Proc. Natl. Acad. Sci. USAVol. 93, pp. 11609-11614, October 1996Biophysics

Intrinsic compressibility and volume compression in solvatedproteins by molecular dynamics simulation at high pressure

(volume fluctuations/static structure factor)

EMANUELE PACI AND MASSIMO MARCHI*Section de Biophysique des Proteines et des Membranes, Commissariat a l'Energie Atomique, Centre d'Etudes, Saclay,91191 Gif-sur-Yvette Cedex, France

Communicated by David Chandler, University of California, Berkeley, CA, June 20, 1996 (received for review May 1, 1996)

ABSTRACT Constant pressure and temperature molecu-lar dynamics techniques have been employed to investigate thechanges in structure and volumes of two globular proteins,superoxide dismutase and lysozyme, under pressure. Com-pression (the relative changes in the proteins' volumes),computed with the Voronoi technique, is closely related withthe so-called protein intrinsic compressibility, estimated bysound velocity measurements. In particular, compressioncomputed with Voronoi volumes predicts, in agreement withexperimental estimates, a negative bound water contributionto the apparent protein compression. While the use of van derWaals and molecular volumes underestimates the intrinsiccompressibilities of proteins, Voronoi volumes produce resultscloser to experimental estimates. Remarkably, for two glob-ular proteins of very different secondary structures, we com-pute identical (within statistical error) protein intrinsic com-pressions, as predicted by recent experimental studies.Changes in the protein interatomic distances under compres-sion are also investigated. It is found that, on average, shortdistances compress less than longer ones. This nonuniformcontraction underlines the peculiar nature of the structuralchanges due to pressure in contrast with temperature effects,which instead produce spatially uniform changes in proteins.The structural effects observed in the simulations at highpressure can explain protein compressibility measurementscarried out by fluorimetric and hole burning techniques.Finally, the calculation of the proteins static structure factorshows significant shifts in the peaks at short wavenumber aspressure changes. These effects might provide an alternativeway to obtain information concerning compressibilities ofselected protein regions.

During the past few years, considerable effort has developedto understand the origin of the pressure effects on the structureand volume of proteins (1-3) and to elucidate the mechanismfor protein denaturation at high pressure (4, 5). Many exper-imental techniques have been used to study pressure inducedchanges in proteins, but only in rare instances has computersimulation been employed. Here, we present a moleculardynamics (MD) investigation of the microscopic compressionof two solvated proteins. By simulating these systems in theNPT ensemble at room temperature and at increasing pres-sures (from 0.1 to 2000 MPa), we were able to computeaverages and fluctuations of the protein volumes and relatethese microscopic properties to the experimental thermody-namic compressibilities.

Experimentally, the effects of pressure on the proteinstructure are investigated by probing the correspondingchanges in volume and by measuring compressibility, the latterbeing related to volume fluctuations, protein flexibility and,indirectly, protein functionality (6). Although crystallography

(7), fluorescence spectroscopy (8), NMR (9), and hole burning(10) experiments have all been used to make estimates ofprotein compressibility, most of the available compressibilitydata come from sound velocity measurements (11, 12). Criticalinvestigations of these experimental results have indicated thatthe compressibility of a protein can be divided into at least twocomponents of opposite sign. Due to compressible cavities andvoids in the protein interior, the first component of the proteinintrinsic compressibility, p13, is positive. The remaining term,fHyd, is the contribution to the compressibility due to hydrationand bound water. As the compressibility of single amino acidsand small peptides in solution is negative (13), tHyd is thoughtto be negative. Most recent work (12) has indicated that theisothermal intrinsic compressibility of the protein, defined asthe compressibility of the protein interior, is unique for allproteins and is about 3 times smaller than that of water. Todistinguish, at a microscopic level, the protein intrinsic com-pressibility from contributions due to the solvents, the behaviorof alternative definitions of protein compressions derived fromcomputer simulation are juxtaposed here with the general ex-perimental results. Although a few simulations of proteins at highpressure (14, 15) have been reported in the past, we carry out herea systematic correlation between experimental protein compress-ibilities and the simulation results.

Additional issues concerning the intrinsic compressibility ofproteins are raised by high pressure experiments on proteinscarried out with techniques other than velocimetry. Althoughmeasurements of B3p from hole burning experiments are inagreement with sound velocity estimates, fluorimetric deter-mination of distances between trytophan and heme groups indifferent heme proteins have led to intrinsic compressibilitieshigher than that of water. In contrast, x-ray crystallographyapplied to atmospheric and high pressure crystals of lysozymeresulted in an overall 3Pp that is one order of magnitude lowerthan that of water and one-third of that estimated from soundvelocity experiments. Thus, an additional goal of this study isto address these experimental issues by characterizing thepressure effects on the protein interatomic distances derivedfrom constant pressure MD simulations in light of the aboveexperimental results.

MATERIALS AND METHODSMD Simulations. All results reported in this paper were

derived from a series ofMD simulations of solvated superoxidedismutase (SOD) and of the tetragonal crystal of hen egg whitelysozyme at constant temperature and pressure, carried out at300 K and 0.1, 1000, and 2000 MPa. Average pressures andtemperature were kept constant using an extended Lagrangianmethod, the application of which to solvated protein has beendiscussed in ref. 16. The extended Lagrangian equations of

Abbreviations: MD, molecular dynamics; SOD, superoxide dismutase;SPC, simple point charge model.*To whom reprint requests should be addressed.

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The publication costs of this article were defrayed in part by page chargepayment. This article must therefore be hereby marked "advertisement" inaccordance with 18 U.S.C. §1734 solely to indicate this fact.

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motion were integrated using a Verlet-based algorithm (17)modified to treat velocity-dependent forces (18). A time step of1.5 fs was used. Full periodic boundary conditions were used.The solvated SOD protein was prepared and equilibrated for

300 ps in the microcanonical ensemble at 300 K, as describedin ref. 19. The final system consisted of a SOD proteinsurrounded by 1457 water molecules and four Na+ ions forelectroneutrality. The orthorombic box had dimensions a =43.01 A, b = 62.56 A, and c = 35.19 A.The simulation box used for lysozyme contained only one

elementary cell of its tetragonal crystal (7). It was generatedby applying symmetry operations to the x-ray coordinates ofthe crystal asymmetric unit and by adding, at a later time, 1946water molecules and 64 Cl- ions to fill the voids and to reachelectroneutrality, respectively. The final system consisted of atetragonal simulation box of dimensions a = 79.1 A and c =37.9 A containing eight lysozyme molecules, 3154 water mol-ecules, and 64 Cl- ions. As for SOD, lysozyme was equilibratedin the NVE ensemble at about 300 K for 300 ps.

All water molecules contained in the systems were modeledby a simple point charge model (SPC; 19). The CHARMMversion 20 united atom force field (20, 21) was used to handlethe interactions among the protein atoms. Standard Lennard-Jones sum rules were adopted for mixed interactions betweensolvent and solute. We adopted a spherical cutoff at 9.0 A witha third order spline between 8.5 and 9 A. All covalent bondswere kept rigid in the MD simulations using SHAKE (22). Forthe Cu-Zn active site of SOD, the ab initio-based electrostaticmodel of Shen et al. (23) was used.The runs in the NPT ensemble were started by fixing the

external pressure and temperature of the system at the atmo-spheric value of 0.1 MPa and 300 K and reequilibrating for =50ps. For solvated SOD, three MD runs were performed, eachlasting 120 ps, at pressure P = 0.1, 1000, and 2000 MPa andtemperature T = 300 K. For the lysozyme crystal, MD runswere carried out for 120 ps each at pressures P = 0.1 and 1000MPa and temperature T = 300 K. After each modification ofthe pressure, both systems were relaxed for additionally 80 ps.We also performed three simulations of 686 molecules of purewater at the same thermodynamic conditions to compute theprotein apparent volume discussed in this paper. While forSOD and water, only isotropic fluctuations of the volume (24)were allowed, for lysozyme, the cell shape and dimensionscould change (25) during the runs. For details on the imple-mentation of the method, see ref. 16.

Protein Volumes. Early experimental investigations (26) ofchanges in volume upon protein folding prompted, more than20 years ago, the development of techniques to compute thevolumes of protein residues directly from x-ray structures. Theso-called molecular volume was originally proposed by Rich-ards (27). It is based on the static view of a protein as acollection of atomic hard spheres, each having a core radiuscorresponding to the atom van der Waals radius. Thus, themolecular volume is defined as the volume of space enclosedby a molecular surface constructed as the contact surfacebetween the van der Waals surface of the protein and a watermolecule probe, represented by a hard sphere of 1.4-A radius.When the probe is simultaneously in contact with more thanone atom of the protein, the interior-facing part of the probeis used as the contact surface.According to its definition, the molecular volume takes into

account solvent effects on rigid protein structures withoutexplicitly including solvent molecules in the calculation. Thiswas a clear computational advantage in early times, as com-pared with the calculation of Voronoi volumes (28), whichrequires, instead, the knowledge of the position of the solventmolecules around the protein to determine the volume asso-ciated with the surface residues. Indeed, computing the uniqueVoronoi polyhedra associated to a given distribution of atomscorresponds in spirit to constructing a Wigner-Seitz cell for

each atom. This can be carried out in a meaningful way onlyfor homogeneous and infinite systems. As shown by earlierstudies on liquids (29, 30), the use of Voronoi volumes canprovide unambiguous structural information. The Voronoiapproach has also been applied to static protein structurederived from x-ray crystallography to determine the volume ofinterior residues (31).

In this study, we have used the MD-generated configurationsof the solvated proteins at different pressures to directlycompute van der Waals, molecular, and Voronoi volumes forthe concerned part of the system. Standard software was usedto compute van der Waals (32) and molecular (33) volumes forthe set of van der Waals radii given in ref. 34. As for theVoronoi volumes, their calculation was carried out using therecursive algorithm described in ref. 35. No explicit hydrogenswere included in the Voronoi volume calculations.

RESULTSApparent and Intrinsic Compressibility. The adiabatic com-

pressibility of a solution is determined experimentally from theLaplace equation:

1j3= U2' [1]

Here, p is the solution density and u is the sound velocity in themedium. Any change in concentration or structure of thesolution is reflected by a change in the sound velocity. Exper-imentally, the partial adiabatic compressibility of the solute isobtained by measuring the sound velocity and the density ofthe solution at different solute concentration and extrapolatingto zero concentration, namely (11):

1 (ai71\ 1 (3/3w-V)}3P = -- ,9 = - lim [2]

where ,3 and ,B are the adiabatic compressibilities of thesolution and of the pure solvent, respectively, c is the concen-tration of the solute, V is the partial specific volume of thesolute, and vo is the apparent volume fraction of the solvent insolution. For solvated proteins, 83p includes contributions bothfrom the protein interior and from the surface residuesinteracting with the solvent. Thus, even in the limit of smallsolute concentration, 3,P is known experimentally as the pro-tein apparent compressibility.For most of the globular proteins investigated, }3p is positive.

At least two contributions of opposing signs contribute to /3P:the compressibility of the protein interior, or protein intrinsiccompressibility (3,p), and the change in compressibility due towater-protein interaction (13Hyd)-i.e.,

3p = (p + IHyd - [3]

Solvation effects and water penetration at the protein surfacecause the latter contribution to be negative, while pressureeffects on voids and cavities in the protein globule produce,instead, a positive (3p.A relation between intrinsic and apparent compressibilities

has been used in the experimental literature by writing ,3, theadiabatic compressibility of the system, as the sum of thepartial compressibilities of its components multiplied by theirvolume fractions. Here, we consider a contributions from theprotein, 3,p, the bulk water, A3,, and the water bound to thesurface residues, Op,,,. Namely (36):

p = fw + vp[3p - /3w(A + 1) + Appw], [4]

where vp is the protein fractional volume and A is the ratiobetween the bound water fractional volume and vp. It must be

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1000

Pressure (MPa)FIG. 1. Volumes of the SOD dimer as a function of pressure. Different protein volumes are compared.

realized that although Eq. 4 cannot be rigorously derived, it hasthe advantage of providing an expression for the apparentprotein compressibility which can be used to easily extract (,pfrom sound velocimetry results. Indeed, when solvation effectsare neglected (i.e., A = 0), ,'P is equal to Pp,. Thus, the hydrationcontribution to 13p' in Eq. 3 is:

1Hyd = -A(Ow 'pw) *[5]

Since hydration effects reduce the compressibility of boundwater with respect to bulk water, a consequence of Eq. 3 is thatprotein intrinsic compressibility is higher than its apparentvalue. The same equation also holds for isothermal compress-ibilities (12).

In this study, we did not attempt to compute partial prop-erties such as protein partial volume or compressibilities as

such calculations, by requiring a series of simulations atincreasingly smaller protein concentration, would be compu-tationally too heavy. For sake of comparison with properties ofwater, we have instead computed the relative compression ofthe protein with respect to simulated SPC water. As shown inref. 16, the compressibility of SPC water is very close to theexperimental one. The compression, k, is related to compress-ibility in the limit of infinitely small change in pressure by:

kp213 = lim

Pi [61P2-~Pl P2

where kpi2 is the fractional change in volume due to a pressurechange P2 - P1, namely:

p (V), - (V)2[7

where ( ) stands for statistical average in the chosen ensemble(NPT in our case).

In Fig. 1 the protein volumes for SOD computed at 0.1, 1000,and 2000 MPa are shown. As expected, van der Waals volumeis almost constant with changes in pressure. In addition,

although at atmospheric pressure molecular, Voronoi andapparent volume are very close, their properties with respectto changes in pressure are strikingly different. This is bettershown in Table 1, where we present the volume compressionscalculated for the hydrated SOD and the lysozyme crystal.Their relative values with respect to compressions of SPCwater are also included. Typically, errorst on Voronoi andapparent compressions are close to 5 and 15%, respectively.Since changes in molecular volumes are smaller than forVoronoi and apparent volumes, the corresponding errors are

larger and on the order of 20%. The observed decorrelationtime for volumes was 2 ps, which is small compared with thelength of our simulations (120 ps). More details on volumeconvergence and errors are reported in ref. 16.We first observe that the compression of the proteins

molecular volumes is consistently smaller than that of theapparent volumes at all pressures. On the contrary, k com-puted from the Voronoi volume is always larger for bothproteins. Thus, the hydration contribution to the apparent kvalues computed from the Voronoi volumes, kvyd in the table,is negative, as predicted from experimental estimates of thedifference between apparent and intrinsic compressibilities(11, 12). On the contrary, a positive contribution is obtainedif molecular volume is used instead of Voronoi volume (seecolumn labeled I4md in Table 1).The similarity of behavior between protein intrinsic and

Voronoi volumes with respect to changes in pressure is alsosupported by the calculation of the protein hydration or theratio between the weight of the bound water and the weight ofthe protein. Indeed, if we adopt the Voronoi compression asthe intrinsic protein compression and use a value of the boundwater compression within zero and one-third of that of bulkwater (47), protein hydration can be estimated directly fromEqs. 3 and 5. While for SOD, A is within 0.12-0.19 g of H20per g of protein, the estimated hydration for the lysozymecrystal is higher and ranges between 0.21 and 0.31 g ofH20 per

tUnless otherwise stated, by error we refer to the maximum error-i.e., three times the standard deviation o-.

38000

clfo<

E>

33000

sss~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-__~~~~~~~~~~~~~~~~~~~% A___.-

%+_t-.".

G-(OMolecularE3-E Van der Waals$- -*Voronoi

. -AApparent

S

280000

L J1

2000

1--- L-i -- --di

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Table 1. Protein compression of SOD and lysozyme

Compressions

Apparent Molecular Voronoi

Absolute Relative Absolute Relative Absolute Relative I'Md kvYdLysozyme

klo.0l° 0.04 0.19 0.031 0.15 0.091 0.44 0.04 -0.25SOD

klo°l° 0.055 0.26 0.022 0.10 0.098 0.47 0.16 -0.21k1882 0.037 0.41 0.021 0.23 0.048 0.53 0.18 -0.12The protein compressions are computed from apparent, molecular, and Voronoi volumes. The values in the first column

of each item are the absolute compressions. In the second column, we report the compression relative to SPC water at thesame pressures. Items km d and kHyd are the hydration contributions relative to SPC water compression obtained usingmolecular and Voronoi volumes, respectively. Mean SPC water volume at 0.1, 1000, and 2000 MPa was computed fromadditional simulations described in ref. 18.

g of protein. This trend is consistent with the experimentalestimates of 0.23 and 0.33 for solvated SOD (37) and lysozyme(7), respectively.To provide an additional corroboration to the intimate

relation between intrinsic and Voronoi volumes, we notice thatthe Voronoi compressions ofSOD and of the lysozyme crystal,between 0.1 and 1000 MPa, are identical within simulationerror. In light of the structural differences between the twoproteins, this result is remarkable. Indeed, SOD is a (-sheetdimeric protein while lysozyme is monomeric and predomi-nantly a-helical. This finding is consistent with recent studieson the intrinsic compressibility of proteins and small peptidesby the regression method (12). Based on the analysis of thecorrelation between isothermal partial compressibilities andmolecular surface area, it was found that the intrinsic com-pressibility of globular proteins depends very weakly on thesize and structural characteristics of the proteins. This suggeststhat the value of the protein intrinsic compressibility is com-mon to all globular proteins and is about one-third that ofwater. This compares with our computed protein compression,which was half that of water.

Since the constant pressure MD simulation techniques usedin this study sample directly from the NPT ensemble (16, 38),we have been able to compute not only volume averages, butalso volume fluctuations. In the past, protein volume fluctu-ations were estimated by relating them to protein isothermalcompressibility, {3p, with the equation (39):

<8V2> =VkBT(p, [8]

where kB is the Boltzmann constant, T is the temperature, andV is the protein volume. This equation follows directly byconsidering either that the protein alone samples the distri-bution characteristic of a NPT ensemble or that the couplingbetween the protein and the solvent volume is negligible.Unfortunately, these assumptions are not true in general. Weverified indeed that the coupling between protein and solventVoronoi volumes was not at all negligible and that the crossfluctuations were of the same order of magnitude as thefluctuations of the two volumes alone. We present in Table 2the calculated fluctuations of the cell volume and the protein

Table 2. Volume fluctuations

P, MPa 6Vc/Vc, xo100 Vv/Vv, x 100

Lysozyme 0.1 0.23 0.271000 0.26 0.22

SOD 0.1 0.43 0.371000 0.23 0.262000 0.19 0.20

Relative volume fluctuations for the simulation cell volume (Vc)and the protein Voronoi volume (Vv) were computed from constantpressure simulations described in the text. P is the external pressureused given in MPa.

Voronoi volume. Statistical errors on volume fluctuations arehigher than those reported for compressions and are in theorder of 30%. Although no estimate exists for SOD, thecomputed volume fluctuation of lysozyme is close to theexperimental estimate of -0.3% obtained in ref. 7. In addition,volume fluctuations in SOD are larger that in lysozyme. Thiscontrasts with the common experimental view (40) that,3-sheet domains are less flexible than a-helical domains undercompression.

Distance Changes Under Pressure and Compressibility.Protein intrinsic compressibility can also be estimated fromtechniques sensitive to changes in distances between atoms ofthe protein. In particular, fluorescence spectroscopy (8) hasbeen used to probe donor-acceptor distances in heme proteinsas a function of pressure, and the resulting underlying com-pressibility was found to be comparable to that of bulk water.To interpret the difference of this result with those obtainedby sound velocimetry, we have investigated how distancesamong the Ca of SOD change with pressure. In Fig. 2, wereport the value of the volume compression between 0.1 and1000 MPa for each pair of Ca, assuming isotropic compression.We first notice that for all distances, the compression greatlyfluctuates and can assume both positive and negative values.These fluctuations are larger for short Ca-Ca distances below20 A and become narrower as the distances increase. Therange of compressions obtained in this way lays between -1.5and 0.7. It is clear from this result that a single distance

0.7

C:0(na)

00ciJE-5

0.0

-0.7

-1.4

0.12

0.08

0.04

0

0 20 40

0 20 40

20 49C0,-CU, distance (A)

60

FIG. 2. Volume compression between 0.1 and 1000 MPa computedfrom the contraction of the Ca-Ca distances. Each point in the plotcorresponds to a pair of Ca atoms. (Inset) The average compressionversus the Ca-Ca distance.

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contraction can never provide an accurate estimate of proteincompression.On the other hand, if the average of compression is plotted

as a function of the Ca-Ca distance, the behavior shown in theInset of Fig. 2 is observed. Protein compression computed fromaverage distance contraction is always positive, or near zero forshort distances, and is also nonuniform within the whole rangeof interactions. Indeed, compression is smaller for shorterdistances and larger for longer ones. Thus, experiments suchas photochemical hole shift measurements, which probe aver-age nearest neighbor distances, are likely to underestimate theintrinsic compressibility of proteins. The results presented inFig. 2 were also confirmed by the high pressure simulations ofthe lysozyme crystal (data not shown).The protein nonuniform dilation revealed by our high-

pressure simulations is in contrast with x-ray crystallographyexperiments on the protein thermal expansion (41), whichfound instead a uniform dilation of interatomic distances.Thus, our result emphasizes the difference in the physicalnature of the structural effects induced by pressure withrespect to those due to temperature changes.Evidence of nonuniform dilation can also be gathered by

computing the protein static structure factor, S(q), at differentpressures. We have evaluated S(q) from the pair correlationfunction, g(r), as:

S(q= + Fr2p sinqrS(q) = 1 + 4'irlr2pg(r) ~~dr, [9]Jqr

where N is the total number of atoms, rij is the distancebetween every pair of atoms i and j, and p is the average

10fl N.

with

p N N

pg(r) = N 1: IS,(r-i j4i

S(q)5

0

[10]

number density of the protein. We included in the calculationof pg(r) only contributions from nonhydrogen atoms.We have computed the pair correlation functions of SOD at

the three pressures. At small distances, no significant shift inthe position of the peaks is noticeable. In addition, beyond 6A, the pair correlation functions become featureless. Moreuseful to our analysis is the behavior under pressure of S(q),shown in Fig. 3 for P = 0.1 and 1000 MPa. Since some of thestatic structure factor peaks are due to the periodic distancescharacteristic of the protein secondary structure, pressureinduced peak shift is an indirect measure of the compressionof the corresponding secondary structure motifs. Assumingisotropic compression, if s1 and S2 are the position of the peakat pressure P1 and P2, respectively, the volume compression kpP2is computed as:

[11]-3 -3Pi,- S-3S1

In Fig. 3, the shifts of the peaks are appreciable mainly in theregion of small and medium wavenumber q (q < 1.5 A-1). Inagreement with results of Fig. 2, we find that compression isnot uniform over various length scales. Indeed, a first peakobserved near 0.3 A-1 at 0.1 MPa shows avolume compressionof 0.12 between 0.1 and 1000 MPa, while for the peak at 1.27A-1, we compute a compression of only 0.06 (see Insets in Fig.3). The former peak is related to the protein radius of gyration(42, 43). Its larger compression agrees well with the compres-sion computed from the Ca-Ca distance changes in Fig. 2 inthe limit of large distances. On the other hand, the peak at 1.27A-1 is due to a space periodicity of 5 A, which is typical of thedistance between the a carbons of two adjacent 13 strands.Thus, the calculation of the pressure effects on this peak canprovide estimates of the underlying compressibility of the SOD(3-sheets, which are the principal motif of the protein second-ary structure. This result indicates that measurements of thestatic structure factor of solvated proteins could, in principle,

1 0 1 2 3q(A)

FIG. 3. Protein static structure factor, S(q), at atmospheric and 1000 MPa pressure. (Insets) On a different scale, the peaks at q 0.3 andq - 1.28 A. Statistical error is within the thickness of the lines.

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be used to investigate the intrinsic compressibility of proteinssecondary structure. Nevertheless, although experimental de-termination of S(q) by coherent elastic neutron scattering ofliquid water at high pressure (44) and of a hydrated proteinpowder at atmospheric pressure (45) has been carried out inthe past, such a technique has not yet been applied to solvatedproteins at high pressure.

CONCLUSIONTo conclude, our study has emphasized the importance ofcomputer simulations to investigate pressure induced struc-tural modifications in solvated proteins. Hopefully, constantpressure MD combined with Voronoi volume analysis will beused in the future to address some of the long-standing issuesconcerned with high-pressure studies of proteins, such asanomalous protein compressibilities, enzymatic activity, andprotein denaturation. We also hope that our results on thestatic structure factor of proteins at high pressure will stimu-late experimental work in the field.

We thank Piero Procacci for useful discussions and for helping us inthe calculation of the Voronoi volumes.

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