Water Models and Hydrogen Bonds

Albert-Ludwigs-UniversitätFreiburg FakultätfürMathematik und Physik

Water models and hydrogen bonds

Dissertation zur Erlangung des Doktorgrades der FakultätfürMathematik und Physik der Albert-Ludwigs-UniversitätFreiburg 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 R˚uˇziˇcka Prodekan : Prof. Dr. Andreas Buchleitner Leiter der Arbeit : Prof. Dr. Gerhard Stock Referent : Prof. Dr. Gerhard Stock Koreferent : PD Dr. Thomas Wellens Datum der m¨undlichen Pr¨ufung : 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 trajectories 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 studies of liquid water, particularly the temperature response of some of the most popular water models, including their hydrogen bond network structure. 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 network 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 dynamics simulations is provided. The most commonly used approaches for temperature and pressure coupling is described as well as the difference 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 described 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 included 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 simulation 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 simulations, 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 simulations packages such as GROMACS [31], NAMD [32] and LAMMPS [33] allow 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 certain 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:

V U(r) = K (r − r )2 + K (θ − θ )2 + n [1 + cos(nχ − δ)]+ b 0 a 0 2 bonds angles dihedrals X X X 2 + Kijkl(S − S0) + ULJ (r) + UE(r) (1.1) impr.dih. X

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 parameters 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:

σ σ U (r) = 4 [ ij − ij ], (1.2) LJ r12 r6 i

q q U (r) = i j . (1.3) E 4π r i

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:

r¨i = 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 r˙i 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 interactions 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 energy of the system fluctuates within a certain distribution characterized by certain temperature and pressure. Here we briefly introduce the most common 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 probability ∆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 components 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 T − T = 0 (1.5) dt τ 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 1 sampling will be incorrect. This error scales with N , 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

dt KK0 dW dK = (K0 − K) + 2 √ , (1.6) τT s Nf τT 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:

2 d ri fi pξ dri 2 = − , (1.7) dt mi Q dt where Q is a constant of the coupling and the equation of the motion for the heat bath is: dp ξ = T − T , (1.8) dt 0 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:

r¨i = f i/mi − γir˙i + 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 1 1 1 − α kBT 2 G v(t + ∆t) = αv(t − ∆t) + F(t) + (1 − α ri (1.10) 2 2 mγ r m

1 r(t + ∆t) = r(t) + ∆tv(t + ∆t), (1.11) 2 where γ∆t α = (1 − ). (1.12) m

G Here ri 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 simulation 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 P − P = 0 (1.13) dt τp

The matrix µ is deﬁned as

n∆t µij = δij − βijP0ij − Pij(t), (1.14) 3τp

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 = V W−1b0−1(P − P ) (1.15) dt2 ref 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:

2 d ri Fi dri 2 = − M , (1.16) dt mi dt where M is:

db0 db M = b−1[b + b0]b0−1 (1.17) dt dt 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:

2 −1 4π βij (W )ij = 2 (1.18) 3τp L

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 reproduce specific features of real water instead of others, like the correct temperature 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 water 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 described 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 (/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].

o Water dOH [A]˚ H-O-H[ ] σ[A]˚ (/kB)[K] qH [A]˚ dOM [A]˚ model 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 potentials 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 properties, 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 thermostat [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 coupling can easily produce a correct canonical ensemble. None-covalent interactions 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 investigation 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 density 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 respect to another solids, where breaking bonds upon melting requires more space and therefore the density decreases [80]. It’s worth to note that this situation does not happen with high-pressure ices (III,VI,VII), which expand on melting [81]. It can be expected that due to large cavities in hydrogen bond network dV water should have a high isothermal compressibility (kT = −[ dP ]T /V ]). In −1 fact, water has unusually low compressibility (0.46 GPa , 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 towards 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

kB T temperature D = 6πηr (here D is diffusion coefficient, η is viscosity and r is approximate radius of the particle) violates for water. At low temperatures the diffusion dependence on temperature can be fitted with Arrenhius lax while at high temperatures it behaves accordingly to empirical Vogel-

kTo Fulcher-Tamman relation D = D0exp( ), where D0 and T0 are ﬁtting T −T0 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 coefficient 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 networks 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 example, 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 amorphous 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 nucleation [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 attention [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 supercooled 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 magnitude 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

3 4 3 1 2 q = 1 − cos ψ + , (3.1) i 8 jik 3 j=1 k=j+1 X X 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 single 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

1 4 qi = Y m(φ θ ), −l ≤ m ≤ l (3.2) l,m 4 l ij ij j∈ni X m where the sum is over four nearest neighbors, ni. Yl is the l, m spherical harmonic function associated with the angular coordinates of the vector ~ri−~rj joining 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 N i Ql,m = ql,m (3.3) i=1 X and then contracted along the m axis to produce a parameter that is invariant with respect to the orientation of the arbitrary external frame,

l 1 ∗ 1 Q = ( Q Q ) 2 (3.4) l N l,m l,m m=−l X 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.

-54 [kJ/mol] p E -55 B 1.01 ] -3 0.99 [g cm ρ 0.97

C 0.88

0.86 T Q 0.84

D 0.82

0.03 6 Q

0.01

0 1000 2000 3000 Time [ns]

Figure 3.2: Time series for the 3 µs trajectory. (A) potential energy; (B) density; (C) tetrahedral order parameter QT ; (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- −6 ergy was found (r = 10 ). 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

1020 1020 -53.5

1010 1010 ] ] -3 1000 -3 1000 -54

990 990

980 980 -54.5 Energy [kJ/mol] Density [kg m Density [kg m 970 970

960 960 -55 -55 -54.5 -54 -53.5 0.82 0.84 0.86 0.88 0.82 0.84 0.86 0.88

Energy [kJ/mol] QT QT

-53.5 1020 0.9

1010 ] 0.88 -54 -3 1000

990 T 0.86 Q

-54.5 980 Energy [kJ/mol] Density [kg m 0.84 970

-55 960 0.82 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05 0 0.01 0.02 0.03 0.04 0.05

Q6 Q6 Q6

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

2 MSD(t) = h(ri(t) − ri(0)) i, (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 nucleation. 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

35 101

0 ] 10 2

10-1 MSD [nm

10-2

10-3 10-2 100 102 104 Time [ns]

Figure 3.4: Oxygen mean square displacement (MSD). The dashed and dotted 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 parameters are conventionally used for this purposes [115, 116]. Some of the conventional order parameters commonly used to measure the structure of liquids 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 different 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 molecular simulations

A network is a set of items, which we will call nodes, with connections between 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 dynamics trajectories

(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 between two states with the same value of an order parameter. (b) Network clusterization techniques allow the lumping of kinetically homogeneous regions 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 dynamics. 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 dynamics trajectories

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 determines 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 maximum 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 definition 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 dynamics trajectories

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 analysis [137,139]. All these approaches aim to clusterize the network into kinetically and structurally well defined basins of attraction. In enthalpy driven free-energy landscapes, of which proteins are an archetypal 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 pathways [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 probability 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:

(i) Zout = Zout (4.1) i X Structural inhomogeneities, i.e., conﬁgurational selection, emerge because

(i) (i) max Zin max Zout , (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 environment 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 temperature [16].

Zin

Zout

Figure 4.4: Conﬁguration-space-networks. Pictorial representation of the relative balance between intra-basin (Zin) and inter-basins (Zout) transition probabilities from the point of view of a node (in blue). Gray regions 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 interaction 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 dynamics trajectories 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 binding [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 directional. 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 thermodynamic 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 definitions in several different conditions, including temperature, water model and cutoff dependence. Here, we present an assessment of most used hydrogen bond definitions based on the analysis of molecular dynamics simulations 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, including 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 cutoffs on inter-atomic distances and angles while the latter mostly avoid this problem using topological criteria. A brief description of the definitions follows.

rOO Θ

rOH

geometrical topological

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 dynamics trajectories

(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 hydrogen 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 dynamics trajectories

3.8

3.6

3.4

3.2

N, HBs per molecule 3

2.8

240 280 320 360 400 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 topological 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 hydrogen 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 calculated 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 0.8 Θ rOH rOO Sk 0.6

0.4

Probability 0.2

0.8 DΘ DA TP 0.6

0.4

Probability 0.2

0 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 N hb N hb N hb 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

(P3) and two or less (P210) bonds with a generic ﬁrst solvation shell. In the ∗ calculation of P4 the propensity of P4 was subtracted. A more comprehen-

50 Chapter 4: New strategies for the analysis of molecular dynamics trajectories

A 10-1

-2 10 -2

10 10-3

10-4 0 0.1 0.2

Probability 10-5

10-6

10-7 0 5 10 15 20 25 30 time [ps] B 101

100 [ps] τ

10-1 2.5 3 3.5 4 4.5 1000/T [K-1]

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.

P4 conﬁguration

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 P4 . 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 dynamics trajectories

0.8 Θ * rOH rOO Sk P 0.6 4

0.4 P3 P4 Population 0.2 P210

0.0

0.8 DΘ DA TP 0.6

0.4

Population 0.2

0.0 240 280 320 360 400 240 280 320 360 400 240 280 320 360 400 Temperature [K] Temperature [K] Temperature [K]

Figure 4.11: Hydrogen bond propensities including the second solvation shell ∗ for temperatures between 220 K and 400 K. P4, P4 , 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 distribution 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 temperature 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 dynamics trajectories

rOH rOO Sk

240K

280K 320K

g(r) g(r)

Probability

1 2 3 4 2 3 4 5 0 0.01 0.02 0.03 0.04 r [Å] r [Å] Occupancy [N] 3.5 0.012 2.424 ] ]

Å Å 3.4 0.010 2.422

3.3 0.008

distance [ 2.420 distance [ Occupancy [N]

2.418 3.2 0.006 240 280 320 240 280 320 240 280 320 Temperature [K] Temperature [K] Temperature [K] 4.0 4.0 4.0 3.8 3.8 3.8 3.6 3.6 3.6 3.4 3.4 3.4 3.2 3.2 3.2

Number of HB 3.0 Number of HB 3.0 Number of HB 3.0 2.8 2.8 2.8 240 280 320 240 280 320 240 280 320 Temperature [K] Temperature [K] 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 results 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 temperature for different water models. According to the original prescription [148],

55 0.012 TIP3P SPC TIP4P SPC/E 0.010 TIP4P-Ew 0.008 TIP4P/2005 N min 0.006

0.004 240 280 320 Temperature [K]

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 dynamics trajectories

SPC−E TIP3P

80 2 60 1.5 Ψ 40 1 20 0.5 0 0 1.5 2 2.5 3 1.5 2 2.5 3 r [Å] r [Å]

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 molecular 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 ambient 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 repulsion 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 accuracy [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 water 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 biomolecular simulations. For example, Best and collaborators showed that predicted 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 compared to conventional water models [165]. In both studies, the improved behavior 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 temperature 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 simulations 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.

ref Water model TMD TMD ∆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 population 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:

3 3 4 1 q = 1 − cos ψ + 2, (5.2) i 8 jik 3 j=1 k=j+1 X X 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 capabilities 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 Pi : the fully coordinated 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 (P4); 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 behavior 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 entropic stabilization takes place at a model dependent temperature. All four water configurations fall into one of these three main classes. The population 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 (P4, filled red circles). For this configuration the population increases with decreasing temperature until it reaches a maximum in correspondence 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 P P * 4 4

13 16 17 13 14 16 17 4 5 4 5 12 15 12 15 1 1 8 11 8 11 2 3 2 3

6 7 9 10 6 7 9 10

P3 P2

13 14 13 14 16 17

4 4 5

12 12 15

1 1

8 11 8 11

2 3 2 3

6 7 9 10 6 7 9 10

Figure 5.1: Schematic representation of the four possible configurations of ∗ water solvation shells (P4,P4 ,P3,P2 population, see text). Dashed lines represent 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 temperature 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 temperatures. 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 P4 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 comparing 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

0.8 0.8 0.8 spc/e spc tip3p 0.6 0.6 0.6 0.4 0.4 0.4

Population 0.2 0.2 0.2 0 0 0 200 250 300 350 200 250 300 350 200 250 300 350 0.8 0.8 0.8 tip4p/2005 tip4p-ew tip4p 0.6 0.6 0.6 0.4 0.4 0.4

Population 0.2 0.2 0.2 0 0 0 200 250 300 350 200 250 300 350 200 250 300 350 0.8 0.8 tip5p P4 * 0.6 0.6 P 4

0.4 0.4 P3

P Population 0.2 Population 0.2 210 0 0 200 250 300 350 200 250 300 350 Temperature Temperature

Figure 5.2: Temperature dependence of water structure populations for seven ∗ classical water models. P4,P4,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 A B

60 80 60 s 40 T

Temperature 20

Energy [kJ/mol] Energy 20 * 0 0 D C spc tip3p spc-e tip4p tip5p

tip4p-ew 3 0.9 tip4p/2005

2 Figure 5.3: Structural temperature shift ∆Ts with respect to the TIP4P/2005 model. TIP5P was excluded 0.8 from the superposition analysis (see text for 1

details) . [kJ/mol] Energy

Tetrahedral parameter 0.7 0

spc spc tip3p tip4p tip5p tip3p factor (∆Ts) exists among the models.spc-e TIP4P/2005 was chosen as referencespc-e tip4p tip5p

tip4p-ew tip4p-ew for its ability to reproduce the density curve [159].tip4p/2005 Using TIP4P/2005 as a tip4p/2005 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 presence of ions [170] but in this case we could superimpose all models onto a master curve. Unfortunately, TIP5P had to be excluded from the superposition 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

A 5 4 3 2 F [kJ/mol] 1 0 B -350

-400 E [kJ/mol]

C -450

0.9 Q 0.8

0.7

spc tip3p spc-e tip4p tip5p

tip4p-ew tip4p/2005

Figure 5.4: Comparison of water models with respect to the fully coordinated 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 90

60 s T ∆ 30

0 0 30 60 90 ∆ 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 independent 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 improved stabilization of the fully coordinated configuration. (Alternatively, it can be said that these models destabilize poorly hydrogen-bonded configurations). 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 correlate well neither with the free energy nor with the structural temperature

66 Chapter 5: Applications

0.8 P4 * 0.6 P 4 0.4 P3

Population 0.2 P210 0 200 250 300 350 Temperature

Figure 5.6: Overlap of the Pi states when a Skinner deﬁnition for the hydrogen 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 temperature 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 between 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 P4 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 P4 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 overlap 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 hydrogen 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 models, recently new water models with explicitly introduced polarizability appeared [171, 172]. The introduction of polarizability in classical molecular simulations holds the promise to increase the accuracy as well as the prediction power of computer modeling. One promising strategy to introduce polarizability 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 electronic induction is represented by a classical negatively charged Drude particle attached to the positively charged oxygen by a harmonic spring [171]. Tests at the ambient conditions showed better agreement with the experimental 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 important 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 potential 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.

-10

250K

-11 200 K

-12 Energy [kcal/mol] 170 K

0 10 20 30 40 50 time [ns]

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.

1050 ] 3

1000

Density [kg/m 950

200 300 400 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 classical 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 (P4 ), 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, respectively. 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 density 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

0.8

P4 P * 0.6 4

0.4

Probability P 0.2 210 P3 0 150 200 250 300 350 400 temperature [K]

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 temperature 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 instead 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 0.9

0.8

0.7

0.6

Tetrahedral parameter Q 0.5 150 200 250 300 350 400 Temperature [K] 0.03

0.02

Probability 0.01

0 0 0.2 0.4 0.6 0.8 1 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 corresponded 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 polarizable 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 simulations. 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 configurations 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 configurations 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

13 14 16 17 13 14 16 17

4 5 4 5

12 15 12 15

1 1

8 11 8 11

2 3 2 3

6 7 9 10 6 7 9 10

12 02

13 14 16 17 13 14 16 17

4 5 4 5

12 15 12 15

1 1

8 11 8 11

2 3 2 3

6 7 9 10 6 7 9 10

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 1.5 ] 5 1

0.5 Num. Nodes [10

0 200 250 300 350 Temperature

Figure 5.12: Number of nodes of the complex network at diﬀerent temperatures. 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

225 Tm 285 1 22 0.8

0.6

0.4 21 Population

0.2 12 02 0 200 250 300 350 Temperature [K]

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].

0.6

190 K 0.4

240 K 0.2 Probability 320 K

0 0 0.2 0.4 0.6 0.8 1 Q

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 organization 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 microstate 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 transition 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 transition probability for many nodes which were acting as attractors at higher (i) temperatures. Fig. 5.16a shows the temperature at which Z22 = max Z 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) i.e. Z22 > i Z (gray bars in Fig. 5.16a). In other words, every time a water molecule assumes a configuration different from 22, the probability to P 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 populations is remarkably similar with the presence of the three liquid regimes. On

81 16 ] ] 2 4 30 _ 12 d 8 20 4 a 10 c nodes [x10 0 degree [x10 0

15 ] 3 100 10

basins 5 b 95 d

0 density [cg/cm 200 250 300 350 200 250 300 350 Temperature Temperature

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 discussion on the quality of hydrogen-bond definitions is still open [148, 153] (See

82 Chapter 5: Applications

a 350

300 (i) ˜ 285K Z22 > max Z Zi

250 ⇣(i) ⌘ Z22 > Z Z22 225K

Temperature [K] X 200 22 21 12 microstate

b 22 22 22

homogeneous homogeneous inhomogeneous c low-density

Figure 5.16: Schematic representations of the three regimes of liquid water; (a) temperatures at which there is a change in the transition probability maximum 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 a 1 22 TIP3P TIP5P 22 0.8

0.6

21 0.4 Population 21 0.2 12 12 02 02 0 b 200 250 300 350 1 Temperature [K] Rθ 22 0.8

0.6

0.4 Population 21 0.2 12 02 0 200 250 300 350 Temperature [K]

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 conditions, 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 properties, 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 inhomogeneities 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 coerces the latter not to return to its initial state, but rather centralize the hydronium 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 H5O2 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 corresponds 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 collective motion controls water dynamics in bulk liquid water [183], with correlations 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 proteins [186], between photoacid and base molecules in bulk water [187, 188], or during hydronium/hydroxide neutralization [189], proton transport becomes 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 valence 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 microstates 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 2D 2A 2D 2A

1A 2D 1D d a 2A

D + A 2D a 2A d 1D 1A

2D 2D 2A 2A

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.

12% daad

2512 2154 36% 20% 9% 9% 5401 1036 2238 255 R ddaa dada babaadad aadd T 1651 12% 3320

adda

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 appreciable) 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 distributions 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 distance, 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 between 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. 0.2

0.15 D:ddaa

A:ddaa 0.1

Probability 0.05

0 1 1.2 1.4 1.6 Oxygen-Proton distance [A]

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 distributions of acceptor molecule A further accentuates, commensurate with the role of these two nodes in reﬂecting or transmitting the proton, respectively. 0.15

0.1

0.05 Probability

0 1 1.2 1.4 1.6 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 configuration ’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 rearrangements in driving the proton transfer process. Finally, hydrogen bond histograms for the 1D and 1A molecules were calculated 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 possible 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 configurations (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 4 3 2 3 1

2 2 4 6 g(r) 1

0 2.4 2.8 3.2 3.6 r [A]

4 3 2 3 1

2 2 4 6 g(r) 1

0 2.4 2.8 3.2 3.6 r [A]

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

Transmission Reﬂection Bulk 0.7 0.6 0.5 0.4 0.3 Population 0.2 0.1 0 1D 1A 1D 1A X ddaa aadd 0.6 0.5 0.4 0.3

Population 0.2 0.1 0 1D 1A 1D 1A

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 analysis 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 without the introduction of the second shell connectivity. Before, conventional methods focused their attention on the first shell only. The network framework 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 concepts 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 represents 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 consensus 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 specifically, 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 providing a more complete picture of the fundamental mechanism behind such biomolecular processes.

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122 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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