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8 0 4 R. C. SRIVASTAVA AND P. K. AVASTHI

A New Equation for Microbial Growth R . C . SRIVASTAVA a n d P . K . AVASTHI

Chemistry Department, Harcourt Butler Technological Institute, Kanpur-2 (India) (Z. Naturforsch. 26 b, 804—806 [19711; received Mardi 3, 1971)

A new equation for microbial growth has been proposed and its implications discussed. The proposed equation has a distinct advantage over B e r t a l a n f f y ' s in being an explicit relationship between growth parameter and time. Attempt has been made to read the proposed equation in the light of H i n s h e 1 w o o d's auto-synthetic model. The constants occurring in the proposed equa-tion have been shown to be intimately related to the kinetic constants in the H i n s h e l w o o d ' s model.

If one wants to know the general rate law govern-ing the kinetics of microbial growth the must know the explicit relationship between mass or dimen-sions or the number of viable counts or anyother parameter measuring the growth of micro-organisms and time. Although many attempts in this direction are recorded in 1. c . 2 the growth equations ob -tained hitherto, suffer mainly on two accounts. Either these equations do not represent all the regions of the growth curve i. e. the regions of lag phase, exponential phase and the stationary phase or they do not represent an explicit relationship between growth parameter and time. Even B e r t a l a n f f y ' s equat i on 3 which can be con-sidered as one of the most successful attempts, is not an explicit relationship between growth parameter and time.

Since no explicit relationship between growth parameter and time f o r all the regions of growth curve, is available in literature, our attempt in the present communicat ion is to suggest one and to discuss the implications thereof.

The proposed equation

If we do not consider the phase of decline, the microbial growth curve in almost all cases is of the f o r m of curve 1 in Fig. 1. For the purpose of pre-sent discussion let us break up the total growth curve i. e. curve 1 of Fig . 1 in two parts A B C and CDE. The part A B C of the curve 1 in Fig . 1 starts f r o m origin, bends upwards and then approaches a straight line which extrapolates to an intercept

Reprints request to Dr. R. C. SRIVASTAVA, Chemistry De-partment, Harcourt Butler Technolog. Institute, Kanpur-2 (Indien).

1 H . G . BRAY and K. WHITE , Kinetics and Thermodynamics in Biochemistry, J. A. Churchill Ltd., London 1966.

x Theoretical Points

o Experimental Points

50 100 150 200 250 300 350 400 450 430 Time in Minutes

Variation of the number of viable counts with time in case of Salmonella Pullorum.

on the X-axis . t0 can be taken to be an index of the lag time. T o describe the growth curve in this region i. e. in the region depicted by the part A B C of curve 1 in Fig. 1, we propose the fo l lowing equa-t ion. g I

log ~jr = L t-t0 j 1

In equation ( 1 ) G stands f o r the growth parameter and t stands f o r time. L is a constant and G1 is the value of G at t = 0 and is constant f o r a particular

8 C. LAMANNA and M . F. MALLETTE, Basic Bacteriology — Its Biological and Chemical Background, The Williams and Wilkins Company, Baltimore 1964.

3 L . VON BERTALANFFY, Theoretische Biologie, vol. I I , Stoff-wechsel, Wachstum. Gebrüder Borntrager, Berlin 1942. — See reference 1, p. 396.

-exp(~i) ( i )

A NEW EQUATION FOR MICROBIAL GROWTH 805

system. It is obvious f r o m equation ( 1 ) that when t0 is zero the microorganisms will fo l low the ideal growth curve of the type shown by curve 2 in Fig . 1.

It is also obvious that when assumes such

high values that the term, e x p | becomes very

small in comparison to unity, equation ( 1 ) reduces to g

log =L(t-t0) , (2)

which is the equation of a straight line of slope L and intercept t0 on X-axis . The equation ( 2 ) thus depicts the portion BC of the curve 1 in Fig . 1 and it is obv i ous f r o m equation ( 2 ) , that in this region a first order rate law

(3)

would be obeyed. In order to test the validity of the proposed equa-

tion ( 1 ) WILSON'S 1> 4 data for the growth of Salmo-nella PulloTum has been made use of . The data has been plotted in Fig. 1 curve 1, f r om which the value of t0 c o m e s out to be equal to 68 minutes. The value of L has been calculated f rom the slope of the straight line part of the curve 1 Fig. 1 i. e. BC, and comes out to be equal to 0 .01 . Using the values of t0 and L thus obtained the curve depicted by equation ( 1 ) was traced by giving hypothetical value to t and calculating the corresponding values of

Q It can be seen f r o m the curve 1 of Fig . 1,

agreement is excellent. The theoretical points and experimental fall on the same curve i. e. the part A B C of curve 1 in Fig. 1.

Having demonstrated the validity of equation ( 1 ) f o r the region ABC of the growth curve 1 in Fig . 1, let us turn our attention to the portion B C D E of the curve 1 Fig. 1.. For the portion BCDE we pro-pose the fo l lowing equatior

G

that the

log Gr

= L(t — t0) +L1(t-t0)2 + (4)

where Lx etc. are constants. If we consider upto square power terms only, the equation

(log G) =L + 2 Ll(t —10) (5)

fo l lows f r o m equation ( 4 ) . Equation ( 5 ) implies

that the plot of - j - ( l og G) against (t —10) would

be a straight line of slope 2 Lx and intercept on the

Y-axis equal to L. The values of ( log G) at various

values of t have been obtained from the values of

the slopes at various points on the non-linear part C D E of the curve 1 in Fig . 1. The values of

— ( l og G) thus obtained have been plotted against

(t — J0) in Fig . 2 and it can be seen that the graph is a straight line which if extended will have an intercept equal to 0 . 0 1 3 8 on the Y-axis. The values of L± and L obtained f r o m the slope and the inter-cept on the Y-axis , o f the straight line in Fig . 2 are respectively - 1 . 7 7 x l 0 ~ * and 0 . 0 1 3 8 . It can be seen that the value of L obtained with the help of equation ( 5 ) compares favourably with the one obtained f r o m the slope of the straight line part BC of curve 1 in Fig . 1. The fact that the graph be-tween ( log G) and (t — 20) in Fig. 2 is a straight

line, not only proves the validity of equation ( 5 ) but also decides that only the first two terms on the right hand side of equation ( 4 ) are sufficient to represent the growth data in the region BCDE of curve 1 in Fig . 1.

0.0 H

150 200 250 300 350 400 ( t - t 0 ) Minutes

Fig. 2. Test of equation (5).

A s a consequence of the analysis given in the f o rego ing lines, the growth equation f o r the entire region, A to E, of the growth curve 1 in Fig. 1 can be written as

log = L t — f0 1 — exp R ) (6)

+ Ll(t~t0)2+-- - .

The terms L1(t — t0)2 etc. in equation ( 6 ) are signi-ficant only in the region C D E and can be neglected in the region A B C . Thus, equations ( 1 ) , ( 2 ) and ( 4 ) are only particular cases of the generalised equation ( 6 ) . The number of significant terms on the right hand side of equation ( 4 ) can always be

4 G . S. WILSON, J. Bacteriol. 7 , 4 0 5 [ 1 9 2 2 ] , 3 4 9

See ref. 2, p.

8 0 6 A NEW EQUATION FOR MICROBIAL GROWTH 806

determined f r o m the actual growth data of a parti-cular organism. The overall rate law f o r the micro-bial growth can now be obtained b y differentiating equation ( 6 ) with respect to time t. The rate law governing the kinetics of growth in the region A B C of curve 1 in Fig . 1, as obtained f r o m equation ( 1 ) can be writetn as

d2 „ I d (log G)-— = 0 .

In (7)

while in the region BC of the curve 1 in Fig . 1, a first order rate law is expected to be obeyed .

The proposed equation and the auto-synthetic model

In this section attempt will be made to read the proposed equation in the light of H i n s h e 1 -w o o d 's auto-synthetic m o d e l 5 f o r a growing system.

Suppose an enzyme system I produces a dif-fusible substance which is used by the second enzyme system II. Let the total amounts of I and II at any time be Xx and X2 respectively, contained in n cells. If II is the component upon the formation of which cell division must wait, then accord ing to H i n s h e l w o o d ' s model

n — ß X2 .

dn „ d Yo = ß dt dt

f r o m which the relationship

1 dn n d*

1 dX2

X, dr

(8)

(9)

(10)

automatically fo l lows, ß in equations ( 8 ) and ( 9 ) dJ

is a constant. Since — ~ depends upon the ratio

, H i n s h e l w o o d postulates that —j - must also 2 X X

depend on the ratio . The ratio must ob-viously increase with time and finally attain a steady value. Let the ratio

dv dt

Xt

1 dXA

X, dt " X, 1 dX2

Xa Xo df

Since the rate of increase of I is given by

dXx

dt = kXx.

( I D

(12)

(13)

the equation ( 1 2 ) with the help of equation ( 1 0 ) can be re-written as

( X. \ 1 dn ,.( XA dv I t i t - * ( ( 1 4 )

The rate constant k in equations ( 1 3 ) and ( 1 4 ) refers to the enzyme system I. At the time when v reaches a steady value and where-upon no further

change in the value of v occurs i. e. — — = 0, the

equation ( 1 4 ) reduces to

1 dn n dt (15)

which is exactly the same as equation ( 3 ) . Com-parison of equation (15 ) with ( 3 ) further reveals that the constant L in the proposed equation ( 1 ) is nothing but the rate constant k f o r the enzyme system I.

It appears therefore that the lag time is related %

to the time required for the ratio to attain the A 2

steady value and it is during this time the growth curve fol lows the exponential character. In other words it could also be said that the portion A B of the growth curve of Fig. 1 indicates the fashion in

X which ratio approaches steady state and once

A 2

the ratio has attained a steady value, the micro -A 2

biol growth fo l l ows first order rate law as indicated by the straight line portion BC of the curve 1 in Fig . 1.

A n attempt can also be made to obtain a co-rela-tion between the lag time index and the kinetic constants in the H i n s h e 1 w o o d's model . Putting t = t0 in equation ( 1 ) we can write

V - f ' T (16)

W h e r e A' is the value of log at t = t0 and is

constant f o r a particular system. Since L has been shown to be the same thing as k the rate constant f o r the enzyme system I, it can be safely concluded f r o m equation ( 1 6 ) that the lag time index is in-versely proport ional to k, the rate constant f o r the enzyme system I. It may be emphasised here that this conclusion is in keeping with the conclusion already drawn earlier in 1. c. 6 .

5 C. N. HINSHELWOOD, J. chem. Soc. [London] 1 9 5 3 , 1947. 6 See ref. 1, p. 382.