000 rpm, 10 °C and 10 minutes and the blue supernatant was

collected. The absorbance was measured at wavelength of 615 nm or 620 nm and 652 nm using

a spectrophotometer (Specord 210-A, Shimadzu) and the phycocyanin concentration was cal-

culated according to the following formula (Patel et al. 2005):

Equation 27: Pc =OD615−0.474 ×OD652

5.34

5.4.4 Exopolysaccharides (EPS) quantifications

Exopolysaccharide measurements were done as following: samples were centrifuged at (4000

rpm, 25) and the supernatant was used to determinate EPS photometric with the anthrone assay

proposed by (Laurentin and Edwards 2003) using galactose solution as a standard reference.

Figure 5-3 shows a 96 well plate with the galactose solution (columns 1 to 3) and various su-

pernatant treated with anthrone in different trails (columns 4 to 12).

Figure 5-3 96-well plate with reference and culture supernatant in the different experiments.

- 44 -

5.5 Experimental plan

In order to test the model predictions, an experimental plan that changes one of the culture

conditions (carbon dioxide, flow rate and light) and leaves the others parameters fixed has been

developed Table 5-1. It is important to highlight that these experiments were planned in order

to test the model fitting performance. All the experiments were done by duplicate.

Table 5-1 Experimental plan

Parameter

Experiment

Carbon

dioxide

(%)

Flow

rate

(vvm)

Light

(µmol m-2 s-1)

Nitrate

(g l-1)

Bicarbonate/

Carbonate

(g l-1)

1. Flow rate 0.035 1 and 5 60 1.8 9.88/1.7

2. Carbon

dioxide 0.8 and 3 1 60 1.8

9.88/1.7

9.88/1.7

9.88/1.7

3. Light 3 1 60 and 120 1.8 9.88/1.7

4. Total nitrate

depletion 1.4 1 60 and 400 1.8 9.88/1.7

5. Nitrate vali-

dation 1.4 1 600 0.9 9.88/1.7

6. Nitrate Fed-

batch additions 1.4 1 600

0.9 + addi-

tions

9.88/1.7

- 45 -

6 Results

6.1 Photobioreactor computational fluid dynamics characterization:

Energy dissipation rates and liquid velocities

The simulation of computational fluid dynamics (CFD) in the photobioreactor (Figure 6-1)

shows that the average energy dissipation rate 𝜀 has slightly increased from 0.025 to 0.03 m2 s-

3. This is the result of the increase in the gas flow rate from 1 to 2 vvm. Subsequently, 𝜀 shows

a drastic increase to 0.15 m2 s-3 for a flow rate of 5 vvm.

Figure 6-1 Average energy dissipation rates and average fluid velocities for different gas flow rates

in the photobioreactor. Each data point is an average from all of the local values in the

mesh.

One of the advantages of using CFD is the possibility to visualize the local hydrodynamic val-

ues, which supply detailed information at every point of the photobioreactor instead of an av-

erage global value. Figure 6-2 shows the local 𝜀 in a front view of the reactor for volumetric

flow rates of 1 vvm and 5 vvm. As we see, at 1 vvm all the local values are lower than 0.1 m2

s-3. However, at 5 vvm there are two zones: one with local values less than 0.1 m2 s-3 and another

with 𝜀 values that range from 0.1 to 0.5 m2 s-3. In other words, by using 5 vvm, the higher 𝜀

seems to be associated with the bubble plume. These results reflect the literature, which states

that in bubble columns 𝜀 commonly varies within the range of 0.1 to 0.4 m2 s-3(Rodríguez et al.

1 2 3 4 50

0.05

0.1

0.15

vvm

1 2 3 4 50.05

0.1

0.15

0.2

Energy dissipation rate m2 s

-3

Liquid velocity m s-1

- 46 -

2009) and in stirred tanks the local 𝜀 is higher in the vicinity of impellers (Rodríguez et al.

2009) than in the bulk of the reactor.

Figure 6-2 Local energy dissipation rates in a front view of the in the photobioreactor for different

gas flow rates (1 vvm and 5 vvm) after 40 s of simulation time.

The damage to the algae is caused by the shear stress which expresses a parallel force on the

surface of the cell. This force originates from the movement transfer (turbulence) and its value

can be calculated using the formula presented above in section 4.1 that for the energy dissipa-

tion rate 𝜀, micro eddy length (𝜆) and cyanobacteria size (𝑑𝑝). The cyanobacterium size was

assumed to be 200 µm. Using 1 vvm, the maximum shear stress calculated with these equations

was 0.3 Pa, which is below the critical shear stress reported in A. platensis (see section 2.4.1).

As seen in Figure 6-2, in the green, yellow and red zones, 5 vvm generates higher energy dissi-

pation rates, which imply higher shear stresses. The values for the shear stress range from 0.3

to 1.34 Pa in the bubble plume area. This result is analyzed later in this work in the cultivation

section.

- 47 -

Along with better mass gas transfer, one of the advantages of increasing the flow rate is that the

cyanobacteria respond better to the light and dark cycles in the reactor. Assuming that cyano-

bacteria are particles that move with the fluid, the liquid velocities and algae trajectories deter-

mine the time of the transition of cyanobacteria between the illuminated and dark zones. Fur-

thermore, the availability of light to the entire cell wall is guaranteed by higher liquid velocities,

which trigger a faster algal rotation. Figure 6-3 shows that from the lowest flow rate, 1 vvm, to

the highest, 5 vvm, the average liquid velocities range from 0.05 to 0.2 m s-1, and displays the

local fluid velocities in a front view for the volumetric flow rates of 1 vvm and 5 vvm. This

hydrodynamic information (energy dissipation rates and liquid velocities) from the photobiore-

actor-screening module (Figure 5-1) is discussed later along with the cyanobacteria cultivation

results in the section “influence of flow rate on biomass”.

Figure 6-3 Local liquid velocities in 1 vvm and 5 vvm in a front view of the in the photobioreactor

for different gas flow rates (1 vvm and 5 vvm) after 40 s of simulation time.

- 48 -

6.2 Biomass cultivation results

Mathematical models are useful tools to predict the behavior of complex systems such as cya-

nobacteria batch cultivation. In this type of cultivation, the nutrients are depleted during the

process, leading to different cyanobacteria compositions. Nonetheless, before a mathematical

model can be applied in practice, it should first be verified against experimental data. In this

work, the proposed mathematical model was tested with the experimental biomass concentra-

tions for the culture conditions indicated in Table 5-1

After the model was validated, it was used to investigate phycocyanin and exopolysaccharide

formation during batch cultivation. It was also used to formulate a possible strategy to enhance

phycocyanin production.

6.2.1 Gas flow rate experiments and model validation

The flow rate validation was carried out with a fixed carbon dioxide concentration of 0.035%

and an illumination of 60 µmol m2 s-1. Two gas flow rates (1 vvm and 5 vvm) were experimen-

tally tested and the experimental biomass concentrations were used to test the model fitting

performance. The same experimental conditions presented in Table 5-1 were used as initial

conditions in the simulations. The increase in flow rate from 1vvm to 5 vvm resulted in an

increase in the growth rate and final biomass of A. platensis (Figure 6-4).

The model simulations (solid and dotted lines) are able to capture this trend. In addition, Figure

6-5 shows the predicted internal light in the reactor. As we see, after 100 hours of cultivation

there is a dark zone that represents almost 60% of the total reactor volume and an illuminated

area that represents the remaining 40% of the reactor volume. Figure 6-5 shows that at 5 vvm

the light is slightly more attenuated as a result of higher growth compared to the 1 vvm reactors.

The mass transfer coefficient (kla) expresses the rate at which CO2 is converted from the gas

phase to the liquid phase. An increase in the mass transfer coefficient along with the increase

of the flow rate is observed in bubble columns (Kantarci et al. 2005). Furthermore, Figure 6-6

shows that the pH increased during the cultivation and moved out of the optimal range (8.5-

9.5). However, this increase was less pronounced with the volumetric flow at 5 vvm.

- 49 -

Figure 6-4 Simulated and experimental growth of A. platensis in different gas flow rates. The lines

show the results from the simulation of the following conditions: 60 µmol m-2 s-1, 0.035 %

carbon dioxide, 30°C, which corresponds to the same experimental conditions implement

in the PSM cultivation.

Figure 6-5 Light attenuation after 100 hours of cultivation in different gas flow rates. The lines show

the results from the simulation of the following conditions: 60 µmol m-2 s-1, 0.035 %

carbon dioxide, 30°C, which corresponds to the same experimental conditions imple-

ment in the PSM cultivation.

- 50 -

Figure 6-6 Simulated and experimental pH in different gas flow rates. . The lines show the results

from the simulation of the following conditions: 60 µmol m-2 s-1, 0.035 % carbon di-

oxide, 30°C, which corresponds to the same experimental conditions implement in the

PSM cultivation.

6.2.2 Carbon dioxide experiments and model validation

The carbon dioxide validation was carried out with a fixed volumetric flow rate of 1 vvm and

60 µmol m-2 s-1. In the first trial, a CO2 concentration of 3 % and 0.035 % was simulated and

tested against experimental biomass concentrations (Figure 6-7). There was found to be a good

correlation between the model and the experimental data. Figure 6-7shows a faster growth and

two-fold increase in the final biomass in the cultivation with 3% CO2 compared to 0.035 %

CO2. This can be explained by the high pH in the cultivation with 0.035 % CO2 (Figure 6-7)

compared to that in the 3 % concentration.

- 51 -

Figure 6-7 Simulated and experimental biomass in 3 % and 0.035 % carbon dioxide (Top) Simulated

and experimental pH at 3 % and 0.035 % carbon dioxide (Bottom). The lines show the

results from the simulation of the following conditions: 60 µmol m-2 s-1, 1 vvm, 30°C,

which corresponds to the same experimental conditions implement in the PSM cultiva-

tion.

After further simulations, an optimal carbon dioxide concentration of 0.8 % was chosen for the

next experiment, since the model forecast that this concentration would lead to an optimal pH

of approximately 9.3.

Figure 6-8 shows the growth curve using 3 % and 0.8 % CO2. The same biomass growth was

found in both cases. In addition, Figure 6-8 shows a pH of approximately 9.5 at 0.8 % and 8.5

at 3 %. Figure 6-9 shows the predicted dissolved carbon dioxide using 3 % and 0.8 %. In both

cases, there is an increase until it reaches a steady state.

- 52 -

Figure 6-8 Simulated and experimental biomass in 3 % and 0.8 % of carbon dioxide (Top) Simulated

and experimental pH at 3 % and 0.8 % carbon dioxide (Bottom). The lines show the

results from the simulation of the following conditions: 60 µmol m-2 s-1, 1 vvm, 30°C,

which corresponds to the same experimental conditions implement in the PSM cultivation

Figure 6-9 Simulated dissolved carbon dioxide in 3 % and 0.8 % of carbon dioxide. The lines show

the results from the simulation of the following conditions: 60 µmol m-2 s-1, 1 vvm,

30°C, which corresponds to the same experimental conditions implement in the PSM

cultivation.

- 53 -

6.2.3 Light intensity experiments and model validation

In order to test the model’s capability and to predict the A. platensis growth when the light

increases, the following experiments were conducted. A simulation and experimental trial with

light intensities of 60 and 120 µmol m-2 s-1 were carried out and an optimal carbon dioxide

concentration of 1.4% was calculated in order to maintain theoretically enough carbon and a

pH of 9.0. Figure 6-10 shows the predictive capability of the model based on the results from

the increase in light intensity. At light intensities of 120 µmol m-2 s-1, the biomass productivity

was two times greater than the experimental results at 60 µmol m-2 s-1.

Figure 6-10 Simulated and experimental growth at different incident light intensity on PSM surface.

The lines show the results from the simulation of the following conditions: 1.4 % carbon

dioxide, 1 vvm, 30°C, which corresponds to the same experimental conditions implement

in the PSM cultivation.

Figure 6-11 shows the simulated results from light attenuation using different incidents light

intensity on PBR surface. At the end of the cultivation, it can be observed that the reactor with

120 µmol m-2 s-1 has slightly increased in light, perhaps as a consequence of the nitrate deple-

tion, which may generate changes in the cyanobacteria harvesting proteins. These phenomena

- 54 -

will be addressed in more detail in the section regarding phycocyanin biosynthesis. This un-

physical rise in the light supply generates the unreal increment in the simulated biomass by

using 120 µmol m-2 s-1 (Figure 6-10). In sum, the use of high light intensities leads to higher

growth rates coupled with faster nutrient consumption.

Figure 6-11 Internal light intensity at different incident light intensity on PSM surface. The lines show

the results from the simulation of the following conditions: 1.4 % carbon dioxide, 1 vvm,

30°C, which corresponds to the same experimental conditions implemented in the PSM

cultivation.

Figure 6-12 shows the predicted nitrate consumption in the cultivation. The main nitrogen

source for A. platensis is the nitrate (NO3-) ion, which is 1.8 g l-1 (29 mM) in the standard

spirulina medium. According to the model, nitrate depletion occurs after 250 hours in the cul-

tivation with 120 µmol m-2 s-1, whereas in the cultivation with 60 µmol m-2 s-1 some nitrate still

remains after 300 hours of cultivation. Different nutrient concentrations over the course of cul-

tivation can lead to different cyanobacteria compositions. This is discussed below in the section

about phycocyanin formation and degradation.

- 55 -

Figure 6-12 Simulated nitrate at different incident light intensity on PSM surface. The lines show the

results from the simulation of the following conditions: 1.4 % carbon dioxide, 1 vvm,

30°C, which corresponds to the same experimental conditions implement in the PSM

cultivation.

6.3 Product formation results

Figure 6-13 shows a comparable phycocyanin mass fraction using various concentrations of

3 %, 0.8 %, 0.035 % CO2 in the experiments conducted at 60 µmol m-2 s-1. The figure shows

that when using low light intensities (60 µmol m2 s-1) phycocyanin increases in the cell until

150 hours, at which point it reaches a steady state. Furthermore, all carbon dioxide concentra-

tions present the same trend in the growth curve.

It is important to highlight that at the beginning of the cultivation some phycocyanin degrada-

tion must be accounted for by the photoadaptation process described above. Thus, collecting

data at the beginning of the experiment impeded the analysis in the first 50 hours. However,

this issue was overcome in a further experiment that is presented and discussed in sections 6.5

and 7.7.

- 56 -

Figure 6-13 Experimental phycocyanin mass fractions at different carbon dioxide concentrations. Ex-

perimental conditions implemented in the PSM cultivation: 1 vvm, 30°C.

A. platensis was cultivated to examine the effect of this factor on growth and product generation

at a higher light intensity. A trial experiment with a light intensity of 400 µmol m-2 s-1 was set

up. Figure 6-14 shows a comparable phycocyanin mass fraction using 60 µmol m-2 s-1 and 400

µmol m-2 s-1, with a higher phycocyanin mass fraction in the first case. At 60 µmol m-2 s-1, the

figure shows that phycocyanin content increased in the cell and after 180 hours its production

stopped and remained stable until the conclusion of the cultivation. Meanwhile, with a light

intensity of 400 µmol m-2 s-1, phycocyanin also built up in the cell, but after 96 hours its pro-

duction stopped it began to degrade.

- 57 -

Figure 6-14 Experimental phycocyanin mass fractions at different incident light intensity on PSM

surface. Experimental conditions implemented in the PSM cultivation: 1.4 % carbon di-

oxide, 1 vvm, 30°C.

Figure 6-15 shows that at light intensities of 400 µmol m-2 s-1, the biomass productivity was four

times greater than the experimental results at 60 µmol m-2 s-1. It is important to highlight that

(Xie et al. 2015) reported photoinhibition in the cultivation of A. platensis at 450 µmol m-2 s-1.

This was supported by an observation of a change in color from green to white during the first

three days of the cultivation. However, in this work, there was no evidence of photoinhibition.

The discrepancy in these results can be explained by the different initial cell densities used in

both cases. Figure 6-15 also shows the predicted nitrate consumption in the cultivation using

the model. According to the model, nitrate depletion occurs after 100 hours in the cultivation at

400 µmol m-2 s-1, whereas in the cultivation at 60 µmol m-2 s-1 some nitrate still remains after

250 hours of cultivation. In order to verify the hypothesis concerning nitrate limitation, a new

experiment was set up and is described in the next section.

- 58 -

Figure 6-15 Experimental biomass at different incident light intensity on PSM surface. (Top) Simu-

lated nitrate at different incident light intensity on PSM surface. (Bottom). The lines show


30°C, which corresponds to the same experimental conditions implement in the PSM cul-

tivation

6.4 Nitrate model validation

In order to test the model’s prediction capability regarding nitrate consumption, an experiment

with 600 µmol m-2 s-1 of light intensity and an initial nitrate concentration of 0.9 g l-1 was set

up. Figure 6-16 shows the experimental and predicted biomass results for cultivation with at

600 µmol m-2 s-1. The model shows good agreement with the experimental data at the beginning

of the cultivation. Furthermore, the model demonstrates good agreement when predicting the

nitrate concentration in the medium. Figure 6-16 shows that nitrate depletion occurs after 50

hours in the cultivation. The theoretical yield of 0.48 g NO3- per g biomass is accurate to calcu-

late the nitrate consumption.

- 59 -

Figure 6-16 Experimental and simulated biomass in 600 µmol m-2 s-1 (Top) Experimental and Sim-

ulated nitrate in 600 µmol m-2 s-1 (Bottom). The lines show the results from the simulation

of the following conditions: 1.4 % carbon dioxide, 1 vvm, 30°C, 0.9 g l-1 initial NO3-,

which corresponds to the same experimental conditions implement in the PSM cultiva-

tion.

6.5 Effects of nitrate on phycocyanin and exopolysaccharides production

A new experiment was set up with a light intensity of 600 µmol m2 s-1 and two nitrate condi-

tions. The first was named ‘the control’ and had an initial nitrate concentration of 0.9 g l-1. The

second had the same initial nitrate concentration but different nitrate feeds at 29, 52, 76, 99

hours. Nitrate, phycocyanin and exopolysaccharides were measured during the cultivation.

Figure 6-17 shows the biomass in the control and in the fed batch cultivation. Between 50 to

150 hours, a higher biomass was found in the control, although the measurements show that the

nitrate was completely depleted. Moreover, in the control experiment the nitrate depletion oc-

curs after 50 hours in the cultivation (Figure 6-17).

- 60 -

Figure 6-17 Experimental biomass in control and nitrate Fed batch experiment (Top) Experimental

and simulated nitrate in control and nitrate Fed batch experiment (Bottom) The lines show

the results from the simulation of the following conditions: 600 µmol m-2 s-1, 1.4 % carbon

dioxide, 1 vvm, 30°C, 0.9 g l-1 initial NO3-, which corresponds to the same experimental

conditions implement in the PSM cultivation.

In addition, the slightly lower biomass concentration in the fed batch was probably also as a

result of the higher phycocyanin concentration (Figure 6-18). According to (Cornet et al. 1992)

the higher the phycocyanin mass fraction, the higher the light absorption, which consequently

leads the light being highly attenuated. In the nitrate fed batch experiment, the phycocyanin

mass fraction was higher than in the control experiment between 50 to 20 hours of cultivation

(Figure 6-18). Therefore, the higher light attenuation can be the explanation to the lower bio-

mass in the fed batch experiment between these period of time.

- 61 -

Figure 6-18 Phycocyanin mass fractions in control and nitrate fed-batch experiment. Experimental

conditions implemented in the PSM cultivation: 600 µmol m-2 s-1,1.4 % carbon dioxide,

1 vvm, 30°C, 0.9 g l-1 initial NO3-.

With the nitrate additions, the phycocyanin mass fraction was expected to remain at least con-

stant during the whole cultivation. However, the phycocyanin decreased after the third addition

(76 hours), but at a slower rate than in the control experiment (Figure 6-18). Limitations of other

macronutrients such as carbon may be the reason for the phycocyanin decrease. This is dis-

cussed further in section 7.5. Nonetheless, phycocyanin mass fractions and productivities were

enhanced by the nitrate additions as compared to the control cultivation.

Concerning the exopolysaccharide (EPS) formation, the nitrate addition experiment (Figure

6-19) shows a greater EPS mass fraction than in the control experiment. In the control experi-

ment the EPS mass fraction (around 20 mg g-1 of biomass) remains stable throughout the culti-

vation, whereas in the nitrate addition experiment there was a slight increase from 25 mg g-1

to 50 mg g-1 until 220 hours of cultivation. After this time, there was an incremental increase

in the EPS from 50 mg g-1 to 100 mg g-1. Multiplying the mass fraction by the biomass in the

reactor shows the same trends as in the mass fraction results. A maximum space-time yield of

(32 mg l-1 d-1)) was reached in the experiment with nitrate additions.

- 62 -

Figure 6-19 Exopolysaccharides mass fractions in control and nitrate fed-batch experiment. Experi-

mental conditions implemented in the PSM cultivation: 600 µmol m-2 s-1,1.4 % carbon

dioxide, 1 vvm, 30°C, 0.9 g l-1 initial NO3-

6.6 Phycocyanin kinetic model

Figure 6-20 shows the simulations and the experimental results for the phycocyanin mass frac-

tion in different light intensity and nitrate conditions. In all trials, simulations show a decrease

in phycocyanin up to 50 hours as result of high illumination at the beginning of the cultivation.

After 50 hours, there is an increase in the mass fraction that concludes when the nitrate and/or

carbon limitation appears. The kinetic mechanism proposed in this work for phycocyanin for-

mation simulated the experimental results in a suitable manner. A detailed discussion of the

kinetic model is presented in section 7.7.

- 63 -

Figure 6-20 Experimental and simulated phycocyanin mass fractions in different experimental condi-

tions. The lines show the results from the simulation of the following conditions:,1.4 %

carbon dioxide, 1 vvm, 30°C, which corresponds to the same experimental conditions

implement in the PSM cultivation.

6.7 Literature data simulations

The model was used to simulate two experiments from the literature. The first, a pond cultiva-

tion (d = 5 m), was performed in a greenhouse facility at the Fisheries Science and Technology

Center, Pukyong National University, Goseong, South Korea (FSTC) from May 21st, 2014 to

June 24th, 2014. The microalgae were grown in SOT media at total volume of 15,000 L, inoc-

ulated with 1000 L of subculture from hanging bag cultivation, at an initial pH of 8.68 under

- 64 -

sunlight with an average of approximately 110 µmol m−2 s−1 (a sunny day), at 20 – 35 °C with

an initial OD at 560 nm of 0.05 and a stirring speed of 10 - 15 min−1.(Reichert 2016). Figure

6-21 shows the experimental results compared to the simulation data. Although the model is

able to capture the growth curve trend, some experimental points between 200 hours and 500

hours are out of the range of prediction. It is important to highlight that the model works with

constant values of light and temperature, which can explain the deviation with the experimental

results in outdoor cultivations. Furthermore, the model does not include water evaporation,

which may also have influenced the experiment results.

Figure 6-21 Simulated and experimental biomass from A. platensis (data from Reichert 2016). The

experimental points correspond to cultivation in an open pond with a diameter of 5 m.

The lines show the results from the simulation of the same experimental conditions ap-

plied by Reichert (2016).

The second experiment was a cultivation under control conditions performed by (Xie et al.

2015). The photobioreactor (PBR) used to cultivate the A. platensis was a 1-L glass vessel (15.5

cm in length and 9.5 cm in diameter) equipped with an external light source (14 W TL5 tungsten

filament lamps, Philips Co., Taipei, Taiwan) mounted on both sides of the PBR. The microalga

was pre-cultured and inoculated into the PBR with an inoculum size of 0.08-0.24 g l-1. The

- 65 -

PBRs were operated at 28 °C, pH 9.0, and an agitation rate of 400 rpm under a light intensity

of approximately 75-450 µmol m2 s-1. Serving as the sole carbon source, 2.5 % CO2 at 0.2 vvm

was fed into the microalgal culture continuously during cultivation (Xie et al. 2015). The bio-

mass simulation results (Figure 6-22) show good agreement until 350 hours of cultivation. After

that point, the model predicts an unphysical steep rise as result of the phycocyanin degradation.

The model was also able to predict the nitrate consumption (Figure 6-22) with a good correlation

between the experimental data and the model simulations. In addition, the model was used to

describe phycocyanin formation. Figure 6-23 shows an increase in the phycocyanin concentra-

tion up to 300 hours. After this point, phycocyanin starts to decrease as a result of the nitrate

limitations.

Figure 6-22 Simulated and experimental biomass from A. platensis (Experimental data from Jing

2015) (Top) Simulated and experimental nitrate concentrations (Experimental data from

Jing 2015) (Bottom). The lines show the results from the simulation of the same experi-

mental conditions applied by Jing (2015).

- 66 -

Figure 6-23 Simulated and experimental phycocyanin concentration from A. platensis (Experimental

data from Jing 2015). The lines show the results from the simulation of the same experi-

mental conditions applied by Jing (2015).

- 67 -

7 Discussion

7.1 Influence of flow rate on biomass

An increase in flow rate results in an increase in the growth rate and final biomass of A. platensis

(Figure 6-4). This might be caused by the equilibrium between the negative effects of shear

damage and the positive influence of the flow rate increase on light availability and mass trans-

fer of CO2, with the positive influence being predominant.

Even if the cyanobacteria are exposed to higher shear stresses at 5 vvm, this exposure is inter-

mittent. Figure 6-2 shows that there are two zones with different 𝜀 values: one that ranges from

0 to 0.08 m2 s-3 and another that displays the bubble plume zone with higher values. Therefore,

cyanobacteria are exposed to an intermittent 𝜀 due to the transition between these zones. Since

Molina et al. have found better growth of Protoceratium reticulatum with intermittent rather

than continuous 𝜀 exposure (García Camacho et al. 2007), this might result in an advantage for

A. plantesis as well. In sum, at values up to 1.34 Pa in cultivations of A. platensis, the biochem-

ical process in the cells may be affected by turbulence but without any serious damage. Never-

theless, light availability and pH control play a more important role within this process, mini-

mizing the effects of the possible damage caused by the turbulence.

When a culture is light-limited, higher liquid velocities can help the cyanobacteria move more

quickly through the light gradients. Consequently, by improving the fluid velocities, a faster

transition between the dark and the light zone is facilitated (Degen et al. 2001; Trujillo et al.

2008). In other words, it is possible that at high flow rates there is more illumination available

per cell than at lower flow rates. This analysis may explain the better growth results found in

the reactor with 5 vvm than in the one with a volumetric flow rate of 1, in which higher liquid

velocities are predicted (Figure 6-1 and Figure 6-2).

The frequency, which is the inverse of the time cycle through the light and dark zones (Figure

6-5), together with the time fraction in the light zone, have been regarded as the two main factors

in the light increase per cell in cyanobacteria culture, which leads to better growth. This effect

has been demonstrated by several experiments with laboratory equipment that control the fre-

quency and light time fraction (i.e. tours photobioreactor (Pruvost et al. 2008)). However, it is

challenging to characterize the frequency and the light time fraction in bubble columns due to

their chaotic hydrodynamic behavior.

- 68 -

Luo et al. used Computer Automated Radioactive Particle Tracking (CARPT) to track the light

history of particles (Luo and Al-Dahhan 2004). This work found an enhancement in the light

intensity per cell with the increase in the flow rate. This results from high frequency 𝑓, coupled

with higher times in the light zone ∅. In sum, a greater frequency and time fraction in the light

zone can lead to better growth at a flow rate of 5 vvm, as the specific growth rate seems to

depend on the light available per cell.

In addition to better light usage at 5 vvm, high flow rates also have positive effects on the pH

(Figure 6-6) by maintaining it close to optimal, as discussed in more detail below. In cyanobac-

teria, after the bicarbonate has passed through the cell membrane, the bicarbonate is reduced by

the enzyme carbon anhydrase (Figure 7-1). As a result, CO2 is incorporated into the algal me-

tabolism and hydroxyl (OH-) is excreted into the medium. A one-to-one stoichiometric rela-

tionship between the formation of OH- and the removal of (HCO3-) by the blue green-algae has

been reported (Miller and Colman 1980). Therefore, due to the removal of bicarbonate, the pH

increases when the dissolved carbon dioxide is low (Uusitalo 1996).

Figure 7-1 Graphical representation of the medium alkalization mechanism.

Adding carbon dioxide to the medium can shift the bicarbonate equilibrium equation to the left,

but first it must be dissolved in the medium and converted from the gas phase to the liquid

phase. High kla values mean faster dissolved carbon equilibrium, which can generate positive

effects in medium alkalization.

The high pH in both cases (Figure 6-6 ) can be explained by the low carbon dioxide concentra-

tion used in those experiments (0.035%). The same trend was also found by Zeng et al., who

found increases in the culture pH as a result of a continuous shift in the bicarbonate equilibrium

system (Zeng et al. 2012) by atmospheric carbon dioxide . Furthermore, the different alkaliza-

tion rate (Figure 6-6) between the two flow rates can be explained by a higher mass transfer and

better growth in the cultivation at 5 vvm. Higher mass transfer leads to higher dissolved carbon

dioxide and a good growth rate generates faster bicarbonate consumption. These two conditions

may be the reason for low velocities of medium alkalization at high flow rates (5 vvm).In sum,

high flow rates favor biomass growth in cultivations which are light-limited and/or cultivated

- 69 -

in atmospheric carbon dioxide concentrations (0.035%). This results from better light usage and

from maintaining the pH close to optimal.

Although increasing the flow rates seems to have a positive effect on growth, some of the bot-

tlenecks in the cyanobacteria production in closed photobioreactors are the power consumption

necessary for mixing and mass transfer, as well as the high investment capital necessary for the

photobioreactor. Increasing the flow rate from 1 to 5 vvm represents an increase in the energy

from 75 W m-3 to 250 W m-3 (Norsker et al. 2011). The energy costs for mixing have a strong

influence on the economics of microalgal biomass production in photobioreactors. For these

reasons, different strategies have been proposed to maintain a constant flow rate while also

increasing the cyanobacteria circulation or mass transfer coefficient. For instance, adapting a

static mixer has been reported to increase the biomass productivity of chlorella by 37%. This

can be explained by shorter light/dark cycles, which result from adapting the static mixer (Wu

et al. 2010). Furthermore, the use of membranes to increase the mass transfer coefficient has

also been examined as a possible strategy to achieve these goals for algae mass growth and cost

efficiency. Last but not least, another strategy is the use of incremental increases in the flow

rates, which has also been reported to have positive effects in cyanobacteria cultivations (Zeng

et al. 2012)

7.2 Influence of carbon dioxide on biomass

During cultivation with the normal fraction of carbon dioxide in the air (0.035%), slow growth

and low final biomass were found due to the high pH. As already explained, this high pH is a

consequence of medium alkalinisation. However, by using 3% CO2 the pH decreases (see Fi-

gure 6-7), perhaps as a result of the high partial pressure (Figure 6-9). Although the growth in

this case is better than in the experiment at 0.035%, the pH is lower than the optimal pH reported

in A. platensis cultivations.


et al. 1998). In the standard spirulina medium, the bicarbonate concentration in the medium is

approximately 9.8 g l-1. This concentration seems to be sufficient to support growth, but carbon

dioxide must be used to maintain the optimal pH. The modification of the bicarbonate/carbonate

equilibrium has an impact on the cyanobacteria pH, which can influence the cyanobacteria

growth rate, as seen below in Table 7-1:

- 70 -

Table 7-1 Experimental growth rates for different pH in Arthrospira platensis.

Carbon dioxide

Concentration pH µ (h-1)

3 % 8.5 0.0080

0.035 % 11 0.0055

0.8 % 9.3 0.0095

After further simulations, an optimal carbon dioxide concentration of 0.8% was chosen for sub-

sequent experiments, since this concentration leads to an optimal pH of approximately 9.3 In

practice, this concentration actually leads to a marginal growth increase compared to the exper-

iment in which 3 % CO2 was used (Figure 6-8). These results confirm that the simulation of the

CO2 partial pressure in the cultivation can be used to optimize it without the need for test ex-

periments and online measurements. The use of higher CO2 concentrations in the inlet leads to

further reductions in the pH, which cause a decrease in the growth rate and increase the carbon

dioxide waste. For example, concentrations of 6 % (v/v) carbon dioxide have been tested with

A. platensis by using a carbon-free medium, with decreases in the pH down to 7.5 and non-real

carbon dioxide biofixation (De Morais and Costa 2007). In conclusion, an optimal carbon di-

oxide concentration must be established in order maintain the optimal pH and carbon supply

but without waste of carbon dioxide. A mathematical model can be helpful for this purpose.

7.3 Influence of light intensity on biomass and nitrate consumption

Among the environmental factors affecting the growth rate of cyanobacteria, light is frequently

limited. Figure 6-10 and Figure 6-15 show a better growth rate and final biomass concentration

with the increase in the light intensity. As was expected, high photon flux density leads to rapid

production of ATP and NADPH, and therefore to faster metabolic pathway rates (Klanchui et

al. 2012). Consequently, faster growth also generates faster nutrient consumption. As nitrate

compositions are linked to the growth rate by the yield of consumption, faster nitrate consump-

tion is expected.

According to the model (Figure 6-15), nitrate depletion occurs after 100 hours in the cultivation

with 400 µmol m-2 s-1, whereas in the cultivation with 60 µmol m-2 s-1 some nitrate still remains

after 250 hours of cultivation. By using 600 µmol m-2 s-1 and an initial nitrate concentration of

0.9 g l-1, the limitation begins after 50 hours (Figure 6-16). Different nutrient concentrations

- 71 -

during cultivation can also lead to different cyanobacteria compositions. This is discussed be-

low in the section on phycocyanin formation and degradation.

7.4 Model fitting performance for biomass

The model fitting performance was tested against several culture conditions. The model was

able to confirm external literature data and predict the biomass growth curve of A. Platensis

after the variations in the flow rate, initial carbon dioxide concentration and Incident light in-

tensity on PBR surface up to 600 µmol m-2 s-1. Table 7-2 shows the main output in biomass

formation with the modifications of flow rates, carbon dioxide concentrations and light input,

as well as the model mechanisms that control these outputs. The proposed model (Equations 18

to 26) adequately describes the biomass production and phycocyanin formation.

All Monod half saturation constants, Ki, Kip, Kb, Kpc and Kn, as well as the maximum specific

growth rate µmax were validated. The light intensity mechanism in the active biomass equation

(Equation 23) 𝐼

𝐼+𝐾𝑖+𝐼2

𝐾𝑖𝑝

properly describes in correct way the influence of light. In order to use

the model in a large scale photobioreactor, the light input (I) in each time-step must be corre-

lated with the cyanobacteria trajectories and not average values along the radius as was done in

this work. The absorption coefficient Ea is affected by phycocyanin and chlorophyll A, and in

the simulation in this work described the phycocyanin changes over time. However, chlorophyll

A was assumed to be constant throughout the cultivation. The model’s predictive capability can

be improved by including these additional factors.

Although only the theoretical yield of consumption for nitrate yn/x, was validated, it can be as-

sumed that the others, for bicarbonate yb/x and phosphate yp/x, are correctly calculated. The the-

oretical yields of consumption to enable the biomass calculation help in the overall prediction

of these macronutrients over the course of time.

It is important to highlight that in batch cultivations the kla can decrease over time, as the

viscosity in the medium increases. An increase in the viscosity due to the production of exopol-

ysaccharides in A. platensis cultivation has been reported. Unfortunately, the model proposed

in this work assumed a constant kla during the cultivation.

After depletion of nutrients, especially nitrate, a change in the biomass composition strongly

affects the predictions. Glycogen accumulation and gradual exopolysaccharide increase must

be accounted in to the biomass formation model. In this work, this effect is introduced in the

- 72 -

terms ( 𝑍𝑝𝑐

𝑍𝑝𝑐+𝐾𝑝𝑐×

𝐾𝑛

𝑛+𝐾𝑛 ) in Equation 23. The term

𝐾𝑛

𝑛+𝐾𝑛 correctly described the effect on biomass

growth due to the accumulation of glycogen and EPS in the cell wall as a result of the nitrogen

depletion. However, if cells do not divide, growth stops when cells reach their maximum size.

This is described by the term 𝑍𝑝𝑐

𝑍𝑝𝑐+𝐾𝑝𝑐 in Equation 23.

Table 7-2 Final remarks on model fitting performance.

Parameter Impact on biomass Mechanism

Increase of flow rates

up to 5 vvm Increase

Generating more light

availability per cell

and helping to stabi-

lize the pH

Increase of carbon diox-

ide concentrations up to

3%

Increase Helping to stabilize

the pH

Incident light intensity

on PBR surface up to

600 µmol m-2 s-1

Increase Higher internal light

intensities

7.5 Phycocyanin production: biosynthesis, steady state and degradation

For biotechnological application, the formation of the putative product is of great importance.

Thus, the evaluation of the parameters that influence product formation is the main goal of the

present work. The phycocyanin mass fraction in A. platensis cells varies from 5 to 20 mg g-1,

with productivities ranging from 10 to 125 mg l-1 d-1 (Chen et al. 2013; Xie et al. 2015). In

phototrophic cultivations, the light intensity strongly influences the cell growth of cyanobacte-

ria and it also changes the levels of light-related molecules in photosynthetic systems. Changes

in the nutrient conditions (light, nitrate, etc.) can lead to three possible stages in the phycocyanin

biochemical pathways: biosynthesis, degradation and steady state.

Figure 6-13 shows that phycocyanin is biosynthesized in the cell, probably due to the low light

conditions. Light intensity is one the most significant factors that influences the light harvesting

complex (phycobilisomes). However, its role is not yet fully understood, but perhaps is a re-

sponse of the photoadaptation processes in order to reach a new steady state. Nonetheless, an

alteration of the photosynthetic units at low or high light intensities has been reported (Rubio

et al. 2003).

Biosynthesis occurred in the experiment with 60 µmol m-2 s-1, perhaps due to the low light

intensities. Two conditions have been reported that trigger phycocyanin synthesis in the cell:

low nitrate (Xie et al. 2013)) and/or low light conditions (Takano et al. 1995). It has been shown

- 73 -

that low light intensities benefit phycocyanin formation. Takano et al. found that the maximum

phycocyanin mass fraction from the cyanobacterium synechococcus sp was at 25 µmol m-2 s-1

(Takano et al. 1995). In addition, they found that phycocyanin content decreases when the light

intensity passes this limit. However, the precise values of low light or low nitrate are not given

for A. platensis. If we make a comparison to the cyanobacterium synechococcus sp, a light value

between 5 and 25 µmol m-2 s-1 can be expected as the range of values that trigger the metabolic

pathway in phycocyanin biosynthesis.

As was mentioned before, phycocyanin was biosynthesized in the cell between 60 hours and

180 hours. After 180 hours the phycocyanin production stopped and the level remained stable

until the end of the cultivation. As the model predicts (Figure 6-12), nitrate is not depleted during

the course of the cultivation at 60 µmol m-2 s-1. Consequently, nitrate limitation can be elimi-

nated as a possible cause of the cessation of phycocyanin production.

According to the simulations, after 50 hours the light inside the reactor is lower than 10 µmol

m-2 s-1 (Figure 6-11). In addition, simulations also confirm that after 180 hours of cultivation,

80% of the internal light in the reactor is lower than the compensation point (4.5 µmol m-2 s-1)

for A. platensis. This may be the reason for the cessation of phycocyanin production and, there-

fore, of the photoautotrophic biomass production in the experiments with low light intensity

(Figure 6-13).

Some researchers have found that phycocyanin is biosynthesized during the exponential growth

phase (Chen et al. 2013; Xie et al. 2015), while others have found an entirely steady state in the

phycocyanin mass fraction. However, As Figure 6-13 shows, the phycocyanin mass fraction

was biosynthesized even during the exponential phase in the cultivation at 60 µmol m-2 s-1.

Phycocyanin was also built up in the cell during the experiment with high light intensity (400

µmol m-2 s-1) (Figure 6-14). However, in this case, the reason may have been that formation was

stimulated when a low level of nitrate was reached. The explanation for this is that at a low

concentration of nitrate, the nitrogen must first be used to synthesize light harvesting molecules

(Xie et al. 2015). Similarly, Xie et al. also reported that a higher lutein content was obtained in

a fed batch culture with a relatively lower concentration of nitrogen (Xie et al. 2013). In addi-

tion, Csőgör et al. found the same trend in the formation of phycoerythrin by Porphyridium

purpureum; in their research the pigment concentration rises when the specific growth rate be-

comes lower (Csőgör et al. 2001). This suggests that by maintaining a low but non-limiting

level of nitrogen, the accumulation of phycocyanin is enhanced.

- 74 -

Nevertheless, the phycocyanin production inside the cell ceased after 100 hours of cultivation

and the concentration began to decrease (Figure 6-14). Under complete nitrate limitation condi-

tions, it has been reported that phycocyanin is degraded. According to simulations made with

the model at 400 µmol m-2 s-1, nitrate should be entirely depleted and phycocyanin decline

should occur.

In experiments with the cyanobacterium synechococcus, Lau et al. found a loss of spectropho-

tometrically measurable phycocyanin, which began soon after the resuspension in a nitrate-free

medium (Lau et al. 1977). During nitrogen starvation, cyanobacteria may consume internal

stores of nitrogen to prolong their growth. In support of this, previous studies have shown that

glycogen is accumulated during nitrogen starvation. For example, Joseph et al. found that Syn-

echocystis sp. stores glycogen and degrades nitrogen-rich phycobilisomes, which results in the

loss of the pigment phycocyanin, a condition referred to as bleaching (Joseph et al. 2014).

Moreover, Hasunuma et al. also found an increase in internal glycogen from 10% to 60% and

a reduction in protein content from 42% to 15% of dry weight after 72 hours (Hasunuma et al.

2013). Additionally, they discovered that the glycogen produced during the depletion is bio-

synthesized with carbon atoms from proteins rather than CO2. Furthermore, Cornet et al. found

the same trend, confirming that as soon as nitrate is exhausted, phycocyanin begins to degrade

and is used to synthesize energy storage products such as glycogen. The size of cells continues

to increase but they no longer divide, which explains the continued increase in biomass (Cornet

et al. 1998).

The degradation of phycocyanin after nitrate limitation explains the results in the experiment at

400 µmol m-2 s-1, in which the biomass production was accelerated (Figure 6-15) and the phy-

cocyanin declined (Figure 6-14). In addition, in the fed batch and control experiment, phycocy-

anin degradation (Figure 6-18) took place due to the nitrate-limitation conditions, as predicted.

This degradation can explain the higher biomass in the control experiment (between 50 and 200

hours) compared to the experiment with nitrate addition (Figure 6-17). In addition, the slightly

lower biomass concentration (Figure 6-17) in the fed batch was probably also a result of the

higher phycocyanin concentration (Figure 6-18). According to (Cornet et al. 1998), the higher

the phycocyanin mass fraction, the higher the light absorption, which consequently leads to

more light being highly attenuated.

With the nitrate addition, the phycocyanin mass fraction was expected to remain at least con-

stant throughout the cultivation. However, the phycocyanin decreased after the third addition

(76 hours), but at a slower rate than in the control experiment (Figure 6-18). Based on the model,

- 75 -

bicarbonate limitations are proposed to be the reason for this decrease. This is discussed in more

detail in the next section. However, in the nitrate fed batch experiment, the phycocyanin mass

fraction and therefore the productivity remained at a higher concentration compared to the ex-

periment without nitrate addition (Figure 6-18).

7.6 Phycocyanin kinetic model

The wide range of phycocyanin productivity in the literature and in this work is explained by

the different experimental conditions. A mathematical model that comprises growth and phy-

cocyanin biosynthesis can explain the conditions that lead to these productivity levels. How-

ever, no kinetic model for phycocyanin biosynthesis currently exists.

Advanced structural models consider that the biomass can change its composition during the

time of cultivation as a result of changes in substrates. In particular, the phycocyanin mass

fraction 𝑧𝑝𝑐 has been shown in this work not to be constant, instead changing as a consequence

of the metabolic pathway shift. Structural models describe not only biomass kinetics, but also,

in particular, the product formation kinetics for transient operation, using a small set of param-

eters which often have a biological meaning. The semi-empirical kinetic law for phycocyanin

mass fraction (Equation 28) was used to simulate its formation.

Equation 28: 𝑑𝑧𝑝𝑐

𝑑𝑡= 𝑅𝑝𝑐 × (

Kli

𝐼+Kli − (

𝐼

𝐼+𝐾𝑙𝑖+

𝐾𝑛

𝑛+𝐾𝑛+

𝐾𝑏

𝑏+𝐾𝑏) ) × 𝑧𝑝𝑐

This kinetic model was linked to the core model and solved simultaneously. In this equation,

the maximum phycocyanin formation rate Rpc (0.02 mg g-1 h-1) is negatively affected by the

following factors:

1. High and low light intensities

2. Nitrate deprivation

3. Bicarbonate deprivation

The first two effects were verified in this work. By cultivating in high light intensities (Figure

6-20) (400-600 µmol m-2 s-1), this mechanism (𝑲𝒍𝒊

𝑰+𝑲𝒍𝒊) is able to capture photoadaptation pro-

cesses due to the high illumination. The positive effects of low light intensities are modelled

through this law (𝑰

𝑰+𝑲𝒍𝒊). The nitrate mechanism

𝑲𝒏

𝒏+𝑲𝒏 describes the well-documented phenom-

enon involving the degradation of phycocyanin due to nitrate limitations.

Bicarbonate results from the simulations performed with the same conditions as in the experi-

ments (Figure 7-2) show bicarbonate depletion before 100 hours of cultivation, which matches

- 76 -

the time at which phycocyanin began to decrease in the fed batch experiment. These results

came from the Equation 19, which includes the theoretical bicarbonate yield.

This work includes the depletion of carbon in the mechanisms that negatively affect the phy-

cocyanin mass fraction (Equation 28). It is important to highlight that this work is the first to

describe this phenomenon quantitatively. Further validation of the bicarbonate in the medium

is necessary to draw final conclusions. However, this is a first insight into how bicarbonate also

affects phycocyanin formation.

Figure 7-2 Simulated bicarbonate concentrations in the nitrate fed batch experiment. The line shows



tivation

7.7 Exopolysaccharide production

Production of exocellular polysaccharides by cyanobacteria is known to respond to changes in

several external factors, such as nitrogen concentration and irradiance. The information in the

literature about the effect of nitrate on exopolysaccharide production is ambiguous. Some re-

searchers have described nitrogen starvation as a condition that enhances EPS synthesis, prob-

ably because this contributes to the increase in the C/N ratio, thus promoting the incorporation

- 77 -

of carbon into polymers. Meanwhile, others have reported that the presence of an abundant

nitrogen source in the culture medium resulted in an increase in EPS synthesis, probably due to

the lower energy requirement necessary for the assimilation of combined nitrogen compared

with the energy needed for nitrogen fixation (Pereira et al. 2009).

In addition, it has been shown that exopolysaccharides can be delivered as result of an overflow

in metabolism (Staats et al. 2000), which generates a drain of the excess of ATP (Cogne et al.

2003) by the exopolysaccharides. For example, (Abd El Baky et al. 2014) found the best ex-

opolysaccharide production at initial nitrate concentrations between 0.2 g l-1 to 0.5 g l-1. The

nitrate addition experiment shows an incremental increase in EPS at the end of the cultivation

(Figure 6-19) and, in this case, the rise in EPS corresponds to the nitrate depletion (Figure 6-17).

However, in the control experiment, in which the nitrate depletion occurred after 50 hours, there

was no increase in EPS formation. In addition, the EPS mass fraction remains lower than the

level in the nitrate addition experiment (Figure 6-19). These findings show that nitrate limitation

is not the only triggering factor in EPS formation.

A possible explanation for this behavior it is that not limitations not only of nitrate are necessary

to trigger EPS formation, but also of another macro-element (e.g. phosphate) or micro-element.

Phosphate predictions using the model show low levels of phosphate after 200 hours (Figure

7-3). It has also been reported that phosphate starvation or low levels of phosphate induced an

increase in EPS production. However, in C. capsulata the absence of phosphate had no signif-

icant effect, and in Anabaena spp. and Phormidium sp., it significantly decreased EPS produc-

tion (Nicolaus et al., 1999). Phosphorus starvation may also induce carbohydrate accumulation

over protein accumulation. In the case of A. platensis, carbohydrate accumulation was reported

to amount to approximately 63% of cell dry weight following phosphate starvation. However,

in this work, phosphate limitations as a possible triggering factor for EPS cannot be considered

a final conclusion, as the phosphate predictions have not yet been validated. Finally, this work

has demonstrated that feeding nitrate into the culture can help yield reasonable productivity of

EPS (32 mg l-1 d-1)

- 78 -

Figure 7-3 Simulated phosphate concentrations in different culture conditions The lines show the

results from the simulation of the following conditions: 1.4 % carbon dioxide, 1 vvm,


tivation

- 79 -

8 Possible experimental errors

For the experimental work, the initialization with pre-cultures with different biomass composi-

tions, e.g. phycocyanin concentration, can lead to small changes in the biomass growth curves.

Therefore, a rigorous protocol must be established in order to produce a pre-culture with the

same features, even if the experiments are performed at different periods of time.

As a result of the sampling process, there are changes in the volume of the culture. This outcome

may create different hydrodynamic conditions that could also affect growth. Although all rota-

meters and light bulbs were calibrated before use, the precision of such systems is not optimal,

so there may have been an error in the flow rate or in the incident light intensity on PBR surface

adjustment. Additionally, at the end of the cultivations a cyanobacteria aggregation generated

difficulties in obtaining perfectly homogenous samples and this may have led to errors in the

analytical measurements. After a biomass concentration of 4 g l-1, biomass aggregations and

cell growth in the wall in the current set-up must be resolved in order to obtain a perfectly

homogenous sample.

Finally, human errors in quantification of dry biomass, phycocyanin and exopolysaccharides

may have occurred during handling of the samples. However, all precautions were taken for the

biomass weight measurements and sample preparation in the analysis of phycocyanin.

- 80 -

9 Model limitations

The semi-empirical model proposed in this work is based on Monod type kinetics, which is

based on empirical constants. Some of the assumed constants come from the experiments con-

ducted within this work, while others are deduced from other literature. For example, the yield

of consumptions is derived from theoretical stoichiometric equations for A. platensis and Mich-

aelis-Menten constants were obtained from other studies. In addition, the mass transfer coeffi-

cient was calculated with an empirical correlation and was assumed to be constant during the

course of the cultivation. However, it is not completely precise, as the viscosity increased with

the growth in exopolysaccharide production and, therefore, the mass transfer decreased during

the cultivation.

Determination of parameters for models taking into account multiple factors may not be easy

because it is difficult to simulate co-limitation conditions (e.g. experimental design and setup).

Furthermore, due to the many parameters (10 empirical parameters in this work) that must be

fitted with experimental data, such models often result in overfitting issues. Complex models

with many parameters often suffer from overfitting because they are too specific or sensitive to

the dataset used to develop the model. Even so, the model was able to predict the growth curve

of A. Platensis under different culture conditions (section 6.2).

In this work, the kinetic mechanism for product formation was validated for nitrate over time,

while other medium components were addressed in a largely speculative way in this mechanism

as possible triggering agents in the biosynthesis and degradation of exopolysaccharides and

phycocyanin. However, further bicarbonate and phosphate validations are necessary to verify

the assumptions made in the mechanism.

In sum, due to its semi-empirical nature, the extrapolation of the model to experimental condi-

tions other than those validated in this work must be performed with caution. However, know-

ing the limitations, the model can help predict and understand the behavior of batch cultivation

of A. platensis under various culture conditions.

- 81 -

10 Final remarks and further study

In batches, the rate of overproduction of metabolites by cyanobacteria is limited or activated by

the depletion of required substrates or by the accumulation of metabolic products and inhibitors.

In addition, different nutrient concentrations during cultivation can lead to different cyanobac-

teria compositions. In other words, controlling nutrient conditions can enhance the production

of key compounds by cyanobacteria. Therefore, it is essential to identify the parameters that

have a significant impact on product formation and to create a mathematical model that can

assist the investigation, optimization and increase of A. platensis growth and phycocyanin and

exopolysaccharide production in batch cultivation. However, this requires not only having a

complete understanding of the mechanisms involved in the limiting process, but also studying

them at different levels (metabolism and physiology).

This work has shown that the use of mathematical models in cyanobacteria cultures can help

interpret experimental results and to predict future behaviors after modifying the culture condi-

tions. In particular, this research has demonstrated that controlling nutrient concentrations (e.g.

nitrate) is a viable strategy for improving production, in this case of phycocyanin and exopoly-

saccharides.

The effect of nitrate concentration on the growth kinetics of A. platensis and its phycocyanin

content was quantitatively interpreted in this work. This was then used to propose a feeding

approach in order to keep this molecule constant during cultivation. Furthermore, the mathe-

matical model was able to predict the nitrate consumption in A. platensis cultivations. Under a

light intensity of 600 µmol m-2 s-1, rapid growth leads to nitrate depletion after 50 hours of

cultivation and this led to the observation of rapid phycocyanin degradation. Consequently, a

nitrate fed batch strategy was proposed for the purpose of lowering the phycocyanin degrada-

tion. Nitrate additions during the cultivation help to keep constant this molecule until new

macro-element limitation appear. According to the model, bicarbonate is this limitation. There-

fore, a kinetic law for phycocyanin formation which includes this phenomenon was established.

The effect of nitrate on exopolysaccharide production was also addressed, and in a largely spec-

ulative way, low phosphate levels were proposed as a potential triggering mechanism is EPS

formation. However, further experimental measurements are required with regard to phosphate

and bicarbonate consumption and its influence on product formation.

Finally, an approach based on comprehensive mechanistic relationships could help us under-

stand how cells interact with their culture medium over time with regard to culture behavior,

- 82 -

dynamic approaches being more appropriate for developing an in silico platform. In the future

we will need to start producing large-scale kinetic models. This work thus paves the way toward

an in silico platform making it possible to assess the performance of different culture media and

fed-batch strategies. As a first approach, the nitrate fed batch strategy proposed in the mathe-

matical model in this work leads to a phycocyanin productivity of 38 mg l-1 d-1 and an exopol-

ysaccharide productivity of 33 mg l-1 d-1.

- 83 -

11 Annex

function process2 clear all, clc, close all global umax A Bi k Kn Ynx Zpc Zch R L Is de KHCO Yco H kla pc k1 k2 kw c fc

Kpc Rpc Ipc Kni fa Rpcd Ypx kip pc2 klao m i inini1 %% % _*%%%%%%%%%%%%%%%%%%parameters which can be modified%%%%%%%%%%%%%%*_ Is1=[600,400,70,600]; %set light intensity (µmol/m2*s-1) for these yields

between (10-120) 46 cornet klao=29; % Flowrate=90; %flow rate (l/h) R1=[0.025, 0.025, 0.025, 0.025]; %reactor radius (m) P=101325; % pascal (Absolute Total Working Pressure) yc1=[1.4,1.4,1.4,1.4]; ph=[9,10.5,9.5,9]; u=0.0693 %u=0.0923; 0.07625 umax1=[u,u,u, u]; fa1=[1,1,1, 1]; inibi1=[0.6,0.66,0.3, 0.60]; kti=[200,200,200,200]; %178 Ynx1=[0.50,0.50,0.50,0.50]; Ipc1=[0.10, 0.06, 0.04, 0.09]; %0.065 ka1=[3,3,3,3]; fc1=[1,1,1,1]; %factor 20% more light due to mixing inini1=[0.9,1.8,1.8,0.9]; for m=1:length(yc1) R=R1(1,m); kla=ka1(1,m); %overall mass transfer coeficient (h^(-1)) 6-7 2 fc=fc1(1,m); yc=yc1(1,m); % Percentage of % CO2 in feed gas (%) Is=Is1(1,m); umax=umax1(1,m); fa=fa1(1,m); k=kti(1,m); Ipc=Ipc1(1,m); H=3412; %Hery coeficient at 25°c (Pa*m3/mol) (30.04 bar*L/mol) 3412

(Pa*m3/mol) at 25°c %%%%%%initial cocentrations Spirulina Medium Concentra-

tion%%%%%%%%%%%(kg/m3) inib=inibi1(1,m); %biomass (kg/mt3) inini=inini1(1,m); %Nitrate (kg/mt3) 1.3 % inisul=0.3; %sulfate (kg/mt3) 0.3 inipho=0.2; %phosphate (kg/mt3) 0.2 bic=9.88; %%Bicarbonate (kg/mt3) 7.2 carbon=1.7; %%carbonate (kg/mt3) 1.7 Tsim=320; %%%simulation time (h) %fc=1.0; %factor 20% more light due to mixing %% % _*%%%%%%%%%%%%%%%%%%parameters which can be modified_They are specifc for % ever microalgae( i.e Arthrospira platensis) %%%%%%%%%%%%%%*_ %maximuspecifgrowth(h-1)

%k=3; %MichaeliseMenten constant of light intensity umol/m2s-1 %%%%%%%%%%%% this yields are assuming 90% of active biomass 10% EPS 100 %%%%%%%%%%%% umol/m2s-1 Kn=4*10^-3 ; %MichaeliseMenten constant of Nitrate (kg NO3/m3) Ynx=Ynx1(1,m); %Yield of Nitrate (kg NO3/kg total biomas (active+EPS) mod-

ified form 0.4 to 0.9 kip=800; % Ks=2.5*10^-4 ; %MichaeliseMenten constant of Sulfate(kg SO3/m3)

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% Ysx=0.024; %Yield of Sulfate(kg SO3/kg total biomas (ac-

tive+EPS) % Kp=2.7*10^-4 ; %MichaeliseMenten constant of phosphate (kg PO3/m3) Ypx=0.024; %Yield of phosphate(kg PO3/kg total biomas (ac-

tive+EPS) Yco=2.8; %Yield of total carbon (kg /m3) (kg carbon/m3) (1.47-4.4)

(2.571) KHCO=8*10^-3; %MichaeliseMenten constant of total carbon Kpc=0.12; %MichaeliseMenten constant of phyco (kg/m3) Rpc=0.02; %h-1 %Rpc=0.02; Rpcd=0.000;

%%%%%%%%%%%%%%%%%%%%%product formation yield by cornet%%%%%%%%%%%%%%%%% % Zpr=0.5; %kgpr/kgbiomas %protein formation %Zpc=0.13; %kgpc/kgbiomas %phycocyanin %kgpc/kgbiomas %phycocyanin Zch=0.009; %kgCLh/kgbiomas %Chlorophyll a 0.0089-0.015 % %%%%%%%%%%species equlibrium constants for pH calculations %%%%%%%%%% k1=10^-6.381; k2=10^-10.37; kw=10^-14; options = optimset('LargeScale','off','Display','off','TolFun',1e-16); %% %%%%%%%%%%%load experimental data %ex(:,:)= xlsread('fabiandata.xlsx'); % ex(:,:)= xlsread('newdata.xlsx'); % %ex1(:,:)= xlsread('newdata.xlsx'); % ex1(:,:)= xlsread('newdata2.xlsx'); ex1(:,:)= xlsread('Fabian20162x.xlsx'); ex(:,:)= xlsread('new3data.xlsx'); %% %%%%%%%one parametre modification sensitive analysis % kla1=[2.5,12,17,28]; % for m=1:length(kla1) % % kla=kla1(1,m); %overall mass transfer coeficient (h^(-1)) 6-7 27 % % end %%%%%%%%%%%solving the model%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tspam=linspace(0,Tsim,100); x4_0 = zeros(9,1); x4_0(1) = inib; x4_0(4) = inini; % x4_0(4) = inisul; x4_0(9) = inipho; de=x4_0(1); x4_0(5) = x4_0(1)*Zch; %chla (kg/mt3) x4_0(3) = Ipc; %phico (kg/mt3) %x4_0(5) = Ipc; %phico (kg/mt3) % x4_0(8) = x4_0(1)*Zpr; %proteins (kg/mt3) x4_0(2) = x4_0(1); %biomass+glucogen (kg/mt3) pc=P*(yc/100); %partial presure yce=0.035; %exit pc2=P*(yce/100); %exit %disol=(((pc/(H))*44)/1000); x4_0(7) = (bic)*fa; %total disolved carbon (kg/mt3) x4_0(6) = (((pc2/(H))*44)/1000); x4_0(8) = 0.986; x4_0(10) = 0.896;

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x4_0(11) = 0.811; x4_0(12) = 1.227; %%%%%%%%%%%%%%%solving%pH%model%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% c=pc/H; %mol/m3 xo = [1; 1]; % Make a starting guess at the solution [y]=fsolve(@pHmodel,xo,options); pH=-log(y(1)); %pH wil be calculated as fuction of bicarbonate (117

mol/m3)(pH=9.2) %%%%%%%%%%%%% [t_traj, x_traj] = ode15s(@Cinetica,tspam,x4_0);

s1(:,:,m)=x_traj; %%%%pH calculation

% L1(:,:)=x_traj'; % %%%%%%%%%%%%%%%solvingPH when co2 model is active%%%%%%%%%%%%%%%%%%%%%%%%

end

%% %%%%%%%%%%%%%%%%%%%Data extraction to plot %%%%%%%%%%%%%%%%%%%%%% for m=1:length(yc1) a=(s1(:,7,m)); a1=(a*1000)/60; %%bicrbonate b=s1(:,6,m); b1=(b*1000)/44; for i=1:length(b1) pkac1=5.5; pH(i,m)=pkac1+log10(a1(i)/b1(i)); end end

for i=1:length(x_traj)

for m=1:length(yc1)

%if s1(i,3,m)>=1

A11=linspace(0.0001,R,1000); A1=A11/R; alfa(i,:,m)=((A*(s1(i,3,m)+0.009))/(A*(s1(i,3,m)+0.009)+Bi))^(1/2); gama(i,:,m)=(A*(s1(i,3,m)+0.009)+Bi)*alfa(i,:,m)*s1(i,2,m)*R; per(i,:,m)=gama(i,:,m)*A1; TO=1./A1; al(i,:,m)=((2*cosh(per(i,:,m)))); bp(i,:,m)=(cosh(gama(i,:,m))+alfa(i,:,m)*sinh(gama(i,:,m))); AK(i,:,m)=al(i,:,m)/bp(i,:,m); IM(i,:,m)=(Is1(1,m)*TO).*AK(i,:,m); Ir(i,:,m)=fc*mean(IM(i,:,m));

% else % zt(:,1)=A.*x_traj(:,2,m); % w(:,1,m)=Is./zt(:,2,m); % It(:,1,m)=1-exp(-zt(:,1,m));

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% Ir(:,1,m)=fc*w(:,1).*It(:,1,m); % end

end end

for i=1:length(tspam) t=tspam'; pr=s1(i,:,1)'; y= feval(@Cinetica,t(i), pr); y1=y'; y2(i,:)=y1; end %% %%%%Ploting comands%%%%%%%%%%%%%%%%%%%%%%%

figure(1); subplot(2,1,1) a2=plot(t_traj,s1(:,2,4)) set(a2,'LineStyle','-' ,'LineWidth',1,'Marker','None','MarkerSize',8) set(a2,'Color',[0,0,0]) hold on

a1=plot((ex1(:,1)),ex1(:,5)) set(a1,'LineStyle','None','LineWidth',0.5,'Marker','diamond','Mark-

erSize',4) set(a1,'Color',[0,0,0]) ylabel('Biomass (g L^{-1})','fontsize',14) xlabel('Time (hours)','fontsize',14) % a1=plot((ex(:,3).*24),ex(:,4)) % set(a1,'LineStyle','None','LineWidth',0.5,'Marker','o','MarkerSize',4) % set(a1,'Color',[0,0,0]) hold off % set(a1,'LineStyle','None','LineWidth',0.5,'Marker','square','Mark-

erSize',4) % set(a1,'Color',[0,0,0]) %set(a9,'YColor',[0,0,0]) ylim([0 10]); xlim([0 270])

% hold off

% h = legend('Fed-batch-strategy','Fed-batch-strategy',0); % legend('boxoff') % hold off % xlabel('input'); % text(0,11,'A', 'Interpreter', 'latex', 'fontsize',14);

%figure(2) subplot(2,1,2)

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a1=plot((ex1(:,1)),ex1(:,6)) set(a1,'LineStyle','None','LineWidth',0.5,'Marker','diamond','Mark-

erSize',4) set(a1,'Color',[0,0,0]) hold on ylim([0 1.5]); xlim([0 270]); ylabel('Nitrate (g L^{-1})','fontsize',14) xlabel('Time (hours)','fontsize',14) a2=plot(t_traj,s1(:,4,4)) set(a2,'LineStyle','-' ,'LineWidth',1,'Marker','None','MarkerSize',8) set(a2,'Color',[0,0,0]) ylim([0 2]); xlim([0 270]); hold off

figure (6)

a10=plot(t_traj,(s1(:,3,4)*1000)) ylabel('Phycocyanin (mg g^{-1})','fontsize',14) xlabel('Time (hours)','fontsize',14) set(a10,'LineStyle','-.','LineWidth',1,'Marker','None','MarkerSize',8) set(a10,'Color',[0,0,0]) ylim([0 130]); xlim([0 270]);

hold on

% a4=plot((ex(:,3).*24),ex(:,5)*10) % set(a4,'LineStyle','None','LineWidth',0.5,'Marker','o','MarkerSize',5)

%3% % set(a4,'Color',[0,0,0]) a1=plot((ex1(:,1)),ex1(:,7)*10); set(a1,'LineStyle','None','LineWidth',0.5,'Marker','diamond','Mark-

erSize',4); set(a1,'Color',[0,0,0]) % ylabel('Phycocyanin (mg g^{-1})','fontsize',14)

hold off % a1=plot(t_traj,s1(:,7,1)); % ylabel('Bicarbonate (g L^{-1})','fontsize',14) % xlabel('Time (hours)','fontsize',14) % set(a1,'LineStyle','-','LineWidth',1,'Marker','None','MarkerSize',8) % set(a1,'Color',[0,0,0]) % %set(a9,'YColor',[0,0,0]) % ylim([0 10]); % xlim([0 320]);

% figure(3);

a1=plot(t_traj,Ir(:,1,1))

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ylabel('light µmol m^{-2} s^{-1}','fontsize',14) xlabel('Time (hours)','fontsize',14) %set(a9,'LineStyle','None','LineWidth',2,'Marker','*','MarkerSize',8) set(a1,'Color',[0,0,0]) set(a1,'LineStyle','-','LineWidth',1,'Marker','None','MarkerSize',8) hold on a1=plot(t_traj,Ir(:,1,2)) ylabel('light µmol m^{-2} s^{-1}','fontsize',14) xlabel('Time (hours)','fontsize',14) %set(a9,'LineStyle','None','LineWidth',2,'Marker','*','MarkerSize',8) set(a1,'Color',[0,0,0]) set(a1,'LineStyle','--','LineWidth',1,'Marker','None','MarkerSize',8) a1=plot(t_traj,Ir(:,1,4)) ylabel('light µmol m^{-2} s^{-1}','fontsize',14) xlabel('Time (hours)','fontsize',14) %set(a9,'LineStyle','None','LineWidth',2,'Marker','*','MarkerSize',8) set(a1,'Color',[0,0,0]) set(a1,'LineStyle','-.','LineWidth',1,'Marker','None','MarkerSize',8) %set(a9,'YColor',[0,0,0]) hold off ylim([0 1000]); xlim([0 270]); % hold on % a2=plot(t_traj,Ir(:,1,2)) % set(a2,'Color',[0,0,0]) % set(a2,'LineStyle','--','LineWidth',1,'Marker','None','MarkerSize',8) % hold off % h = legend('Fed-batch-strategy',0); % legend('boxoff')

figure (7) a1=plot(t_traj,s1(:,7,2)); ylabel('Bicarbonate (g L^{-1})','fontsize',14) xlabel('Time (hours)','fontsize',14) set(a1,'LineStyle','--','LineWidth',1,'Marker','None','MarkerSize',8) set(a1,'Color',[0,0,0]) hold on a1=plot(t_traj,s1(:,7,4)); ylabel('Bicarbonate (g L^{-1})','fontsize',14) xlabel('Time (hours)','fontsize',14) set(a1,'LineStyle','-.','LineWidth',1,'Marker','None','MarkerSize',8) set(a1,'Color',[0,0,0]) a1=plot(t_traj,s1(:,7,1)); ylabel('Bicarbonate (g L^{-1})','fontsize',14) xlabel('Time (hours)','fontsize',14) set(a1,'LineStyle','-','LineWidth',1,'Marker','None','MarkerSize',8) set(a1,'Color',[0,0,0]) %set(a9,'YColor',[0,0,0]) ylim([0 10]); xlim([0 270]); hold off

figure(10) a1=plot(t_traj,s1(:,9,1)); ylabel('Phosphate (g/l)','fontsize',14) xlabel('Time (hours)','fontsize',14) set(a1,'LineStyle','-','LineWidth',1,'Marker','None','MarkerSize',8) set(a1,'Color',[0,0,0]) %set(a9,'YColor',[0,0,0])

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ylim([0 0.2]); xlim([0 270]);

figure (9);

a1=plot(t_traj,pH(:,1)) ylabel('pH','fontsize',14) xlabel('Time (hours)','fontsize',14) set(a1,'LineStyle','-' ,'LineWidth',1,'Marker','None','MarkerSize',8) set(a1,'Color',[0,0,0]) %set(a9,'YColor',[0,0,0]) ylim([2 10]); xlim([0 320]); hold on

figure (10) subplot(2,1,1) a1=plot((ex1(:,1)),ex1(:,9)*10); set(a1,'LineStyle','None','LineWidth',0.5,'Marker','square','Mark-

erSize',4); ylabel('Exopolysaccharides (mg g^{-1})','fontsize',14) %set(a9,'YColor',[0,0,0]) ylim([0 110]); xlim([0 240]);

hold on a1=plot((ex1(:,1)),ex1(:,8)*10); set(a1,'LineStyle','None','LineWidth',0.5,'Marker','diamond','Mark-

erSize',4); set(a1,'Color',[0,0,0]) ylabel('Exopolysaccharides (mg g^{-1})','fontsize',14) ylim([0 120]); xlim([0 270]); hold off % a1=plot(t_traj,s1(:,7,1)); % ylabel('Bicarbonate (g L^{-1})','fontsize',14) % xlabel('Time (hours)','fontsize',14) % set(a1,'LineStyle','-','LineWidth',1,'Marker','None','MarkerSize',8) % set(a1,'Color',[0,0,0]) % %set(a9,'YColor',[0,0,0]) % ylim([0 10]); % xlim([0 320]);

subplot(2,1,2) a1=plot(t_traj(1:9),s1(1:9,4,1)); set(a1,'LineStyle','-','LineWidth',1,'Marker','None','MarkerSize',8) set(a1,'Color',[0,0,0]) hold on a2=plot(t_traj(9:16),s1(9:16,8,1)); set(a2,'LineStyle','-','LineWidth',1,'Marker','None','MarkerSize',8) set(a2,'Color',[0,0,0]) % ylabel('Nitrate g L^{-1}','fontsize',14) % xlabel('Time (hours)','fontsize',14) % set(a2,'LineStyle','-','LineWidth',1,'Marker','None','MarkerSize',8) % set(a2,'Color',[0,0,0]) a3=plot(t_traj(16:24),s1(16:24,10,1)); ylabel('Nitrate (g L^{-1})','fontsize',14) xlabel('Time (hours)','fontsize',14) set(a3,'LineStyle','-','LineWidth',1,'Marker','None','MarkerSize',8) set(a3,'Color',[0,0,0]) a4=plot(t_traj(24:31),s1(24:31,11,1));

- 90 -

set(a4,'LineStyle','-','LineWidth',1,'Marker','None','MarkerSize',8) set(a4,'Color',[0,0,0]) a5=plot(t_traj(31:100),s1(31:100,12,1)); set(a5,'LineStyle','-','LineWidth',1,'Marker','None','MarkerSize',8) set(a5,'Color',[0,0,0]) %set(a9,'YColor',[0,0,0]) a1=plot((ex1(:,1)),ex1(:,3)) set(a1,'LineStyle','None','LineWidth',0.5,'Marker','square','MarkerSize',4) ylim([0 2]); xlim([0 240]); a1=plot((ex1(:,1)),ex1(:,6)) set(a1,'LineStyle','None','LineWidth',0.5,'Marker','diamond','Mark-

erSize',4) set(a1,'Color',[0,0,0]) ylim([0 1.5]); xlim([0 270]);

hold off

figure(11) a1=plot(t_traj,s1(:,9,1)); ylabel('Phosphate (g/l)','fontsize',14) xlabel('Time (hours)','fontsize',14) set(a1,'LineStyle','-','LineWidth',1,'Marker','None','MarkerSize',8) set(a1,'Color',[0,0,0]) hold on a1=plot(t_traj,s1(:,9,2)); set(a1,'LineStyle','--','LineWidth',1,'Marker','None','MarkerSize',8) set(a1,'Color',[0,0,0])

a1=plot(t_traj,s1(:,9,4)); set(a1,'LineStyle','-.','LineWidth',1,'Marker','None','MarkerSize',8) set(a1,'Color',[0,0,0])

hold off %set(a9,'YColor',[0,0,0]) ylim([0 0.2]); xlim([0 270]); %% return %% function [f] = Cinetica(t,x) global umax A Bi k Kn Ynx Zch Zpc R Is de Yco KHCO H kla pc fc Kpc Rpc Ipc

Kni kli fa Rpcd Ypx kip P pc2 yce klao m inini1 f = zeros(12,1); %variables %kli=12.5; X=x(1); %biomass without glycogen (g/l) XT= x(2); %biomass+glycogen(g/l)

if m==1 %if t<80 if t<=28 N= x(4); %Nitrate (g/l) else if t<=52 N=x(8); else if t<=76 N=x(10);

- 91 -

else if t<=76 N=x(11); else N=x(12); end end end end else N= x(4) end

Ch=x(5); %chlorophyll (g/l) Pc= x(3); %phyco (gphyco/gbiomass) %Pc=Pcg*X; % Pr= x(8); %Protein (g/l) Dis=x(6); B=x(7); %bicarbonate (g/l)

%%%%%%%%%%%%viscosity and microalgaemovmentproblmes%%%%%%%%%%%%%%%%%%%

% if t>=139 % fc=0.20; % else % if t>=118 % fc= 1; % else % % fc=fc; % end % end %end %end

%%%%%%%%%%%%%%%%%%%%%%%%%% % if N<0.1; % A=7300; %extinction or absorption coeficient m2/kg % Bi=200; % else % A=7300; %extinction or absorption coeficient m2/kg % Bi=200; % end

A=1300; %extinction or absorption coeficient m2/kg Bi=200;

% % if N<0.1; % A=2200; %extinction or absorption coeficient m2/kg % Bi=200; %scatering coeficient m2/kg % else % A=1800; %extinction or absorption coeficient m2/kg % Bi=730; %scatering coeficient m2/kg % end

- 92 -

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if XT>1 A11=linspace(0.0001,R,1000); A1=A11/R; alfa=((A*(Pc+0.009))/((A*(Pc+0.009))+Bi))^(1/2); gama=((A*(Pc+0.009))+Bi)*alfa*XT*R; per=gama*A1; TO=1./A1; al=((2*cosh(per))); bp=(cosh(gama)+alfa*sinh(gama)); AK=al/bp; IM=((Is*TO).*AK); I1=fc*mean(IM);

%%%%%%%%%%%%beer lambert else AY=30; A11=linspace(0.0001,R,1000); IM =Is*exp(-AY*XT*A11); I1=fc*mean(IM); I=I1; end

if I1<6; I=2; else I=I1; end

% end

kli=90; %%%%%%kinectic model%%%%%%% f(1) = umax*(N/(Kn+N))*(I/(I+k+((I^2)/kip)))*(B/(KHCO+B))*X; %biomass with-

out glycogen production %f(2) =

umax*(I/(I+k))*((B/(KHCO+B))*(N/(Kn+N))+(Pc/(Kpc+Pc))*(Kn/(Kn+N)))*XT; f(2) =

umax*((I/(I+k+((I^2)/kip)))*(B/(KHCO+B))*(N/(Kn+N))+((Pc/(Kpc+Pc))*(B/(KHCO

+B))*(Kn/(Kn+N))))*XT;

if I>12; f(3)=Rpc*((kli/(I+kli))-((I/(I+kli))+(Kn/(N+Kn))+(KHCO/(B+KHCO))))*Pc; % else f(3)=Rpc*-((Kn/(N+Kn)))*Pc; %f(3)=0; end

if m==1; if N<0.000005 f(4)=0; f(8)=0; f(10)=0; f(11)=0; f(12)=0; N==0.000005;

- 93 -

else if t<=28 f(4) =-Ynx*f(2); else if t<=52 f(8) =-Ynx*f(2); else if t<=76 f(10) =-Ynx*f(2); else if t<=100 f(11) =-Ynx*f(2); else f(12) =-Ynx*f(2); end end end end end else if N<0.000005 f(4)=0; else f(4) =-Ynx*f(2); end end

% end

% f(4) =-Ysx*f(1); %Sulphate comsuption f(9) =-Ypx*f(2); %phosphaspe comsuption (%%%%%intracelluar phosphaete

model) f(5)=Zch*f(1); %chlorophyll generation ti=(((pc/(H))*44)/1000); %from mol/me to g/L tio=(((pc2/(H))*44)/1000); f(6) =kla*(ti-Dis)+klao*(tio-Dis)-1*f(2); %co2 transfer to the medium -

co2cconvert into bicarboante - and co2 desortion to mantain enquilibrium

if B<0.004 %Yco=0; %B==0; f(7)=0;

else

f(7) =-Yco*f(2); %bicarboante consume by the algae + bicarbonate produce in

the reaction CO2 + OH

end return % function [f] = pHmodel(y) global c k1 k2 kw f = zeros(2,1); %%%%% co2 concentration b = y(1); %b=b%h+ cocentration CT = y(2); %c=c%co2 concentration %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% f(1)= ((1+(k1/b)+((k1*k2)/(b^2)))*c)-c*((k1/(b^2))+((2*k1*k2)/(b^3)))*b-CT;

- 94 -

f(2)=

(((k1/(b))+2*(k1*k2/(b^2)))/1+(kw/(b^2))+((k1*c)/(b^2))+((4*k1*k2*c)/(b^3))

)*c-b; return;

- 95 -

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