Molecular dynamics study of the hydration structure of an antigen

13864

J. Phys. Chem. 1993, 97, 13864-13876

Molecular Dynamics Study of the Hydration Structure of an Antigen-Antibody Complex Fabienne Alary,' Jean Dump, and Yves-Henri Sanejouand Luboratoire de Physique Quantique,f IRSAMC, 118 route de Narbonne, 31062 Toulouse, France Received: May 17, 1993; In Final Form: October 4, 1993'

Molecular dynamics simulations of the hen egg-white lysozymeFab D1.3 complex were performed, starting from the X-ray crystallographic coordinates obtained by Fischmann et al. [J. Biol. Chem. 1991, 266, 12915129201, and including counterions and explicit random water molecules. Both the crystal state and the complex in solution were studied, the former as the asymmetric unit cell in interaction with all its neighbors, and the latter with a new model (the "egg") consisting of a truncated ellipsoid containing the complex and solvent molecules, a first shell with additional solvent, and two further shells with virtual solvent molecules derived from those of the first shell by a geometrical correspondence. The hydration structure of the complex is analyzed from the data obtained after equilibration and fluctuation dynamics of these two systems. The water density around the protein is found to increase, up to a maximum of 1.5 for thecomplex in solution, so that the integrated density largely exceeds unity. Therefore additional water molecules were steadily included into the "egg" model. In the crystal cell model an average density of unity was conserved. Detailed analyses are given of the pair correlation functions, coordination numbers, local water density, and orientations of water molecules. Two hydration shells are observed around the complex in solution, the inner one with molecular orientations very dependent on the local character of the protein surface (either nonpolar, or positively or negatively charged), whereas the second shell which extends continuously toward the bulk is essentially homogeneous, apart from slight residual orientational preferences of the dipole moments in the first few A of this shell when the closest protein atom is positively or negatively charged. Comparisons of the root-mean-square fluctuations of the protein backbone atoms from the simulations of the complex in the crystal state and in solution with experimental B factors from the X-ray crystallographers show the importance of a correct description of the water density around the protein. It is argued that the absence in the crystal cell of a bulklike aqueous phase may lead to a higher mobility of the protein chains in the crystal than in solution.

Introduction The main characters of protein hydration shells have been known for long (for recent reviews, see ref 4). Data from various experimental techniques allow us to distinguish roughly two classes of water molecules related toa protein surface (letting alone buried molecules): (i) ca. lo2 tightly bound molecules, with residence times over 10-lO s, and (ii) one or two5 layers of molecules with translational and rotational relaxation times larger than in pure water by 1 or 2 orders of magnitude only and thus being in the 10-ll-lO-lo -s range. Physically of course the inner layer includes all class (i) molecules. Whether additional layers of water molecules, with physical properties identical to those of bulk water, still affect protein function and, possibly, dynamics, is not clear (see, e.g., Figure 1, curve g of ref 2). We shall briefly review the available evidence for classes i and ii. (i) From dielectric measurements Bone and Pethig6 found an amount of tightly bound water ranging in a set of proteins from 0.04 to 0.08 g/g of protein, similar in order of magnitude to that required, from infrared spectroscopy and other for hydration of the polar groups in lysozyme. High-resolution X-ray crystallography typically provides the locations of lo2 tightly bound water molecules, whose binding sites therefore may be considered as being well defined (see ref 9 for a recent overview, and ref 10 for water bound to lysozyme crystals). Neutron diffraction is a more recent technique, which was developed in particular by Cheng and Schoenbord to obtain accurate data on hydrogen bonding of water molecules to specific residues of carbon monoxymyoglobin. Proton nuclear magnetic resonance (NMR) was used with the spin-echotechnique by Bourret and Parelloll who found a number ~~

t Universit6 Paul Sabatier and Unit6 Assmi& 505 of the C.N.R.S. e Abstract published in Aduonce ACS Absiracrs, December 1, 1993.

of tightly bound water molecules in lysozyme very close to that obtained by X-ray diffraction. However in a high-resolution nuclear Overhauser effect (NOE) NMR study of the bovine pancreatic trypsin inhibitor, including both laboratory-frame (NOESY) and rotating-frame (ROESY) experiments, Otting, Liepinsch and WiithrichIz showed that many water molecules have residence times of at least a few tenths of a nanosecond in solution although they are not seen by X-ray diffraction in the crystal state, and they pointed out that in the latter technique the reference sites are space fixed, whereas in NMR they are defined by protein atom locations. Furthermore, Qian et al.I3 observed that water molecules located at the interface of a protein-DNA complex have longer residence times (> 1 ns) than those at the protein and DNA surfaces. Using the ROESY IsN-lH multiple quantum coherence technique, Clore, Gronenborn, and cow o r k e r ~also ' ~ directly observed bound water moleculeswith over nanosecond lifetimes, in particular on the surface of the immunoglobulin binding domain of streptococcal protein G in solution. Finally, from the analysis of 2H NMR longitudinal ( T I )relaxation times, associated with a theoretical estimate of the rate constant for hydrogen bond rupture, Koenig et al.I5proposed to distinguish threeclassesof interfacialwater molecules in both protein solutions and in vivo tissues: a first class with very long residence times ( ~ 1 0 - 6s), corresponding to water molecules held by multiple hydrogen bonds (typically 4) and representing ca. 1% of a hydration layer, a second class with residence times around 3 X 1@ l o s, corresponding to molecules attached by two hydrogen bonds and representing 10-20% of a hydration layer [those two classes constituting together our class i], and a third class with shorter residence times, corresponding to our class ii, now to be discussed. (ii) Using calorimetric data Rupley et aL2gave a value of 0.38 g H2O/g of protein as the amount necessary for full hydration of lysozyme, defined by the equality between the heat capacity of added water and that of bulkwater. This value was considered

0022-3654/93/2097- 13864%04.00/0 0 1993 American Chemical Society

An Antigen-Antibody Complex by these authors as representative of a monolayer of water. A very similar amount of water (0.3 g/g) was determined, also for full hydration of lysozyme, by cross-polarization magic-angle spinning I3CNMR spectroscopyby Gregory et al. l 6 In an earlier study of hydrated lysozyme by pulsed Kerr dielectric relaxation Bourret and Parelloll had obtained the dimension of the hydrodynamically equivalent ellipsoid which corresponded to a water monolayer of 3.2-A thickness. A careful analysis of scattering densities and liquidity factors in neutron diffraction by carbon monoxymyoglobin crystals led Cheng and Schoenborns to distinguish two hydration layers, the inner one with low mobility and located at ca. 2.3 A from the hydrogen atoms of the protein surface, and the outer one located at 3.74.0 A. The observed variation of scattering density with distance from the protein indicated that the mass density of water Very recently was higher in the inner layer than further Bellissent-Funel et al.I7 found by neutron powder diffraction of the fully deuterated protein C-phycocyanin that at a high level of hydration (0.365 g of DzO g of protein) a first hydration shell appears at a distance of 3.5 between protein and water atoms. Numerous proton NMR experiments showed the reduced mobility of water molecules surrounding the protein.18-z0As early as in 1969 Kuntz et a1.I8 reported for various proteins at -35 OC reduced-mobility water amounts ranging from 0.31 to 0.45 g per g of protein. Halle et al.zOfound around two water layers with reorientation time about 20 ps, as compared to 2.40 ps for pure water. Reorientation correlation times of 50-100 ps at room temperature were observed by Usha et a1.,21,zz h i n g ZH NMR, for the water of various proteins in the crystal state. In an earlier dynamical study by NMR relaxation Bryant and Shirley238 controversially23b questioned the characterization of protein hydration structure as “icelike”(first by Frank and Evans in 194Y4and later by other authors), noting that the translational motion of first-shell water molecules, although much slower than in the bulk, is still faster by several orders of magnitude than in the solid Using electron spin probes, Polnaszek and Bryantzs measured a translational diffusion rate of (3 f 1) X 10-6 cm2 s-l for water within the first 10 A of the surface of bovine serum albumine, compared with (18 f 4) X 10-6 cm2 s-l in pure water. More recently Piculell and Halle26 showed that the lateral and radial diffusivitiesofwater moleculeslyingwithinca. lOAof thesurface of lysozymeor of haemocyanin are reduced by 1-2 and 2-3 orders of magnitude, respectively,compared with bulk water. In a recent work Bellissent-Funelet aLZ7reported quasielastic neutron data on C-phycocyanin in vivo which they interpreted, using Volino and Dianouxz*jump diffusion model, as evidence for molecules with residence times &lo times larger than those of bulk water.27 Detailed experimental information on the hydration structure of various solutes have recently appeared. Heteronuclear Overhauser effect measurements by Canet et alez9yielded an average distance between water molecules and charged micellar surfaces. In an experiment by neutron specular reflection Lee et al.30found oscillationsin water density near a surfactant layer. Using neutron diffraction Langan et al.31 observed the first hydration layer of A-DNA, well ordered along the major groove (with the same 1 1-fold symmetry as the double helix itself) in contrast with the “hydration spine” in the minor grooveof B-DNA earlier observed by X-ray crystall~graphy.~~ Other data from neutron diffraction showed various hydration structures around small solutes;33 however in studies of urea,34tetramethylammonium chloride,3s and dimethyl sulfoxide36solutions. Finney, Soper, et al.34pointed out that the degree of structural ordering of the observed water cage, as it appears, e.g., from the H-H pair correlation functions, is no more and no less than in pure water. These conclusions may be related to microwave dielectric studies: no changes were observed in the relaxation properties of water in the 10-I I-1 0-lo-s range on introduction as a solute of glucose, which can replace

d

The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 13865 a hexagonal cluster of HzO moleculeswithout structural changes, whereas other solutes such as L-ascorbic acid which break that cluster give rise to specific relaxation peaks.37 Finally, small-angle neutron scattering was recently shown by Bezzabotnov et aL3*to provide detailed information on water molecules around tetramethylurea. A number of molecular dynamics (MD) or Monte Carlo (MC) simulationshave been performed on the structure of water around a c a ~ i t y ,or~ around ~ - ~ ~ ions4143and various hydrophobic42-44.45 or h y d r o p h i l i ~ solutes, ~ , ~ ~ ~including small peptides49 and micelles or related me so phase^,^^*^^ or between hydrophobicSz-S7 or hydrophilics8walls, or within a hydrophobic spherical boundor near the surfaces of lipid bilayers,60-63 ion channels,64 and nucleic a ~ i d s . ~ ~The , ~ 5features around nonpolar species (includinga hydrophilic nonpolar solute, 1,4dioxaneM)were often characterized as clathrate-like structure^,^^ and those near a hydrophobic wall as ice-Mikes6ones. As reminded above, this analogy is pertinent as regards average structure, but not Conversely, as pointed out by Remerie et a1.,40selfdiffusional and reorientational slowing down of the solvent around a solute (or an empty cavity) is not necessarily associated with an increase in structure. Simulation of the solvent structurearound a protein was first performed by Hagler and using the MC method, with explicit water molecules. Later, considerable work has been devoted to MD simulations of proteins in their aqueous environment. Explicit handling of water molecules filling the free space of the crystal unit cell was pioneered by van Gunsteren et al.,67-72followed by other a ~ t h o r s . ~ ~Studies - ’ ~ with a protein immersed in the solvent in a rectangular,s0*67*70,76-92 or truncated octahedra1,69-93.94 or p r i s m a t i ~ box ~ ~ ,with ~ ~ periodic conditions are comparable to crystal cell models as regards the protocols used. An alternative method rests upon a partition of the space surrounding the protein into an essential part, with explicit water molecules, and a buffer zone handled by Langevin dynamics with stochasticboundaryconditi~ns.~~-~~~ Finally, several authors used a model without any boundary c o n d i t i ~ n , ~sometimes ~ J ~ ~ restraining the motion of the water molecules by constraints applied to their d i s p l a c e m e n t ~ , ~or~ even ~ J ~ ~not restraining them and therefore allowing the possibility for some water molecules to yleave” the mode1.106J07 Another wealth of results have been obtained by using continuous models for the solvent (for comprehensive reviews, see ref 108;see also refs 88 and 109). These studies,which provide important results on the behavior of the proteins in the aqueous phase, will not be further commented since they are out of the scope of the present paper. A few similar studies were carried out on protein complexes, with the solvent either handled explicitly’1° or as a continuous medium,IlI or both.Il2 Among the large number of papers on MD studies of proteins with explicit water molecules, several have been concerned with the structure of the hydration s h e 1 1 . 5 0 ~ 7 4 ~ 7 6 ~ 7 9 ~ 8 0 ~ 8 z ~ g ~ ~ g 2 ~ ~ ~ The present work is part of an MD study of antigen-antibody recognition, which uses as a model case the hen eggwhite lysozyme (HEL)-Fab D l .3 complex, whose atomic coordinates were determined by X-ray crystallography by Fischmann, Poljak, and their colleaguesof Institut Pasteur in Paris.Il3 An Fab fragment is the part of an immunoglobulin containing a full light chain (variable and constant domains), and the variable domain and the first constant domain of the heavy chain, linked together by disulfide bonds and nonbonded interactions. Both variable domains vary among antibodies, in particular the three distinct small regions of each chain named complementarity-determining regions (CDR), which are responsible for antigen recognition. The Fab D1.3 is a protein with 432 residues. Starting from Fischmann et ala’s data113 we successively developed models of the complex in the crystal state1I4and in

Alary et al.

13866 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993

TABLE I: Partial Charges and Lennard-Jones Potential Parameters Used for Water-Solute and Water-Water Pair Interactions’ Lmnard-Jonespotential parameters Charge (elementary water-solute pairs water-water pairs atom charge units) e (kcal/mol) u (A) c (kcal/mol) u (A) 0

H

-0.834b*c 0.417b*c

0.1591d 0.0498d

2.8509d 1.4254d

0.1521b*c 3.1506b3‘ 0.04598‘ 0.4oooC

The c and u parameters of the Lennard-Jonespotential V(r)= 4c[ ( u / r)I2 - ( ~ / r )are ~ ]derived for each atom pair from the usual rules CAB ( ~ A c B ) ’UAB / ~ , = (UA erence 117.

+ UB)/2.

Lysozyme

FabD1.3

Reference 119. Reference 120. Ref-

solution, both with explicit water molecules and counterions. We paid a large attention to the behavior of the water molecules, partly as a check of our protocols and partly for its own interest. This complex indeed is an excellent test case since Poljak et al. showed on the analogous HEL-FV D1.3 complex that water molecules play a significant role in the binding of the partners to each other.’” On another hand a thermodynamic analysis of the binding in the HEL-Fab D1.3 complex was performed by Novotnf et al. using empirical functions incorporating solvent effects.’I6

Figure 1. Illustrationof the “egg”model of the protein complex in solution, with either an ellipsoid or a truncated ellipsoid as the central zone.

Modelings of the Protein Complex in the Crystal State and in Solution All the codes we used for energy calculations, for dynamics, and for various sorting facilities were from the CHARMM-20 version of the CHARMM program117with a minor modification of the parameters,l14consisting in assigning a -0.01 charge to the aromatic CH groups of the phenylalanine and tyrosine residues and compensating it by a 0.06 charge on the &carbon. These changes were necessary for describing hydrogen bonds with an aromatic ring as an acceptor, an example of which occurs in our complex (H bond between Fab D1.3 light-chain Tyr 32 and lysozyme Gln 121); this kind of hydrogen bond was included in the crystallographers’ refinement program. A modified TIP3P model was used for H2O molecules, with parameters’’* listed in Table I. The “extended atom” model1I7 was used for the CH, CH2, and CH3 groups. A cutoff at 7.5 A was applied to nonbonding interactions, with a switchingfunction between 6.5 and 7.5 A for the van der Waals interactions and the useof “shifted electrostatics”for chargeinteractions.’” Nospecial potential function was used for the description of the hydrogen bonds. In modeling the 1ysozymeFab D 1.3complex both in the crystal and in solution we added, in front of the charged residues exposed to the solvent, counterions identical with those used by the crystallographers as a buffer, namely, HzPO4- and K+ ions. Specific HzO molecules were placed around the charged atoms, and thereafter we filled the free space with water molecules at a density of 0.996 g/cm3, a few of them however being replaced with additional ions accounting for the 0.1 M concentration of the buffer. In the crystal model we took into account all interactions between the asymmetric unit cell and its 26 neighbors. To describe the complexin solution, we introduced a new model, represented in Figure 1, which we designate as the “egg”. Its heart is the smallest possible truncated ellipsoid, with as axes the principal inertia axes of the protein complex, and containing this complex. In the present case the full lengths of the three axes of the ellipsoid are 154, 67, and 63 A, and after truncation respectively 113, 59, and 53 A, the principal one being almost exactly parallel to the line of centers of mass of the two partners. The interior of this truncated ellipsoid,which contains the protein complex and the immediately surrounding solvent, will be designated as zone 1. Around it, a larger pseudo-ellipsoidis built by adding a fixed distance Ar to all radii vectors joining the

Figure 2. Geometrical operations applied for deriving virtual solvent molecules in zones 3 and 4 from actual ones of zone 2: A‘, B’, and A”, B” are the images of atoms A, B in zones 3 and 4, respectively. The arrows schematically indicate how the velocity vectors transform.

center to the surface of the truncated ellipsoid (it may be shown algebraically that the figure thus built around an ellipsoid is slightlysmaller than the ellipsoid which would have been obtained by simply adding Ar to all half-axes of the inner one, and is tangent to it at its six tips). We first set Ar to 4.5 A, and later in the dynamics reduced it to 4 A. The shell between these two surfaces, which we name zone 2, was also filled with explicit solvent molecules. Zones 1 and 2 initially included 9154 HzO molecules and 16 ion pairs in the bulk (compared to 1864 and 4, respectively, in the crystal model114),in addition to the 5348 atoms of the protein complex and to the 40 HzP04- and 22 K+ used as counterions. Two more solvent shells (zones 3 and 4) were added to complete the simulation of the environment of the proteins. They were occupied by virtual water molecules and ions whose locations were derived from those of zone 2 by an inversion procedure illustrated in Figure 2. It is designed in such a way that (i) those virtual molecules in interaction with zone-2 “actual” ones are the images of actual molecules located far away from the latter ones, on the opposite side of the model, and (ii) the images in zone 3 of those zone-2 molecules located close to the protein complex are close to the outer surface of zone 3 and therefore experience little interaction with zones 1 and 2. The widths of image zones

An Antigen-Antibody Complex 3 and 4 were calculated for yielding the same average solvent density as that of zone 2. In this model, the interactions within image zones 3 and 4 are not included in the calculation of energy and forces. The nonbonding interaction energies between explicit and implicit zones are halved, and the corresponding forces are derived in such a way that the whole system keeps mathematically Hamiltonian. The detailed algebra of this model will be published elsewhere.lZi Since this "egg" model was original, we carefully checked that no significant artifact was introduced by this procedure. All major trends of the dynamics were found to be preserved if the weighting of the interactions with image zones 3 and 4 was altered. In particular, suppression of zone 4 from the calculations induced very small changes, and therefore we did not feel it necessary to add further outer zones. The list of nonbonded interactions in the whole model was updated every 50 fs, the time step of the dynamics being 1 fs. Simultaneously we updated the repartition of solvent molecules between zone 1 and zone 2; finally those having escaped from zone 2 to zone 3 (always by less than an angstrom, owing to the frequency of that updating), and whose images therefore had entered zone 2 on the opposite side of the model, were simply replaced with their images and vice versa. Main Features of Water Dynamics in the Models of the Crystal and of the Solution A first verification of the correctness of the dynamics and of the achievementof equilibrium in the solvent, after 32 ps of thermal equilibration and 30 ps of regular fluctuation dynamics at 300 K in the crystal, and correspondingly 77 and 16 ps, respectively, with the "egg" model, rested upon the distribution functions of the 0-0, 0-H, and H-H distances between atoms belonging to different water molecules, which will be discussed in detail in the next section. Thesefunctions, as well as the correlatedcoordinated numbers, provide information on short-range order around each water molecule, and the quality of their agreement with experimental data is essentially dependent on the quality of the model and potentials used for the water molecule. Long-range order, on the other hand, is typical of the system under study, and its correct description depends on the overall quality of the model. It includes the variation of water density around the protein complex and the orientational correlations of water molecules. This again will be discussed in detail in next section. In both simulations we started from a water density of 0.996 g/cm3 in the whole free space external to the van der Waals envelope of the protein. In the case of the complex in solution we observed during the dynamics simulation a drift of water toward the protein, with the formation of a layer with higher density, and accordingly a decrease of the water density far from the protein surface. We then looked for the appearance of cavities in the bulk. For this purpose we used a three-dimensional mesh with 1-8, steps and defined a cauity (in the same way as a "territory" in the Go game) as any set, contiguous along the three directions of space, of mesh nodes at points free from any atom at a distance under itsvander Waals radius. Wecarefully checked for the appearance of such cavities under conditions where the solvent is homogeneous. As expected, solvent fluctuations induce the presence of cavities but with volumes of no more than a few hundred In the "egg" model after the first 67 ps of equilibration dynamics there had appeared much larger cavities, the largest having a volume of 4 X lo4 R.3 This large cavity however was not local but long, thin, and sinuous, and in fact would not have appeared as a cavity if we had used, e.g., a 3-8, mesh. No such cauity appeared in the crystal simulation under the same conditions, although a slight migration of water toward the protein is not excluded.

The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 13867 1.55

A

0

E

1.16

0

\

U J

v

Q

0.77

0.39

Figure 3. Variation of water density as a function of the distance to the van der Waals envelope of the proteins (or counterions). The statistics is performed over the nodes of a three-dimensional 1-A grid. Solid line: protein complex in solution; dashed line: crystal.

We verified that no inwards drift of water occurred when the "egg" model was tested with pure water, which proved that the observed migration was not an artefact of the model. The drift of water molecules toward the protein, giving rise to a cavity at the periphery of the model, had been described by Brooks and KarplusloO(and earlier mentioned by Briinger et These authors used the stochastic boundary conditions. Later Komeiji et a1.85also reported cavity formation in an MD simulation using a box with periodic conditions. The present results thus demonstrate that this feature is not model dependent. To pursue the dynamics simulation in solution we then filled the cavities with water molecules and periodically added new molecules by the following procedure: we moved over the 'egg" a 22-8, cubic box containing H20 molecules (which had been previously equilibrated a t 300 K) by successive steps equal to the edge vectors of the box and released a water molecule into the model whenever the location of its 0 atom was distant from the van der Waals sphere of any other atom by more than 1.6 A, taken as the approximate van der Waals radius of an HzO molecule.122Becauseof thenatural density fluctuations of water, this procedure makes sense only up to the moment where the normal density of water would have been restored in the outermost parts of the model. Actually we stopped adding water before that moment, partly because the model became exceedingly large (over 10 000 water molecules), partly because our next goal was to simulate a forced separation of our protein complex,i23and we found it appropriate to do it inside a slightly underloaded environment. The fact that no cavity appeared in the crystal model, although the density far from the protein complex decreased during the simulation down to 0.8 as shown in Figure 3, is probably due to the fact that each water molecule "sees" protein atoms in all directions, and this would explain why most previous workers using a solvent box with periodic conditions did not report similar observations as ours. It must be pointed out that in our "egg" some water molecules of the explicit part of the model, in zone 2, are as far as 15 8,from any protein atom or counterion, whereas in the crystal this maximum distance is only 7 A, and the packing is so tight that 50% of the free space around the proteins lies within 1.6 8, from their van der Waals envelope. We did not apply the same "rehydration" procedure in the crystal as in the "egg" model. The reason was that we lack experimental data on the density of the water surrounding the proteins in the ~rystal.12~The crystallization process actually was achieved with poly(ethy1ene glycol) (PEG)as a precipitating agent.!I3 This very hydrophilic compound pumps water away from the protein surface, permitting contacts between neighboring

Alary et al.

13868 The Journal of Physical Chemistry, Vol. 97, No. S I , 1993 3.5

TABLE 11: Locations and Magnitudes of the Extrema in the Pair Correlation Functions and Coordination Numbers for Water Pairs 0-0 and 0-H present work experimental water crystal solution data (water)O 3.41 3.09 goo firstmax ht 4.19 2.75 2.75 2.875 2.825 location (A) 2.83 3.30 3.325 3.33 first min location (A) 3.4 NO0 gOH

first max ht

location (A) first min location (A) Non

5.08 1.243 1.85 2.45 1.94

5.8 2.53 1.85 2.35 2.25

4.8 1.72 1.85 2.30 2.06

3.0 2.5

2.0

8 cn

4.50 1.385 1.85 2.35 1.80

1.o

I

0.5

Reference 125. protein complexes to take place, thus facilitating crystallization, while otherwise the corresponding areas would be exposed to the solvent. We had not included PEG in our model, and it is difficult to assess what water density around the proteins would best simulate their actual environment in the crystal. We shall come back to this point in the last two sections of this paper.

2.0

1.6

Hydration Structure of the Protein Complex

1.2

6 cn

The structure of pure water or of a hydration shell is often visualized by a set of normalized atom-atom distance distributions or pair correlation functions (pcf) denoted by g,a(r,a) where the indexes a and B refer to a pair of atoms belonging to different molecules and rnB is their internuclear distance. The coordination number Nab of an a atom with the p atoms is related to the integral of g,a(rop) up to some radius rmin:

0.8

0.4

0.0

Nap = p p p g a d ( r )4 r r 2d r wherer,k is usually taken as the location of the minimum following the first maximum of g,a(r,s), and pa is the number density of /3 atoms (in our derivations pp was set to a water density of 1 g/cm3). N,o measures the average number of @ atoms in a sphere of radius r,,, drawn about a given a atom. Table I1 presents a comparison of our results, including a simulation of pure water using the same potential, with experimental data'25 as regards the locations and magnitudes of the extrema of the pcf s and the coordination numbers resulting from their integration up to the first minimum. The goo, gOH, gHH pcfsobtained fromour simulations and from Soper andPhillips'Iz5 neutron diffraction data on pure water are shown in Figures 4 and 5 for the egg and the crystal models, respectively. In all cases the shapes of our curves are similar to the experimental ones. Nevertheless there are some discrepancies concerning the heights and to a lesser extent the locations of the peaks. The goo(r00) obtain for the complex in solution with the egg model (g&) differ in two significant ways at large roo from the experimental data and from simulations of pure water either from earlier authors using the TIP3P model, or from the present work with the modified TIP3 potentia1118(see Table I and Figure 6 ) , whereas those obtained with the crystal model (gLo) are in closer agreement except for the peak intensity. First, as well-known the TIP3P model, either original1l9or as modified by Pettitt and Rossky,Iz6does not reproduce the second and third water shells,'27 even with our modified potential as shown in Figure 6,whereas they appear clearly in our gbo curve (Figure 4a). Second, our gbo second and third peaks are shifted toward longer distances as compared to the experimental peaks: 5.0 and 7.2 A instead of 4.5 and 6.7 A, respectively. On another hand the intensity of our first peak is too high especially in the crystal simulation (gbo, Figure sa). For the g&, pcfs the slight discrepancy with respect to the experimental peak height and the appearance of the second and third maxip:su * the inhomogeneity of water

1.5

1.6

0.8

0.4 0.0

L

a

1

I,,;

( , , , I , , , , , , ,

0

2

4

r(A)

6

l , , , j 8

10

Figure 4. Pair correlation functions (pcf) of water molecules. Solid line: simulationof the protein complexin solution. Dashed line: experimental results of Soper and Phillips'25for pure water. (a) g h , (b)g&, (c)

A". density (see Figure 3), as theoretically described by Ben-Naim,12s who showed that when the density of a Lennard-Jones fluid increases the location of the first peak in the pcf is essentially unchanged, but its height steadily increases while more and more pronounced new peaks emerge at higher distances. This was observed in theoretical studies of pure water by Pettitt and CalefIz7 using the reference interaction site model (RISM) with the TIPS potential'29 modified by Pettitt and Rossky:'26 increasing water density from 1.00 to 1.35 g/cm3 induced the emergence of the second peak in goo with its maximum located around 5.5 A. The similarity of the first part of the curves in Figure 4a indicates that the presence of the protein in the solution does not disrupt the first coordination shell of most water oxygen atoms.

The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 13869

An Antigen-Antibody Complex

4

-

3

-

4.00 3.00

en

2.00

-

4

2

t

1.00

1 -

0

1 tk 4

"

8 :

en

2

I i

1

2

3

4

5

6

-

~*

1

7

r(A)

en6 2 . 0 1.5

1.o

os

1 11 0

4

8

12

16

20

dtAl Figure 7. Variation of the coordination number No0 as a function of the distance to the van der Waals envelope of the proteins (or counterions). Solid line: protein complex in solution; dashed line: crystal. 0.8

0.4

0.0

Figure 5. Pair correlation functions (pfc) of water molecules. Solid line: simulationof the proteincomplexin thecrystal. Dashed line: experimental results of Soper and Phillips'2s for pure water. (a) g&, (b) (c) &H*

They see on average 4.8 neighboring 0 atoms (see Table 11), a value which also appears from Figure 7 which shows the behavior of the coordination number No0 as a function of the distance to the van der Waals envelope of the protein. Finally, the slight shift of the first peak in do0 toward shorter distances as compared to experimental data on pure water (Figure 4a) is probably again a consequenceof a higher average density of water in our simulation of the complex in solution. In the case of the crystal simulation, the higher intensity of the first peak of g&, (Figure Sa) would lead through eq 1 to an overestimation of the coordination number (No0 = 5.8, see Table 11) with respect to that directly recorded (4.2, see Figure 7),

which is closer to the experimental value for pure water (4.5125). The too high value of No0 obtained by integrating g&, up to its first minimum is an artifact of the normalizationprocedure,which consists in dividing the raw pcf s by the factor 4 r p 8 and adjusting the result to unity at infinite roo. This procedure is correct if the solid angle occupied by the solvent at distance roo from any oxygen atom is, in average, independent of roo. This contention is approximately valid in the case of the solution (our egg model), but not at all in the crystal simulation where, because of the periodic conditions, each water molecule sees proteins in all directions, so that the protein-free solid angle steadily decreases with increasing roo. Our pcfs for 0-H distances (Figures 4b and Sb) also are consistent with the experimental data. The coordination number NOHdefined by eq 1 with integration up to the first minimum of goH gives the mean number of protons in the first coordination sphere of a given water oxygen, in other words the mean number of hydrogen bonds per 0 atom acting as an acceptor. The values we obtained in both simulations are in agreement with experimental data on pure water and indicate that the tetrahedral hydrogen bond structure is not disrupted by the presence of the protein. The H-H pcfs (Figures 4c and Sc) also are in good agreement with neutron diffraction data on pure water. We shall now proceed to a more detailed analysis of the shell structure of water around the macromolecules in solution, again

13870 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993

Alary et al. 150

t

500

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0

0 0

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d(A) Figure 8. Total number of water 0 atoms (dashed line) and H atoms (fullline) in the simulationof the protein complex in solution,as a function of the distance d of the 0 or H atom to the van der Waals envelope of the proteins (and counterions). Each point represents a slice of 0.2-A width in d.

TABLE 111: Repartition of Water Molecules into Three classes no. of molecules' water class type of nearest atom total first layeP second layerb I CH3, CH2, CHc 1764 515 533 I1 C,d H,CCH/K+ 4179 1069 1269 1241 :181 3978 111 O,S, CHg a In the model of the complexin solution. 0-3 and 3 - 6 A, respectively, from the van der Waals envelope of the proteins and counterions. C'Extended atoms".ll' C atoms of the peptide bond, and proline C. atoms. e Polar H atoms. f C, H "extended atoms". g Aromatic C atoms and aromatic CH *extended atoms". from the set of coordinates obtained after 77-ps equilibration and 26-ps dynamics with the egg model. Figures 8 and 3 (full line) respectively show the number of oxygen or hydrogen atoms and the local density of water molecules as functions of the distance to the van der Waals envelope of the protein (including counterions). Figure 8 is obtained from the ensemble of all water molecules, whereas Figure 3 rests upon a statistics over water oxygen atoms present in the spheres of 1-A radius centered on all knots, from a three-dimensional grid with 1-A mesh size, in the free space outside the protein complex. Figure 3 (full line) demonstrates the strong interaction between the solvent and the protein surface. There is a packing of water molecules in the vicinity of the macromolecules, resulting in an increase of the water density up to a maximum value of almost 1.5 g/cm3 at a distance of 1.5 A from the protein van der Waals envelope. Figures 8 and 3 both clearly show the first hydration layer around the proteins. This first layer has a thickness of 3.0 f 0.2 A, which is consistent with the experimental results of Bourret and Parello.]' The characters of the second layer will be discussed a t the end of this section. Finally, to analyze the orientations of the water molecules belonging to the first hydration layer, we plotted the distributions of (OA, OH') and (OA, OH") angles, where for each H'OH" molecule A is the protein (or counterion) atom closest to 0,and H' is closer to A than H" (Figure 9S,in the supplementary material). Oneobserves a structure in the (OA,OH') distribution which will become clear from the next to come analysis. To achieve a better understanding of the arrangement of water molecules within each layer, we performed separate statistics according to the nature of the protein (or counterion) atoms closest to any water molecule. For this purpose we divided the water molecules into three classes as shown in Table 111. Class I, class 11, and class 111, respectively, include those water molecules with

300 c

-.I

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0 0

5

10

d

15

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(A)

250 200 150 100 50

0 0

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d(A) Figure 10. Sameas Figure 8, for each class of water molecules. (a) Class I molecules: 0 atom closer to a nonpolar protein atom; (b) class I1 molecules: 0 atom closer to a positively charged protein or counterion atom; (c) class 111 molecules: 0 atom closer to a negatively charged protein or counterion atom.

their oxygen atom closer to a nonpolar protein atom, to a protein (or counterion) atom with a full or partial positive charge, and with a full or partial negative charge. A similar separation had been performed by Levitt and Sharon,8Owho recordedthedensities of water oxygen atoms as functions of their distances to either polar or nonpolar parts of the protein surface, and by Brooks and Karplus,lm who performed a detailed study of water around specific residues. Figure 10, to be compared with Figure 8, shows the variations of the numbers of oxygen and hydrogen atoms, from water molecules of each class, as functions of the distance of the oxygen

An Antigen-Antibody Complex atom to thevander Waals envelopeof the protein (i.e., theenvelope of the spheres centered on each atom and with a radius equal to the van der Waals radius of the atom as defined by half the u parameter used in the Lennard-Jones potential). Figure 11s (in the supplementary material) gives for each of the three above-defined classes of water molecules, but limited to those belonging to the first hydration layer (defined by a distance of 0-3 A to the van der Waals envelope of the protein and counterions), thedistributions of orientationsas earlier defined for Figure 9s. Similar data are gathered in Figure 12s for the first slice of the second hydration layer (defined similarly by a distance of 3-6 A). Finally Figure 13 shows, for each of the three classes of water molecules, the cosine distributions of the orientation of the water dipole with respect to the OA vector (where again 0 is the water oxygen and A the closest protein atom) for both first and second hydration layers in the sense just defined. It is obvious from Figures 10 and 13 that the arrangement of first-shell water molecules strongly depends on the local nature of the protein surface. Figures 10a and 13a (full circles) show that, in the vicinity of nonpolar protein atoms, the water molecules preferentially have both their 0-H bonds approximately parallel to the protein surface, since the locations of 0 and H first peaks in Figure 10a are identical and the areas of these peaks are in the 1:2 ratio. However, as apparent from Figure 1lSa, neither of the histograms is peaking around 90': one of the 0-H bonds is in average slightly directed toward the protein surface, while the second 0-H bond is rather directed outwards with respect to that surface. From Figures 10b and 11Sb it appears that near protein atoms with positive partial charges the water molecules have preferentially one of their hydrogen atoms slightly closer to the protein surface than theoxygen atom, and theother hydrogenatom further away. This indicates that class I1 water molecules of the first layer are oriented in such a way that one of their 0-H bonds is directed roughly parallel to the protein surface whereas the other 0-H bond points toward the second hydration layer. The slightly inward orientation of the former 0-H bond is probably due to the fact that the H atom is attracted by any atom with a partial negative charge bonded to the pitivelycharged one which defined the belonging of the molecule to class 11. From Figures 1Oc and 13c (full circles), in contrast, one sees that water molecules close to negatively charged atoms have one of their 0-H bonds directed almost perpendicularly toward the van der Waals envelope of the protein. Consequently, the second 0-H bond is oriented in an oblique way toward the second hydration layer, in such a way to respect the canonical valence angle is 104.9', and the dipole moment, as apparent from Figure 13c, is strongly peaked at 52'. Now Figure 10and even more Figures 12s and 13 (open circles) show that the dependence of the orientational ordering of the water molecules with respect to the local nature of the protein surface tends to disappear in thefirst slice of thesecond hydration layer as defined by a 3-6-A distance range. There is still a slight anisotropy of the water molecules close to positively charged atoms, with their dipole moments preferentially oriented away from the protein surface (Figure 13b, open circles), and of those close to negativelycharged atoms, with their dipolemoments preferentially oriented towards the protein (Figure 13c, open circles). This slight anisotropy disappears in further slices. This is an indication that the perturbation of the water structure by the protein essentially does not extend beyond one water layer. Our results compare well with earlier MD studies of water around proteins. In a simulation of bovine pancreatic trypsin inhibitor Levitt and Sharoneoobserved density oscillationsshowing two hydration layers. The water density increased up to 1.25 g/cm3 at 3-45 A from the protein surface (defined by the closest non-hydrogen atom of the protein). In an MD study of

The Journal of Physical Chemistry, Vol. 97, No. 51, 1993 13871

5

le50 1.12

0.00 -1.00

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0.00

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cos0 1.20

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

b

0.80 0.60 0.40 0.20 !-

-I

4

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-0.50

0.00

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1.OO

cose 1.50

1.00

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-0.50

0.00

cose Figure 13. Cosine normalized distributionof the orientation of the dipole moment of water molecules of each class with respect to the OA vector (0, water ox gen; A, closest protein atom). Solid circles: first hydration Open circles: first slice of the second hydration layer layer (0-3 (3-6 A). (a) Class I molecules: nonpolar A. (b) Class I1 molecules: positively charged A. (c) Class 111 molecules: negatively charged A.

i),

parvalbumin Ahlstram et al.'9 also reported an increased density of water, in the range of 2-4 A from the closest protein atom. The water molecules in this distance interval had their dipole moment preferentially parallel to the protein surface, whereas those further away from the protein were isotropically oriented. Finally, Paraket al.74in an MC simulation of myoglobin observed water density oscillations with a maximum at 1.35 A from the van der Waals envelope of the protein, while Lounnas et al.91 from an MD study of the samemoleculepresented water isodensity

13872

The Journal of Physical Chemistry, Vol. 97, No. 51, 1993

TABLE IV: Comparison between L o c p t i ~of~Protein First Hydration Layer from Different Authors (A) first max of water

first max of distance distribution

densitv

wlaP

nonwlaf

Simulations Levitt and Sharonso

3Sb

Brooks and Karpluslw Parak et aI.l4

1.3Sd lSd

present work

2.6 2.7-2.95‘

3.5 3.45

2.8

3.6

Experimental 2.Y Cheng and Schoenborng Thanki er 2.8-3. lcJ Walshaw and Go~dfeIlow~~~ z3.98 Bellissent-Funeler aLi7 3.9 a Distance of water oxygen atoms to the closest (polar or nonpolar) protein atom. Distance to non-hydrogen atoms of the protein surface. c Sets of values for specific residues. d Distance to the van der Waals surfaceof the protein. e Distanceto hydrogen atoms of the protein surface. f3.1 for H-bonddonors. 8 Estimated from the figures pertinent to water 0 atoms with nonpolar closest protein atom. * Distance to neutronscattering averaged protein surface.

0.40

0.20 0.00 0

5

10

I5

d(A) Figure 14. Variation of water density as a function of the distance to the van der Waals sphereoftheclosestprotein (or counterion)atom,according to its character. Solid line: closest atom nonpolar. Dashed line: closest atom polar. The statistics is performed over the nodes of a threedimensional 1-A grid.

contours (up to a maximum of twice the bulk density) illustrating these oscillations. Table IV gives some quantitative comparisons between these various results from M D or M C simulations, along with experimental data. In view of the different definitions of the protein surface the overall picture is consistent. Furthermore, our results on the distributions of distances between water oxygen atoms and closest polar or nonpolar protein atoms are in good agreement with those of Levitt and Sharonso and of Brooks and Karpluslm and are comparable with the values obtained by Goodfellow and c o - w o r k e r ~ by ~ ~inspection ~ J ~ ~ of X-ray crystallographic data. The evidence relative to the second hydration layer, in our simulation as well as in Levitt and Sharon’s?O rests upon the presence of a weak second maximum of water density, which appears in Figure 3 (full line) as well as in Figure 14 where we plotted the density as a function of the distance to the van der Waals sphereof theclosest protein (or counterion) atomseparately when the latter is polar or nonpolar. But there is no evidence in either our or Levitt and Sharon’s simulations of the existence of a second minimum, which would define the width of this second layer. The same holds for the experimental results of Cheng and S~hoenborn:~ they find two minima of their ‘liquidity factor” (or B factor) which is similar to the “temperature factor” of X-ray crystallographers and thus is related to water mobility, so that

Alary et al. these minima may correspond to density maxima, but they show no evidence of a second maximum after the second minimum of the B factor. Thus the second layer may be considered as extending continuously toward the bulk, even though the properties of its innermost region, namely, higher water density and slight vicinitydependent anisotropy, are still different from pure water prop erties. We shall now compare our results with simulations by previous authors of various kinds of solutes and with experimental data whenever available. In this discussion we shall limit ourselves to the essential features. More detailed comparisons may be done using the extensive literature we have quoted. Nonpolar solutes as well as hydrophobic walls give rise from all earlier s t ~ d i e s to~ oscillations ~ ~ ~ ~of water ~ ~ density, ~ ~ ~ ~ - ~ ~ with a first maximum ranging from l.156 to 1.952g/cm3 and located a t 1.354to 2.055A from the van der Waals surface of the solute, depending on the system considered. The corresponding data from our results (Figure 14) are 1.4 g/cm3 and 1.9 A. According to almost all authors, in this first hydration layer of a nonpolar solute or hydrophobic wall (or around an empty cavity40) most water molecules are oriented in a tangential way, with their dipole moments dominantly parallel to the van der Waals surface of the solute; one or both 0-H bonds often point slightly towards the solute, at 60’ or more from the normal to the surface, giving rise to a clathrate-like structure; a minor number of water molecules have one of their 0-H bonds directed away from the solute, toward the second hydration layer. Our results (Figures loa, llSa, and 13a, full circles) are in full agreement with this description. Lee et al.56however found that a part of the water molecules closest to a hydrophobic surface have one 0-H bond directed toward the solute, the other molecules essentially following the above-described scheme. This probably arises from the planeity of their surface, contrasting with the cases of small solute molecules or of a rough surface. Around ionic solutes the well-known successivehydration layers have been observed in all M D or M C studies.4143 A less pronounced structure, which may be described as semiclathratelike,132appears around partially charged atoms of polar neutral s o l ~ t e s . Our ~ ~ results ~ * ~ (Figures ~ lObc, llSbc, and 13bc, full circles) again reproduce the essential features of these studies of positively or negatively charged solutes. These data by themselvesbring no information about a possible inhomogeneity of the density of water close to the protein surface, which requires distribution functions around arbitrary points of space such as those we used for deriving the curves in Figures 3 and 14. Density oscillations like those we obtained near charged protein (or counterion) atoms were reported in a few previous simulations of charged interfaces. In MD studies of a model negatively charged membrane Schlenkrich et a1.62and Nicklas et al.63observed oscillations of the densities of water oxygen and hydrogen atoms, both peaking with three times the bulk density at respectively 2 and 3 A from the plane of carboxylate C atoms.62 Similarly, in an M C simulation of solvent and counterions around a negatively charged helicoidal polyion Gordon and Goldman5lb observed water oxygen density oscillations with a first maximum slightly above twice the bulk density. These maximum densities are very high compared to ours (1.43 g/cm3 at 1.8 A from the van der Waals sphere of charged atoms, cf. Figure 14), and we think that they are due to cooperative effects of the very uniform charged surfaces in these authors’ models. To our knowledge only one experimental result on this question has been published: in a study by neutron specular reflection Lee et 01.30 observed an increase by 9% of the water density near a decylammonium bromide layer. Effect of the Hydration Structure on Protein Motion Many years ago Miissbauer spectroscopy and dielectric relaxation experiments have led Singh et ~ 1 . tol the ~ ~conclusion

The Journal of Physical Chemistry, Vo1. 97, No. 51, 1993 13873

An Antigen-Antibody Complex llght chain 3.00 It

2.50

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Residue number Figure 15. Root-mean-square (rms) fluctuations of the coordinates of the backbone atoms of the protein complex. The asterisks indicate the locations of the complementarity-determining regions in Fab D1.3 and of the epitopes in lysozyme. (a) Solid line: from experimental DebyeWaller factors.Il3 Dashed line: simulation of the complex in the crystal state. (b) Dashed line: simulation of the complex in the crystal state. Solid line: simulation of the complex in solution.

that adsorbed water "imposes its dynamics to the protein"; stating it in a slightly different way, Doster et al.3 from calorimetry and infrared spectroscopy at low temperatures, suggested "a cross correlationbetween structural fluctuationsand the thermal motion of crystal water". Various authors have compared the structure and dynamics of proteins in the crystal state (or in solution) and in vacuo, from MD s t ~ d i e s . ~ Among ~ ~ ~ ~the , ~main ~ J trends ~ there is an increase in the root-mean-square (rms) fluctuations of the atomic coordinates on h y d r a t i ~ n , ~in~agreement - ~ ~ J ~ ~with inelasticneutron scattering data.8J34 Few comparisonshave been performed between MD simulations in the crystal state and in sol~tion.~~,~~ In Figure 15 we compare, from the last 30 ps of the dynamics in the model of the crystal and in the model of the complex in solution, the rms fluctuations of the coordinates of the backbone atoms as functions of residue number. The first 214 residues form the light chain of the Fab D1.3 antibody fragment, the residues number 215-432 the heavy chain, and the last residues, from number 433 to 561, the hen-egg lysozyme antigen. The computed rms fluctuations are compared with those derived from the experimental crystallographic Debye-Waller factors B by the

First of all it appears from Figure 1Sa that the overall features of the crystal simulntcd and observed rms fluctuations are in

qualitative agreement. In particular, the complementaritydeterminingregions (CDR) of the antibody,as well as the epitopa of the antigen, denoted by stars in Figure 15, appear as regions of lower mobility, in agreement with their function of ensuring the antigen-antibod y association. Second, the computed fluctuation amplitudes are consistently smaller than the experimental ones, an observation which was already made in earlier MD studies including the solvent,83 especially for large proteins,75~88~89~96~~os~~07 and which may be explained by two reasons: the simulation time is insufficient for accounting for conformationalchanges occurring at a nanosecond, or higher, time scale, and a possible microheterogeneityof protein conformations in the crystal will contribute to the experimental rms fluctuations (by about one-third13'j),whereas we only are modeling some unique conformation among the actual 0ne9.83J07 This analysis is not inconsistent with the good agreement found by several authors between rms fluctuations from MD in solution and Debye-Waller factors in the crystal for small proteins.68-70-71~82.86,90~93J37 However, a good agreement was also observed recently by Heiner et al.72 between computed and experimental rms fluctuations in a larger-size protein, subtilisin (275 residues, to compare to 432 in Fab 01.3). Third, Figure 15b shows that therms fluctuations in the model of the complex in solution are significantly smaller than in the crystal model. No such trend was observed by authors comparing rms fluctuations from simulations of the crystal cell and of the protein in a large box with periodic condition^.^^*^^ Since we used identical potentials in our two simulations except for the addition of water before the reference period in the simulation of the complex, it is quite likely that the difference observed in the rms's arises from a larger water density near the protein surface in the solution than in the crystal model; another related effect could be that the conformational sampling of the proteins in a dynamics simulation of a given duration is hampered by a higher water density. If this is true, one may wonder whether the difference between the crystal model and the experimental data is not due to a third reason independent from those recalled above, namely, that in the real crystal the amount of water dynamically coupled to the protein would be even smaller than in our crystal model. Discussion and Conclusions The overall agreement of our results on the arrangement of water molecules of thefirst hydration layer of the complex in solution, according to the local character of the protein surface, with the MD simulations performed by many authors on various kinds of solutes indicates that the hydration structure in the immediate vicinity of any residue might be derived from e ~ p e r i m e n t a lor ' ~ theoretical48 ~ data on this residue as a solute. The oscillations of the water density with respect to the distance to the protein surface and the locations of the first maxima are in agreement with earlier MD resultss0and are consistent with the experimental data available. As regards the second hydration layer, from our results it extends continuouslyup to bulk behavior, even though we observe a weak second maximum of water density and a slight residual anisotropy (also dependent on the local character of the protein surface) of the orientation of the water molecules in the first slice of this second layer. These findings are consistent with the obserr ation by various experimentalists of a reduced water mob. y in a region extending several angstroms beyond the first ' ,dr, ::E layer,zo*2s**6 as reported also from MD simulation.'6 A more detailed analysis is in progress on the individual fate of the water molecules of the first hydration layers of our complex.123 The density oscillations of a fluid in the vicinity of a wall, on the scale of molecular dimensions, have been known for long time from We reviewed in the previous sectionsthe available

13874 The Journal of Physical Chemistry, Vol. 97, No. 51, 1993

experimental evidence of this phenomenon as regards surfactant layer30 and pr~teins.~JI Our results on a protein complex not only support these observations,but they also indicate an increased integrated density of water in the vicinity of the protein surface, with respect to pure water. The maximum water density in the first hydration layer is still an open question. Our value of 1.5 g/cm3 (see Figure 3) is comparable to the density of ice VI1 or ice VI11 (1.5140). The difference of course is that our hydration layer reaches this high local density only across a width of an angstrom, in contrast with a bulk material like ice. In fact the density we observe might be slightly underestimated since, as explained earlier, in our simulation with the egg model of the protein complex in solution we did not add water up to the point where the density far from the protein would have attained unity. The computational effort necessary for achieving full equilibrium of a protein in solution within the grand-canonical ensemble would be worthwhile only if using more elaborate potentials including water polarizability,I4l which is presently impracticable unless dealing with a small peptide. On another hand, the subtlety of the structural equilibrium of liquid water implies that the radius of curvature of the solute, which controls interfacial tension, be correctly described. Thus it seems wise to wait for new experimental data until undertaking ruch very large-scale simulations. [The widespread use of a cutoff for electrostatic interactions in MD simulations, which we followed in this work, was recently questionned by Schreiber and S t e i n h a u ~ e r especially ~~* for the study of solvated biomolecules. These authors found that a heptadecapeptide helix would unfold or not during a 100-ps simulation according to the choice of the cutoff radius but keeps folded within a t least 225 ps if no cutoff is applied (using the Ewald furthermore, they observed that artifacts in the oriental correlation functionsappear at H20-HzO distances equal to the cutoff radius. These findings indeed deserve future careful studies. However, their cutoff scheme induced a discontinuity in the potential, which does not occur with the “shifted electrostatics”I17which we use.] In the discussion of our results on the rms fluctuations of backbone protein atoms we considered the possibility that the density of water around the protein be actually smaller in the crystal than in solution. From the thermodynamic viewpoint the increased density of water in the hydration shell derives from a lower standardchemical potential of water (i.e,, chemical potential at unit density) in the field of protein atoms than in the bulk: the requirement of equilibrium, and therefore of equality of the chemical potential of water around the protein and in the bulk, then leads to a higher actiuity (or density) of water near the protein surface. In the crystal however, most water molecules in the reference cell belong to the hydration structureof the protein molecules of the reference and surrounding cells: from our simulation ca. 75% belong to the first hydration layer (distance range 0-3 A) and 23% to the next 3-Aslice. It appearslikely that the few remaining water molecules and part of the latter slice form the hydration layers of the poly(ethylene glycol) molecules and of possible free ions. If so, there will be no bulk water phase inside the crystal, which indicates that the separation between protein molecules of neighboring cells is not large enough to allow any bulk behavior in the free space, as earlier noted by Usha et a1.,22and therefore the chemical potential (and, thus, the density) of the hydration water might be lower than for the protein in solution. This probably is the requirement for protein molecules to make stable contacts between them in the crystal instead of being kept apart from each other by “hydration forces”.144 If this analysis is correct, one may expect a larger mobility of the protein chain in the crystal than in solution. There does not seem to be any technique available for making such a comparison directly.

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