Albert-Ludwigs-Universitat Freiburg

Fakultat fur Mathematik und Physik

Water models and hydrogen

bonds

Dissertation zur Erlangung des Doktorgrades der

Fakultat fur Mathematik und Physik

der Albert-Ludwigs-Universitat Freiburg im Breisgau

Freiburg Institute for Advanced Studies

vorgelegt von Roman Shevchuk

betreut durch Prof. Dr. Gerhard Stock / Dr. Francesco Rao

Freiburg, 2014

Dekan : Prof. Dr. Michal Ruzicka

Prodekan : Prof. Dr. Andreas Buchleitner

Leiter der Arbeit : Prof. Dr. Gerhard Stock

Referent : Prof. Dr. Gerhard Stock

Koreferent : PD Dr. Thomas Wellens

Datum der mundlichen Prufung : 08.05.2014

Contents

Introduction 4

1 Molecular simulations 9

1.1 Force fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Newtonian dynamics . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Thermostats . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Barostats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Water models in molecular dynamics . . . . . . . . . . . . . . 16

1.6 Simulation details . . . . . . . . . . . . . . . . . . . . . . . . . 20

2 Water phase diagram and water anomalies 22

2.1 Water phase diagram . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Water anomalies . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Water supercooling and freezing 28

3.1 General perspective . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Test of water freezing . . . . . . . . . . . . . . . . . . . . . . . 30

4 Complex network approach for molecular dynamics trajec-

tories and hydrogen bond as an order parameter 36

4.1 Complex network as a tool to study molecular simulations . . 37

4.2 Hydrogen bond criteria . . . . . . . . . . . . . . . . . . . . . . 43

3

4 CONTENTS

5 Applications 57

5.1 Study of classical water models at ambient pressure . . . . . . 57

5.2 Effect of polarizability . . . . . . . . . . . . . . . . . . . . . . 67

5.3 Free energy landscape of water . . . . . . . . . . . . . . . . . . 74

5.4 Proton transfer . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Conclusions 97

Bibliography 100

Acknowledgment 122

Introduction

For every phenomenon, however

complex, someone will

eventually come up with a

simple and elegant theory. This

theory will be wrong.

Rotschild’s Rule

Water is the most important element for all living organisms on Earth.

About 80 percents of all living cells consist of water [1]. It plays a role of

solvent and thermoregulator, being the environment for the vast majority of

all biochemical processes. At the fundamental level, water directly influences

several biologically relevant processes including protein folding [2], protein-

protein association [2–5] and amyloid aggregation [6].

A single water molecule consists of two hydrogens and an oxygen atom

forming a V-shaped molecule with an angle of about 106◦. Because oxygen

has a higher electronegativity than hydrogen, the side of the molecule with

the oxygen is partially negative and the hydrogen end is partially positive.

Consequently, the direction of the dipole moment points from the oxygen

towards the center of the hydrogens. This charge difference causes water

molecules to be attracted to each other through highly directional hydrogen

bonds (the relatively positive areas being attracted to the relatively negative

areas) as well as to other polar molecules [7].

One of most interesting properties of water is its polyamorphism. At

5

least 15 crystalline forms of ice are known [8]. For example the number

of crystalline modifications of Si or Ge is comparable, but their structural

diversity is connected with the transition from semiconductors to metals,

on the other hand, the nature of intermolecular interactions in water ice

is the same. Water molecules keep their individuality and what changes is

the order and structure of the hydrogen bond network [9]. Since there are

so many possible crystal structures of water, two questions spontaneously

emerge: (i) is there any residual structure in liquid water? (ii) how does

water crystallize into ice?

To address these questions the concept of network of hydrogen bonds

which is continuous in space was proposed by Bernal and Fowler [10]. With all

modern experimental and computational techniques there is no doubt that at

normal conditions water molecules are connected through three-dimensional

network of hydrogen bonds [11, 12]. Many interesting results were obtained

by simulations [13–17] and experiments [18,19]. But the problem is that even

nowadays none of the experimental methods can track the motion of single

water molecules in bulk liquid or explicitly detect all hydrogen bonds in the

bulk. This is where computer simulations come into play.

The first computer simulation of water was done at the end of the 60s

[20,21]. At that time it was possible to simulate a system of a few hundreds

of water molecules, where van der Waals interactions were described with

a Lennard-Jones potential [22]. With the rise of computational power, the

number of simulated molecules increased by several orders of magnitude [23]

as well as new refined (and more complex) water models appeared, including

molecular flexibility and polarizability [24–27].

In this thesis we will focus on several aspects of molecular dynamics stud-

ies of liquid water, particularly the temperature response of some of the

most popular water models, including their hydrogen bond network struc-

ture. Apart from commonly used thermodynamical measurements here we

apply a recently developed complex network framework [16,28]. Within this

framework the system is described by a discrete set of a microstates evolv-

6

Introduction

ing in time. Microstates represent the nodes of a transition network where

a link is placed between two microstates if the system jumped from one to

the other one along the molecular dynamics trajectory. Thanks to the net-

work analyzing such as cluster structure it is possible to characterize both

thermodynamics and kinetics of the system. Combining a complex network

framework with more conventional tools like radial distribution function, a

detailed description of liquid water is achieved.

A short overview of this thesis is presented below:

• In Chapter 1 an introduction of the basic principles of molecular dy-

namics simulations is provided. The most commonly used approaches

for temperature and pressure coupling is described as well as the dif-

ference between classical molecular dynamics and Langevin dynamics.

• In Chapter 2 the picture of the phase diagram of water is given as well

as the description of some of water’s properties and so called anomalies.

In particular, the water density and thermodynamic anomalies such as

presence of the maximum of the density above melting temperature and

anomalous increase of viscosity at supercooled region is highlighted.

• In Chapter 3 we briefly describe the problems related to supercooled

water. The results of the microsecond-long simulation of water in this

region are shown, where the correlation between water energy, density

and structural order as well as possible scenarios of water freezing were

discussed.

• In Chapter 4 we give an analysis of the molecular dynamics trajectories

via the complex network approach. The detailed description of complex

network building for the case of liquid water is provided. In the second

section of this chapter the hydrogen bond definitions commonly used

in molecular dynamics are analyzed in detail.

• In Chapter 5 the applications of above described methods and tools are

provided. In particular, the free-energy landscape of water in 220K <

7

T < 340K temperature range is studied via complex network analysis.

We present the comparative analysis of seven classical water models

as well as the polarizable SWM4-NDP water model. Moreover, the

simplified complex network analysis for the case of proton transfer in

bulk water is presented.

All molecular simulations presented in this thesis (except the ones de-

scribed in section 5.4) have been prepared, launched and analyzed by

myself. The statistical tools and algorithms used for the analysis have

been coded by me in collaboration with Dr. D. Prada-Gracia and in-

cluded in a software library called AQUAlab (GPL license, available at

raolab.com).

8

Introduction

Some results of this thesis were published

in the following papers:

– R. Shevchuk, D. Prada-Gracia, and F. Rao. Water structure-

forming capabilities are temperature shifted for different models.

J. Phys. Chem. B., 116(25):7538–7543, 2012.

– R. Shevchuk and F. Rao. Note: Microsecond long atomistic sim-

ulation of supercooled water. J. Chem. Phys., 137:036101, 2012.

– D. Prada-Gracia, R. Shevchuk, P. Hamm, and F. Rao. Towards a

microscopic description of the free-energy landscape of water. J.

Chem. Phys., 137:144504, 2012.

– D. Prada-Gracia*, R. Shevchuk* and F. Rao. The quest for

self-consistency in hydrogen bond definitions. J. Chem. Phys.,

139:084501, 2013.

* authors contributed equally to this work.

9

Chapter 1

Molecular simulations

In the recent years along with traditional experiments, computer simulations

became a useful tool to elucidate some physical and chemical processes on

the molecular level. Here we mainly use classical molecular dynamics sim-

ulations, which are a tool that allows to simulate the microscopic system

with all-atom resolution using simple Newtonian equations of motion. There

are multiple applications of molecular dynamics: they are used for refine-

ment of molecular structure from the experiments (crystallography, NMR or

electronic microscopy), for the interpretation of the experimental data, for

the prediction of functional properties of biological systems and for sampling

the regions of phase space which are unreachable in the experiments [29].

First molecular simulations of water were made around forty years ago and

were able to calculate the trajectory of few hundreds of atoms for several

picoseconds [30]. Since that time the increase of computational power allows

simulations to be significantly larger in size and longer in time. Several simu-

lations packages such as GROMACS [31], NAMD [32] and LAMMPS [33] al-

low to use modern hardware and multiclustering algorithms. Here we briefly

describe the basic concepts of molecular dynamics simulations.

10

Chapter 1: Molecular simulations

1.1 Force fields

In classical molecular dynamics all the covalent bonds can not be broken. In

the classical form, the potential energy the potential energy of the system

U(r) depends on the positions of all N atoms of the system r = (r1, r2, ..., rN).

Moreover, the system is characterized by the mass of each atom mi and cer-

tain boundary conditions. In practice the molecular simulation is performed

with one of the available potentials (force fields) such as CHARMM [34],

AMBER [35], OPLS [36], where the potential typically has such a form:

U(r) =∑

bonds

Kb(r− r0)2 +∑

angles

Ka(θ− θ0)2 +∑

dihedrals

Vn2

[1 + cos(nχ− δ)]+

+∑

impr.dih.

Kijkl(S − S0)2 + ULJ(r) + UE(r) (1.1)

where l is the length of a bond, θ is bond angle, χ is the dihedral angle, rij

is the distance between two atoms and all the other variables are the param-

eters of the model, which numerical values can be different in different force

fields. Here, the coefficients Ki for each term are fitted from ab initio data

or are empirical and calculated in a way that better match the experimental

behavior of studied system.

Lennard-Jones potential is representative for repulsion and van der Waals

forces [22] and is defined as:

ULJ(r) = 4ε∑

i<j

[σijr12ij− σijr6ij

], (1.2)

and electrostatic potential is:

UE(r) =∑

i<j

qiqj4πε0rij

. (1.3)

It is worth to note that in classical molecular dynamics all positive and

negative charges are presented as point charges.

11

Figure 1.1: Schematic illustration of terms of bonded potential energy in

molecular dynamics simulations.

1.2 Newtonian dynamics

In classical mechanics, the time evolution of the system is governed by the

classical Newton equations:

ri = fi/mi, (1.4)

where fi is the potential force acting on the i-th atom: fi = ∂U/∂ri It is

assumed that the system occupies the volume of appropriate shape, so the

periodic boundary conditions can be applied. In numerical simulation, the

system moves with a discrete steps of a small time interval ∆t. The value

of ∆t has to be smaller than the fastest vibrations of the systems in order

to obtain reasonable trajectory. The moves are performed with a numerical

algorithms [37–40] that allows to obtain the coordinates of each atom ri and

velocities ri at the next timestep t0 + ∆t, provided that these values are

known at time t0. The most common practice is to apply periodic boundary

conditions and calculate the energy of the long-ranged electrostatic inter-

actions via particle-mesh Ewald method [39]. For improving the efficiency,

12

Chapter 1: Molecular simulations

the constrains for the covalent bonds are applied. This approach introduces

additional forces that act on the atoms along their bonds. Hence the bond

between atoms i and j gives rise to a pair of forces: the force gij = λij(ri−rj)

acting on atom i and the force gji = λji(rj − ri) acting on j atom, where the

coefficients λij and λji are equal [29]. The Newtonian dynamics require that

the system keeps its total energy constant and moves in a way predefined

by its initial conditions (i.e. starting positions of the atoms). However, the

real systems involve some stochastic degrees of freedom via coupling to the

external environment which acts as a heat bath. In this case the total en-

ergy of the system fluctuates within a certain distribution characterized by

certain temperature and pressure. Here we briefly introduce the most com-

mon algorithms to introduce temperature and pressure coupling in molecular

dynamics.

1.3 Thermostats

1.3.1 Andersen thermostat

The easy way to obtain a temperature coupling is to periodically redefine the

velocities of each particle from a Maxwell-Boltzmann distribution [41]. This

can either be done by randomizing all the velocities simultaneously every

τT/∆t steps, or by randomizing every particle with some small probabil-

ity ∆t/τ every timestep, where ∆t is the timestep and τT is characteristic

coupling time.

This algorithm avoids some of the ergodicity issues of other algorithms, as

energy cannot flow back and forth between energetically decoupled compo-

nents of the system as in velocity scaling motions. However, it can slow down

the kinetics of system by randomizing correlated motions of the system.

13

1.3.2 Berendsen thermostat

The Berendsen algorithm mimics weak coupling with first-order kinetics to

an external heat bath with given temperature T0 [42]. The effect of this

algorithm is that a deviation of the system temperature from T0 is slowly

corrected according to:

dT

dt=T0 − Tτ

(1.5)

which means that a temperature deviation decays exponentially with a time

constant τ . This method of coupling has the advantage that the strength of

the coupling can be varied and adapted to the specific system. The Berendsen

thermostat suppresses the fluctuations of the kinetic energy. This means

that one does not generate a proper canonical ensemble, so rigorously, the

sampling will be incorrect. This error scales with 1N

, so for very large

systems most ensemble averages will not be affected significantly, except for

the distribution of the kinetic energy itself. However, fluctuation properties,

such as the heat capacity, will be affected [31].

1.3.3 Velocity-rescaling thermostat

The velocity-rescaling thermostat [43] is similar to a Berendsen thermostat

but has an additional stochastic term that ensures a correct kinetic energy

distribution by modifying it according to

dK = (K0 −K)dt

τT+ 2

√KK0

Nf

dW√τT, (1.6)

where K is the kinetic energy, Nf is the number of degrees of freedom and

dW a Wiener process. This thermostat produces a correct canonical ensemble

and still has the advantage of the Berendsen thermostat: first order decay of

temperature deviations and no oscillations.

14

Chapter 1: Molecular simulations

1.3.4 Nose-Hoover thermostat

In the Nose-Hoover scheme the system Hamiltonian extended by introducing

a thermal reservoir and a friction term in the equations of motion [44, 45].

The friction force is proportional to the product of each particle velocity and

a friction parameter, ξ. This parameter is a dynamic quantity with its own

momentum and equation of motion and the time derivative is calculated

from the difference between the current kinetic energy and the reference

temperature [31]. In this case the Newtonian equation has an additional

term:

d2ridt2

=fimi

− pξ

Q

dridt, (1.7)

where Q is a constant of the coupling and the equation of the motion for

the heat bath is:dpξdt

= T − T0, (1.8)

where T0 is the reference temperature and T is the current temperature of

the system.

1.3.5 Langevin dynamics

Another way to introduce stochastic degrees of freedom to the system is

to introduce random forces and to compensate for their overheating effect

using phenomenological friction terms [46]. In this way the modified Newton

equation will take a form:

ri = f i/mi − γiri + Fi/mi, (1.9)

where the force Fi is a random function of time which fluctuates very rapidly

in comparison with integration timestep ∆t. This force does not depend on

positions and velocities of the atoms. Then, the integrators of the system

can be written as:

15

v(t+1

2∆t) = αv(t− 1

2∆t) +

1− αmγ

F(t) +

√kBT

m(1− α2rGi (1.10)

r(t+ ∆t) = r(t) + ∆tv(t+1

2∆t), (1.11)

where

α = (1− γ∆t

m). (1.12)

Here rGi is Gaussian distributed noise with µ = 0, σ = 1.

1.4 Barostats

1.4.1 Berendsen barostat

The Berendsen barostat rescales the coordinates and the size of the simula-

tion system every step [31, 42], or every n steps, with a matrix µ which has

the effect of a first-order kinetic relaxation of the pressure towards a given

reference pressure P0 according to

dP

dt=P0 − Pτp

(1.13)

The matrix µ is defined as

µij = δij −n∆t

3τpβijP0ij − Pij(t), (1.14)

where β is the isothermal compressibility of the system. It is worth to

note that Berendsen barostat does not give the exact NPT ensemble but is

just an approximation.

16

Chapter 1: Molecular simulations

1.4.2 Parinello-Rahman barostat

Parinello-Rahman pressure coupling scheme is similar to to the Nose-Hoover

thermostat [31, 45, 47, 48]. With the Parrinello-Rahman barostat, the box

vectors as represented by the matrix b obey the matrix equation of motion:

db2

dt2= VW−1b′−1(P−Pref ) (1.15)

Here, the volume of the system is denoted as V and W is a matrix

parameter that determines the strength of the coupling (similarly to ξ in

Nose-Hoover scheme). The matrices P and Pref are the current and reference

pressures.

The equations of motion also have to be modified:

d2ridt2

=Fi

mi

−Mdridt, (1.16)

where M is:

M = b−1[bdb′

dt+db

dtb′]b′−1 (1.17)

The mass parameter W−1 determines the strength of the coupling and

possible deformation of the simulation box. It depends on the isothermal

compressibilities β, pressure coupling time τp and the largest matrix element

of simulation box L:

(W−1)ij =4π2βij3τ 2pL

(1.18)

1.5 Water models in molecular dynamics

1.5.1 Classical water models

Computer simulations of water started from the pioneering paper by Rah-

man and Stillinger about forty years ago [21]. Most important issue when

17

performing water simulations is the choice of the potential model used to

describe the interaction between molecules [49,50]. A large number of water

models exists for molecular simulations. They differ in the ability to repro-

duce specific features of real water instead of others, like the correct temper-

ature for the density maximum or the melting temperature. The mostly used

”classical” water potentials are simple rigid non-polarizable models such as

TIP3P,SPC,TIP4P,TIP4P/2005 [51–55]. However, with the increase of the

computational power new polarizable and flexible potentials begin to ap-

pear [26, 56]. The simplest water models have the positive charge on the

hydrogen atoms and a Lennard-Jones interaction site and negative charge on

the position of the oxygen. Classical water models differ in three significant

aspects: (i) the geometry of the molecule, i.e. length of OH bond and H-O-

H angle; (ii) the charge position (the negative charge of the oxygen can be

placed not in the center of oxygen atom or even can be splitted); (iii) target

properties, i.e. some properties of real water which the model is fitted to

reproduce. The parameters of Lennard-Jones potential as well as geometry

for the most used classical water models are shown in Table 1.1.

a b c

Figure 1.2: Schematic representation of three (a), four (b) and five-site wa-

ter models. All parameters can vary depending on particular water model.

Figure is adapted from Ref. [57].

All the water models were developed to reproduce certain water prop-

18

Chapter 1: Molecular simulations

erties. So as consequence, while focused on one single property they show

different results. Such an example is shown on Fig. 1.3 for the case of density.

Figure 1.3: Maximum in density for several water models at atmospheric

pressure. Filled circles: experimental results, lines: simulation results. Fig-

ure is adapted from Ref. [50].

1.5.2 Non-classical water models

With recent increase of computational power it becomes possible to simulate

relatively big systems with the potentials which explicitly takes into account

such an effects as polarizability or flexibility.Generally rigid water models give

excessive stabilization of the dimer compared with polarizable models [58].

Although the simulation time needed to simulate polarizable water model

is approximately one order of magnitude higher than rigid-body water de-

scribed above, it should increase the accuracy of the simulation results and

shed the light upon the role of polarization in the water anomalies. Polar-

izability is the ability of changing the distribution of the electronic cloud

of the atom in the presence of the external field. In classical rigid water

19

Table 1.1: Potential parameters of the classical water models. The distance

between the oxygen and hydrogen is denoted as dOH . The angle formed

by hydrogen, oxygen and the other hydrogen atom is denoted as H-O-H.

The parameters of Lennard-Jones potential is denoted as σ and (ε/kB). The

charge of oxygen is qH .All the models (except TIP5P) place the negative

charge in a point M at a distance dOM from the oxygen along the H-O-H

bisector. For TIP5P, dOM is the distance between the oxygen and the L sites

placed at the lone electron pairs. Schematic picture of different water models

is given of Fig.1.2. The table is adapted from Ref. [50].

Water

model

dOH [A] H-O-H[o] σ[A] (ε/kB)[K] qH [A] dOM [A]

SPC 1.0 109.47 3.1656 78.20 0.41 0

SPC/E 1.0 109.47 3.1656 78.20 0.423 0

TIP3P 0.9572 104.52 3.1506 76.52 0.417 0

TIP4P 0.9572 104.52 3.1540 78.02 0.52 0.15

TIP4P/2005 0.9572 104.52 3.1589 93.2 0.5564 0.1546

TIP5P 0.9572 104.52 3.1200 80.51 0.241 0.70

models this effect was not implemented due to its computational cost. Ob-

viously in this case the polarization effects are neglected and this fact can

be a source of errors and deviations from the experimental data. However,

recently several polarizable water model such as BK, SWM4, AMOEBA were

developed [25, 26, 59]. There are different ways to implement polarization.

For example, in AMOEBA force field polarization effects are treated via mu-

tual induction of dipoles at atomic centers where atomic polarizabilities were

derived from the experimental data. In terms of computational time such

approach is 8 times slower that the simulation of classical rigid-body water

model. Also it’s worth to mention that for vdW interactions AMOEBA uses

14-7 potential [60] with repulsion-dispersion parameters placed on both oxy-

20

Chapter 1: Molecular simulations

gens and hydrogens instead commonly used Lennard-Jones potential which

is used only for oxygen atoms. Another way to introduce polarization is to

use Drude oscillator potential. In this case the point charge is connected via

classical spring to the oxygen atom. In the absence of external field the spring

particle remains on the oxygen site and net charge on the oxygen is zero and

to balance the positive charges of the hydrogen the charge of hydrogens the

dummy particle with negative charge is introduced. However, the description

of some processes, such as proton transfer, requires breaking and formation

of the covalent bonds [61]. For these purposes more complex water poten-

tials are used [62]. These potentials use ab initio calculations to represent the

reacting fragments, while the remainder of the system is treated classically.

One of the simplest methods is Empirical-Valence-Body method in which

the ab initio potential energy surface is fit with an analytic form [63]. In the

same time there are attempts to create a coarse-grained potential to mimic

the behavior of water [64]. The aim of this model is to qualitatively good

description of the water properties and remain fast in terms of computational

speed. In general such models can be tuned to calculate some water prop-

erties, such as density, but lack of fully atomic description gives the error in

other properties which depend on reoriental movement of hydrogens.

1.6 Simulation details

All the simulations of bulk water in this work if not specified elsewhere were

done as following. GROMACS simulation package was used to handle the

molecular dynamics [31]. The Berendsen barostat [42], velocity rescale ther-

mostat [43] and Particle-Mesh-Ewald [39] were used for pressure coupling,

temperature coupling and long-range electrostatics calculation, respectively.

Coupling times for the barostat and thermostat were set to τP=1.0 ps and

τT=1.0 ps, respectively. This combination of pressure and temperature cou-

pling can easily produce a correct canonical ensemble. None-covalent inter-

actions were treated with 1.2 nm cut-off. The integration time-step was set

21

to 2 fs. Such value was chosen in order to monitor the kinetics of a single

hydrogen bond which lifetime is on a similar timescale. All the simulations

were done at atmospheric pressure and periodic boundary conditions. The

data was obtained over 25000 snapshots obtained from a 100 ps long run

after a 10 ns equilibration in the same conditions. Such simulation length

was chosen to equilibrate the system at low temperatures. In all cases of

bulk water simulations the box contains 1024 water molecules.

22

Chapter 2

Water phase diagram and

water anomalies

2.1 Water phase diagram

Water is present on Earth as a gas, a liquid and a solid. Its properties are of

great interest of researchers from various fields because of following reasons.

First, water plays the main role in biological properties and studying the

dynamical and kinetical properties of water molecules can help in investiga-

tion of role of water around biomolecules. Second, water is one of the most

prevalent substances in the universe and investigation of its properties can

shed some light upon composition and behavior of objects in outer space.

Third, water has reach phase diagram and many different crystalline forms,

and studying its properties and structure can help to investigate general laws

of phase transition, properties of amorphous, liquid and crystal substances.

H2O ice is characterized by one of the most complex phase diagrams: at least

16 different crystalline and amorphous modifications are observed at different

pressures P and temperatures T [65, 66]. Some of this crystalline forms are

stable, others (IC ,IV,IX,XII) exist only in metastable form. In crystal phases

of normal pressure the water local structure is close to perfect tetrahedral

while at high pressures it becomes distorted [67]. And at the pressures higher

23

than 5 katm the two independent interpenetrating hydrogen bond networks

are created (Ice VI,VII,VII) [68–70]. In general, all these possible phases

of water can occur in nature due to the restructurization of water hydrogen

network [71].

However, because phase transitions are on longer timescales than are

accessible by molecular dynamics simulations, the direct observation of the

crystallization is impossible. For this purpose the methods based on the

energy calculations of beforehand constructed structures are used [72, 73].

On Fig. 2.1 experimental phase diagram and the results for TIP4P water

model [73, 74]. Although two diagrams quantitatively are not the same,

TIP4P model is able to capture the main features of water phase diagram.

Figure 2.1: Phase diagrams of water. Left panel: simulation results from

TIP4P water model. Right panel: experimental phase diagram. Only stable

phases of ice are shown. Adapted from Ref. [73].

2.2 Water anomalies

The anomalies of water are properties where the behavior of liquid water is

different from what is found with other liquids [75]. In the following section

24

Chapter 2: Water phase diagram and water anomalies

we highlight some of the anomalous properties of water.

At atmospheric pressure after passing the melting point water density

increases, reaches its maximum at 277 K and only after that going down,

while in other liquids the density always decreases with the increasing of

temperature [76,77]. Such a maximum is the only one occurring in liquids in

their stable liquid phases just above the melting point [77]. The high density

of liquid water is due mainly to the cohesive nature of the hydrogen-bonded

network, with each water molecule capable of forming four hydrogen bonds.

This reduces the free volume and ensures a relatively high-density, partially

compensating for the open nature of the hydrogen-bonded network. The

anomalous temperature-density behavior of water can be explained utilizing

the range of environments within whole or partially formed clusters with

differing degrees of dodecahedral puckering [78,79].

Another interesting property related to the water density is that the den-

sity of liquid water is higher than the density of ice. It is usual for liquids to

contract on freezing and expand on melting. This is because the molecules

are in fixed positions within the solid but require more space to move around

within the liquid [80]. The structure of ice Ih is open with a low pack-

ing efficiency where all the water molecules are involved in four directed

tetrahedrally-oriented hydrogen bonds and passing the melting point some

of these bonds break and some become distorted, what is different with re-

spect to another solids, where breaking bonds upon melting requires more

space and therefore the density decreases [80]. It’s worth to note that this sit-

uation does not happen with high-pressure ices (III,V I,V II), which expand

on melting [81].

It can be expected that due to large cavities in hydrogen bond network

water should have a high isothermal compressibility (kT = −[dVdP

]T/V ]). In

fact, water has unusually low compressibility (0.46 GPa−1, compare to CCl4

1.05 GPa−1 at 300 K) [82, 83]. The low compressibility of water is due to

the cohesive nature of its hydrogen bonds. This means that in fact there’s

not so many free space as it can be expected. Also, the compressibility

25

behavior in temperature space is different with respect to typical liquids. In

a typical liquid the compressibility increases with increase of the temperature

(the structure becomes less compact). But because water structure becomes

more open at lower temperatures, the capacity to be compressed increases

[84–86]. At sufficiently low temperatures, where the liquid-amorphous phase

transition occurs the compressibility reaches its maximum [86] (see Fig. 2.2).

Figure 2.2: Isothermal compressibility of water. Solid lines are data from

Ref. [86], symbols represents the data from Ref. [84,85,87]. Figure is adapted

from Ref. [86].

Water has the highest specific heat of all liquids except ammonia. This

occurs because as water is heated, the increased movement of water causes

the hydrogen bonds to bend and break. As the energy absorbed in these

processes is not available to increase the kinetic energy of the water, it takes

considerable heat to raise water’s temperature. Also, as water is a light

molecule there are more molecules per gram, than most similar molecules,

to absorb this energy [57,76]. However the occurrence of a maximum in the

26

Chapter 2: Water phase diagram and water anomalies

specific heat as the pressure or temperature is varied across the extension

of the coexistence line is well documented. This is understood by definition

of the ’Widom line’ a term introduced to define the locus of maximum

correlation length that extends into the single fluid phase beyond the critical

point [88].

Another striking property of water is anomalous increase of viscosity with

lowering the temperature [89, 90]. The water cluster equilibrium shifts to-

wards the more open structure as the temperature is lowered. This structure

is formed by stronger hydrogen bonding. This creates larger clusters and

reduces the ability to move or in other words increases viscosity [57]. It is

also interesting that Einstein-Stokes relation which connects viscosity and

temperature D = kBT6πηr

(here D is diffusion coefficient, η is viscosity and r

is approximate radius of the particle) violates for water. At low tempera-

tures the diffusion dependence on temperature can be fitted with Arrenhius

lax while at high temperatures it behaves accordingly to empirical Vogel-

Fulcher-Tamman relation D = D0exp(kToT−T0 ), where D0 and T0 are fitting

coefficients). The example of such a behavior is shown on Fig. 2.3 [90–92].

Figure 2.3: The temperature dependence of the inverse of self-diffusion coef-

ficient of water. Red line is fit to the Vogel-Fulcher-Tamman relation, dashed

line is fit to the Arrhenius law. Figure is adapted from Ref. [90]

27

Here we explain only some unusual properties of water, but it’s evi-

dent that its properties are strongly correlated with its hydrogen bond local

structure. In order to study structure and dynamics of hydrogen bond net-

works various experiments were made [71, 73, 90, 93–95] and theories were

proposed [9, 10, 13, 16, 66], but yet the whole picture is unclear. For exam-

ple, there is still open question about inhomogeneties of liquid water and its

structure in general [16].

28

Chapter 3

Water supercooling and

freezing

3.1 General perspective

Water freezing is not simply the reverse of ice melting . Melting is a single

step process that occurs at the melting point as ice is heated whereas freezing

of liquid water on cooling requires ice crystal nucleation and crystal growth

that generally is initiated a few degrees below the melting point even for pure

water [96]. Here we refer to the liquid water below its melting temperature

as to supercooled water. Liquid water may be easily supercooled to 248 K

and with more difficulty to the temperature of homogeneous nucleation TH ≈225 K at atmospheric pressure [84, 97]. Supercooled water is a metastable

phase of liquid water below the melting temperature [66]. In this regime, the

transition to the solid phase is irreversible once the process is activated.

At low temperatures water is a liquid, but glassy water - also called amor-

phous ice - can exist when the temperature drops below the glass transition

temperature Tg (about 130 K at 1 atm). Although glassy water is a solid,

its structure exhibits a disordered liquid-like arrangement [66]. This state of

water is known for many years and calls low-density amorphous ice. Around

thirty years ago another form of amorphous ice with much higher density

29

Figure 3.1: Schematic illustration indicating the various phases of liquid

water. Figure is adapted from Ref. [97].

(High-density amorphous ice, HDA) was obtained experimentally [98] (See

Fig. 3.1).

Low-density ice originally was obtained by depositing water vapor upon a

cold plate [99] or by rapid cooling of small water droplets [100]. Upon heating

up to 130K this form of ice transforms to a highly viscous liquid [101]. On

the other hand, high-density ice was obtained by compressing hexagonal ice

IH below temperatures of 150K [66,98,102]. After further compression HDA

crystallizes into high-density crystalline ice [103]. Moreover, with changing

pressure this two forms (LDA and HDA) can interconvert with volume change

30

Chapter 3: Water supercooling and freezing

of about 20%. Thus it remains unresolved whether one considers HDA to be

a glassy state of liquid water or to be a collapsed crystal state . Recently

it was hypothesized that at higher temperatures LDA and HDA will turn

into low-density liquid and high-density liquid phases respectively [13, 66].

However, the possible liquid-liquid critical point lays in so called ”no man’s

land”, the region almost unreachable for the experiments because supercooled

water freezes at such temperatures.

An interesting discussion recently developed on the relationship between

crystallization rate and the time scales of equilibration within the liquid

phase [104, 105]. Calculations using a coarse grained monoatomic model of

water, the mW model, suggested that equilibration of the liquid below the

temperature of homogeneous nucleation TH ≈ 225 K is slower than ice nu-

cleation [105]. This observation has important consequences to a proposed

theory of water anomalies, predicting a second critical point below TH where

a liquid-liquid phase transition occurs [13]. Although it has attracted at-

tention [106–109], this theory is not without problems. If the speed of ice

nucleation is faster than liquid relaxation, the liquid-liquid transition would

loose sense from a thermodynamical point of view, being the liquid phase

not equilibrated [104]. It is worth to note that during the whole history

of the molecular dynamics simulations of water there’s still no evidence of

systematic water nucleation so far [14].

3.2 Test of water freezing

To investigate the relaxation properties of an atomistic model in the super-

cooled region below TH , a 3 µs long molecular dynamics simulation of the

TIP4P-Ew water model. The length of this calculation is one order of magni-

tude larger than the 350 ns used to study freezing with the mW model [105].

The simulation was run at 190 K and 1250 atm. These values are close to

the estimated liquid-liquid critical point for the TIP4P-Ew [15], congruous

with recent calculations on the similar TIP4P/2005 model [107].

31

The structural parameters are designed to distinguish between different

phases by analyzing the geometrical structure. Here we used two different

approaches to estimate the structural order of water molecules. First one,

the tetrahedral order parameter which takes into account the configuration

of four nearest neighbors of the water molecule i, qi. It was calculated as

qi = 1− 3

8

3∑

j=1

4∑

k=j+1

(cosψjik +

1

3

)2, (3.1)

where ψjik is the angle formed by their oxygens [71]. The averaged value

of this order parameter over an ensemble of water molecules for each sin-

gle timestep is denoted as QT . The second parameter we used is bond-

orientational parameter Q6 developed by Steinhardt et. al. [110]. This

parameter is a function of a projection of the density field into averaged

spherical harmonic components. To calculate Q6 we need to calculate the

set of quantities

qil,m =1

4

4∑

j∈ni

Y ml (φijθij), −l ≤ m ≤ l (3.2)

where the sum is over four nearest neighbors, ni. Y ml is the l,m spherical

harmonic function associated with the angular coordinates of the vector ~ri−~rjjoining molecules i and j, measured with respect to an arbitrary external

frame. These quantities are then summed over all particles to obtain a global

metric

Ql,m =N∑

i=1

qil,m (3.3)

and then contracted along the m axis to produce a parameter that is invariant

with respect to the orientation of the arbitrary external frame,

Ql =1

N(

l∑

m=−l

Ql,mQ∗l,m)

12 (3.4)

The most probable value of Ql for an amorphous phase approaches zero in

the thermodynamic limit, while it is finite for a crystalline phase [104]. We

32

Chapter 3: Water supercooling and freezing

used l = 6 because it was found empirically that it is useful for distinguishing

liquid water and ice [104, 111]. It is worth to note that the main difference

between these order parameters is that Q6 is the measure of the crystalline

order for the whole system. On the other hand QT describes tetrahedral order

of the single water molecule and can vary for the different water molecules

showing at the same time moment that some waters keep tetrahedral ice-like

structure while another have distorted liquidlike structure.

-55

-54

Ep

[kJ/

mol

]

A

B

C

D

0.97

0.99

1.01

ρ [g

cm

-3]

0.82

0.84

0.86

0.88

QT

0.01

0.03

0 1000 2000 3000

Q6

Time [ns]

Figure 3.2: Time series for the 3 µs trajectory. (A) potential energy; (B)

density; (C) tetrahedral order parameterQT ; (D)Q6 parameter. Right panels

show the probability distribution of the respective quantities.

33

In the simulated conditions, water freezing was not observed as shown

by the timeseries of the potential energy Ep (Fig. 3.2A). Fluctuations are

of the order of 0.5 kJ/mol per molecule with no systematic drift. It has

been observed that once freezing is activated the energy drifts very quickly

to low values of the potential energy, with large energy changes (e.g. roughly

5 and 2 kJ/mol per molecule for TIP4P at 230 K [14] and TIP4P/2005 at

242 K [112], respectively).

The time series of the density ρ and the tetrahedral order parameter QT

[71] are shown in Fig. 3.2B-C. They respectively correlate and anticorrelate

with the potential energy (Pearson correlation coefficient r = 0.69 and -0.86)

(see upper panel of Fig. 3.3). The distributions of both ρ and QT show

an appreciable bump at one of the tails (see right panel of Fig. 3.2B-C),

suggesting the presence of a subpopulation. For the case of the tetrahedral

order parameter, the subpopulation emerges at values around 0.873 (red

dashed line and right side of Fig. 3.2C). This fluctuation is localized in a

time window between 2.3 and 2.6 µs in correspondence to a decreasing of

both the density and the potential energy. It is interesting to note that

density subpopulations have been interpreted by some [111] as a signature of

the aforementioned liquid-liquid transition.

To check whether this fluctuation corresponded to an ice nucleation at-

tempt, the Q6 order parameter [104, 110, 113] was calculated (Fig. 3.2D). In

the time window between 2.3-2.6 µs the value of the parameter is around

0.025, with no signs of ice nucleation. Moreover, no correlation with the en-

ergy was found (r = 10−6). With a value of Q6 for hexagonal ice expected to

be one order of magnitude larger [113], no evidence for ice nucleation is found

in the present trajectory. Moreover, nor correlation neither anticorrelation

between Q6 and any other of calculated parameters was observed (bottom

panel of Fig. 3.3).

Also to check the fact that at studied conditions the water molecules can

move we calculated the oxygen mean-square-displacement (MSD) as:

34

Chapter 3: Water supercooling and freezing

960

970

980

990

1000

1010

1020

-55 -54.5 -54 -53.5

De

nsity [

kg

m-3

]

Energy [kJ/mol]

960

970

980

990

1000

1010

1020

-55 -54.5 -54 -53.5

De

nsity [

kg

m-3

]

Energy [kJ/mol]

960

970

980

990

1000

1010

1020

0.82 0.84 0.86 0.88D

en

sity [

kg

m-3

]QT

960

970

980

990

1000

1010

1020

0.82 0.84 0.86 0.88D

en

sity [

kg

m-3

]QT

-55

-54.5

-54

-53.5

0.82 0.84 0.86 0.88

En

erg

y [

kJ/m

ol]

QT

-55

-54.5

-54

-53.5

0.82 0.84 0.86 0.88

En

erg

y [

kJ/m

ol]

QT

-55

-54.5

-54

-53.5

0 0.01 0.02 0.03 0.04 0.05

En

erg

y [

kJ/m

ol]

Q6

960

970

980

990

1000

1010

1020

0 0.01 0.02 0.03 0.04 0.05

De

nsity [

kg

m-3

]

Q6

0.82

0.84

0.86

0.88

0.9

0 0.01 0.02 0.03 0.04 0.05

QT

Q6

Figure 3.3: Instant relationship between Q6,QT , density and potential energy.

MSD(t) = 〈(ri(t)− ri(0))2〉, (3.5)

where ri is the coordinates of single atom (Fig. 3.4). At timescales shorter

than one ns, water shows a subdiffusive behavior (dotted line in Fig. 3.4).

For larger times the system enters a diffusive regime, following the linear

relationship MSD ≈ t (dashed line), with a maximum average displacement

of 3.47 nm after 3 µs. Taking into account that the molecular diameter

is around 0.3 nm, water molecules have diffused for about 11.5 molecular

diameters (the average box side length is of 3.14 nm).

With these results the evidence is provided that the liquid phase of the

TIP4P-Ew model is at equilibrium in the supercooled regime before ice nu-

cleation. This result is in agreement with another µs long simulation of

supercooled water with a 5-site model [111], suggesting that equilibration of

the liquid phase below TH is a common feature of atomistic models. The mW

35

10-3

10-2

10-1

100

101

MS

D [n

m2 ]

10-3

10-2

10-1

100

101

10-2 100 102 104

MS

D [n

m2 ]

Time [ns]

Figure 3.4: Oxygen mean square displacement (MSD). The dashed and dot-

ted lines represent a linear and a power-law (exponent equal to 0.1) regres-

sion, respectively. The diffusion coefficient extracted from the linear regime

is of 6.6× 10−9cm2/s. The g msd function of GROMACS was used with 150

windows to improve statistics.

model has shown to reproduce several properties of water, including density

and phase diagram [114]. But the lack of hydrogens, and consequently of

molecular reorientations [17], might considerably speed up the time scales.

Probably, the differences in the relaxation kinetics between atomistic models

and the mW model are due to the lack of molecular reorientations in the

latter. Clearly, further experimental validation is needed to clarify which

proposed mechanism (if any) is closer to real water.

36

Chapter 4

Complex network approach for

molecular dynamics trajectories

and hydrogen bond as an order

parameter

Molecular dynamics simulations can give the important information about

thermodynamics and kinetics of the simulated systems [28]. Order param-

eters are conventionally used for this purposes [115, 116]. Some of the con-

ventional order parameters commonly used to measure the structure of liq-

uids were described in previous chapter. Unfortunately, it is known that

reduced descriptions based on order parameters in many cases are inaccu-

rate [28, 115, 117–120]. The description based on order parameter can not

clearly define to which state belong the certain value of an order parameter.

Moreover, in some cases kinetic description based on the order parameter is

wrong. The example of such a problem is a stochastic two state model, which

was studied in Ref. [115] (see Fig. 4.1). The origin of the failure is due to

overlaps in the order parameter distribution, i.e., configurations with differ-

ent properties corresponding to the same value of the coordinate, making the

discrimination between states almost impossible [121, 122]. To improve this

37

situation a new arsenal of tools emerged making use of complex networks

and the theory of stochastic processes [28, 123–125] as it described in the

following.

4.1 Complex network as a tool to study molec-

ular simulations

A network is a set of items, which we will call nodes, with connections be-

tween them, called edges. Systems taking the form of networks abound in

the world [126]. Here we will call “complex network” the network with non-

trivial topological properties. Surprisingly such networks can be obtained

from many sociological [127], biological [128] or technological systems [129].

From analysis of the networks built from the real systems one can obtain

many useful information. For example with network analysis possible to

detect the most vulnerable nodes, destroying which the connectivity of the

network would be highly reduced. Another useful property of the networks is

community (cluster) structure i.e., groups of nodes that have a high density

of edges within them, with a lower connectivity between these groups.

It is obvious that social complex networks split in a groups along certain

interests, friends, age, occupation. The same happens with the complex

networks built from other systems. But in the case of some systems splitting

into communities is not so easy. For this purpose many algorithms were

proposed [130–133]. Some of them are fast but not precisely accurate, some

are better in predicting cluster structure but require more computational

time. Another important aspect is that the output of algorithm depends on

the structure of the complex network. However, for the typical analysis of

molecular dynamics trajectory not all the conventional algorithms are able

to properly map the free-energy landscape [124,134].

Here we describe one of the complex networks approaches to map the free-

energy landscape of the system from the molecular dynamics simulation. The

basic idea behind this approach is to map a dynamical system into a discrete

38

Chapter 4: New strategies for the analysis of molecular dynamicstrajectories

(a)

(b)

Figure 4.1: Timeseries of an artificial order parameter of stochastic two-state

model. (a) The conventional histogram method is unable to distinguish be-

tween two states with the same value of an order parameter. (b) Network

clusterization techniques allow the lumping of kinetically homogeneous re-

gions of the network into states and build a model of the original process.

Figure is adapted from Ref. [115].

set of microstates, and their interconvertion rates as calculated from the

original trajectory. The advantage of this approach is that it allows to merge

different parameters into a single order parameter. To obtain the transition

network from molecular dynamics trajectory the following procedure has to

be done. For the snapshot at time t for each water molecule we define a

microstate based on some order parameter. In the case of water the most

natural parameter is a hydrogen bond structure of its solvation shells [16].

This microstate represents a single node of a transition network. Then we

39

Figure 4.2: The example of complex network obtained from molecular dy-

namics. Here, microstates were defined as different conformations of protein.

On the upper panel the whole complex network is shown, on the lower panel

nodes which belong to the same clusters were merged together. Figure is

adapted from Ref. [135].

can obtain the value for the order parameter at the next snapshot t+∆t and

get the corresponding microstate. If two microstates i and j are different

40

Chapter 4: New strategies for the analysis of molecular dynamicstrajectories

the link with weight Wij=1 is put into the transition network, for the case

when microstate remained the same, the selflink Wii is put in the network.

If certain transition occured second time the link weight has to be increased:

Wij+=1. Doing this procedure for all the snapshots in the trajectory one

can obtain the transition network. At equilibrium the obtained weight of

the certain node is equal to its probability and link between two nodes is

proportional to the transition probability [28]

In case of liquid water the definition of the microstate has to mimic the

topology of hydrogen-bond network around a given water molecule that de-

termines the structural and dynamical properties of the bulk. However, the

binding partners to any central water molecule are not predefined but keep

exchanging on a fast picosecond time scale [136]. Therefore, any approach

to define a microstate must be invariant to interchanging water molecules,

as well as binding sites [16]. To simplify the definition of the microstate it is

useful to make an approximation that each water molecule can have at max-

imum four hydrogen bonds (two on the oxygen and one on each hydrogen).

In some cases all of four possible hydrogen bonds are formed, but in others

there are broken bonds and distorted loops (See Fig.4.3). The microstate def-

inition describes each of possible structures by a unique string that encodes

the connectivity through hydrogen bonds. For each molecule the search of

a hydrogen bond partners is performed. After finding this molecules which

form the first solvation shell, the search expands in a treelike manner. Each

subsequent solvation shell is a new generation and follows, in order, in the

microstate string, numbered by their position in the fully hydrogen-bonded

tree up to the second solvation shell [16].

From an operative point of view, the algorithm works on a per-node

basis by deleting all the links (transitions) but the most visited one (which

represents the local direction of the gradient). When applied to the whole

network, the algorithm provides a set of disconnected trees, each of them

representing a collective pathway of relaxation to the bottom of the local

free-energy basin of attraction (gradient-cluster, gray regions in Fig. 4.4).

41

[h!]

Figure 4.3: Water microstates. (a) Conformation in which all four hydrogen-

bonding sites of each water molecule connect to new water molecules, and the

corresponding microstate string. Water molecules are numbered according

to their appearance in the tree search, and water molecules from subsequent

generations are placed next to each other. (b) If a hydrogen-bonding site is

empty (e.g., molecule 5), it is labeled as 0, as are all subsequent entries down

the tree. Small loops, such as 1-2-3, are included in a natural fashion. Figure

is adapted from Ref. [16].

42

Chapter 4: New strategies for the analysis of molecular dynamicstrajectories

Each gradient-cluster represents a structurally and kinetically well defined

molecular arrangement with an extension of up to two solvation shells [16].

The application of the conformational network technique is shown in Chapter

V of this work.

As observed elsewhere [119, 124, 135, 137], the transition network syn-

thetically encodes the complex organization of the underlying free-energy

landscape. Specifically, densely connected regions of the network correspond

to free-energy basins, i.e., metastable regions of the configuration space. Sev-

eral algorithms can be used to extract this information, including the max

flow theorem [119], random walks [124, 138] or transition gradient analy-

sis [137,139]. All these approaches aim to clusterize the network into kineti-

cally and structurally well defined basins of attraction.

In enthalpy driven free-energy landscapes, of which proteins are an archety-

pal example, the transition probability to stay inside a given basin Zin is

much larger then the probability to hop outside Zout [119, 135]. That is,

basin hoping is a rare event. Moreover, the number of neighboring basins

is usually very limited, with the emergence of well defined transition path-

ways [28, 125, 135]. This is not the case for water [16]. Being a liquid, it

is mainly characterized by entropic basins of attraction. As illustrated in

Fig. 4.4, Zin and Zout become comparable because the cumulative of the

many small inter-basin transition probabilities (Zout) is similar to the few

highly populated intra-basin relaxations (Zin). In other words, the probabil-

ity to leave the basin i is similar to stay in it. This observation would lead

to the conclusion that, at the atomic level, water does not have any type of

configurational selection. However, this is not true when considering all the

contributions to Zout separately:

Zout =∑

i

Z(i)out (4.1)

Structural inhomogeneities, i.e., configurational selection, emerge because

max(Z

(i)in

)� max

(Z

(i)out

), (4.2)

43

meaning that the probability of an intra-basin transition is larger than

hoping to any other specific basin. When this condition holds, the environ-

ment of a given water molecule alternatively adopts a number of different

configurations, each of them characterized by a specific free-energy basin

of attraction. This is an emergent property of water at ambient tempera-

ture [16].

Zout

Zin

Figure 4.4: Configuration-space-networks. Pictorial representation of the

relative balance between intra-basin (Zin) and inter-basins (Zout) transi-

tion probabilities from the point of view of a node (in blue). Gray re-

gions represent free-energy basins of attraction as detected by the gradient-

algorithm [137,139].

4.2 Hydrogen bond criteria

Hydrogen bond is one of the possible order parameters which can be used

to obtain free-energy landscape of water. It represents a fundamental in-

teraction in molecular systems [140]. Its peculiarity resides in the common

aspects it has with both covalent bonds and van der Waals interactions. In

hexagonal ice the energy of the hydrogen bond is part electrostatic (90 %)

44

Chapter 4: New strategies for the analysis of molecular dynamicstrajectories

and part covalent(10 %) [141], however it is not clear if this is the case for

the liquid water. The strong directionality together with the ease of being

formed and broken at ambient conditions makes it an important ingredient in

water structure and dynamics [142], protein stability [143] and ligand bind-

ing [144]. Notwithstanding, a universal definition of this interaction is still

missing [145]. The case is even more difficult for molecular dynamics where

the different potentials for water are used [136].

Hydrogen bonds are formed between two polar atoms via a hydrogen

which is covalently bound to one of the two. This interaction is highly direc-

tional. For example, in bulk water at 300 K the angle OH-O is mostly below

30 degrees [146], while the donor-acceptor distance is of around 3.5 A [147].

Despite the apparent simplicity, the presence of thermal fluctuations as well

as the non-trivial effects of the environment made the development of an

operative definition of this bond difficult.

In the last decades, several definitions were proposed based on computer

simulations [136]. The most popular ones look at bond formation by using a

mixture of distances and angles between the two partners [148–150]. Others

tried to avoid altogether cutoffs by proposing topology-based definitions [151–

153]. Given the many degrees of freedom involved in molecular association,

it is now clear that all definitions retain some degree of arbitrariness [154].

In most cases, hydrogen bond definitions were developed at specific ther-

modynamic conditions. However, not much is known on the behavior of

those definitions as a function of temperature and water model. This section

is an effort to present a transparent comparison between hydrogen bond def-

initions in several different conditions, including temperature, water model

and cutoff dependence. Here, we present an assessment of most used hy-

drogen bond definitions based on the analysis of molecular dynamics simu-

lations of water in a temperature range from 220 K to 400 K. Six among

the most widespread classical water models were used in the analysis, in-

cluding SPC [52], SPC/E [53], TIP3P [54], TIP4P [74], TIP4P-Ew [51] and

TIP4P/2005 [55]. Comparison of this water models per se is presented in

45

Chapter V.

Six hydrogen bond definitions were considered. Here we distinguish two

broad classes of hydrogen bond definitions: geometrical and topological (Fig. 4.5).

The difference between them is that geometrical definitions make use of cut-

offs on inter-atomic distances and angles while the latter mostly avoid this

problem using topological criteria. A brief description of the definitions fol-

lows.

geometrical topological

rOO

rOH

Θ

Figure 4.5: Hydrogen bond definitions can be roughly partitioned into two

classes: geometrical and topological.

Geometrical definitions

1. rOH . In this definition the oxygen-hydrogen distance (rOH) is used

as criterion (Fig. 4.5A) [149]. In the original work, a cutoff of 2.3 A

was proposed by simulating amorphous ice at T=10 K with the TIPS2

potential [155]. The distance cutoff value is related with the position of

the first minimum in the oxygen-hydrogen radial distribution function.

2. rOOΘ. This definition makes use of both the oxygen-oxygen distance

46

Chapter 4: New strategies for the analysis of molecular dynamicstrajectories

(rOO) and the ∠OOH angle (Θ) between two water molecules. In the

original work, a bond was considered formed when rOO and Θ were

smaller than 3.5 A and 30 degrees, respectively [150]. The distance

cutoff was taken from the position of the first minimum in the oxygen-

oxygen radial distribution function. Missing a clear signature of the

bond state in the distribution of the angle Θ, the cutoff value was

taken from experimental data [146,147].

3. Sk. The hydrogen bond definition of Skinner and collaborators is based

on an empirical correlation between the occupancy N of the O · · ·H σ∗

orbital and the geometries observed in molecular dynamics simulations

[148]. Two water molecules were considered bonded if the value of N

is higher than a certain cutoff which is taken in correspondence to the

position of the first minimum in the distribution of N . In the original

paper N was defined as:

N = exp(−r/0.343)(7.1− 0.05φ+ 0.00021φ2), (4.3)

where φ is the angle bewteen water molecule bisector and a vector

between oxygen of a water molecule and hydrogen of a possible partner

(See Fig. 4.6). A cutoff equal to 0.0085 was chosen by analyzing MD

simulations of the SPC/E model at ambient conditions.

Topological definitions

4. DΘ. A hydrogen bond is formed between a hydrogen atom and its

nearest oxygen not covalently bound. An additional restriction was

imposed: the angle Θ had to be lower than π/3. In the original work

[152], this definition was applied to the study of the SPC/E water model

for temperatures ranging from 273 to 373 K.

5. DA. Two criteria for the hydrogen bond were used: (i) the acceptor

is defined as the closest oxygen to a donating hydrogen and (ii) this

hydrogen is the first or second nearest neighbor of the oxygen. As a

47

Figure 4.6: Pictorial representation of the distances and angles used for hy-

drogen bond definitions. The z axis is perpendicular to the molecular plane.

Figure is adapted from Ref. [148].

consequence, the total number of hydrogen bonds per water is limited

to four. This definition was proposed with simulations of the EMP

water model at 292 K [151].

6. TP. A hydrogen bond is formed between a hydrogen and its closest

oxygen. When more than one hydrogen bond between the two water

molecules is found, the one with the shortest oxygen-hydrogen distance

is considered to be formed [153]. This definition was mainly evaluated

at ambient conditions using the TIP4P/2005 water model.

To analyze difference between hydrogen bond definitions described before

we analyzed the number of hydrogen bonds per molecule. Here we performed

analysis over SPC/E model since results of its simulations were used to define

the most recent Sk hydrogen bond criterium [148]. Discrepancies were found

in the distribution of the number of bonded partners (Fig. 4.8). At 300 K

geometrical definitions were quite consistent among each other, with a larger

fraction of three hydrogen bonded configurations for Sk. For the topological

case, DA and TP agreed on the number of four coordinated molecules. How-

ever, the former detected a larger fraction of three and two bonded molecules

while TP presented a non-negligible fraction of cases with five partners and

48

Chapter 4: New strategies for the analysis of molecular dynamicstrajectories

2.8

3

3.2

3.4

3.6

3.8

4

240 280 320 360 400

N,

HB

s p

er

mo

lecu

le

Temperature [K]

Figure 4.7: Average number of hydrogen bonds per water molecule for the

six hydrogen bond definitions: rOH (orange), rOOΘ (green), Sk (red), DΘ

(cyan), DA (blue) and TP (purple)

no evidence for two bonded molecules. This scenario changes when a topo-

logical definition is coupled with an angle cutoff (DΘ). In this case, almost

identical results as the conventional rOOΘ were found with an agreement that

persists in the entire temperature range as shown in Fig. 4.7 (green and cyan

data).

Kinetics was analyzed in terms of hydrogen bond lifetime distributions.

The lifetime was calculated as follows. For each definition, pairwise hydro-

gen bonds among all water molecules were calculated for every frame. For

each of the water pairs that formed a bond, the time span for how long that

particular bond lasted is called lifetime. The distribution was then calcu-

lated by building an histogram of all the lifetimes collected in the molecular

trajectory. The average lifetime was denoted with the symbol τ .

Distributions for the six hydrogen bond definitions at 300 K are shown

in Fig. 4.9A. Fastest decays (i.e. shorter life times) were observed for rOOΘ

(green) and DΘ (cyan), strongly suggesting that fluctuations along the Θ an-

49

0

0.2

0.4

0.6

0.8

Pro

ba

bili

ty

rOH rOOΘ Sk

0

0.2

0.4

0.6

0.8

1 2 3 4 5

Pro

ba

bili

ty

N hb

DΘ

1 2 3 4 5

N hb

DA

1 2 3 4 5

N hb

TP

Figure 4.8: Average number of bonded partners for the six hydrogen bond

definitions at 300K.

gle represent the major responsible for the faster kinetics. On the other hand,

the largest lifetimes were found with the TP definition. At very short times

(<200 fs) purely topological approaches provided the best results (inset of

Fig. 4.9A). In fact, both DA (blue) and TP (purple) showed a smooth decay,

in contrast to all the other definitions which provided a debatable oscillating

behavior [156]. This observation strongly suggests that those fluctuations

are an artifact of the use of cutoffs.

For the average lifetime τ , Arrhenius behavior in the range 260 K<T<400 K

was found, breaking down for lower temperatures (Fig. 4.9B). rOOΘ, DΘ

and rOH , DA, TP provided fastest and slowest timescales, respectively. Sk

shifted from one group to the other while changing temperature (red data in

Fig. 4.9B).

Hydrogen bond propensities up to the second solvation shell were ob-

tained by calculating the following parameters: the probability P4 to have a

water molecule with a fully-coordinated first and second solvation shells, for

a total of 16 bonds ( Fig. 4.10); and the probability to have four (P ∗4 ), three

(P3) and two or less (P210) bonds with a generic first solvation shell. In the

calculation of P ∗4 the propensity of P4 was subtracted. A more comprehen-

50

Chapter 4: New strategies for the analysis of molecular dynamicstrajectories

10-7

10-6

10-5

10-4

10-3

10-2

10-1

0 5 10 15 20 25 30

Pro

babi

lity

time [ps]

A

10-1

100

101

2.5 3 3.5 4 4.5

τ [p

s]

1000/T [K-1]

B

10-2

0 0.1 0.2

Figure 4.9: Hydrogen bond kinetics for the six different definitions: color-

code is the same as in Fig 4.7. (A) The lifetime distribution at T=300K is

shown. (B) the average hydrogen bond lifetime versus 1/T is plotted (error

bars are smaller than the symbol size). The Arrhenius behavior is observed

in the range of temperatures from 260 to 400K.

51

sive study of these four propensities, including temperature and water model

dependence is presented in Chapter V of this thesis.

configurationP4

Figure 4.10: Graphical representation of fully coordinated molecule, P4.

In Fig. 4.11 hydrogen bond propensities including the second solvation

shell are presented. The behavior of these propensities strongly depend on the

hydrogen bond definition taken into account. Consistency was found within

two groups. The first one includes rOH , rOOΘ and DΘ and the second one

TP and DA. Sk did not match very well any of them. The value of P4,

i.e., the probability to have a four-coordinated water molecule with a fully

coordinated first and second shells (Fig. 4.5B), was equal to 0.34 and 0.58 at

220K for Sk and TP , respectively (red data). As temperature was increased

this difference became even more pronounced. A similar disagreement was

also observed for the other three propensities.

An interesting case is given by P ∗4 . This quantity reports on four-coordinated

water molecules with an arbitrarily disordered second solvation shell. For all

definitions this quantity presented a peak. However, TP and DA made an ex-

ception being the maximum much more shallow and at a higher temperature

with respect to the other approaches. This leads to an over estimation of four

coordinated water molecules which are predicted to be the most abundant

52

Chapter 4: New strategies for the analysis of molecular dynamicstrajectories

0.0

0.2

0.4

0.6

0.8

Popula

tion

P4

P*4

P3

P210

rOH

rOOΘ

Sk

0.0

0.2

0.4

0.6

0.8

240 280 320 360 400

Popula

tion

Temperature [K]

DΘ

240 280 320 360 400

Temperature [K]

DA

240 280 320 360 400

Temperature [K]

TP

Figure 4.11: Hydrogen bond propensities including the second solvation shell

for temperatures between 220 K and 400 K. P4, P∗4 , P3 and P210 are shown

in red, blue, light blue and very light blue, respectively.

configuration at temperatures as high as 400 K. This result is counter intu-

itive as waters with three or less hydrogen bonds would have been expected

to represent a larger fraction of the sample at a such high temperature. Sub-

stantial discrepancies among definitions were also found in the case of P210

(water molecules with two bonds or less). For the case of TP this probability

was essentially zero at all temperatures while it grew with temperature in all

the other cases.

Hydrogen bonds were described so far on the base of propensities and

kinetics. Now, we investigate the robustness of the geometrical definitions

with cutoff choice. The aim of the following analysis is to understand what

53

is the influence of temperature and water model on the distributions relevant

to cutoff choice. In fact, default cutoff values were originally proposed from

experiments and calculations at specific temperatures and water models. Al-

though in most cases prescriptions were given to properly choose the cutoffs,

default values were often applied in conditions far away from the original

works.

To check this temperature dependence was investigated with conventinal

radial distribution functions (RDF). Here and in the following text we define

RDF as:

g(r) = 4πr2ρdr, (4.4)

where ρ is the density of the system and ρ is the number of particles over

volume.

For rOH , the distribution that matters is the oxygen-hydrogen radial dis-

tribution function (g(r), left column in Fig. 4.12). The plot shows that the

first minimum becomes less pronounced with temperature while its position

gets closer to the origin (from 2.424 to 2.419 A, left middle row of Fig. 4.12).

Choosing the cutoff according to the position of the minimum, the average

number of hydrogen bonds per molecule was significantly affected despite

the small change of the cutoff value. In the bottom left panel of Fig. 4.12

the difference between a “standard” cutoff approach (empty circles) and a

temperature dependent cutoff (filled circles) is shown.

Similar results were obtained for the other two geometrical definitions.

For these cases the value of the cutoff was chosen according to the position

of the first minimum of the oxygen-oxygen radial distribution function and

the distribution of the occupancy N for the case of rOOΘ and Sk, respectively

(second and third columns of Fig. 4.12). Lacking of a bimodal behavior we

intentionally avoided the study of the angle Θ cutoff dependence.

Interestingly, radial distribution functions depend not only on temper-

ature but also on the water model under study. This suggests a further

dependence on cutoff choice. To verify this idea, we ran MD simulations of

54

Chapter 4: New strategies for the analysis of molecular dynamicstrajectories

1 2 3 4

g(r

)

r [Å]

rOH

240K

280K

320K

2 3 4 5g(r

)

r [Å]

rOO

0 0.01 0.02 0.03 0.04

Pro

babili

ty

Occupancy [N]

Sk

2.418

2.420

2.422

2.424

240 280 320

dis

tance [Å

]

Temperature [K]

2.418

2.420

2.422

2.424

240 280 320

dis

tance [Å

]

Temperature [K]

3.2

3.3

3.4

3.5

240 280 320

dis

tance [Å

]

Temperature [K]

3.2

3.3

3.4

3.5

240 280 320

dis

tance [Å

]

Temperature [K]

0.006

0.008

0.010

0.012

240 280 320

Occupancy [N

]Temperature [K]

2.8

3.0

3.2

3.4

3.6

3.8

4.0

240 280 320

Num

ber

of H

B

Temperature [K]

2.8

3.0

3.2

3.4

3.6

3.8

4.0

240 280 320

Num

ber

of H

B

Temperature [K]

2.8

3.0

3.2

3.4

3.6

3.8

4.0

240 280 320

Num

ber

of H

B

Temperature [K]

Figure 4.12: Temperature dependence for cutoff choice. Data relative to the

rOH , rOOΘ and Sk definitions are shown in the first, second and third column,

respectively. (Top) The oxygen-hydrogen, oxygen-oxygen radial distribution

functions and the occupancy distribution are displayed from left to right.

(Middle) Cutoff dependence as a function of temperature. (Bottom) Average

number of hydrogen bonds with fixed and variable cutoffs are shown as empty

and filled circles, respectively.

six of the most commonly used water models. In Fig. 4.13 and Fig. 4.14 re-

sults for the Sk definition are presented. The data reports on the position of

the first minimum in the occupancy N distribution as a function of tempera-

ture for different water models. According to the original prescription [148],

55

0.004

0.006

0.008

0.010

0.012

240 280 320

N m

in

Temperature [K]

SPC/E

TIP4P/2005

SPCTIP3P

TIP4P-Ew

TIP4P

Figure 4.13: The position of the first minimum of the occupancy distribution

relative to the Sk definition for different water models. Red line refer to the

SPC/E model which was used for the rest of the analysis presented in this

section.

the hydrogen bond cutoff should be taken as the position of this minimum.

The plot shows that this value strongly depends on both water model and

temperature. Similar conclusions can be drawn for the case of rOH and rOO.

To study Sk definition more precisely we built a contour map for SPC-

E and TIP3P water models in similar way as in original study [148] (see

Fig. 4.14). While for the case of SPC-E the default value of a cut-off overlaps

with the obtained minimum, for TIP3P water model this two values are

different. However, it is also clear that even usage of a minimum of occupancy

does not perfectly separate bound and unbound states.

Overall, analysis presented here put in evidence a number of limitations

in current approaches, highlighting a general lack of consensus among them.

Somewhat surprising was to find that two of the most recent definitions, Sk

and TP , were the ones to agree the least with each other. This certainly

motivates the exploration of alternative routes, like the use of multi-body

definitions going beyond the classical pairwise models [157,158].

56

Chapter 4: New strategies for the analysis of molecular dynamicstrajectories

SPC−E

1.5 2 2.5 3

r [Å]

0

20

40

60

80

Ψ

SPC−E

1.5 2 2.5 3

r [Å]

0

20

40

60

80

Ψ

SPC−E

1.5 2 2.5 3

r [Å]

0

20

40

60

80

Ψ

TIP3P

1.5 2 2.5 3

r [Å]

0

0.5

1

1.5

2

TIP3P

1.5 2 2.5 3

r [Å]

0

0.5

1

1.5

2

TIP3P

1.5 2 2.5 3

r [Å]

0

0.5

1

1.5

2

TIP3P

1.5 2 2.5 3

r [Å]

0

0.5

1

1.5

2

Figure 4.14: Contour map with the parameters used for Sk definition for

SPC-E and TIP3P water models at 300 K. On the left figure default value

of the occupancy cutoff overlaps with the minimum we found (yellow line).

With the TIP3P water model the default value of occupancy (black line)

differs from the minimum we found (dotted line).

57

Chapter 5

Applications

In the previous chapters we described some of the water anomalies as well

as the problem of the hydrogen bond definition in molecular dynamics. Also

the description of complex network framework was provided for the analysis

of molecular dynamics trajectories. In this chapter the results of the molec-

ular dynamics simulation of water is presented. In particular, we compare

seven widely used classical rigid-body water models in terms of their local

structure at wide range of temperatures. Also we present similar analysis of

the polarizable SWM4-NDP water model. The complex network approach

is presented in order to build free energy landscape of water. And at last

we show the analysis of proton transfer events in bulk water with simplified

network approach.

5.1 Study of classical water models at ambi-

ent pressure

The simplest water models in molecular dynamics consists only of three atoms

connected with unbreakable covalent bonds. Its potential is composed with

only two pieces: Lennard-Jones potential for van der Vaals force and repul-

sion and the simple electrostatic potential [22]. Surprisingly, this relatively

58

Chapter 5: Applications

simple models with fixed charges and geometry are able to reproduce the

phase diagram as well as many of the anomalies of water with good accu-

racy [159, 160]. For example, all popular classical water models present a

density maximum [55,161]. However, only those that explicitly included this

information in the fitting of the potential are able to correctly reproduce the

experimental value located at around 277 K at ambient pressure [162].

Due to their improved speed, biomolecular simulations in explicit wa-

ter were traditionally run with TIP3P [54] or SPC [52]. Nowadays, more

elaborated models can be easily used and their impact on the calculation

assessed [163]. Optimized four site models reproducing the experimental

temperature of maximum density seem to improve the accuracy of biomolec-

ular simulations. For example, Best and collaborators showed that pre-

dicted helical propensities are in better agreement with experiments when

a TIP4P/2005 water model is chosen in place of the traditional TIP3P [164].

Others reported that TIP4P-Ew provides better free-energy estimations com-

pared to conventional water models [165]. In both studies, the improved be-

havior was not connected to a clear microscopic property of the water model.

To this aim, one limitation is the lack of a common framework to compare

the structural behavior of liquid water at the atomic level.

Here, seven most popular classical water models, namely SPC [52], SPC/E

[53], TIP3P [54], TIP4P [74], TIP4P-Ew [51], TIP4P/2005 [55] and TIP5P

[24] were investigated in terms of their local structure forming capabilities.

That is, their ability to form structured or partially structured environments

of the size of up to two solvation shells through hydrogen bonds.

The simulations of a classical water models were performed in tempera-

ture range from 210 K to 350 K with steps of 10 K. TIP5P data was collected

from 230 K, just before the approaching of the glass-transition [166].

The location of the maximum density was obtained from 1 ns long sim-

ulations after 10 ns of equilibration. The temperature of maximum density

was extracted by polynomial fitting around the maximum. Variations from

the literature (see Table 5.1) may be due to size effects and a different treat-

59

Table 5.1: Temperature of maximum density calculated from our simulations

(TMD), as found in the literature (TMDref ) and the structural temperature

shift (∆Ts) for the seven water models investigated in this section.

Water model TMD TMDref ∆Ts T

TIP3P 199 182 [161] 65 229

SPC 226 228 [161] 42 247

SPC/E 250 241 [167] 18 275

TIP4P 256 248 [54] 20 268

TIP4P/2005 280 278 [55] 0 287

TIP4P-Ew 273 274 [51] 6 281

TIP5P 282 285 [161,168] n.a. 269

ment of the electrostatics. The location of the TIP3P density maximum was

obtained by running further simulations at lower temperatures.

The free energy of a configuration i is given by

∆Fi = −kBT log(Pi), (5.1)

where kB is the Boltzmann factor, T the temperature and Pi the popu-

lation of the selected configuration. The enthalpy is estimated by summing

up all pairwise contributions to the enthalpy between the water molecules

belonging to the same configuration (i.e. sum of the Lennard-Jones and

electrostatic interactions).

The tetrahedral order parameter [71] for a water molecule i was calculated

in the same way as for the microsecond simulation described in Chapter III

of this work:

qi = 1− 3

8

3∑

j=1

4∑

k=j+1

(cosψjik +

1

3

)2, (5.2)

60

Chapter 5: Applications

Here, we focus on water structural propensities which were used to analyze

hydrogen bond definitions in Chapter IV. Water structure forming capabili-

ties were investigated by analyzing the hydrogen-bond network of each water

molecule in the simulation box together with its first and second solvation

shells. A maximum of four hydrogen-bonds per molecule was considered. A

bond is formed when the distance between oxygens and the angle O-H-O is

smaller than 3.5 A and 30 degrees, respectively [150]. Water structures were

grouped into four archetypal configurations of population P(∗)i : the fully co-

ordinated first and second solvation shells for a total of 16 hydrogen-bonds

(P4, see Fig. 5.1 for a schematic representation); the fully coordinated first

shell, in which one or more hydrogen bonds between the first and the sec-

ond shells are missing or loops are formed (P∗4); the three coordinated water

molecule (P3) and the rest (P210). Within this representation the sum over

the four populations is always equal to one for each temperature.

In Fig. 5.2, the temperature dependence of the four microscopic water

structures is shown. Among the different water models, the qualitative be-

havior is strikingly similar. Three main types of temperature scalings were

observed: increasing population with decreasing temperature (enthalpically

stabilized); increasing population with increasing temperature (entropically

stabilized); with a maximum, where a turnover between enthalpic and en-

tropic stabilization takes place at a model dependent temperature. All four

water configurations fall into one of these three main classes. The popula-

tion of the fully ordered structure, P4, increases with decreasing temperature

(Fig. 5.2, red empty circles). Consequently, this configuration is enthalpically

stabilized. This is not the case when defects in the hydrogen bond structure

are introduced (P∗4, filled red circles). For this configuration the population

increases with decreasing temperature until it reaches a maximum in cor-

respondence to a rapid increase of the population of the fully-coordinated

configuration. The maximum is located in a temperature range close to the

temperature of maximum density of the model under consideration (dashed

vertical line). Finally, both P3 and P210 are mainly entropically stabilized,

61

1

2 3

4 5

6 7 9 10

8 11

12

13 14 16 17

15

1

2 3

4 5

6 7 9 10

8 11

12

13 14 16 17

15

1

2 3

4

6 7 9 10

8 11

12

13 14

P4

1

2 3

4 5

6 7 9 10

8 11

12

13 16 17

15

P4*

P3 P2

Figure 5.1: Schematic representation of the four possible configurations of

water solvation shells (P4,P4∗,P3,P2 population, see text). Dashed lines rep-

resent hydrogen bonds. For clarity, all water molecules are labeled with

numbers.

showing larger populations at higher temperatures. Taken together, these

results indicated that specific water configurations dominate at each temper-

ature range: full-coordination extending to at least two solvation shells at

low temperatures, four-coordinated configurations with no spatial extension

at intermediate temperatures and mainly disordered ones at higher temper-

atures.

Despite these similarities, an important difference among the models is

the temperature range at which the relative configurations become dominant.

For example, the maximum population of P∗4 for the SPC model was observed

around 245 K. This is not the case for TIP4P/2005, where the maximum is

located at a 40 K larger temperature. The same behavior was observed com-

paring the temperatures at which P4 and P3 are equal (e.g., around 270 K

for TIP4P/2005). These observations suggested that a temperature shift

62

Chapter 5: Applications

0

0.2

0.4

0.6

0.8

200 250 300 350

Popula

tion

Temperature

tip5p

0

0.2

0.4

0.6

0.8

200 250 300 350

Popula

tion

Temperature

P4

P*4

P3

P210

0

0.2

0.4

0.6

0.8

200 250 300 350

Popula

tion

tip4p/2005

0

0.2

0.4

0.6

0.8

200 250 300 350

tip4p-ew

0

0.2

0.4

0.6

0.8

200 250 300 350

tip4p

0

0.2

0.4

0.6

0.8

200 250 300 350

Popula

tion

spc/e

0

0.2

0.4

0.6

0.8

200 250 300 350

spc

0

0.2

0.4

0.6

0.8

200 250 300 350

tip3p

Figure 5.2: Temperature dependence of water structure populations for seven

classical water models. P4, P∗4, P3, and P210 are shown in red empty, filled

red, blue, and cyan circles, respectively (see text for details). The gray stretch

highlights the temperature difference between the calculated position of the

temperature of maximum density (vertical dashed line, see also Table 5.1) and

the experimental value at 277 K (solid line). The bottom right monochrome

plot shows the superposition of all models after temperature shifting each

data set (TIP4P/2005 data was used as reference). For each temperature,

the sum over the 4 groups is equal to one.

63

Tem

pera

ture

Ener

gy [k

J/m

ol]

Ener

gy [k

J/m

ol]

Tetra

hedr

al p

aram

eter

A B

CD 0

20

40

60

0

20

40

60

80

0.7

0.8

0.9

0

1

2

3

tip3p spc

spc-e

tip4p

tip4p-ew

tip4p/2005

tip5p

tip3p spc

spc-e

tip4p

tip4p-ew

tip4p/2005

tip5p

tip3p spc

spc-e

tip4p

tip4p-ew

tip4p/2005

tip5p

�T

s*

Figure 5.3: Structural temperature shift ∆Ts with respect to the TIP4P/2005

model. TIP5P was excluded from the superposition analysis (see text for

details) .

factor (∆Ts) exists among the models. TIP4P/2005 was chosen as reference

for its ability to reproduce the density curve [159]. Using TIP4P/2005 as a

reference, we found a temperature shift factor for each model ranging from

65 K to 6 K (see Fig. 5.3 and Table 5.1). Applying this shift to the data

allowed the superposition of all models onto four master curves, one for each

structural configuration, as shown in the monochrome plot at the bottom

right of Fig. 5.2. Our observation is consistent with previously found phase

diagram shifts among different water models [73, 169] as well as in the pres-

ence of ions [170] but in this case we could superimpose all models onto a

master curve. Unfortunately, TIP5P had to be excluded from the super-

position because all points show an increased curvature with respect to the

other models, consistent with the increased curvature of the isobaric density

at 1 atm [161].

The structural temperature shift is larger for three-site models (yellow

bars in Fig. 5.3) with a spread of up to 65 K for TIP3P. On the other hand,

64

Chapter 5: Applications

0 1 2 3 4 5

6F

[kJ/

mol

]

A

-450

-400

-350

6E

[kJ/

mol

]

B

0.7

0.8

0.9

Q

C

tip3p spc

spc-e

tip4p

tip4p-ew

tip4p/2005

tip5p

Figure 5.4: Comparison of water models with respect to the fully coordi-

nated configuration at 230 K. (A) The value of the free energy. (B) Average

enthalpy. (C) Average value of the tetrahedral order parameter.

four-site models deviate less. Both SPC-E and TIP4P are characterized by

a temperature shift with respect to TIP4P/2005 of around 20 K. In general,

models providing a better estimation of the position of the density maximum

deviate less.

To check the robustness of the Pi overlap with the hydrogen bond defini-

65

0

30

60

90

0 30 60 90∆

Ts

∆Tdensity

Figure 5.5: Comparison between the structural temperature shifts (∆Ts)

and the position of the density maximum (∆Tdensity). Four-site models were

compared to TIP4P/2005 (filled circles). Three-site models were compared

to SPC/E (empty circles). Crosses refer to the case when TIP4P/2005 was

used as reference for the three-site models.

tion, the recent definition of Skinner [148] which was discussed in the Chapter

IV was applied. Fig. 5.6 shows that the overlap between the curves is in-

dependent from the hydrogen bond definition. Moreover, the temperature

shifts calculated in this case are very similar to the ones reported in Table 5.1.

At all temperatures, water models with smaller shifts provide an im-

proved stabilization of the fully coordinated configuration. (Alternatively, it

can be said that these models destabilize poorly hydrogen-bonded configu-

rations). To make this point clearer, the free energy of the fully coordinated

configuration at 230 K was calculated (Fig. 5.4A). At this temperature P4

is appreciably large for all water models. Comparison with the temperature

shifts of Fig. 5.3 indicates a remarkable correlation where even the small

differences between SPC-E and TIP4P are respected. This is not the case

when looking at the enthalpy alone. In Fig. 5.4B, the average value of the

enthalpy for the same configuration is shown. Interestingly, it does not cor-

relate well neither with the free energy nor with the structural temperature

66

Chapter 5: Applications

0

0.2

0.4

0.6

0.8

200 250 300 350

Popula

tion

Temperature

P4

P*4

P3

P210

Figure 5.6: Overlap of the Pi states when a Skinner definition for the hydro-

gen bond was used.

shifts. On the other hand, enthalpy and free energy correlate within the same

model family. This is particularly clear when looking at three sites models

(i.e., the trend for TIP3P, SPC and SPC/E), suggesting a different entropic

contribution between three and four sites models which is systematic.

Finally, the average value of the tetrahedral order parameter [71] of the

fully coordinated configuration calculated at the same temperature is shown

in Fig. 5.4C. In first approximation, the parameter correlates well with the

structural shift although not as good as the free energy.

It is worth commenting on the relation between the structural temper-

ature shifts found in this section and the model-dependent temperature of

maximum density. As shown in Fig. 5.5 and Table 5.1, the relationship be-

tween the structural ∆Ts and the density ∆Tdensity temperature shifts is linear

within the three or the four-sites models (filled and empty circles in Fig. 5.5).

However, when comparing all models together using TIP4P/2005 as refer-

67

ence a small systematic deviation is observed (filled circles and crosses in the

figure). This is due to the relation that exists between the populations Pi and

the density. To make this point clearer, it is noted that the relative position

of the P∗4 maximum with respect to the temperature of maximum density

(dashed line in Fig. 5.2) depends on the model family. For four-sites models

the two temperatures are identical, while for three and five-sites models the

maximum of P∗4 is found at a higher and a lower temperature, respectively.

This behavior might be connected with the systematic deviations between

free energy and enthalpy for the different water models (Fig. 5.4A-B).

In conclusion, we found that seven among the most used classical water

models are characterized by very similar hydrogen-bond structure-forming

capabilities up to a temperature shift. All models but TIP5P perfectly over-

lap onto a master curve when this shift is applied. This behavior does not

depend on the hydrogen-bond definition. Our findings suggest that model

reparametrization acts as an effective shift in temperature space. On the

other hand, changes in the geometry or the number of sites cannot be fully

reconducted to temperature shifts alone as shown by the analysis of the

density as well as the radial distribution function. As such, although the hy-

drogen bond topology is universal when applying a certain temperature shift,

this is not the case for the structure, each model family being characterized

by its own signature.

5.2 Effect of polarizability

Apart from described in previous section classical rigid-body water mod-

els, recently new water models with explicitly introduced polarizability ap-

peared [171, 172]. The introduction of polarizability in classical molecular

simulations holds the promise to increase the accuracy as well as the predic-

tion power of computer modeling. One promising strategy to introduce polar-

izability in a straightforward way is based on Drude particles: dummy atoms

whose displacement mimics polarizability. The SWM4-NDP is a Drude-based

68

Chapter 5: Applications

water model which is simple to implement, being compatible right away with

conventional pairwise force-fields. Here, molecular dynamics simulations of

SWM4-NDP were performed for a wide range of temperatures going from

170 K to 340 K.

One of the recent polarizable water potentials is SWM4-NDP where elec-

tronic induction is represented by a classical negatively charged Drude par-

ticle attached to the positively charged oxygen by a harmonic spring [171].

Tests at the ambient conditions showed better agreement with the experi-

mental value of viscosity and hydration free energy than rigid-body water

models. This means that SWM4-NDP in principle should produce more

correct results for the dynamical processes. Here we present the study of

SWM4-NDP water model at low temperatures range. One of the most im-

portant points of water model is the position of the density peak. Building

the density curve in temperature space and comparing it with rigid-body

model it becomes possible to find the effect of polarization on the structure

of the liquid. Moreover, the calculations of a structural parameter, such as

radial distribution function or the tetrahedral order Q [71] can shed the light

on the influence of the polarization on the structure of liquid water. The

detailed description of structural order parameters is provided in Chapter II

of this thesis.

All molecular dynamics simulations of SWM4-NDP water model were

run with NAMD program package [32]. Temperature and pressure were

controlled with Langevin thermostat and Berendsen barostat with 1 ps and

100 fs relaxation time respectively. The temperature of the Drude particles

were set to 1 K at all conditions. Such temperature allows Drude-particle to

reproduce polarization effect with good accuracy [171]. The simulations of

of 50 ns length were made at temperatures from 170 K to 260 K with step of

10 K at pressure equal 1 atmosphere. At higher temperatures(from 260 K up

to 340 K) the simulations length was 10 ns. Recently Kiss and coworkers [172]

showed that this model presents no density maximum for temperatures as low

as 180 K. Independently from them we were also looking at similar properties

69

of the same model. One important difference in the present simulations is

that simulations were run for much longer times: 50 ns per trajectory opposed

to 5 ns in their case. For comparison with the classical water model we used

the data obtained from TIP4P/2005 simulation described in previous section.

To study SWM4-NDP in the deeply supercooled regime longer runs are

mandatory. This becomes clear when looking at the time series of the po-

tential energy. In Fig. 5.7 traces for different temperatures from 250 K to

170 K are shown. It was found that for temperatures lower than 200 K the

relaxation time is dramatically slowed down. The red line corresponding to

the 170 K case shows that the system required at least 20 ns to equilibrate

(gray region). This is a longer time with respect to what was presented in

Ref. [172], indicating that their data in the supercooled region was affected by

the partially equilibrated system. This is particularly relevant when studying

the density.

-12

-11

-10

0 10 20 30 40 50

En

erg

y [

kca

l/m

ol]

time [ns]

170 K

200 K

250K

-12

-10

Figure 5.7: Timeseries of the potential energy of SWM4-NDP water model

for three different temperatures.

With the longer trajectories at hand, the density curve for the model

70

Chapter 5: Applications

did present a maximum at 200K (red points in Fig. 5.8). However, this

maximum is not as clear as in experiments (black line) or in other classical

models, e.g. TIP4P/2005 (orange points). In fact, density grows again at

lower temperatures (T < 190K), making the density peak difficult to step

out from the statistical error especially for short trajectories. The density

growing at the very low temperatures is a feature of several water models.

For example, this happens as well for the TIP4P/2005 model below 220 K.

What makes the case of SWM4-NDP peculiar is the fact that the values of

the density in this regime become higher than the density maximum. The

density curve per se sets this water model apart from all the classical models

investigated . Even TIP3P which is known to have a density maximum at a

similar temperature (182 K [161], see also section 1.6) does not present such

an increase in density as for this Drude-based polarizable model.

950

1000

1050

200 300 400

De

nsity [

kg

/m3]

Temperature [K]

Figure 5.8: Experimental density curve (black line) and density values from

MD simulations for SWM4-NDP and TIP4P/2005 water models in red and

orange respectively.

Complementary information was obtained by investigating hydrogen-bond

71

propensities. As done in the first section of this chapter for seven classi-

cal water models we calculated the probability to form fully coordinated

hydrogen-bond configurations up to the second shell (P4) as well as fully co-

ordinated first shells with a disordered second shell (P ∗4 ), three coordinated

(P3) and less (P210) first solvation shells. Results for the SWM4-NDP and

TIP4P/2005 are shown in Fig.5.9 as filled circles and empty squares, respec-

tively. Contrary to the density analysis, hydrogen-bond propensities between

the two models look much more similar. The two sets of curves seem to be

nicely overlapping if a shift of approximately 20 K would be applied to the

data. This observation suggests that while spatial rearrangement responsible

for the density is dramatically different between the two models (and when

compared to experiments), hydrogen-bond connectivity is similar. In section

5.1 of this thesis a similar difference was already observed when comparing

three-sites with four-sites models where there was a 10 K difference between

temperature shifts estimated from hydrogen bonds or the position of the den-

sity maximum. But in this case the discrepancy is much larger being these

temperature shifts respectively of 20 K and 80 K.

Similar temperature shifts were observed as well when calculating the

average value of the tetrahedral order parameter QT . Fig. 5.10 shows this

quantity as a function of temperature for both SWM4-NDP (red line) and

TIP4P/2005 (orange). As for the case of hydrogen-bond propensities, the

two models do not differ very much. At ambient condition the temperature-

shift is of about 30 K, a number that is in line for what observed in the

hydrogen-bond case. For the sake of comparison the distribution of Q at

300K for the two models is shown in panel B of the same figure. As it could

have been expected from the behavior of the average value of the tetrahedral

order parameter, the TIP4P/2005 has a slightly larger fraction of molecules

in a tetrahedral configuration but the overall shape of the distribution stays

the same for the two cases.

Summarizing this section, we performed molecular dynamics simulations

of the Drude-based polarizable water model SWM4-NDP as a function of

72

Chapter 5: Applications

0

0.2

0.4

0.6

0.8

150 200 250 300 350 400

Pro

ba

bili

ty

temperature [K]

P4P4

*

P3

P210

Figure 5.9: Microstates for SWM4-NDP (circles) and TIP4P/2005 (squares)

models.

temperature. Contrary to what was reported in a recent paper [172], it was

found that the model do present a density maximum which was found to be

around 200 K. This point was not trivial to find because of the tremendous

slowing down of the system for temperatures lower than 200 K. To overcome

this problem simulation runs of 50 ns each have been performed, finding that

at temperatures as low as 170 K the system requires at least 20 ns to have

the potential energy relaxing to a stationary average value without drifts.

However, the density maximum we found is not as pronounced as other

classical water models. This was somewhat unexpected. As system temper-

ature was lowered below 190 K, the density started to increase again. This is

in principle similar to what was observed for other models, like for example

TIP4P/2005. But in this case the density value suddenly increased to a value

larger than the density maximum, making the latter a relative maximum in-

stead of an absolute one. The raising of the density at such low temperature

is probably due to some sort of frustration into the system leading to glassy

behavior. This explanation would also explain the dramatic slowing down of

73

0.5

0.6

0.7

0.8

0.9

150 200 250 300 350 400

Te

tra

he

dra

l p

ara

me

ter

Q

Temperature [K]

0

0.01

0.02

0.03

0 0.2 0.4 0.6 0.8 1

Pro

ba

bili

ty

Tetrahedral parameter Q

Figure 5.10: Average value of tetrahedral parameter in temperature space.

Red and orange colors stay for SWM4-NDP and TIP4P/2005 water models

the relaxation kinetics of the model below 200 K.

In comparison to other classical models, SWM4-NDP performed very

poorly in reproducing the density curve. This is somewhat disappointing

given the success of other models in this field, especially the reparametrized

versions of the four-site model, TIP4P/2005 [55] and TIP4P-Ew [51].

What really set apart SWM4-NDP with other non-polarizable classical

models was the fact that despite the position of the density maximum is off

74

Chapter 5: Applications

by roughly 80 K, the behavior of the hydrogen-bond propensities and tetra-

hedrality are very well in line to what the best models in the field predict.

Such separation in the behavior is new for us because what we found in the

past that a temperature-shift in the position of the density maximum corre-

sponded to a similar shift in the hydrogen-bond propensities. The presence

of polarizability instead completely decouples these two aspects, giving in

principle a wider space to match experimental data, at least in principle.

Our analysis shed some light on the behavior of SWM4-NDP polariz-

able model in temperature space. Apparently, model parametrization is still

needed to match up with other polarizable models like AMOEBA which have

been shown to perform quite well in temperature [56]. The great advantage

of SWM4-NDP is clearly its straightforward pairwise interaction model that

can be easily implemented in all modern force-fields for biomolecular simu-

lations. However, to make this model fully effective a new parametrization

to better reproduce the density curve and other quantities in temperature

appears to be required.

5.3 Free energy landscape of water

In previous chapter we described complex network approach which is the

powerful tool to obtain free energy landscape of the system from molecular

dynamics trajectory. Here, the complex network analysis is presented by

running extensive molecular dynamics simulations of the TIP4P/2005 water

model from 340 K to the supercooled regime.

The simulations of TIP4P/2005 [55] were performed for the temperatures

from 190 K to 340 K with the step of 10 K. To check the approach for different

water potentials the simulations of the TIP3P [54] and TIP5P [24] water

models were made for the same temperatures. For temperatures below 240 K

the equilibration time was elongated up to 25 ns. All details of molecular

simulations is presented in Chapter I.

For each temperature we built configuration-space-networks, where the

75

microstate is defined as a set of hydrogen bonds as described in Chapter IV.

For the hydrogen bond definition we used criterium proposed by Skinner [148]

which was discussed in the previous chapter. For each of these networks, we

looked for the free-energy basins characterizing local water arrangements by

means of a gradient-cluster analysis [16, 137, 139]. The structural configu-

rations at the bottom of the most visited free-energy basins are pictorially

represented in Fig. 5.11. In Fig. 5.12 the number of nodes of the complex

network for each temperature is shown. With increasing temperature the

number of nodes linearly increases. The population of the clusters obtained

from the network shows different behavior. In Fig. 5.13, the population of

the most visited gradient-clusters is shown as a function of temperature. At

temperatures larger than 285 K, several free-energy basins of attraction are

found in agreement with previous analysis on the SPC model at 300 K [16].

They correspond to the following hydrogen-bond configurations of the

central water molecule: 2 donors, 1 acceptor (21, dark blue in Fig. 5.13,

population of 0.32 at 300K); 2 donors, 2 acceptors (22, light blue, 0.21); 1

donor, 2 acceptors (12, red, 0.13); 0 donors, 2 acceptors (02, yellow, 0.01).

At the highest temperatures a fifth basin appears being characterized by a 11

first solvation shell (gray). This acceptor/donor representation is adopted for

simplicity but the contribution of the second solvation shell organization is

strictly needed when it comes to correctly characterize the free-energy basins

(e.g. there are basins of attraction with the same first shell but different

second shell [16]).

In this temperature range the liquid is inhomogeneous in the sense that

the local environment of a water molecule interconverts between config-

urations with distinct structural properties. Those represent short-lived

metastable arrangements with sub-ps lifetime [16,173].

Below 285 K this property is lost as shown by the rapid increase of the

population of the 22 gradient-cluster to a value larger than 0.8 (light blue

in Fig. 5.13). As such, all highly populated gradient-clusters collapse to 22,

being the only largely populated free-energy basin. The population of this

76

Chapter 5: Applications

21 22

12 02

1

2 3

4 5

6 7 9 10

8 11

12

13 14 16 17

15

1

2 3

4 5

6 7 9 10

8 11

12

13 14 16 17

15

1

2 3

4 5

6 7 9 10

8 11

12

13 14 16 17

15

1

2 3

4 5

6 7 9 10

8 11

12

13 14 16 17

15

Figure 5.11: Representative water microstates belonging to the four most

populated gradient-clusters at 300 K. Hydrogen bonds are represented as

dashed lines. For simplicity of reference, each of the four configurations is

classified by two numbers, indicating the number of donors and acceptors of

the central water molecule (e.g. 21 stands for two donors and one acceptor).

basin is almost constant until 225 K. In this regime, the liquid is homogeneous

and the free-energy landscape resembles a funnel, with the fully-coordinated

configuration 22 at the bottom of it. The funnel behavior emerges because

22 becomes a global attractor of the dynamics as it is the case for the native

state in protein folding [174]. Still, a cumulative population of 0.2 split into

roughly six basins survives. These configurations are rich of 4-fold hydrogen-

bond loops, slowly interconverting with the fully coordinated configuration.

Below 225 K, i.e. roughly below the temperature of maximum compress-

77

0

0.5

1

1.5

200 250 300 350

Num

. Nod

es [1

05 ]

Temperature

Figure 5.12: Number of nodes of the complex network at different tempera-

tures.

ibility (estimated to be around 230 K [107]), the entire landscape collapses

onto 22 with a much more pronounced funnel behavior. Interestingly, the

temperature of maximum compressibility is considered by some as the Widom

line, i.e. the propagation of a liquid-liquid critical point located at higher

pressure [66, 107, 109]. If this is so, this water regime would be connected

to the mentioned transition. In this temperature region the density assumes

its minimum value (see also Fig. 5.15d). For this reason, we refer to this

temperature segment as the low-density homogeneous regime of the liquid.

It is interesting to compare these regimes with the distribution of the

average tetrahedral order parameter Q [71]. In Fig. 5.14 data for 320 K,

240 K and 190 K is shown. At around ambient conditions the distribution

is bi-modal (light gray), indicating that the liquid assumes both ordered and

disordered atomic arrangements. This property is lost at lower temperatures

where the distribution becomes uni-modal (gray) with a small population for

values close to 0.5. This sub-population disappears below the temperature

of maximum compressibility, resulting in a sharply peaked distribution (dark

gray). The shape shift from bi-modal to uni-modal is in good agreement with

the change from the inhomogeneous regime to the homogeneous one.

78

Chapter 5: Applications

0

0.2

0.4

0.6

0.8

1

200 250 300 350

Pop

ulat

ion

Temperature [K]

21

22

12

02

225 285Tm

Figure 5.13: Population of the most visited gradient-clusters as a function of

temperature for the TIP4P/2005 model. Vertical lines correspond to 225 K

and 285 K. Tm indicates the melting temperature of the model at around

250 K [55].

0

0.2

0.4

0.6

0 0.2 0.4 0.6 0.8 1

Pro

babi

lity

Q

190 K

240 K

320 K

Figure 5.14: Distribution of the tetrahedral order parameter Q for the three

regimes.

79

Summarizing this section, three different regimes for the liquid phase of

water were found. Each of these regions is characterized by a specific organi-

zation of the underlying free-energy landscape as shown by the temperature

dependence of the populations of the major free-energy basins (Fig. 5.13).

As a function of temperature, the number of visited microstates (i.e.

nodes) is increasing monotonously, as shown by Fig. 5.15a. That is, the

higher the temperature the larger the portion of the configuration space

visited by the molecular dynamics simulation. Above 225 K the relation is

linear but below this temperature the number of microstates changes in a

non-linear way, visiting in proportion a smaller fraction of the configuration

space. This behavior might be related to the breakdown of the Einstein

diffusion relationship below the temperature of maximum compressibility as

observed for the ST2 model [175].

In Fig. 5.15b, the number of gradient-clusters with a population larger

than 0.01 as a function of temperature is shown. The data presents a step

wise behavior, correlating very well with the presence of the three regimes.

Interestingly, the number of gradient-clusters is mostly constant in the two

low-temperature regimes with only one free-energy basin below 225 K.

From a network topology point of view, the number of connections per

node (degree) grows with temperature, going from an average value d of 14.47

to 29.55 at 190 K and 340 K, respectively (dashed line in Fig. 5.15c). This is

not the case for the node degree of the microstate 22. As shown in Fig. 5.15c,

the degree increases up to around 285 K. Then it starts to decrease where

the liquid changes from the homogeneous regime to the inhomogeneous one.

Comparison with density (Fig. 5.15d) shows a remarkable correlation.

With a Pearson coefficient of 0.98, the behavior of the node degree of mi-

crostate 22 correlates with the density anomaly. This seems an interesting

fact connecting an ensemble property like the density to a purely microscopic

quantity, i.e. the number of accessible transitions from the fully coordinated

configuration 22.

In this section, characteristic properties of the transition network cor-

80

Chapter 5: Applications

responding to the three regimes are illustrated. In the higher temperatures

regime, structural inhomogeneities emerge because the maximum of the tran-

sition probability max(Z(i))

points towards the attractor of the basin (e.g.

Z in the pictorial representation of Fig. 5.16a). This is not the case below

285 K where the transition to 22 (Z22) becomes the maximum of the tran-

sition probability for many nodes which were acting as attractors at higher

temperatures. Fig. 5.16a shows the temperature at which Z22 = max(Z(i))

for the relevant microstates 21 and 12 (dark gray bars). Relaxing directly to

22, they do not build basins of attraction anymore. For nodes not directly

connected to 22 the relaxation process to it goes through two or more steps

like for 02. Consequently, a free-energy landscape characterized by a single

predominant minimum (22) develops. This type of landscapes recall the well-

known funnel-landscape paradigm applied to protein folding [174,176,177].

Below 225 K, Z22 drives the dynamics in an even stronger way being the

corresponding transition larger than the cumulative of all other transitions,

i.e. Z22 >∑

i Z(i) (gray bars in Fig. 5.16a). In other words, every time a

water molecule assumes a configuration different from 22, the probability to

go back to 22 is larger with respect to the cumulative of any other transition.

From a qualitative point of view the three regimes of the free-energy

landscape are represented in Fig. 5.16b (in panel c a pictorial representation

of the underlying network).

The origin of the temperature shift is related to the relative hydrogen-

bond strength differences of the various models (See section 5.1). Conse-

quently, it is expected that artificial modifications of the hydrogen-bond

strength due to more (or less) conservative bond definitions might shift the

three regimes as well. This is so because water microstates are based on

hydrogen-bond connectivity and its propensity. To check this behavior, the

whole analysis was repeated by using another definition of hydrogen-bond

based on the classical inter-oxygen distance and donor-acceptor angle Rθ.

As shown in Fig. 5.17b, the overall behavior of the gradient-cluster popula-

tions is remarkably similar with the presence of the three liquid regimes. On

81

0

5

10

15

200 250 300 350

basi

ns

Temperature

b

0

10

20

30

degr

ee [x

102 ]

d_

c

0

4

8

12

16

node

s [x

104 ]

a

95

100

200 250 300 350de

nsity

[cg/

cm3 ]

Temperature

d

Figure 5.15: Topology of the configuration-space-network as a function of

temperature. (a) Number of nodes; (b) Number of gradient-clusters with

a population larger than 0.01; (c) Number of connections of the 22 node.

For comparison, the average number of connections per node d is shown as

a dashed line (in this case the multiplicative factor is 1 and not ×102); (d)

density. Vertical lines correspond to 225 K and 285 K.

the other hand, the expected temperature shift is present. We found that

Rθ predicts a larger number of hydrogen-bonds than the Skinner definition.

For the former definition, 3.8 and 3.6 average number of hydrogen bonds per

molecule are found at temperatures of 250 K and 300 K, respectively. These

numbers decrease to 3.7 and 3.3 when the Skinner definition is used. Us-

ing a less conservative definition like Rθ, effectively increases hydrogen-bond

strength. As a consequence, the population of the fully-coordinated gradient-

cluster is over estimated, giving in turn a temperature shift. Since the discus-

sion on the quality of hydrogen-bond definitions is still open [148, 153] (See

82

Chapter 5: Applications

homogeneous low-density homogeneous inhomogeneous

22 2222

22

a

b

c

Zi

Z22

22

Z22 > max⇣Z(i)

⌘

Z22 >X

Z(i)

200

250

300

350

21 12

Tem

pera

ture

[K]

microstate

285K

225K

Figure 5.16: Schematic representations of the three regimes of liquid water;

(a) temperatures at which there is a change in the transition probability max-

imum for the 21 and 12 microstates; (b) free-energy landscape representation;

(c) network representation.

also section 4.2), we want to remark that the change of the hydrogen-bond

definition would only slightly affect the exact position of the three regimes

but not the existence of them.

83

0

0.2

0.4

0.6

0.8

1

Pop

ulat

ion

21

22

12

02

TIP3P

a

b200 250 300 350

Temperature [K]

21

22

12

02

TIP5P

a

b

0

0.2

0.4

0.6

0.8

1

200 250 300 350

Pop

ulat

ion

Temperature [K]

21

22

12

02

Rθ

a

b

Figure 5.17: Robustness of the gradient-cluster analysis. (a) Gradient-cluster

populations for TIP3P and TIP5P water models and (b) for TIP4P/2005 by

using the Rθ hydrogen-bond definition.

To conclude, from a microscopic point of view, the free-energy landscape

of liquid water is characterized by three major regimes. At ambient con-

ditions, several metastable water configurations with distinct structure and

dynamics are found (inhomogeneous regime). Below 285 K, the free-energy

landscape develops a funnel dominated by the fully coordinated configuration

with an extension of at least two solvation shells (homogeneous regime). By

lowering the temperature below 225 K, the funnel becomes more pronounced,

with the fully-coordinated configuration becoming a global attractor of the

dynamics (homogeneous low-density regime).

84

Chapter 5: Applications

While the three regimes were deducted from water microscopic proper-

ties, the presence of the tree regimes is correlated to the behavior of the

density curve, which is an ensemble property of the system. As such, the

homogeneous low-density regime spans till the density start to grow with a

change in concavity at 225 K; the homogeneous regime is characterized by

the monotonous increase of the density curve up to the density maximum at

around 280 K; finally, the descending section of the density is located into

the inhomogeneous regime.

From an experimental point of view, the presence of structural inhomo-

geneities at ambient temperature is in qualitative agreement with small-angle

X-ray scattering measurements [178] while the presence of multiple kinetics

is in principle accessible to high order non-linear spectroscopy [173].

5.4 Proton transfer

Proton motion in aqueous environments is unusually fast, allowing its par-

ticipation in a myriad of reactions in e.g. oceans, the atmosphere, acidic

rain, metal surfaces and enzymes. Even when a proton is not a reactant or

product, it quite often participates in some intermediate step. In fact, there

is hardly any enzyme without at least one acid- or base-catalyzed step in its

activity cycle. Proton mobility in water is a factor 4.5 faster than the next

most mobile cation (Rb+). This is ascribed to the fact that it is the only

cation whose diffusion requires only hydrogen-bond rearrangement, and not

necessarily mass motion [61,179].

Simulations have shown that the prevalent solvation state of the pro-

ton in liquid water is that of a distorted Eigen cation [180], H3O+(H2O)3,

in which one hydrogen-bond from the central hydronium (H3O+) moiety is

shorter than the other two, thus forming a “special pair” (SP) [179] (See

Fig. 5.18). The identity of the closest water ligand interchanges dynamically

between the three, giving rise to the “special pair dance” [180]. The special

partner is characterized by a loss of an accepted hydrogen bond [61, 180],

85

in “preparation” of transforming into a Zundel cation, H2O...H+...OH2, that

donates (four) hydrogen-bonds to its next shell neighbors, but accepts none.

Eventually the SP O–O distance contracts by an additional 0.1 A, forming

a Zundel cation in which the proton is shared nearly equally between the

two oxygen centers, rapidly rattling between them. The Zundel intermediate

can then return to the initial distorted-Eigen configuration or, more rarely,

transform to a distorted-Eigen cation centered on the special partner. When

this happens, the proton transfer process between the two water molecules

is deemed successful.

Figure 5.18: EigenZundelEigen (EZE) proton mobility mechanism. Hydro-

gen bonds depicted by dashed lines. Figure is adapted from Ref. [180].

What coerces the SP to convert into a Zundel intermediate? What co-

erces the latter not to return to its initial state, but rather centralize the hy-

dronium on the ex-special partner? Simulations [181] and experiment [182]

suggest that this depends on the collective rearrangement of the first two

solvation shells of the Zundel intermediate, involving a cluster of about 20

water molecules. Specifically, out of the two water molecules of the H5O+2

cation, the one that eventually accepts the excess proton should have shorter

donor type (e.g., A 99K 1A and 1A 99K 2A in Fig. 5.19) and longer acceptor

type hydrogen bonds (e.g., 1A L99 2A), both in its first and second solvation

shells [181]. (The arrow indicates the directionality of the hydrogen-bond).

The opposite is true for the water molecule that gives up the proton. This cor-

responds to the general trend of strengthening donor hydrogen-bonds while

weakening the acceptor ones near a protonated water center [157].

The observation that such hydrogen-bond length changes occur collec-

86

Chapter 5: Applications

tively on two hydration shells is in line with earlier observations that col-

lective motion controls water dynamics in bulk liquid water [183], with cor-

relations extending over at least two solvation shells [16]. In this tightly

hydrogen-bonded system, even the rotation of a water molecule during di-

electric relaxation requires pre-organization of a whole water cluster around

it [157].

The mechanism of proton mobility outlined above not only explains why

protons diffuse so fast compared to other cations, but also why they diffuse so

slowly compared to less disordered hydrogen-bonded networks. For example,

when a “water wire” is formed e.g., in carbon nanotubes [184,185], inside pro-

teins [186], between photoacid and base molecules in bulk water [187, 188],

or during hydronium/hydroxide neutralization [189], proton transport be-

comes considerably faster. The slower transport in bulk water is thus due

to the need to wait for a large scale fluctuation that could stabilize the new

microscopic state. In this respect, proton transfer is analogous to ligand

binding to proteins [190], or electron transfer in solution [191], where the

fast ligand/electron motion responds to the much slower protein/solvent re-

organization that prepares the appropriate conformation for accommodating

the product state.

Here, molecular dynamics simulations were made with the empirical va-

lence bond potential [192–196]. The system contains 216 SPC/Fw [195]

water molecules and a single excess proton. Lennard-Jones interactions were

truncated at an atom-atom distance of 0.9 nm. 0.5 ns equilibration run

was performed in constant volume ensemble with timestep of 0.5 fs, a target

temperature of 300 K and pressure of 1 atm, maintained by a Nose-Hoover

thermostat and barostat [45]. Following equilibration, the trajectory was

continued for 2 ns with constant energy.

Here, we characterize the microstates of the Zundel proton-transferring

intermediate that connect the SP to its proton transfer product. The mi-

crostates introduced herein are defined in terms of the length of the four

hydrogen-bonds in the first solvation-shell of the Zundel core (Fig. 5.19).

87

d

da

a

+

1D 1A

1A1D

D A

2D 2D

2D

2D

2D 2D

2A 2A

2A

2A

2A2A

Figure 5.19: Schematic picture of proton at Zundel region. The proton is

transferred from donor water molecule,“D” to the acceptor, “A”. There are

almost always two donor hydrogen bonds emanating from the hydrogens of

D and A, and these are denoted d and a, respectively.

In this study we have focused on Zundel-like segments of the trajectory.

The searching algorithm for these segments was as follows: at each timestep

the first and second closest water molecule to the center of excess charge

define a putative Zundel pair. If the proton rattles between them for at least

100 fs then it is considered a Zundel segment. The closest water molecule to

the proton at the beginning of the segment is called “donor” (D) while the

other one is the “acceptor” (A). A schematic picture of the proton and its

surrounding water molecules is given in Fig. 5.19. With the above method,

around 2000 Zundel segments of total length 327 ps and average length of

about 180 fs were found. They were sorted into two major groups: if at

the end of the trajectory the proton resides on the acceptor the segment

depicts a transmission event, T, otherwise it represents reflection, R, where

the proton remains with the donor. These are our reactive vs. non-reactive

events. We found a R:T ratio of approximately 4 : 1 with average lengths of

around 170 fs and 230 fs for R and T segments, respectively. Hence most of

the time the molecule which held the proton at the beginning of the Zundel

88

Chapter 5: Applications

segment keeps it till the end. Proton fluctuations make it almost impossible

to detect the exact moment of the proton transfer in a statistical manner.

We will now identify the factors responsible for the proton transfer events.

Table 5.2: Average length (in A) of the for O · · ·H distances, ri, in the six

nodes, for Zundel segments in which the R events (top) and T events (bottom

line) are separated.

node r1 r2 r3 r4

ddaa 1.462 1.555 1.669 1.792

1.461 1.554 1.668 1.791

dada 1.480 1.586 1.651 1.772

1.477 1.585 1.649 1.765

daad 1.487 1.595 1.664 1.757

1.489 1.584 1.651 1.748

adda 1.508 1.574 1.644 1.762

1.503 1.574 1.643 1.751

adad 1.504 1.582 1.654 1.754

1.486 1.573 1.645 1.746

aadd 1.481 1.563 1.655 1.764

1.474 1.556 1.653 1.757

To analyze the role of the hydrogen bond environment on proton transfer

the lengths of the four hydrogen bonds in the first solvation shell of the two

water molecules sharing the proton (“a” and “d” bonds in Fig. 5.19) were

calculated for every frame of the Zundel segments. These values were used

to build a sorted array in bond distances, which characterizes the state of the

Zundel complex. For example, when the two bonds on the A side are shorter

than the ones on the D side the array will be ’aadd’. Conversely, ’ddaa’

represents a situation where the first shell hydrogen bonds are shorter for

the donor. There are 3! distinguishable arrays (ddaa,dada, daad and aadd,

adad, adda). Their average O · · ·H distances, which are denoted by r1, r2, r3

and r4 (where ri < ri+1, i = 1..4), are given in Table 5.2, separately for the

R and T segments. As can be seen, the hydrogen-bond length increases by

roughly 0.1 A along each array (i.e., the value of ri−ri−1 is roughly constant),

with relatively small differences between R and T segments. In addition, we

89

include in the scheme nodes labeled R and T if in the last time-frame of

the Zundel complex the proton belonged to the acceptor or the donor water

molecules (A and D labels in Fig. 5.19), respectively. In this way, one may

judge whether the complex led to a proton transfer or not.

The time evolution of the bond sorted arrays was analyzed with the help

of a transition network framework [16], which description is given in Chapter

IV of the thesis.For the present case, there are 8 states in the network (the six

nodes ddaa, aadd etc. plus R and T). The total number of transitions between

any two states (irrespective of directionality) is recorded in a 8 × 8 matrix.

We note that the largest number of transitions is between a state and itself,

and then there are typically only a few other states with appreciable number

of transitions. Given that all the prominent transitions in this network are

characterized by thousand of passages we introduce an arbitrary cutoff of

100 transitions, and discard the connections below this cutoff. A link is then

placed between two nodes if the number of transitions between them exceeds

this cutoff. Results are not sensitive to the exact value of the cutoff provided

that it is smaller than the most visited transitions.

The resulting transition network is depicted in Fig. 5.20. In this picture,

the node size is proportional to the fraction of the total time that the node

was occupied. The link size is proportional to the number of transitions (in

either direction) between the two linked nodes (minimum 100). The network

illustrates some characteristics of proton transfer. First, all six nodes play a

role in the network. Reflection and transmission events are directly mediated

by the ’ddaa’ and ’aadd’ states, respectively (e.g. the link between the node

’aadd’ and T represents around 90% of the total flux to T). Consequently,

proton hopping from D to A is favored by longer hydrogen bond lengths

at the donor site, and shorter ones at the acceptor. Second, the scheme

highlights the presence of preferential pathways for proton transfer through

two alternative routes, progressively making the first shell hydrogen bond

lengths around A shorter. In conclusion, the transition network shows that

the length of hydrogen bonds in the first solvation shell of the Zundel cation is

90

Chapter 5: Applications

a property which strongly affects the behavior of proton transfer: if molecules

in the first shell are closer to one of Zundel core water molecules, that one

becomes more attractive to the proton.

baba TR dada adad aadd

daad

adda

ddaa

5401

2512

3320

2154

1651

2238 2551036

36%

20%

12%

9%

12%

9%

Figure 5.20: Schematic network of proton trajectory during Zundel trajectory

segments. The size of the links and nodes is proportional to their weight.

The R-T effect can be also seen in Table 5.2, where nearly all the hydrogen

bond distances in the T-segments are slightly shorter than the corresponding

ones in the R-segments. There is thus a small hydrogen bond contraction in

reactive trajectories, and this is reminiscent of the (evidently, more appre-

ciable) water-wire contraction recently reported from AIMD simulations of

hydronium-hydroxide neutralization in bulk water [189].

To further characterize the network nodes, the oxygen-proton distance

distributions for the D and A oxygens were calculated for the ’ddaa’ and

’aadd’ configurations (Fig. 5.21). For the ddaa state, the O–H distance dis-

tributions for the D and A oxygen atoms (red and orange curves respectively),

have a very small overlap. Hence in this state the proton is much closer to

the donor molecule ( 1.09 A) than to the acceptor ( 1.32 A). The O–O dis-

tance, which is their sum ( 2.41 A ), corresponds to a SP. This is indeed the

value found in previous MS-EVB3 simulations [180] for the SP that stabilizes

the distorted Eigen cation. Thus the ddaa state, which is the borderline be-

tween Eigen and Zundel segments, is still characteristic of a SP rather than a

91

genuine Zundel complex. Nevertheless, we find that only 70% of our Zundel

segments start in this state.

On the other hand, the two distributions become much closer for the

’aadd’ configuration (light and dark gray curves). In this case, the proton

is already slightly closer to A (1.11 A ) than to D (1.19 A ). The O–O

distance is now 2.30 A, which corresponds to a rather tight Zundel complex

(the most probable O–O distance for a MS-EVB3 Zundel cation is around

2.35 A) [180]. According to the transition network analysis this configuration

is the one prone to transmit the proton, and indeed it has the proton partially

transferred to the A water molecule.

0

0.05

0.1

0.15

0.2

1 1.2 1.4 1.6

Pro

babi

lity

Oxygen-Proton distance [A]

D:ddaa

A:ddaa

Figure 5.21: Oxygen-proton distance distributions. Red and orange colors

stand for D and A molecules within ’ddaa’ node; gray and lightgray colors

stand for D and A molecules in ’aadd’ node.

The effect of separating the Zundel trajectory into R- and T-segments

can be seen in Fig. 5.22 for the acceptor oxygen atom. Its long distance with

the transferring proton in the ddaa state becomes slightly longer when only

92

Chapter 5: Applications

R-trajectories are considered. Likewise, its short distance in the aadd state

becomes slightly shorter for T-trajectories. In both cases the distributions

also sharpen. Thus the difference between the R:ddaa and T:aadd distri-

butions of acceptor molecule A further accentuates, commensurate with the

role of these two nodes in reflecting or transmitting the proton, respectively.

0

0.05

0.1

0.15

1 1.2 1.4 1.6

Pro

babi

lity

Acceptor-Proton distance [A]

Figure 5.22: Acceptor oxygen-proton distance distributions for R- vs. T-

trajectories. Red and orange colors stay for the distribution for the reflection

events in aadd and ddaa states respectively. Gray and lightgray stands for

aadd and ddaa states during transmission events

Characterization of the second solvation shell was done by conventional

oxygen-oxygen radial distribution function (RDF) analysis. To this aim,

RDFs were calculated for the water molecules belonging to the first shell of

D and A (1D and 1A molecules Fig. 5.19). For the reflection mediating con-

figuration ’ddaa’ the RDFs corresponding to the 1D and 1A molecules differ

dramatically as shown respectively by the red and orange curves in Fig. 5.23.

The peak at short distances (2.58A) in g1D(r) (red curve) is predominantly

93

due to the hydrogen bond between the D and D1 water molecules, whereas

its tail to longer distances is contributed by hydrogen bonds to outer water

molecules ( [180]). Taking 1.0A for the covalent OH bond length, we find

from Table 5.2 (for ddaa) that 1 + (r1 + r2)/2 = 2.50 A. The g1D(r) peak

is at somewhat longer distances due to the two outer water molecules. The

peak at long distances (2.68A) in g1A(r) (orange curve) corresponds with

1 + (r3 + r4)/2 = 2.73 A. Thus there is a more compact configuration for the

proton keeping environment.

For the transmission mediating configuration ’aadd’ the RDFs are nearly

identical to those of the ddaa state with interchanged D and A labels. Here

the environment of A becomes more compact, in preparation for accepting the

proton. Interestingly, therefore, while the transferring proton distances do

not show perfect symmetry between the ddaa and aadd states, the first-shell

RDF’s do, reinforcing the importance of first- and second-shell rearrange-

ments in driving the proton transfer process.

Finally, hydrogen bond histograms for the 1D and 1A molecules were cal-

culated separately for the entire transmission and reflection Zundel segments.

In Fig. 5.24 gray, dark gray and light gray bars correspond to hydrogen bond

configuration with two bonds on both the oxygen and hydrogen atoms, one

bond on the oxygen and two on the hydrogens and the sum of all other possi-

ble bond configurations, respectively. For the transmission case, no difference

between the bond configurations of 1D and 1A were observed. This is not

the case for reflection events where a dramatic change between 1D and 1A

was found.

The 1A molecules show a large population for fully coordinated configu-

rations (gray bar), while the 1D molecules present an inversion of trend with

the largest population on configurations with only one hydrogen bond to the

oxygen site (darkgray bar). Such bonding is prevalent for water molecules

close to the protonated site [181], because it is unfavorable to donate a HB

to a water molecule that harbors some of the positive charge. Thus, for

T-segments the second solvation shell of D and A is symmetric, supporting

94

Chapter 5: Applications

a Zundel cation, whereas for R-segments the solvation the second solvation

shell of D and A is asymmetric, corresponding to an Eigen cation on D and

bulk water on 1A. This interpretation is in general agreement with the “pre-

solvation” concept discussed by Tuckerman et al. [197], extended to the 2nd

shell of the Zundel cation, as advocated by Lapid et al [181].

To conclude, we found that the length of the hydrogen bonds in the first

solvation shell of the proton plays crucial role in its transfer from one water

molecule to another. Since from the network analysis follows that proton

motion in Zundel cation and forming new hydronium ion was driven by small

occasional difference in hydrogen bond lengthes we can conclude that there’s

no special condition for only one water to be proton acceptor but the process

of chosing the water molecule for creating the hydronium ion is driven by

competative dynamics of its first and second hydration shells.

95

0

1

2

3

4

2.4 2.8 3.2 3.6

g(r)

r [A]

1

2

3

2 4 6

0

1

2

3

4

2.4 2.8 3.2 3.6

g(r)

r [A]

1

2

3

2 4 6

Figure 5.23: Oxygen-oxygen radial distribution function. Upper panel: Red

and orange colors stand for 1D and 1A molecules within ’ddaa’ node; gray

and lightgray colors stand for 1D and 1A molecules in the ’aadd’ node. Lower

panel: Red and gray for 1D molecule in transmission and reflection, orange

and darkgray for 1A in transmission and reflection. Black line depicts the

RDF in bulk water on both panels.

96

Chapter 5: Applications

00.10.20.30.40.50.60.7

Pop

ulat

ion

1D 1A 1D 1A X

Transmission Reflection Bulk

0

0.1

0.2

0.3

0.4

0.5

0.6

1D 1A 1D 1A

Pop

ulat

ion

ddaa aadd

Figure 5.24: Number of donor and acceptor hydrogen bonds in the first

solvation shell of a Zundel complex. Darkgray stands for 2 donors and 1

acceptor hydrogen bonds, gray stands for 2 donors and 2 acceptor bonds and

lightgray for all the other configurations of the first solvation shell.

97

Conclusions

In this thesis both the structural and dynamical characterization of liquid

water by means of molecular dynamics simulation and complex network anal-

ysis was presented. My main contribution towards this aim was to introduce

a new simplified statistical framework to characterize the elusive structure

forming capabilities of liquid water based on complex network analysis. I

summarize below four main achievements I was able to obtain by applying

this framework to several different aspects of liquid water and its modeling.

First, I investigated seven classical atomistic water models for a wide

range of conditions. For all water models I defined structural propensities

describing the first and second solvation shells of a water molecule. I found

that for all studied models these propensities perfectly overlap once a simple

temperature shift is applied. This result would not have been possible with-

out the introduction of the second shell connectivity. Before, conventional

methods focused their attention on the first shell only. The network frame-

work introduced here on the other hand allowed an exhaustive description

of the structure and kinetics of the second solvation shell, which allowed in

turn to reveal the common behavior underlying the apparently very different

water models. The same principles were also applied to the investigation of

more advanced water models which take explicitly into account polarizability.

My results provided clear evidence of structural differences and similarities

compared to conventional classical water models. An extension of these con-

cepts were also used for the characterization of the free-energy landscape of

water, demonstrating the flexibility and usefulness of this network approach

98

Conclusions

for the study of many body dynamical systems.

Second, the aforementioned network approach was also applied to the

characterization of the hydrogen bonds in liquid water. Notwithstanding the

efforts, this omnipresent interaction is still poorly understood. My work rep-

resents an effort to clarify the robustness and reliability of the most widely

used definitions of the hydrogen bond in a wide range of temperatures and

water models. My observations provide strong evidence for a lack of con-

sensus between different definitions, calling for a new generation of hydrogen

bond definitions based on the behavior of the kinetics. This should be the

most accurate way to determine this elusive bond especially when coupled

with the study of the solvation shell of a biomolecule.

Third, in this thesis I presented the first examples of microsecond long

simulations of supercooled water. Since these calculations were almost two

order of magnitude longer that the state-of-the-art trajectories found in the

literature, I was able to demonstrate that some recent theories of nucleation

based on very simplified models of water do not hold in the fully atomistic

case. This finding motivates the spring of a new generation of studies on

nucleation, calling for the development of new techniques to characterize

and investigate the first steps of nucleation at the microscopic level.

Fourth, a novel approach to investigate proton transfer was introduced.

The idea behind this method is based on both the experience I gained on

complex network analysis and the characterization of hydrogen bonds. By

putting together these two elements, I was able to present a new description

of proton transfer in liquid water, providing a quantitative characterization

of the transfer event.

Along with these promising results presented here there are still several

questions which have to be further investigated and clarified. More specif-

ically, there is a general lack in providing a quantitative description of the

kinetics of processes explicitly including the degree of freedoms of water.

This is an important ingredient for the description of biomolecular processes

where water solvation plays a role, like protein folding and ligand binding.

99

The development presented in this thesis represent a good starting point on

how to explicitly treat the degrees of freedom of the solvent, finally pro-

viding a more complete picture of the fundamental mechanism behind such

biomolecular processes.

100

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Acknowledgment

This work wouldn’t be done without help of many people. First of all I want

to thank Dr. Francesco Rao and Prof. Gerhard Stock for supervising me

during my PhD. I want to point my gratitude to my colleagues Diego Prada-

Gracia, Stefano Mostarda, Anna Berezovska, Nasrollah Moradi and Cheng

Lu who learned me a lot. Also I want to thank Prof. Noam Agmon and

Waldemar Kulig for providing me the trajectory of proton transfer events.

Another thanks to my friends Max Melnik, Alex Simonov and Anastasiia

Anishchenko. And least but not last for my family, especially my wife Helen

who supported me during all years of my study. Financial support provided

by the Excellence Initiative of the German Federal and State Governments.

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