an efficient simulation technique for electrostatic free energies with applications to azurin

An Efficient Simulation Technique for Electrostatic Free Energies with Applications to Azurin

BO SVENSSON Department of Chemist y, University College, University of New South Wales, ACT 2600, Australia

BO JONSSON* Department of Physical Chemistry 2, Chemical Centre, POB 124, 221 00 Lund, Sweden

Received 17 March 1994; accepted 25 July 1994

We present an efficient technique for Monte Carlo simulation of electrostatic free energy changes in biomolecular systems. It is a development of a recent method for the study of the influence of electrostatic interactions on the ion binding properties and redox potentials of biomolecules. The electrolyte solution is described by the primitive model, in which ions are treated as hard charged spheres and the solvent is replaced by a structureless continuum. The protein is kept fixed in the center of a spherical simulation cell, and the dielectric constant has the solvent value throughout the cell. By a multiparticle perturbation approach, it is possible to obtain a number of free energy changes within one simulation only. The usefulness of the method is illustrated with a study of the copper binding electron-transport protein azurin (from Alcaligenes denitrificans). The change in acidity of the histidine residues upon changing the redox state of the copper ion is calculated. The theoretical predictions agree well with available experiments. 0 1995 by John Wiley & Sons, Inc.

biomolecules, as well as their rates of reactions with charged substrates.’ The increasing appear- ance of genetically engineered biomolecules in which charged residues have been replaced or mutated offers other examples.’r3 The complexity of biomolecular systems makes theoretical modeling a great challenge, and a number of different

The simplest models are based on the solution

Introduction

here are numerous examples of the large in- T fluence of electrostatic interactions on the properties of biomolecular systems. These include titrations, ion binding and redox properties of

*Author to whom correspondence should be addressed.

have appeared.

Journal of Computational Chemistry, Vol. 16, No. 3, 370-377 (1995) 0 1995 by John Wiley & Sons, Inc. CCC 01 92-8651 I95 I C-30370-08

EFFICIENT SIMULATION

of the Poisson-Boltzmann (PB) equation or its lin- earized version (LPB) for a system, in which the biomolecule is assumed to have a simple geometry and the solvent is replaced by a structureless con- t i n~um.~ , ' The most detailed models are based on molecular dynamics simulations with an explicit treatment of all the biomolecular atoms as well as the solvent molecules.6 Two major drawbacks limit the usefulness of the detailed models. First, these simulations are computationally expensive, which may prevent a real-time simulation of more than a nanosecond. This may be far too short to obtain a reliable statistical mechanical average of the system properties. Second, the outcome of the simulation depends on the particular atomic force field in use.

Other models which, in complexity, are in between those mentioned include solution of the PB or LPB equation with an explicit account taken of the detailed biomolecule ~tructure .~ The structure is usually taken to be that of the crystalline molecule. In such calculations, it has become a standard to use a low dielectric constant of the biomolecule, extending out to its van der Waals surface.' The solvent containing a given salt concentration is treated as a structureless continuum with the dielectric constant of the pure solvent. A major problem with these models is that the exact location of the dielectric boundary as well as the assumed dielectric constant of the biomolecule may have a large influence on the outcome of a calculation. In some cases, a shift of t t e discontinuity just a few tenths of an Angstrom (A) may give drastic changes,' and experimentally there is no guide for choosing the exact location of the discontinuity. Furthermore,othe resolution of the crystal structure is around 1 A. In addition, the solution structure, which in general is the relevant one, may be different from the crystalline one.

We have previously developed a Monte Carlo simulation technique to study the effect of mutations of charges as well as the addition of elec- trolytes on the binding free energy of charged species to biomolecules." The model resembles the detailed continuum models just mentioned, in that we explicitly include the protein atoms in the simulation as well as the charged ions, all embed- ded in a uniform dielectric continuum. The electrostatic contribution to the free energy of binding is calculated via a perturbation method. The use of a uniform dielectric constant is a necessity to pre- serve computational simplicity. However, t h s choice also offers other advantages. The electrostatic free energies become insensitive to the choice

of parameters such as atomic sizes, the presence of crystal-bound water molecules, and small changes in the biomolecular structure. Furthermore, even if the interior of a biomolecule is hydrocarbonlike and expected to give a weak dielectric response, the surface may be very hydrophilic because it often contains semiflexible charged groups and thus is likely to give a larger dielectric response than its interior. One also expects the solvent water molecules to penetrate the biomolecular surface to some degree.

The aim of this article is to present a further development of the simulation technique as well as present results for the change in acidity of some histidine residues of the protein azurin (from Al- caligenes denitrificans). This protein acts as a link in physiological electron-transport chains and contains around 130 amino acid residues and one copper ion. The acidity of its histidine residues depends on the valency of the copper, and experimentally this has been studied by means of nu- clear magnetic resonance." The new perturbation approach introduced allows the study of a large number of charge mutations, in this case protonation, in just one simulation. Because the under- lying simulation is the same, each additional mutation studied requires a minimum of extra computational effort.

In the next section we give a brief introduction to the electrostatic binding model as well as the simulation technique. A new simplified procedure to reduce finite volume effects in the chemical potentials is also given. Then a multiparticle perturbation method to free energy shifts is introduced. Its efficiency is demonstrated by calculation of the pK shifts on histidines in azurin. Finally, comparisons are made with experiment as well as with previous LPB calculations on the same system.

Theory

BINDING MODEL

The binding of an ion I to a biomolecule P may be expressed in terms of the binding constant K as

where the brackets denote the concentration. There are several contributions to the binding free energy. These include contributions from direct lig-

JOURNAL OF COMPUTATIONAL CHEMISTRY 371

SVENSSON AND JONSSON

and binding, changes in solvation, structural changes, as well as the electrostatic free energies of the bound and free ion? We will focus on the change in ion binding as the charge of the biomolecule changes due to a mutation several angstroms away from the binding site. By assum- ing that this only affects the electrostatic contribution and the other contributions remain the same, one then has relative to a chosen reference system

where AGe' is the electrostatic free energy change upon binding and p = l/(kT), k being Boltz- mann's constant and T the absolute temperature. The reference may be chosen as, for example, the system in a salt-free solution. In terms of the electrostatic contribution to the excess chemical potential of the bound ( B ) and free ion ( F ) , AGel is given by

AG" = peX(B) - p e X ( F ) (3)

The excess chemical potentials will depend on the concentration of all charged species, including biomolecule, counterions, and added salt. This means, for example, that the screening caused by counterions can be substantial in a solution with a highly charged biomolecule and low salt concentration.12 If, on the other hand, the biomolecular concentration is low or its net charge is low, one may neglect the free ions in calculating the pK shift at a given salt concentration. This is so because peX(F) will then mainly be determined by the surrounding salt solution. Thus, by calculating the chemical potentials for the bound ions in the two systems, one may then estimate the difference in pK.

MODEL SYSTEM

The simulations are performed in the canonical ensemble where the temperature, volume, and the number of particles N are constant. The primitive model is used, in which the ions are treated as charged hard spheres and the solvent is replaced by a structureless continuum.13 The simulation cell is spherical and the biomolecule is placed at its center. The configuration of the biomolecule is kept fixed during the simulation, and its atoms are assigned hard core diameters and charges. A schematic picture of the model system is given in Figure 1. The interaction between any two species

FIGURE 1. A schematic picture of the simulated system. The free ions are shown as open circles with a charge, whereas the fixed protein atoms are shaded.

with the valencies qi and ql separated by the distance r l j is given by

where e is the proton charge, E , is the permittivity of free space, E , is the dielectric constant of the solvent, and (T is the hard core diameter of the ions. The Monte Carlo simulations are performed using the Metropolis a1g0rithm.I~

Azurin consists of 129 amino acid residues. The coordinates were obtained from the Brookhaven Protein Data Bank file 2AZA, which contains 988 protein atoms. The hydrogen atoms are not present in the coordinate list and were neglected in the simulation. The protein charges were obtained from a simple charge model. The carboxylic oxy- gens on aspartate and glutamate residues as well as the C-terminal residue were given a -0.5 unit charge. The lysines were given a unit charge on the 5 nitrogen, whereas the arginines were given a 1/3 unit charge on each of the E , ql, and q2 nitrogens. Finally, the N-terminal amino group was given a unit charge on the nitrogen. The protein containing Cu2+ was electroneutral, and unless otherwise stated 20 salt pairs were added to the cell. The diameter of the cell was chosen to give the desired salt concentration. The hard core diameters of the protein atoms and the ions were taken to be 4 A. It has been previously shown that

372 VOL. 16, NO. 3


for the present model, neither the neglect of hydrc- gen atoms, the exact diameter of the atoms and ions, nor the use of a simple charge distribution on the protein atoms significantly affects the simulated pK shifts.'"

CHEMICAL POTENTIALS

The chemical potentials may be evaluated using Widom's method.I6 By inserting a nonperturbing test particle at the position r, we have

where U(r> is the interaction energy of the test particle with the rest of the system, including hard core contributions, and the angular brackets denote an ensemble average. For the free ions, we have to make random insertion throughout the cell rather than just at one point. Although it is rigor- ously correct for infinite systems, this expression in practice gives unacceptable finite volume effects unless very large systems are used. The reason is that the simulation cell is always electroneutral and therefore cannot respond properly to an inserted charge. It has been shown that most of the error can be removed by using a charge scaling te~hnique.'~ By introducing the scaling parameter A, eq. (5) may be rewritten as

where we have separated the hard core and electrostatic contributions to the energy. The correction is made by artificially scaling the charges of the free ions such that the simulated cell including the test particle is electroneutral at each A. This can be done, for example, by scaling the pair interaction between the inserted test particle charge A9, and the real particles q, by (1 - A9,/( 9, Nf)), where Nf is the total number of free ions. Typically, it is sufficient to evaluate the integrand in eq. (6) at a handful of A values. Although the expression is simple, care must be taken in the scaling procedure upon inserting several test charges simultaneously.'8 It also has the disadvantage that a large number of exponential functions must be evaluated if the excess chemical potentials at several different positions are to be calculated. Therefore,

a simpler but still reliable correction term is desir- able. This can be obtained by making approxima- tions to the aforementioned scaling technique. We approximate the position of the bound ion to the center of the cell, and at each A value we smear out the neutralizing charge - A q , as a uniform background throughout the cell. For a particular A we then have the correction to the energy as

where

(7)

where R and V are the radius and the volume of the cell, respectively, and the integration is made over the volume of the cell. Introducing eqs. (7) and (8) into eq. (61, the configuration independent correction term to the excess chemical potential becomes

pC/ldAA = -3Pq:/(16m,~,X) (9)

Thus, the correction term is proportional to the square of the inserted charge and inversely proportional to the cell radii.

Widom's test particle method may also be applied to the simultaneous introduction of several test charges. With the simple correction method, the excess chemical potential for the M interacting test charges 9t,(rl), qf$r2), . . . , 9t,(rn) becomes

0

This shows that the correction term vanishes in the special case of a neutral combination of inserted charges. In general, the fluctuations in the measured excess chemical potential will increase with the number of particles inserted simultaneously as well as their charges.

In deriving the correction term, one may easily relax the approximation of having the test charge



11

10

X

=I. a.

9 -

in the center of the cell and also in a simplified way take the excluded volume due to the biomolecule into account. For the bound ions, this extra effort would in most cases only make a slight improvement, because in general the size of the biomolecule is small compared to the size of the cell.

A simple correction term for the free ions can also be obtained. As before, we smear out the neutralizing background over the whole cell. The interaction between the inserted test particle and the neutralizing background obviously depends on its position. We therefore approximate the correction term as the average obtained from all positions of the test particle in the cell. Using Gauss's law, one obtains

+

+ + -

+

0

0

Comparing with eq. (9), the correction term for a free ion becomes a factor of 0.8 times the correction term for a bound ion. However, except in cases where effects of the protein concentration are studied, the chemical potentials of the free ions are more consistently calculated from a uniform solution with periodic boundary conditions. This re- duces effects from the spherical boundary because, except at high protein concentration, low salt concentration, or highly charged proteins, a free ion effectively sees a uniform salt solution.12

A simple example may serve to demonstrate the usefulness of the correction derived here compared to the more complex charge scaling procedure used in eq. (6). Figure 2 compares the uncorrected Widom results and the two correction procedures for the binding of two calcium ions to the protein calbindin D,, in a 150-mM KC1 solution. In this particular case, both correction meth- ods are converged for a system containing about 100 charges, in which case the uncorrected method gives an error of several kT. Note that the uncorrected method has an almost linear dependence on NJ-1/3.

MULTIPARTICLE INSERTIONS

Using eq. (21, the shift in proton affinity of the histidine residues upon oxidation of the Cu+ in

FIGURE 2. Finite volume effects in the excess chemical potential for two bound calcium ions to the protein calbindin D,,, in 150 mM KCI solution. The constant contribution from the fixed protein charges has been subtracted off. The number of free ions, N,, is varied from 58 to 408. (+) Widom's method without correction term, (0) simple correction term from eq. (lo), and (0) more rigorous correction term from eq. (6).

azurin may be written as

ApK = p{ pex(His = HisclCu2+)

where the two terms on the right-hand side mean the excess chemical potential for adding a proton to the histidine residue, given that the copper has the valency + 2 or + 1, respectively. With single- particle insertions, this requires two simulations.

Because the chemical potential is a state func- tion, we may rewrite this expression using

pex(His * His+lCu+)

= pex(His - His+;Cu2+* Cu+>

- peX(Cu2+ - Cu'lHis) (13)

where the first term is the excess chemical potential for the simultaneous addition of a positive unit charge to histidine and a negative unit charge to Cu2+. Combining these two equations shows that the necessary free energy changes to obtain a pK shift may all be calculated relative to the protein containing uncharged histidine and Cu2+. Thus, the multiparticle insertion technique allows a

374 VOL. 16, NO. 3


number of pK shifts to be obtained within the same simulation.

Results and Discussion

We have investigated proton binding to three histidine residues: His,,, His35, and His,,. Experi- mentally the pK for His,, could be measured for the reduced form only. It is not known which of the two positions, N, or N,, on the histidines is protonated. Consequently, both positions for all three residues were studied in the simulation. The experimental ionic strength varied between 20 mM and 40 mM. The results from two simulations using multiparticle insertions are compared with experiments in Table I. Each simulation required 8 central processing unit (CPU) seconds on an IBM RISC 6000/550, or 1.3 CPU seconds per pK shift. The first column in Table I gives the predicted shifts at zero salt concentration, which is given by the static contribution from the copper ion. The simple method predicts shifts close to the experimental data. In spite of the short simulations, the standard deviation of the shifts was below 0.002 pK units. The insensitivity of the shifts to the salt concentration is a consequence of the low net charge of the protein. One should also note that the shifts are rather insensitive to which of the two positions is protonated. In the two aforementioned simulations, 1000 moves per ion were attempted, and on average test particle insertions were made at every tenth attempted move. The first 10% of the simulation was used for equilibration, and the

TABLE 1. - Simulated and Experimental pK Shifts of Histidines upon Oxidation of Cu+.

Position ApK, ApK,, mM

His,, N, 0.175 0.088

His,, N, 0.162 0.076 His,, N, 0.371 0.278 His,, N, 0.446 0.326 His,, N, 0.249 0.160

His,, N, 0.215 0.1 26

0.070

0.059 0.257 0.331 0.142

0.109

0.07

0.25

The experimental ionic strength ranges from 20 rnM to 40 mM. At each histidine, both the N, and N, position had been considered as possible sites. The standard deviation in the simulated shifts are below 0.002 pK unit. The p K shifts at zero salt concentration are trivially obtained from the distance between the copper ion and protonation position.

rest was used for accumulation of averages. The standard deviation was obtained from the fluctuations of 10 subaverages.

There are at least two advantages of using the multiple perturbation approach compared to run- ning separate simulations. First, this method allows all the shifts to be obtained within one simulation, whereas using single-particle insertions requires several simulations. In this particular case, two simulations would have been necessary, one for each oxidation state of the copper. However, in a more general case, the gain from using multiple insertions may be much higher. It is illustrative here to compare with calculations using the PB approximation, which at each salt concentration requires separate free energy calculations for each of the 14 involved states of the protein. Second, because the excess chemical potentials are obtained within the same simulation, correlations in their fluctuations bring down the uncertainties in the final pK shifts. Table I1 shows the excess chemical potentials contributing to the pK shifts at N, of His,,. It is seen that the sum of the contributions contains smaller fluctuations than the indi- vidual terms. In fact, to obtain the same fluctuations in the pK shift using separate simulations would require approximately 20 times ( 2 * {0.01/ 0.003}2 = 22) longer runs. One drawback with multiparticle insertions must be pointed out. In general, we expect larger finite volume effects when simultaneous insertions are attempted. A check for finite volume effects shows that increasing the cell volume 10 times gives pK shifts about 0.01 unit larger than those in Table 1. Thus, the predicted numbers in Table I are essentially free of boundary effects. A similar test of pK shifts obtained by single-particle insertions shows that the finite volume effects are less, about 0.005 p K units.

The space outside the proto5ation sites was divided into spherical shells of 2 A thickness. Dur- ing the simulation, the number of ions appearing in these shells was counted to give the concentra-

TABLE II. Different Chemical Potential Contributions to the pK Shift at the N, Position of His,,.

a. ppex(His * His+ICu2+) = - 1.09 f 0.01 b. ppex(His * His+, Cu2+- Cu') = -0.12 f 0.02 c. ppex(Cu2+* Cu+IHis) = 1.13 f 0.02 a - b + c = 0.161 f 0.003

The electrolyte concentration is 0.40 rnM. the quoted standard deviations are estimated as described in the text.

- JOURNAL OF COMPUTATIONAL CHEMISTRY 375


tion profiles. Figure 3 shows the resulting profiles outside the N, of His,, and the copper at a salt concentration of 40 mM. The noise is large due to the short simulation run. It may seem surprising then that the fluctuation in the free energies are small. The reason is that the chemical potential is a global property obtained from an integration over all ionic positions, whereas contributions to the distribution functions only come from ions appearing in each particular shell. Indeed, there is an accurate alternative for obtaining distribution functions based on the calculation of chemical potentials and thermodynamic arguments.” The number of ions within a given distance from a site may be obtained by an integration of the profiles in Figure 3. The results, which contain smaller fluctuations, are shown in Figure 4. It is clear that there are more positive than negative ions around the nitrogen, whereas the opposite holds for the copper, an effect due to the electrostatic moments of the protein.

It is of interest to compare the present results with previous LPB calculations on the same system. Results collected from Juffer et al.19 and Bash- ford et a1.” are shown in Table 111. In these calculations, the dielectric constants of the protein and the solution were taken to be 4 and 80, respectively. Juffer et al. used both GROMOS” and CHARMM’’ atomic parameters for the protein, whereas Bash- ford et al. used CHARMM parameters. Only results for the salt-free systems are shown here, but their observed salt effects are similar to the ones

40

9

g 20

E v

0

0

n - 0 10 20

r (A)

FIGURE 3. The spherically averaged ionic concentration profiles outside N, of His,, and the Cu2+ ion. The electrolyte concentration is 40 mM. Site N,: (0) positive ions and (0) negative ions; site Cu’? ( A )

positive ions and (A) negative ions.

r (A)

FIGURE 4. The number of ions within a distance r from the sites N, of His,, and the Cu2+ ion. The same system and legend as in Figure 3.

presented in this work. It is seen that a range of numbers is obtained for identical protonation sites. We further note that for all residues the differences between the N, and N, positions are enhanced. Averaged over the two positions, the results of Bashford et al. compare as well with experiment as the present simulations, whereas the results of Juffer et al. give a slight overestimation. One should note that for His,,, for which there are unfortunately no experimental results available, there is a difference of up to two orders of magni- tude in the predicted pK shifts, even when the same parameter sets are used. This particular unit is located more internally in the protein as well as closer to the copper atom. As explained in ref. 19,

TABLE 111. The pK Shift at Zero Salt Concentration Predicted by the LPB Approximation.

APK APK APK

Position et a1.19 et a1.I9 et aI.,O

(GROMOS) (CHARMM) (CHARMM) Juffer Juffer Bashford

~

His,, N, 0.27 0.19 0.18 Nc 0.18 0.15 0.16

HIS,, N, 2.68 2.28 1.27 H’%, N, 3.91 3.51 1.78 His,, N, 0.95 0.89 0.54 His,, N, 0.41 0.45 0.28

The protein is given a dielectric constant of 4, whereas 80 is used for the electrolyte region. Juffer et al.” used both GROMOS and CHARMM parameters, whereas Bashford et aLZ0 used CHARMM parameters.

376 VOL. 16, NO. 3


the difference may be attributed to the different techniques used in constructing the surface of the protein. Thus, using a low dielectric interior of the biomolecule may give plenty of room for fitting the theoretical predictions to experimental results. In reality, nonelectrostatic effects (e.g., structural changes and steric interactions) may significantly contribute to the observed pK shifts, and so can titrations of the biomolecule. The nonelectrostatic effects are not explicitly included in continuum treatments using fixed biomolecular structures, and titration effects are generally neglected. The physical significance of the actual parameter values giving the best agreement between continuum modeling and experimental results should therefore be carefully and critically examined. On comparison, the calculations with a uniform dielectric constant are insensitive to changes in the size of the atoms as well as to the exact structure of the biomolecule.

Finally, the speed of the simulations demonstrated here makes it possible in the future to study electrostatic interactions in very large aggre- gates ( - 100,000 atoms) or to try more detailed biomolecular models without making the calculations excessively expensive. For instance, a limited flexibility in the biomolecule may be introduced, or the question of simultaneous titration upon ion binding may be addressed.

Conclusion

We have presented a new, efficient method for Monte Carlo simulations of electrostatic free energy changes in biomolecular solutions, using a multiparticle insertion technique. A simplified correction for finite volume effects in the excess chemical potentials is introduced. The advantage of the multiparticle insertion technique over single-particle insertions is twofold. One can obtain results in one simulation that would otherwise require several separate simulations. In addition, correlat- ed fluctuations in the excess chemical potentials may yield cancellations in the fluctuations of their relevant sums.

The method was applied to the copper-containing redox protein azurin. The change in proton binding at six positions in the protein induced by a change in the oxidation state of the copper was calculated in one simulation using less than 10 CPU seconds on a workstation. The same calcula-

tions using the PB equation would require more than a dozen separate calculations. Comparison with experiment shows that the simulation model with a uniform dielectric constant throughout the system yields results which compares well with experiments.

Acknowledgments

B. S. wishes to acknowledge financial support from the Swedish Natural Science Research Coun- cil (NFR).

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an efficient simulation technique for electrostatic free energies with applications to azurin

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