Food Biophysics (2011) 6:186–198DOI 10.1007/s11483-010-9198-4

SPECIAL ISSUE ARTICLE

Water at Interface with Proteins

Giancarlo Franzese · Valentino Bianco · Svilen Iskrov

Received: 29 September 2010 / Accepted: 15 December 2010 / Published online: 31 December 2010© Springer Science+Business Media, LLC 2010

Abstract Water is essential for the activity of proteins.However, the effect of the properties of water on thebehavior of proteins is only partially understood. Re-cently, several experiments have investigated the rela-tion between the dynamics of the hydration water andthe dynamics of protein. These works have generated alarge amount of data whose interpretation is debated.New experiments measure the dynamics of water atlow temperature on the surface of proteins, finding aqualitative change (crossover) that might be relatedto the slowing down and stop of the protein’s activity(protein glass transition), possibly relevant for the safepreservation of organic material at low temperature. Tobetter understand the experimental data several sce-narios have been discussed. Here, we review these ex-periments and discuss their interpretations in relationwith the anomalous properties of water. We summarizethe results for the thermodynamics and dynamics ofsupercooled water at an interface. We consider alsothe effect of water on protein stability, making a stepin the direction of understanding, by means of MonteCarlo simulations and theoretical calculations, how theinterplay of water cooperativity and hydrogen bondsinterfacial strengthening affects the protein cold denat-uration.

G. Franzese (B) · V. Bianco · S. IskrovDepartament de Física Fonamental,Universitat de Barcelona, Diagonal 647,08028, Barcelona, Spaine-mail: gfranzese@ub.edu

S. IskrovEcole Normale Supérieure de Cachan, Paris, France

Keywords Water · Hydrated proteins · Confinedwater · Biological interfaces

Introduction

Water is ubiquitous in biological systems. It is a majorcomponent of cells and participates in the majority ofthe biological processes. It is usually considered es-sential for life, but it is still under debate why.1 Onepossible reason is that water has many properties thatare unusual with respect to other liquids.2

The anomalous behavior of water is evident in theliquid phase. For example, fluctuations of volume andfluctuations of entropy have a minimum for liquid wa-ter, while in usual liquids they decrease when the tem-perature T is decreased. Volume fluctuations can beobserved by measuring the compressibility KT , definedas how much the volume changes when the pressure Pis changed at constant T, and entropy fluctuations areproportional to the specific heat CP at constant P. Forwater at ambient pressure, KT has a minimum at 46 ◦Cand CP at minimum at 35 ◦C.

The anomalies of water become more evident whenT is decreased toward and below 0 ◦C. For example,water has a maximum in density at 4 ◦C. Normal liquids,such as argon, reduce their density when the tempera-ture decreases and reach their maximum density whenthey solidify in a crystal. Water, instead, below 4 ◦Cexpands. Therefore, liquid water at 0 ◦C has a densitysmaller than water at 4 ◦C. By solidifying into ice,water expands even further, becoming “lighter”. Forthis reason ice cubes float in a glass of water. Thisproperty has a dramatic consequence in processes suchas the cryopreservation of biological cells, because the

Food Biophysics (2011) 6:186–198 187

large amount of water in each cell expands when itforms ice and breaks the cell.

Another property of water that is relevant in theseconditions is that water can stay in its liquid state evenat T < 0 ◦C, i.e., it can be supercooled below its meltingtemperature. Bulk water can be supercooled to −41 ◦Cat atmospheric pressure, but in different conditionswater can remain liquid even at lower temperatures.For example, it can be supercooled down to −47 ◦Cwhen confined in vegetable fibers, or down to −92 ◦Cwhen compressed at 2 kbars. An anti-freezing effectcan be achieved also by dissolving in water polymersor proteins, with several practical applications. Under-standing the details of this phenomenon and how toregulate it could be very relevant for cryopreservation,food storage and refrigeration.3,4

Also in its solid state, water is peculiar. Water is apolymorph with many different crystal phases, morethan 15, some of which are stable only at pressure P >

100 GPa. But water can easily form a solid that is not acrystal. If quenched rapidly below −123 ◦C at ambientpressure, liquid water freezes in a metastable amor-phous state, which is an arrested liquid configuration.2

At low pressure, water forms a low-density amorphous(LDA) state,5 while at high pressure it forms a high-density amorphous (HDA) state,6 separated by a volumediscontinuity of ≈27%, comparable to that betweencrystalline ice I and ice VI. A smaller discontinuity hasbeen observed more recently,7,8 but its interpretation isunder debate.

The discontinuities between amorphous states atdifferent densities and the fact that quantities such asKT or CP largely increase in the supercooled state,show that water has a complex behavior at low T. Thisobservation and the fact that water remains in its liquidstate at very low T when in contact with organic orinorganic interfaces suggest that water could play amain role in phenomena such as the so-called proteinglass transition, or the protein cold denaturation at lowtemperature.

Experiments

To explore how the dynamics of proteins and water arerelated, Chen et al. in 2006 studied by high-resolutionquasi-elastic neutron scattering (QUENS) the struc-ture and dynamics of water molecules in the hydrationlayer surrounding lysozyme proteins at temperaturesaround 220 K (−53.15 ◦C).9 Below this temperature,the protein is in a solid-like “glassy state”, with noconformational flexibility and no biological functions.As the temperature is increased, the protein displays

a harmonic atomic motion that, in hydrated proteins,suddenly becomes anharmonic and liquid-like at about220 K. The change in the protein dynamics is believedto be triggered by the coupling with the hydrationwater through the hydrogen bonds because the hydra-tion water displays a dynamic transition at a similarT. 10,11 Chen interpreted the hydration water dynamictransition as a change in a main structural relaxationand recently extended this interpretation to hydratedproteins at high pressure.

This explanation has been questioned by Swensonet al. 12 By using dielectric measurements on myo-globin in water–glycerol mixtures, they found a dy-namic crossover at about 200 K, and they interpretedit as an evidence that local (secondary) protein mo-tions are controlled (slaved) by the local fluctuationsin the hydration shell, as proposed by Fenimore et al.in 200413 based on Mössbauer and neutron-scatteringexperiments.

On the other hand, Pawlus et al. in 2008, based onmeasurement of conductivity on hydrated lysozyme,found no crossover around 220 K. They also ascribethe apparent crossover observed with QUENS to asecondary relaxation and to a lack of resolution onmain structural relaxation of QUENS.14

More recently nuclear magnetic resonance (NMR)experiments on water hydrating elastin and collagen,performed by Vogel, showed no crossover at 220 Kbut a crossover at about 200 K. These data have beeninterpreted as consistent with thermally activated tetra-hedral jump motion of hydration water.15

These and other experiments, therefore, show thatit is difficult to achieve a clear understanding of theprotein-hydration water coupling solely based on theexperimental data. A possible way to gain further in-sight is offered by the numerical results of detailedcomputer simulations.

Numerical Results

In 2006, Kumar et al., 16 by simulating lysozyme orDNA surrounded by water represented with the TIP5Pmodel, found a dynamic transition of the macromole-cules at about the same temperature of a dynamiccrossover in the diffusivity of hydration water, in therange 242–250 K. They also show numerical indicationsthat this crossover coincides with the maximum of theisobaric specific heat of the whole system and the max-imum fluctuations in tetrahedral order of the hydrationwater.

This observation was confirmed in 2008 by Lagiet al. 17 They found a strong crossover in the water

188 Food Biophysics (2011) 6:186–198

translational (α) relaxation time and in the inverse ofits self-diffusion constant at about 223 K, by simulat-ing hydrated lysozyme with water represented by theTIP4P-Ew model. They corroborate that the crossovercorresponds to the maximum structural change, andthey found a low activation energy, showing consistencywith the neutron scattering and the NMR experiments.

Nevertheless, in 2009 this view was disputed byVogel.18 By simulating elastin or collagen hydratedby SPC-water, he found only a weak crossover in thehydration water correlation time at about 200 K and ahigh activation energy associated to a secondary relax-ation, consistent with activation energies determined indielectric spectroscopy and NMR studies.

It is evident from these and other numerical studiesthat the comparison of experiments with simulationis useful, but is not enough to elucidate the mecha-nisms that regulate the coupling of the water dynam-ics with the biomolecules dynamics. In particular, thedifficulties in finding clear answers to the open ques-tions rise for the fact that experiments and numericalresults are both affected by errors and uncertainties.Therefore, it is natural to look for a theory that couldbe able to find exact relations and make predictions totest in further experiments.

In the next section, we describe a model for a hydra-tion water monolayer that allows to develop a theoryfor these phenomena and to predict properties thatcould be verified in experiments. The model, moreover,allows for efficient simulations whose results comple-ment the theoretical analysis. In “Results for WaterBetween Hydrophobic Plates,” we review results of thistractable model, including new data for the density andenergy distributions at different pressures and very lowtemperature (“Thermodynamics at Very Low Tempe-rature”) and for the hydration percolation (“HydrationPercolation”). In “Discussion: Implications in Food Sci-ence,” we discuss new results about protein stabilityand confined water in the context of food processingand cells in living organism. In “Conclusions,” we giveour conclusive remarks and we discuss possible exten-sions of this model.

A Tractable Model for a Hydration Water Monolayer

We first consider the case of water nanoconfined be-tween two hydrophobic surfaces. To fix the idea, let’sconsider the case with a distance δ = 0.7 nm betweenthe two surfaces. Because the confining surfaces are hy-drophobic, the water molecules will not form hydrogenbonds (HB) with the surfaces. Experiments shows that

in bulk each water molecule is surrounded by four near-est neighbor molecules at a distance of about 3Å witha structure that resembles that of a tetrahedron at lowT, V and P. 19 Hence, one would expect that eachwater molecule between the two hydrophobic plateswill adjust in a way to form a distorted network of HBswith each molecules surrounded by other four. Thishas been indeed found by Kumar et al.20 by simulatingTIP5P-water in these conditions.

To define a tractable model,2,21–25 we coarse grainedthe structure described above, dividing the slab of spaceoccupied by water into cells with volume v = δr2, witha square section of size r ≥ r0, where r0 = 2.9Å26 isthe closest approach (van der Waals) distance betweenwater molecules (Figure 1). If the density of water isρ, and the number of water molecules in N, the watervolume is V ≡ N/ρ. If the system is uniform and wedivide the system in N cells, each cell has on averageone water molecule. More in general if the system isnot uniform, some cells can be empty or each cell canoccupy a different volume vi and the distance rij be-tween two molecules in the cells j and j is the distancebetween their centers. For example, if i and j are theindices of nearest neighbor cells, the distance betweenthe molecules in these cells is rij ≡ (

√vi/δ + √

v j/δ)/2 .

Fig. 1 Schematic representation of a water monolayer. Topview of water molecules with an oxygen atom (red) and twohydrogen atoms (blue) distributed over a surface, between twohydrophobic plates (not represented). Possible hydrogen bondsare represented by gray sticks. The total surface area is dividedin equal-size square cells (dashed lines). In the tractable modeladopted here, the coordinates of each molecule inside a cell arecoarse grained. A configuration of water molecules is representedby the occupancy state of the cell (local density) and the statesof the four bonding indices of each molecule, accounting forthe hydrogen bonds formed with water molecules in the nearestneighbor cells

Food Biophysics (2011) 6:186–198 189

We describe the isotropic (e.g., van der Waals) at-tractive and repulsive interactions between the mole-cules by a standard a Lennard–Jones interaction

U ≡∑

ij

ε

[(r0

rij

)12

−(

r0

rij

)6]

(1)

where ε = 5.8 kJ/mol27 is the attractive energy and thesum is over all the possible pairs of molecules i andj. Some modifications of this interaction, such as theintroduction of a maximum cut-off distance or a hard-core distance, have been adopted in previous analysis ofthis model24,28–33 to simplify the numerical simulations.For the theoretical analysis, instead, this interactionterm has been replaced by a more tractable discreteinteraction.2,21–25,33–35

To take into account the energy and entropy varia-tion when water molecules form HBs, we introduce foreach water molecule i four bonding indices σij, one foreach of the possible HBs with the four nearest neighborwater molecules j. The index can assume q differentstates, σij = 1, . . . , q, where q = 6 is a parameter whosevalue we will discuss in the following. Therefore, eachwater molecule has q4 = 64 = 1296 possible states, thatcan be interpreted as possible rotational configurations.The total number of configurations for the system isq4N , that for N = 105 (the maximum number that wehave considered in our analysis) is an astronomicalnumber (about 3 × 10311260). When a molecule formsa HB, the number of its accessible configurations de-creases, and the energy of the system is reduced.

To estimate the decrease in the number of accessibleconfigurations, we observe that a HB is broken if itdeviates from a linear bond more than ±30◦. Therefore,only one sixth of the whole continuous range of orien-tations (360◦) in the OH–O plane are associated to abonded state. Hence, 5/6 of the possible configurationsare not-bonded. By allowing q = 6 possible states foreach bonding index σij, we can count correctly theentropy loss associated to the formation of a HB ifonly one out of q states corresponds to a HB. This isachieved by allowing the formation of the HB betweenmolecules i and j only if δσij,σ ji = 1, where by definitionδa,b = 1 if a = b and δa,b = 0 otherwise.

Next, we take into account that a HB is broken ifthe OH–O distance is too large.36,37 To simplify theanalysis, we introduce a condition on the O–O distancer, allowing the formation of HB only if r ≤ rmax, withrmax = r0

√2 = 4.10Å. Therefore, considering that the

length of the OH covalent bond is rOH = 0.96Å38, themaximum bond length is rmax − rOH = 3.14Å, consis-tent with other choices in literature.36,37 In particular,we impose this condition by introducing a discrete vari-

able ni for each cell i, with ni = 1 if ri ≤ r0√

2, otherwiseni = 0. A cell with ni = 1 is liquid-like because its (di-mensionless) density ρi ≡ (δr2

0)/(δr2i ) ≥ 1/2, while a cell

with ni = 0 is gas-like because has ρi < 1/2. Hence, wedefine the number of HBs as

NHB ≡∑

〈i, j〉nin jδσij,σ ji (2)

where the symbol∑

〈i, j〉 denotes that the sum is per-formed over nearest neighbor cells i and j.

From the experiments, we know the formation ofHBs leads to an open network of molecules with, onaverage, four neighbors instead of twelve as in argon-like fluids. The resulting volume per molecule with HBsis larger than the volume per molecule with no HBs.This is observable as the anomalous density decreasedescribed in the introduction, that is a consequence ofthe formation of a macroscopic number of HBs. Thiseffect is incorporated in the model by considering thetotal water volume to be given by

V ≡ V0 + NHBvHB, (3)

where V0 is the water volume in absence of HBs,and vHB is the increase of volume per HB. To estimatethe parameter vHB, we consider as reference values theincrease between the density ρIh = 0.92 g/cm3 of theice Ih at atmospheric pressure and ice VI, with den-sity ρVI = 1.31 g/cm3, or ice VIII, with density ρVIII =1.46 g/cm3. Ice Ih is characterized by hexagonal ringsof HBs with an almost perfect tetrahedral structure,while ice VI and ice VIII have a structure consisting oftwo interpenetrating tetrahedral networks of HBs. Therelative increase of density in these cases is equal to 0.39and 0.54, respectively. Hence, in the model we set therelative HB increase of volume per molecule equal toan intermediate value between the previous referencevalues, i.e. vHB/δr2

0 = 0.5.Note that this increase of volume per molecule is due

to a decrease of number of first neighbors and does notimply an increase of distance between molecules. Ourmodel, being coarse grained, does not include all thedetails about the structure, but maintains the increaseof volume per molecule with no effect on the distancer between molecules. In particular, the HB volumeincrease does not affect the calculation of U0(r) of Eq. 1.

The formation of a HB leads to an energy gain,represented in the model by an interaction term

HHB ≡ −JNHB, (4)

where J is the characteristic energy of the covalent(directional) component of the HB. This term only ac-counts for the two-body component of the HB interac-tion. However, in water many-body effects are relevant

190 Food Biophysics (2011) 6:186–198

and, in particular, the three-body term.39,40 This can beobserved from the T-dependence of the O–O–O angledistribution. This distribution becomes sharper aroundthe tetrahedral angle when T decreases.41 Hence, weinclude in the model the many-body (cooperative)effect due to HBs,42–44 which minimizes the energywhen the HBs of nearby molecules optimize the tetra-hedral orientation. This is accomplished by furtheradding to the Hamiltonian in Eqs. 1 and 4 the term

Hcoop = −Jσ

∑

i

ni

∑

(k,�)i

δσik,σi� , (5)

where Jσ is the characteristic energy of the cooperativecomponent of the H bond, and the sum over (k, �)i isperformed over all the six different pairs of the fourbonding indices of molecule i.

As discussed by Stokely et al., 33 Eqs. 4 and 5 havetwo free parameters: J and Jσ , respectively. Experi-ments estimate the HB in ice Ih to be ≈3.0 kJ/molstronger than in liquid water.45 Attributing this in-crease to a cooperative interaction among HBs,46 wecan estimate the value of Jσ in the cell model to be≈1.0 kJ/mol, because for each HB there would be 6/2pairs of Jσ interactions. The optimal HB energy, EHB,has been measured to be ≈23.3 kJ/mol.47 By consid-ering tetrahedral clusters of H-bonded molecules, withHB and van der Waals interactions up to the third near-est neighbor molecules, the value for the directionalcomponent of the HB is estimated by Stokely et al. 33 asJ ≈12.0 kJ/mol. Other experimental estimates suggestthat breaking the directional component of the HBrequires J ≈6.3 kJ/mol.48 It is, therefore, a reasonableestimate to set Jσ /J = 1/10 and to consider J as theonly free parameter of the model.

In the following, we will briefly summarize some re-cent results for this model, to show that it reproduces ina qualitative way the properties of water. An appropri-ate choice of the free parameter J leads to results that,as for more detailed models, can be fairly rescaled onthe known proprieties of a water monolayer. In this re-spect, this model is not better then detailed models, but,it has two features that detailed models have not. (1) Itis less computationally expensive, because it is coarsegrained. This allows to simulate very large number ofwater molecules (about a million) on simple desktopcomputers in a few hours. (2) More importantly anddifferently from the detailed models, this model istractable for theoretical calculations. The trade-off forthese advantages is that the model is coarse grainedand cannot give informations about some properties ofwater, such as the structure. Nevertheless, works is inprogress to overcome this limitation.

Results for Water Between Hydrophobic Plates

We study the model described in the previous sectionby mean-field (MF) analysis and Monte Carlo (MC)simulations. The MF approach follows the Bethe–Peierls and the cavity method,49 by expressing themolar Gibbs free energy in terms of an exact partitionfunction for a portion of the system, and taking intoaccount the effect of all the rest of the system as a meanfield acting on the border of this portion, as describedby Franzese and Stanley2 and Stokely et al. 50

MC simulations are performed at constant N, P,T, allowing the volume V0 in Eq. 3 to fluctuate as astochastic variable. To minimize the boundary effects,we consider periodic boundary conditions in the di-rections parallel to the confining surfaces. To studythe thermodynamic properties of the model we adoptan efficient cluster MC dynamics, defined by Mazzaet al., 31 or a continuous T algorithm, the histogramreweighting method, as by Franzese et al. 24 To studythe dynamics of the HBs, we adopt a standard Metropo-lis algorithm,35 while to study the diffusion propertieswe use the Kawasaki algorithm.29

The Gas and Liquid Phases for Nanoconfined Water:Transport Properties

For water nanoconfined between hydrophobic plates,the model displays a gas–liquid first-order phase tran-sition ending in a critical point C, that qualitativelyresembles the gas–liquid transition for bulk water(Figure 2a). In their work, de los Santos et al. 30 (de losSantos and Franzese, unpublished data) verify thatthe water diffusion constant D decreases when waterchanges from gas to liquid. The change in D is strongfar from the critical point C, and it disappears at C. Inthe liquid phase, de los Santos et al. 30 observe that, asfor bulk water, the nanoconfined system has a region inthe P–T plane where D increases for increasing P. Thisbehavior is an anomaly of water, because in normalfluids D decreases for increasing P. 51,52 This anomalousbehavior is qualitatively rationalized as a consequenceof the HB formation and it has been observed both inbulk and confined water.53

At lower temperature, by decreasing T at con-stant P the model displays the line of temperaturesof maximum density (TMD),29,35 as in the bulk case(Figure 2a). The TMD line in the P–T plane has apositive slope at low P and negative slope at high P.

The nanoconfinement does not allows the formationof crystal ice. Nevertheless, it is possible to calculatewhere in the P–T plane D(P, T) is constant and to

Food Biophysics (2011) 6:186–198 191

0.4 0.6 0.8 1 1.2Temperature T/TC

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Den

sity

ρv 0

0.180.160.140.120.100.080.060.040.02

P [GPa]

(a)

C

0.030.020.01

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0.5

0.6

0.7

Den

sity

ρv 0

0.180.170.160.150.140.130.120.110.10

P [GPa]

(b)

C’

Fig. 2 Density at constant pressure for a water monolayernanoconfined between hydrophobic slabs of infinite section andseparated by a distance δ = 0.7 nm. Results are from Monte Carlosimulations of a system with N = 15625 water molecules. Thepressure P is calculated in GPa, the temperature T is rescaledwith respect to the liquid–gas critical point temperature TC ofthe water model, the density ρ = N/V is rescaled by the volumev0 = r0δ 0.059 nm3, where r0 = 2.9Å is the van der Waalsdistance. a Isobars at high T, around the liquid–gas critical pointC, marked by a full large circle. At P < PC 0.18 GPa a dis-continuity in the isobars denotes the coexistence of the gas (atρv0 < 0.5) with the liquid (at ρv0 > 0.5). In the circled regionfor 0.02 GPa≤ P ≤ 0.06 GPa the isobars reach the temperatureof maximum density at ρv0 0.65 and T/TC 0.35. b At muchlower T, the isobars display another discontinuity in density thatis evident at P = 0.18 GPa, and smoothly disappears, within thecalculation error, for P < 0.13 GPa. This discontinuity denotesthe coexistence of a liquid at higher density with a liquid atlower density. Where the discontinuity disappears, the systemdisplays a liquid–liquid critical point C′ (large open circle, withPC′ 0.13 GPa and TC′ 180 K). In both panels, errors are ofthe order of the size of symbols

show that the lines of constant-D qualitatively resemblethe melting line of bulk water (de los SAntos andFranzese, unpublished data).30

Water Monolayer Compared to Protein HydrationWater

At low T the diffusion constant largely decreasesand, eventually, the monolayer becomes subdiffusive,i.e., the mean square displacement of water moleculesnever reaches the diffusive regime (de los SAntos andFranzese, unpublished data).30 This property has beenobserved experimentally below 320 K by neutron scat-tering in a monolayer of water hydrating a myoglobinsurface at low hydration level (h = 0.35 g H2O/g ofprotein), corresponding to a number of water moleculessufficient to cover the entire protein surface.54

Franzese and de los Santos in 200929 found thata water monolayer partially hydrating a hydrophobicsurface, described by the model considered here, woulddisplay a very slow dynamics for the HBs at low P andT. In these conditions, the HB correlation function C(t)that quantifies how much the HBs are correlated intime, is almost not changing in time, showing that thewater dynamics is completely frozen. Hence, water isin its glassy state at low T and low P, consistent withthe observed freezing of the incoherent intermediatescattering function of water hydrating myoglobin athydration level h = 0.34 g H2O/g of protein at about180 K.55 Glassy water is observed also in bulk, butbelow 150 K.

This slowing down of the dynamics is well under-stood in our model where, at low T and low P, thenumber of HBs largely increases when the T decreases.This progressive building up of the HB network trapsthe water molecules in a percolating network of HBsthat leaves small areas of the hydrophobic surface com-pletely dehydrated.

At higher P the decreased number of HB allows tothe small dry cavities to slowly equilibrate, due to large-scale rearrangements of the HBs and leading to partialdehydration of the surface. In this case water can slowlyflow on the surface because the high pressure reducesthe volume per molecules and disfavors the formationof HBs. Hence, the HB network builds up at lower Twith respect to the low pressure condition.29 In this casethe long time behavior of C(t) is well described by astretched exponential function

C(t) = C0 exp[− (t/τ)β

](6)

where C0, τ and β ≤ 1 are fitting constant. For β = 1the function is exponential, and the more stretched isthe function, the smaller is the exponent 0 < β ≤ 1.

192 Food Biophysics (2011) 6:186–198

Franzese and de los Santos29 predicted that at lowP, by increasing P, (1) the time needed for the HBsto decorrelate decreases, i.e., water can relax morerapidly, and (2) the β exponent decreases going fromβ = 0.8 to β = 0.4 for a pressure approaching a charac-teristic value PC′ . We will discuss further about PC′ inthe following. By increasing the pressure even further,for P > PC′ , the HB correlation function relaxes fasterand the exponent becomes β = 1, i.e., C(t) becomes anexponential function.

The prediction about β is consistent with the experi-mental findings of Settles and Doster showing that, forwater hydrating myoglobin at low hydration level, theincoherent intermediate scattering function at large Qvector, i.e., the dynamics of density fluctuations at shortdistance, are well described by stretched exponentialfunctions with β varying between 0.4 and 0.3 at 320 K.54

We observe that the theoretical lower limit for β isexpected to be 1/356,57 and that 1 − β is a measure ofthe heterogeneity in the system. Therefore, the predic-tion29 that the stretching parameter β approaches itssmallest possible value when the pressure tends to PC′

implies that the water monolayer reaches its maximumin heterogeneity at PC′ . In the following we will discussfurther this point.

Thermodynamics at Very Low Temperature

To understand the origin of the heterogeneity at PC′ ,we recall here that the properties of water are consis-tent with theories that propose different mechanismsand different phase behaviors at very low temperature(approximately 150 K≤ T ≤ 200 K). At these tempera-tures, supercooled bulk water forms ice, while confinedwater in appropriate conditions can be kept liquid.58

The different theories can be summarized in four possi-ble scenarios for the P–T phase diagram.

1. In the stability limit scenario,59 it is hypothesizedthat the limits of stability of superheated-stretchedliquid water changes its slope in the P–T planefrom positive at high T, to negative at low T andnegative P, giving as a consequence the anom-alous increase at low T of quantities such as KT ,CP, and αP (thermal expansion coefficient).

2. In the liquid–liquid critical point scenario,60 it ishypothesized the existence of a first-order phasetransition line in the supercooled liquid region,with negative slope in the P–T plane and termi-nating in a critical point C′. This phase transitionseparates two liquid phases, both metastable withrespect to the crystal phases: one with low density,resembling the LDA disordered ice, and one with

high density, resembling the HDA disordered ice.The low-density-liquid (LDL), high-density-liquid(HDL) critical point C′ has been predicted at pos-itive P60 or negative P, 61 depending on the watermodel adopted in simulations. The anomalies ofwater are the consequence of approaching C′.

3. In the singularity–free scenario,62 it is hypothe-sized that the HBs have no cooperativity. In thiscase it is shown that the anomalous increase ofKT , CP and αP is a consequence of the low-Tanticorrelation between volume and entropy, alsoresponsible for negative slope in the P–T plane ofthe line of TMD.

4. In the critical–point free (CPF) scenario,63 it is hy-pothesized that the LDL-HDL first–order phasetransition line extends to P < 0, reaching the su-perheated limit of stability of liquid water andwith no critical point. As a consequence the HDLhas a superheated-stretched limit of stability simi-lar to that predicted in scenario (1).

As discussed by Stokely et al., 33 it is still unclearwhich of the scenarios best describes water, becausethere is no definitive experimental test. In Stokelyet al., 33 the same tractable model presented in theprevious section is analyzed by means of theoreticalcalculations and numerical simulations, showing thatthe four scenarios (1)–(4) may be mapped in the spaceof the parameters J and Jσ , representing the strengthof the HB directional component and the strength ofthe HB cooperative component, respectively. The re-lation Jσ /J = 1/10 discussed at the end of the previ-ous section, and based on estimates from experimentaldata, supports the prediction of a liquid–liquid criticalpoint C′ at positive pressure for supercooled water(Figure 2b). The model allows to distinguish the LDLphase and the HDL phase by their different densitiesand characteristic energies. The presence of two max-ima in the distributions of these quantities marks thecoexistence of the two phases and corresponds to theoccurrence of two minima in the free energy of thesystem (Figure 3).

At the critical point C′ the cooperativity of the HBsis maximum. Hence, cooperative rearrangements of theHB are necessary to allow the relaxation of the dynam-ics. These rearrangements occur on different length-scales, each associated with a different time-scale. Asa consequence a single time-scale cannot be definedfor the dynamics, resulting in a stretched decay of thecorrelation function, i.e., in heterogeneous dynamics.It is, therefore, C′ the origin of the heterogeneity de-scribed in the previous section and occurring at PC′ ofthe liquid–liquid critical point.

Food Biophysics (2011) 6:186–198 193

(b)(a)

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100000

120000

-1.85 -1.8 -1.75 -1.7 -1.65 -1.6

RESCALED ENERGY

NUMBER OF EVENTS

Fig. 3 The histogram of rescaled density ρv0 and rescaled energy[U + HHB + Hcoop]/ε above (a, b) and below (c, d) the liquid–liquid critical point C′. In the one-phase region, density histogram(a) and energy histogram (b) display one single maximum. At theliquid–liquid phase separation, density histogram (c) and energyhistogram (d) display two maxima separated by a minimum,

corresponding to two coexisting phases with different densitiesand energies. Calculations are from Monte Carlo simulations of amonolayer with N = 15,625 water molecules at T = 175.1 K andP = 0.12 GPa (a, b), and T = 173.8 K and P = 0.13 GPa (c, d).The liquid–liquid critical point C′ is estimated at TC′ 180 K andPC′ 0.13 GPa

Hydration Percolation

It possible to study the cooperative regions, and theirlength-scales, by using a geometrical approach based onthe concept of correlated percolation. 31,64 We define acluster of correlated water molecules as described in thefollowing steps.

• The first step consists in including in the cluster oneof the bonding indices σi, j of a randomly selectedwater molecule i in the hydration shell.

• The second step is to add to the cluster anotherbonding index of the same molecule i withprobability psame ≡ min {1, 1 − exp[−Jσ /(kBT)]}(where kB is the Boltzmann constant),or the facing bonding index σ j,i of the

nearest neighbor molecule j with probabilitypfacing ≡ min

{1, 1 − exp[−J′/(kBT)]} where

J′ ≡ J − PvHB. In this expression, J′ is theP–dependent effective coupling between twofacing indices as results from the enthalpyU + HHB + Hcoop + PV of the system Eqs. 1–5. The quantity J′ can be positive or negativedepending on P. If J′ > 0, a bonding index can beadded, with probability pfacing, to the cluster only ifit is in the same state as the other indices alreadyin the cluster. Instead, if J′ < 0, the index can beadded only if it is in a different state with respect tothe index to whom it will be connected.

• The third step is to randomly select a bonding indexon the border of the cluster and pick at randomone of the indices on the same molecule, or the

194 Food Biophysics (2011) 6:186–198

facing index on a molecule, that is not already inthe cluster, and to include it in the cluster withprobability psame or pfacing, respectively. This step isrepeated until all the possible directions of growthfor the cluster have been considered.

The resulting cluster statistically represents the re-gion of correlated HBs, as can be shown,65 and itscharacteristic linear size statistically corresponds to thecorrelation length of the water molecules. Therefore,in the vicinity of the liquid–liquid critical point, wherethe correlation length increases, it is possible to ob-serve that the size of the clusters of correlated watermolecules increases. At the liquid–liquid critical pointthe correlation length diverges and a cluster of cor-related water molecules spans (percolates) the wholemonolayer. General results on correlated percolationtheory allow to find mathematical relations betweenthermodynamics quantities and percolation quantities.In particular, it can been shown that the mean sizeof the clusters defined above diverges with the samepower-law exponent as the compressibility of water andthat the distribution of number n(s) of finite cluster ofsize s per water molecule is exponential far from theliquid-liquid critical point C′, while follows a power lawwith exponent τ near C′. From general considerations,it is possible to show that τ = 1 + d/DF where d = 2is the effective dimensionality of the monolayer andDF is the fractal dimension of the clusters.66 Prelim-inary estimate of τ 2 suggests that the clusters ofcorrelated water molecules are compact with DF 2(Figure 4; Bianco and Franzese, unpublished data).Since the compressibility is proportional to the density

fluctuations, the clusters allows for a geometrical analy-sis of the diverging density fluctuations near the liquid–liquid critical point.32

Hydrogen Bonds Dynamics on Hydrated Protein

The density fluctuations are observable also far fromthe liquid–liquid critical point C′. In particular, theycan be observed along a line in the P–T phase dia-gram that emanates from C′ into the one-phase regionand marks the maxima of the correlation length. Thisline has been named after Widom2,67,68 and can becharacterized in the study of hydrated protein. Forexample, Kumar et al. 28 used the tractable model de-scribed above to investigate the case of a percolatingmonolayer of water molecules adsorbed on a proteinsurface, with hydration level about h 0.4 g H2O/g dryprotein. Under these conditions the protein is immobileand inhibits the ice crystallization because it forces thewater molecules out of the positions corresponding toa crystal configurations. The authors studied the HBdynamics, regardless if the HBs are formed within thewater molecules or with the surface.

They first locate the Widom line, by Monte Carlosimulations, and observed that it corresponds to thelocus where there is the largest change in the numberof HBs. At P and T above the Widom line, water hasa few HBs, while at P and T below the Widom line,it has a well-developed network of HBs. This change ofstructure is reflected by a maxima in constant-P specificheat along the Widom line. The simulations also revealthe presence of a dynamic crossover in the HB corre-lation function C(t) when the Widom line is crossed

110

-5

10-4

10-3

10-2

10-1

Ave

rage

Num

ber

n(s)

1

Cluster Size s

10-5

10-4

10-3

10-2

10-1

110

-5

10-4

10-3

10-2

10-1(a) )c()b(

101001010

Fig. 4 Histograms of number n(s) of clusters of size s near theliquid–liquid critical point C′ for a monolayer with N = 25,600water molecules. For T < TC′ (a) and T > TC′ (c) the distrib-ution of n(s) is exponential, while for T TC′ (b) it becomes

approximately a power law, whose leading term is n(s) sτ withan estimate τ 2. Lines are fits of the histogram with exponentialfunctions in a and c and with n(s) sτ in (b)

Food Biophysics (2011) 6:186–198 195

at constant P. Using mean field theory and makingthe hypothesis that the dynamics is dominated by therearrangements of the HBs, the authors calculate theactivation energy for the relaxation of the system andshow that it gives the same relaxation time calculatedby Monte Carlo simulations.35 The proposed mecha-nism consists in breaking a HB that does not fit intothe tetrahedral arrangement and reorient the moleculeto optimize locally the tetrahedral configuration. Thismechanism has been confirmed also by simulations ofthe hydration shells of elastin-like and collagen-likepeptides.18 In particular, Kumar et al. 35 predict (1) howthis barrier is affected by the variation of P, (2) howthe crossover T is affected by the variation of P, and(3) that for any P the correlation time at the crossoverT is the same (isochronic crossover). Experiments withhydrated lysozyme, spanning a range of pressures goingfrom ambient pressure up to 1,600 bar, performed bythe group of Chen at MIT, have confirmed these threepredictions.34,69

More recent analysis of lysozyme proteins at a lowerhydration level (h = 0.3 g H2O/g dry protein) revealsanother surprising results. 70 At this very low hydration,dielectric spectroscopy, probing the proton relaxation,displays that at ambient P not only there is at about250 K the dynamic crossover described above, but alsoanother crossover at about 180 K. The study of thetractable model presented here associates this lower-Tcrossover to the saturation of the cooperative orderingof the HB network. Specifically, the HBs rearrangeto maximize the number of tetrahedral orientationsamong the bonds. This ordering is marked by a sec-ond specific heat maxima found in the model at about180 K. Therefore, summarizing, the model predicts abroad specific heat maximum at about 250 K, due to thesaturation of a macroscopic HB network, and a sharperspecific heat maximum at about 180 K, due to thesaturation of the tetrahedral ordering of the HBs. Byincreasing the pressure of the hydrated protein, Mazzaet al. predict that these two maxima merge and thendiverge at the liquid–liquid critical point C′. 71

Discussion: Implications in Food Science

The presence of a liquid–liquid critical point C′ at lowT in confined water could be an undesirable propertyfor the storage of frozen food and, more in general, bi-ological cells. This is because in the vicinity of a criticalpoint between two liquid phases, both metastable withrespect to the crystal phase, large density fluctuationsoccur. These enhanced fluctuations would drastically

changes the pathway for the formation of a crystalnucleus, because the crystal would form from the densefluid, instead that from the low density fluid. As aconsequence, there would be a strong reduction of thecrystal nucleation free-energy barrier and, hence, anincrease by many orders of magnitude of the crystalliza-tion rate, as theoretically predicted by ten Wolde andFrenkel.72

Under this conditions, the enhanced formation ofice could destroy the cells as a consequence of theincrease of volume of ice with respect to the liquidwater.3,4 Hence, the best way to preserve the cellswould be to freeze them at a T that is far away from theliquid–liquid critical temperature. Nevertheless, furtheranalysis show that the situation is even more com-plex. Indeed, simulations of a model with a metastableliquid–liquid critical point display enhanced crystalliza-tion rate not only in the vicinity of the critical point,but in the vicinity of the whole region of liquid–liquidcoexistence73 and possibly also in the one-phase regionabove the critical point along the Widom line.2 Work isin progress to elucidate these implications.

The evaluation of these effects is extremely im-portant to properly include the interaction with theconfining surfaces. We are presently studying how toincorporate these effects in our tractable model. A firststep in this direction is to include a description of thehydrophobic effect. Frank and Evans74 and Silversteinet al. 75 proposed that supercooled water forms highlystructured “ice-like” regions in the hydration shell ofnonpolar solutes. Stillinger76 proposed that HBs in thehydration shell are not significantly perturbed nearsmall hydrophobic solutes, while the HB network isstrongly affected by hydrophobic particles with sizeabove a characteristic value. Chandler estimated thisvalue of the order of 1 nm on the basis of free en-ergy calculations.77 Muller explained the vibrationaland NMR spectroscopy results by suggesting enthalpicstrengthening of the hydration HBs with a simultane-ous entropy increase in the hydration shell. 78 We arepresently including the enthalpic strengthening in ourmodel, and properly accounting for the entropy in-crease for the study of water in confined by hydropho-bic nanoparticles. Our preliminary results79 show asurprising change of thermodynamic fluctuations at lowT, whose implication is a large decrease of compress-ibility also at very low nanoparticles concentration. Thisfinding suggests that adding hydrophobic particles atlow concentration in organic solutions would decreasethe density fluctuations and the formation of ice.

Further confirmation of the validity of our assump-tion about the hydrophobic effect comes from an-other study that we are performing to establish if our

196 Food Biophysics (2011) 6:186–198

(a) (b) (c) (d) (e)

Fig. 5 Folding and unfolding of a coarse-grained protein sus-pended in water at different temperatures T and high pressuresP. Typical configurations in a system in two dimensions arerepresented for the sake of schematic description of the process.The protein is represented as a fully hydrophobic chain (inwhite), surrounded by water molecules (turquoise background).a At high pressure and high T, the protein is unfolded andthe surrounding water has only a few hydrogen bonds (HBs)formed, represented as colored sticks. Different colors of the

HBs correspond to different relative orientations of the HBs. bAt the same pressure but lower T, the protein start to fold in amolten globule state. c At lower T the protein folds, while thesurrounded water has a large number of HBs. d At much lowerT we observe cold denaturation of the protein when the numberof water HBs is largely reduced due to the combined action of Tand P. e At higher P the denaturation effect occurs at higher T,and we observe the protein only in open configurations

tractable model is able to describe the stability of pro-teins with respect to changes of temperature and pres-sure. In this study we consider how the hydrophobicinteraction with water of a coarse-grained protein in-duces hot denaturation, folding, cold denaturation andpressure denaturation. Our preliminary results (Iskrovand Franzese, unpublished data) display a region ofstable folded configurations that, in the P–T phasediagram, has the same qualitative features of the exper-imental stability diagram of myoglobin80 (Figure 5).

Conclusions

We have introduced a tractable model for a watermonolayer hydrating surfaces of proteins and, moregenerally, for confined water. The model includes HBcooperativity and elucidates how the many-body com-ponent of the HB is important to understand the low-Tbehavior of water. In particular, parameters estimatedfrom the experiments suggest the occurrence of a criti-cal point at T 180 K and P 0.13 GPa at the end ofa first-order coexistence line between two liquids withdifferent densities.23,24,33

The model shows that the liquid–liquid critical pointaffects the low-T dehydration of a hydrophobic sur-face.29 The cooperativity of water induces dynamic het-erogeneities that reach their maximum when the clusterof correlated HBs percolates (Bianco and Franzese,unpublished data).31,32

This heterogeneous dynamic behavior is revealedby a strongly non-exponential relaxation of the HBdynamics and by a subdiffusive motion of the water

molecules in the hydration shell, 29 as observed in hy-drated proteins at low T. 54,55

The model predicts that, at low protein hydration,shell water should be characterized by two structuraltransitions. One associated to the macroscopic formationof HBs,28,35,81 occurring at about 250 K for low-hydrated lysozyme, and another associated to the tetra-hedral reordering of the HBs, at about 180 K.70 Thesetwo structural changes are at the origin of two dynamiccrossovers: the one at higher T has been observed byQUENS experiments on hydrated lysozyme,34,69 andboth have been measured by dielectric spectroscopy onhydrated lysozyme.70 The pressure behavior of thesecrossover is consistent with the presence of the liquid–liquid critical point C′ at high P and low T. 34,71

The large increase of density fluctuations in thevicinity of the critical point C′ is expected to enhancethe water crystallization process. As a consequence ofthe expansion of ice, this process could destroy biologi-cal structures in a crowed environment, as for examplein food stored at low temperature.3,4 It is, therefore,relevant to understand if this process is affected byconfinement or if it could be controlled. Our prelimi-nary results79 show that hydrophobic confinement has astrong effect of the thermodynamics of water, suppress-ing the density fluctuations associated to the criticalpoint C′. This result suggests that by dissolving hy-drophobic nanoparticles at low concentration it couldbe possible to control the water compressibility and theformation of large crystals.

These predictions are based on a modelization of thehydrophobic interaction that is able to reproduce a sta-bility diagram for a coarse-grained protein. In particu-lar, the model shows that the protein cold denaturation

Food Biophysics (2011) 6:186–198 197

and the pressure denaturation can be explained as aconsequence of the strengthening of interfacial water–water HBs at the hydrophobic interface and the properaccount of the entropy change due to the presence ofthe interface (Iskrov and Franzese, unpublished data).

Work is in progress to include other features inthe model and to use it to make other predictions indifferent contexts. For example, we are extending themodel in such a way to describe the ice formation andanalyze how the crystallization process is affected byinterfaces. We are also developing the generalization inthree dimensions to allow the study of many layers ofhydration water and to extend the investigation to thebulk case. In this way, our research about protein sta-bility will be performed also in bulk water. All togetherthese generalizations will allow us to explore situationsof possible interest in food science and biology, as forexample the effect on water structure and dynamics ofpreservative agents such as trehalose, or antifreeze pro-teins, or cryoprotectants such as glycerol or dimethylsulphoxide.

Acknowledgements G. Franzese thanks M.C. Barbosa, S.V.Buldyrev, F. Bruni, S.-H. Chen, A. Hernando-Martínez, P.Kumar, G. Malescio, F. Mallamace, M.I. Marqués, M.G. Mazza,A.B. de Oliveira, S. Pagnotta, F. de los Santos, H.E. Stanley,K. Stokely, E.G. Strekalova, and P. Vilaseca for collaborationand helpful discussions, and the Spanish Ministerio de Cienciae Innovación Grants FIS2009-10210 (co-financed FEDER) forsupport.

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