1990, computer simulation of molecular dynamics- methodology, applications, and perspectives in...

7/30/2019 1990, Computer Simulation of Molecular Dynamics- Methodology, Applications, And Perspectives in Chemistry by

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Computer Simulation of Molecular Dynamics:Methodology, Applications, and Perspectives in ChemistryBy Wilfred E van Gunsteren* and Herman J. C. Berendsen*

Dur ing recent decades i t has becom e feasible to simulate the dynam ics of molecular systemso n a computer . T he method of molecular dynamics (MD ) solves Newton's equations of motionfo r a molecular system, which results in trajector ies for all ato ms in the system. Fro m theseatomic trajector ies a var iety of proper t ies can be calculated. The aim of com pute r s imulat ionsof molecular systems is t o c o m p u t e macroscopic behavior f rom microscopic interactions. Themain contr ibutions a microscopic considerat ion can offer are ( 1 ) the under s tand ing and(2) interpretat ion of exper imental results, (3 ) semiquan ti tat ive estimates of exper imental re-sults , an d (4) the capabil i ty to interpolate or extrapolate exper imental data in to regions th atar e only difficultly accessible in the labora tory. O ne of the two basic problems in the field ofmolecular modeling a nd simulat ion is how to ef f iciently search the vast conf igurat ion spacewhich is spanned by al l possible molecular conformations for the global low (f ree) energyregions which will be popu lated by a molecular system in thermal equil ibrium. Th e other basicproblem is the der ivat ion of a suff iciently accurate interact ion energy function or force fieldfor the molecular system of interest . An im porta nt par t of the ar t of com pute r s imulat ion isto choose the un avoidable assum ptions, approximations and simplif icat ions of the molecularmodel an d com puta t iona l procedure such tha t their contr ibutions to the overal l inaccuracy areof compa rable s ize, without af fect ing significantly th e proper ty o f interest . Methodolog y andsome pract ical applicat ions of computer simulation in the field of (bio)chemistry will bereviewed.

1. IntroductionCom puta t ional chemis t ry is a branch of chemistry that

enjoys a growing interest f rom exper imental chem ists . In thisdiscipl ine chem ical problems are resolved by com puta t iona lmethods . A model of the real world is constructed, bothmeasurab le and unm easurab le p roper t ies a r e com puted , andthe former are compared with exper imental ly determinedproper t ies . This compar ison validates or invalidates themodel that is used. In the for me r case the model m ay be usedto study relat ionships between model parameters and as-sumptions or to predic t unknow n or unmeasurable quanti-ties.

Since chemistry concerns the s tudy of proper t ies of sub-stances or molecular systems in terms of atoms, the basicchallenge facing com puta t iona l chemistry is to descr ibe oreven predict1. the s tructure and stabil i ty of a molecular system,2. the ( f ree) energy of dif ferent s tates of a molecular system,3 . reaction processes within molecular systems

[*I Prof. Dr. W. F. van Gunsteren"'Department of Physical ChemistryUniversity of GroningenNijenborgh 16, NL-9747 AG Groningen (The Netherlands)Department of Physics and AslronomyFree UniversityDe Boelelaan 1081, NL-1081 HV Amsterdam (The Netherlands)Prof. Dr. H. J. C. BerendsenDepartment of Physical ChemistryUniversity of GroningenNijenborgh 16, NL-9747 AG Groningen (The Netherlands)Eidgenossische Technische HochschuleETH ZentrumUniversititstrasse 6 , CH-8092 (Switzerland)

['I Present address:

99 2 0 C H Verla~. s~e .~e l l schu/ th H , 0.6940 Weinheim, I990 0570-083

in terms of interact ions a t the a tom ic level. These three basicchallenges are listed acc ording to increasing difficulty. T hefirst challenge concerns prediction of which s tate of a systemhas the lowest energy. The second challenge goes fur ther ; i tinvolves prediction of the relative (free) energy of differentstates. Th e third challeng e involves prediction of t h e d y n am -ic process of change of s tates .

Chemical systems are general ly too inhomogeneous andcomplex to be treated by analyt ical theoret ical methods. T hisis illustrated in Fig ure 1. Th e treatment of molecular systemsin the gas phase by quan tum m echanical methods is s traight-forw ard; i f a classical s tat is t ical mechanical approxim ationis permitted the problem becomes even trivial. This is d u e t othe possibility of reducing the many-particle problem to afew-par ticle one based on the low density of a system in thegas phase. In the crystall ine sol id s tate, t reatment by qua n-tum mechan ica l or classical mechanical methods is madepossible by a reduction of the many-par t icle problem to afew-(quasi)par t icle problem based on symmetry proper t iesof the solid state. Between these two extremes, that is, forl iquids, macromolecules, solut ions, am orph ous solids, etc. ,one is faced with an essentially many-par t icle system. N osimple reduction to a few degrees of freedom is possible, a nda full treatment of many degrees of freedom is required ino r d e r to adequately descr ibe the proper t ies of m olecular sys-tems in th e fluid-like state. This state of affairs has tw o directconsequences when treating fluid-like systems.1. One has to r esor t to numer ica l simulation of the behavior

of the molecular system on a computer , which2. produces a statistical ensemble of conf igurat ions repre-

sen t ing the s ta te o f th e sys tem.If one is only interested in static equilibrium properties, it

suff ices to generate a n ensemble o f equil ibr ium states , which3/90/0909-0992 8 3.50+ 2510 Anger,. Chem. lnr. Ed . Engl. 29 ( I9901 992-I023


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may lack any temporal correlations. To obtain dynamic andnon-equilibrium properties dynamic simulation methodsthat produce trajectories in phase space are to be used. Theconnection between the microscopic behavior and macro-scopic properties of the molecular system is governed by thelaws of statistical mechanics.

Figure I also shows the broad applicability of computersimulation methods in chemistry. For any fluid-like, essen-tially many-particle system, it is the method of choice.

CRYSTALLINE LlOUlD STATE GA S P H A S ESOLID STATE MACROMOLECULES

QUANTUMMECHANICS

CLASSICALSTAT1ST I CA LMECHANICS

s t i l lp o s s / b l e p o s s i b l ei m p o s s i b l e

t r i v i a la s yREDUCTION t REDUCTION

lo7

to f e w d e g r e e s e s s d n t i a l t o f e wof f r e e d o m by - ,,any-particTa r t i c l e s bySYMMETRY s y s t e m DILUTION- AM 360/195

Fig. 1 . Classification of molecular systems. Systems in the shaded area areamenable to treatment by computer simulation.

lo1

The expanding role of computational methods in chemis-try has been fueled by the steady and rapid increase in com-puting power over the last 40 years, as is illustrated in Figure2. The ratio of performance to price has increased an orderof magnitude every 5-7 years, and there is no sign of anyweakening in this trend. The introduction of massive paral-lelism in computer architecture will easily maintain the pres-ent growth rate. This means that more complex molecularsystems may be simulated over longer periods of time, orthat it will be possible to handle more complex interactionfunctions in the decades to come.

The present article is concerned with computer sirnulationof molecular systems. In Section 2 the two basic problemsare formulated, a brief history of dynamic computer simula-tion is presented, the reliability of current simulations is dis-cussed, and the usefulness of simulation studies is consid-ered. Section 3 deals with simulation methodology: choice ofcomputational model and atomic interaction function (Sec-tion 3.1), techniques to search configuration space for lowenergy configurations (Section 3.2), boundary conditions(Section 3.3), types of dynamical simulation methods (Sec-

-

tion 3.4), algorithms for integration of the equations of mo-tion (Section 3.9, and finally equilibration and analysis ofmolecular systems (Section 3.6). In Section 4, a number ofapplications of computer simulation in chemistry are dis-

Flops [Floating Point Operatlons per second 1teraflop'o''lO'O ///////gigaflop /,/NEC sxzwCRAY X -MP .FUITSU V P 2000' CRAY -2. BER-205,RA Y - 1

slope:l o x per6 years

30

Fig. 2. Development of computing power of the most powerful computers.

cussed and examples are given. Finally, future developmentsare considered in Section 5. For other relatively recentmonographs on computer simulation the reader is referredto the literature (see Refs. [l-91).

2. Computer Simulation of Molecular Systems2.1. Two Basic Problems

Two basic problems are encountered in the computer sim-ulation of fluid-like molecular systems:1 . the size of the configurational space that is accessible to the

molecular system, and2. the accuracy of the molecular model or atomic interaction

function or force field that is used to model the molecularsystem.

Wilfred E: van Gunsteren was born in 1947 in Wassenaar {The Netherlands). In 1968 he gaineda B.Sc. in physics at the Free University of Ams terdam; in 1976 he was awarded a "Meester"in Law, andin 1976 a Ph.D. in nuclearphysics. After postdoc years at the University of Groningen(1976-1978) and at Harvard University (1978-1980) he was, from 1980 until 1987, seniorlecturer and, until August 1990, Professor for Physical Chemistry at the University ofcroningen.Since September 1987 he has also been Professor of Computer Physics at the Free UniversityAmsterdam. In 1990 he was offered and accepted a professorship of Computer Chemistry at theETH Ziirich. He is holder of a gold medal for research of the Royal Netherlands ChemicalSociety; in 1988 he was Degussa Guest Professor at the University of Frankfurt. His maininterests center on the physical fundamentals of the structure and function of biomolecules.

Angew. Chem. In r Ed. Engl. 29 (1990) 992-1023 993


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Table 1. Models at different levels of approx imationModel Degrees of freedom Example of

left removed predictable property force field- quan tum mechanical nuclei, electrons nucleons reactions- all atoms, atoms electrons binding charged

polarizable dipoles ligand- all atoms solute +solvent d po1e hydration

atoms- all solute atoms solute atoms solvent gas phase conformation- groups of atoms atom groups individual atoms folding topology

as balls of macromolecule

2.1.1. Size of the Configurational SpaceThe sim ulat ion of molecular systems at non-zero tem pera-

tures requires the gene ration of a statistically rep resentativeset of configura tions, a so-called ensemble. The proper ties o fa system are def ined as ensemble averages or integrals overthe conf igurat ion space (or m ore general ly p hase space) . Fora many-par t icle system the averaging or integrat ion wil l in-volve m any degrees of f reedom, and as a result c an only becarr ied out ov er par t of the conf igurat ion space. The smallerthe conf igurat ion space, the bet ter t he ensemble average orintegral can be approxima ted. W hen choosing a model f romwhich a specific proper ty is to be co mp uted , one would l iketo explicit ly include only those degrees of f reedom on whichthe required proper ty is dependent .

In Table 1 a hierarchy of models is shown, in whichspecific types of degrees of freedom are successively re-moved. Examples are given of proper t ies that can be com-puted at the dif ferent levels of approximation. I f one is inter -ested in chem ical react ions, a quan tum mechan ica l t rea tmentof electronic degrees of f reedom with th e nuclear coordinatesas parameters is mandatory. The intranuclear degrees off reedom (nucleon motion) are lef t out of the model . Thecomputing power required for a quantum mechanical t reat-ment scales at least with the third power of the number ofelectrons N , tha t are considered. Such a treatm ent is onlypossible for a l imited number of degrees of f reedom. At anext level of approximation the electronic degrees of f ree-dom are r emoved f rom the model and approx imated by(point) polar izabil i ties . Such a model , al lowing fo r motionsof polar izable atoms, could, e.g . , be used to compute thebinding proper t ies of polar or charged l igands which wil lpolar ize the receptor . At this level of modeling the com put-ing effort scales with the squ are of the number of atom s N , .This means tha t many more a tom s can be cons idered than a tthe quantum level . Removal of the polar izabil i ty of theatoms yields a next level of approximation, in which onlyatom ic posi t ional degrees of f reedom are considered an d themean polarization is included in the effective interatomicinteract ion function. Such a model al lows for the s tudy ofsolvation p roper t ies of molecules. Wh en studying solut ions,a next level of approximation is reached by omitting thesolvent degrees of f reedom from the model and simulta-neously ada ptin g the interact ion function for the solute suchthat i t includes the mean solvent ef fect . When studyingprotein folding, the conf igurat ion space spanned by all

Coulombionic models [lo]OPLS [ l l ]GROMOS [12]MM2 [13]LW [14] 1

~

increasing:simplicity,speed of computation,search power,time scale

decreasing :complexity, accuracy ofatomic properties

protein atoms is s t i l l far too large to be searched for lowenergy conformations. In this case a fur ther reduction in i tcan be obtained by representing whole groups of atoms, f orexample an am ino acid residue, as one o r two balls (3 o r 6degrees of f reedom).

Fro m this discussion i t is clear that the level of approxima-t ion of the model that wil l be used in the s imulat ion wil ldepend on the specif ic proper ty one is interested in. Thevar ious force f ields tha t ar e available correspond to dif ferentlevels of approximation, as is illustrated in the right half ofTable 1 . There is a hierarchy of force fields.

Once the level of approxima tion has been chosen, th at is ,the types of degrees of f reedom in the model , one m ust decidehow m any degrees of f reedom (electrons, atom s, etc.) are tobe taken into account. How small a system can be chosenwithout seriously affecting a proper representat ion of theproper ty of interest? Th e smaller the s ize of the system thebetter i ts degrees of f reedom ca n be sampled, o r in dynamicterms, the longer the time scale over which it can be simu-lated.

We may sum marize the firs t basic problem in the compu t-er s imulat ion of molecular systems as fol lows: the level ofapproximation of the model should be chosen such thatthose degrees of f reedom that are essential t o a p rope r eval-uation of the quanti ty or prope r ty of interest can be suffi-ciently sa mpled. In pract ice any choice involves a com pro-mise between type and number of degrees of f reedom andextent of the s imulat ion on the on e hand, an d the availablecomput ing power on the o ther .

2.1.2. Accuracy of Molecular Model and Force FieldWhen the degrees of f reedom are infinitely dense or long

sampled the accuracy by which var ious quanti t ies are pre-dicted by a sim ulation will depe nd solely on the qu ality of theassumptions and approximations of the molecular modelan d intera tom ic force field. At the level (Table 1)of q u a n t u mmechanical modeling the basic assumption is the val idi ty ofthe Born-Oppe nheimer approximation separat ing electronicand nuclear motion. The interact ion between point atomsand electrons is descr ibed by Coulombs law and th e Pauliexclusion principle. W hen ex cluding chem ical reactions, lowtemperatures o r detai ls of hydrogen ato m mo tion, i t is rela-tively safe to ass um e that th e system is governed by th e lawsof classical mechanics. The atomic interact ion function iscalled an effective interaction since the average effect of the

994 A n g rw . Chem. I n t . Ed. Eng l . 29 (1990) 992-1023


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omitted (electronic) degrees of freedom has been incorporat-ed in the interaction between the (atomic) degreesof freedomexplicitly present in the model. To each level of approxima-tion in Table 1 there corresponds a type of effective interac-tion or force field. For example, a pair potential with anenhanced dipole moment (enhanced atomic charges) may beused as an effective potential that mimics the average effectof polarizability.

In view of the different levels of approximation of molec-ular models it is not surprising that the literature contains agreat variety of force fields. They can be classified alongdifferent lines:

type of compound that is to be mimicked, e.g. carbohy-drates, sugars, polypeptides, polynucleotides ;type of environment of the compound of interest; e.g. gasphase, aqueous or nonpolar solu tion;range of temperatures covered by the effective interaction;type of interaction terms in the force field; e.g. bondstretching, bond angle bending, torsional terms, two-,three- or many-body nonbonded terms;functional form of the interaction terms; e.g. exponentialor 12th power repulsive nonbonded interaction;type of parameter fitting, that is, to which quantities areparameters fitted, and are experimentally or ab-initio the-oretically obtained values used as target values.We note that the choice of a particular force field should

depend on the system properties one is interested in. Someapplications require more refined force fields than others.Moreover, there should be a balance between the level ofaccuracy or refinement of different parts of a molecularmodel. Otherwise the computing effort put into a very de-tailed and accurate part of the calculation may easily bewasted due to the distorting effect of the cruder parts of themodel.

2.1.3. Assumptions, Approximations and LimitationsWhen using a particular model to predict the properties of

a molecular system, one should be aware of the assumptions,simplifications, approximations and limitations that are irn-plicit in the model. Below, we list the four most importantapproximations and limitations of classical computer simu-lation techniques which should be kept in mind when usingthem.2.1.3.1. Classical Mechanics of Point Masses

A molecular system is described as a system of point mass-es moving in an effective potential field, which is generally aconservative field, i.e. it only depends on the instantaneouscoordinates of the point masses. The motion of the pointmasses is with a sufficient degree of accuracy governed by thelaws of classical mechanics. These assumptions imply thefollowing restrictions to modeling:- low-temperature (0- 10K ) molecular motion is not ade-- detailed motion of light atoms such as hydrogen atoms is- description of chemical reactions lies outside the scope of

quately described;not correctly described even at room temperature;classical simulation methods.

For a short discussion of the inclusion of quantum correc-tions to a classical treatment o r of the quantum (dynamical)simulation techniques that are presently under development,we refer to Section 5.1.

2.1.3.2. System Size or Number of Degrees of Freedomto be IncludedOnly a rather limited number of atoms can be simulated

on a computer. Simulations of liquids typically involve 10-103 atoms, simulations of solutions or crystals of macro-molecules about 103-2 x lo4 atoms. Generally, the systemsize is kept as small as possible in order to allow for a sufti-cient sampling of the degrees of freedom that a re simulated.This means that those degrees of freedom that are not essen-tial to the property one is interested in should be removedfrom the system. The dependence of the property of intereston the size of the system may give a clue to the minimumnumber of degrees of freedom required for an adequate sim-ulation of it. The larger the spatial correlation length of theproperty of interest, the more atoms are to be included in thesimulation.2.1.3.3. Sufliciency of Sampling of Configuration Spaceor Time Scale of Processes

Computer simulation generates an ensemble of configura-tions of the system. Whether the generated set of configura-tions is representative for the state of the system depends onthe extent to which the important (generally low energy)parts of the configuration space (or, more generally formu-lated, phase space) have been sampled. This depends in turnon the sampling algorithm that is applied. This should beable to overcome the multitude of barriers of the multi-dimensional energy surface of the system. In dynamic simu-lations, the time scale of the process that is mimicked islimited. Presently, molecular simulations cover time periodsof 100- 1000 picoseconds. In the case of activated processeslonger time scales can be reached by using specialtricks.[5 9 Essentially slow processes, like the folding of aprotein, are still well out of reach of computer simulation.We note that the observation that a property is independentof the length of a simulation is a necessary but no t a sufficientcondition fo r adequate sampling. The system may just residefor a period longer than the simulation time in a certain areawithout being effected by a nearby region of much lowerenergy from which it is separated by a large energy barrier.2.1.3.4. Accuracy of the Molecular Model and Force Field

As is illustrated in Table 1, there exists a variety of molec-ular models and force fields, differing in the accuracy bywhich different physical quantities are modeled. The choiceof a particular force field will depend on the property andlevel of accuracy one is interested in.

When studying a molecular system by computer simula-tion three factors should be considered (cf. Fig. 3).1. The properties of the molecular system one is interested in

should be listed and the configuration space (or timescale) to be searched for relevant configurations should beestimated.

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O F INTEREST

f/

1 POWER 1Fig. 3. Choice of molecular model, force field and sample size depends on1) the property one is interested in (space to be searched), 2) required accuracyof the prediction, 3) the available computing power to generate the ensemble.

2. The required accuracy of the proper t ies should be speci-fied.

3. Th e avai lab le compu ter t ime shou ld be es t imated .Given these three specif ications the m olecular model a nd

force field should be chosen. As is indicated in Table 1 thereis a t ra de off between accuracy of the force f ield o n the on ehand and the searching power or t ime scale that can beattained on the other . We note tha t in many pract ical casesone should ab and on the idea of t rying to s imulate the systemof interest , s ince the available computing power wil l notallow for a sufficiently accurate simulation.

2.2. History of Dynamic Computer SimulationThe growth in the number o f app l ica t ions o f computer

simulat ion methods in chemistry and physics is direct lycaused by the rapid increase in computing power over thelast four decades (Fig. 2). Table 2 shows the development o f

Table 2. History of computer simulation of molecular dynamics.~

Year System Length of Required cpusimulation time on super-is1 computer [h]

1957 hard two-dimensional disks [17]1964 monatomic liquid [i s] lo-" 0.051971 molecular liquid [19] 5 x lo- ' ' 11971 molten salt [20] 10-11 11975 simple small polymer 1211 10-11 11971 protein in vacuo [22] 2 x 10-'1 41982 simple membrane [23 ] 2 x 10-10 41983 protein in aqueous crystal [24] 2 x o- ' ' 301986 DNA in aqueous solution [25 ] 10-'0 601989 protein-DNA complex in solution [26] lo -' ' 300

large polymers 10-8 103reactions 107macromolecular interactions 10-3 108protein folding lo-' 109

the applicat ion of molecular dynamics s imulat ion in chemis-try. Alder a n d Wainwright pioneered the method using 2-di-mensional hard disks. Rahman,wh o can real ly be consideredthe f a ther of he field, simulated liquid arg on in 1964, liquidwater in 1971, an d super ionic con duc tors n 1978.[271 e aIsomade many con t r ibu t ions to the m ethodology , and s t imula t-

ed the dissemination of the technique in the scientific com-munity. In the seventies , the transi t ion f rom atomic tomolecular liquids was made: rigid molecules like water(1971), flexible alka nes (1975), an d a small protein, trypsininhibitor (1977). The application to molten salts (1971) re-quired the development of methods to handle long-rangeCo ulom b interact ions. The eighties brought s imulat ions ofbiomolecules of increasing size in aqueous solution. For amore thorough review of the past and present of dynamiccom pute r simulat ion in chemistry and physics we refer to them o n o g r ap h s o f McCammon an d Harveyr5I an d Allen an dTildesley.[61

From Table 2 it is also clear that a supercomputer is stillmany orders of magnitude slower than n atur e: a s tate of thear t s imulat ion is abo ut 1015 imes s lower than natu re . Man yinteresting systems and processes still fall far outside thereach of compu ter s imulat ion. Yet, with a growth rate of afactor 10 per 5-7 years for (super)computing power thespeed o f s imulat ion wil l have caught up with that of naturein abou t 100 years .

2.3. Accuracy and Reliability of Computer Simu lationTh e reliabil ity of predict ions ma de o n the basis of com put-

er simulation will basically depend on two factors:1 ) whethe r the molecu lar mod el and force field ar e sufficient-ly accurate, an d 2) whether th e conf igurat ion spa ce accessi-ble to the molecu lar system has been sufficiently thoro ughlysearched for low-energy conf igurat ions. Although the accu -racy of a predict ion may be est imated by consider ing theapproxima tions an d simplif icat ions of the model and com-putational procedure, the final test lies in a comparison oftheoret ical ly predicted and exper imental ly measured p rope r-ties. In order to p rov ide a firm basis fo r the applicat ion ofcomputer s imulat ion methods the results should be com-pared with exper imental data whenever possible.

In this context we would l ike to s tress that good ag reementbetween calculated and exper imental data doe s not necessar-il y mean that the theoret ical model under lying the calcula-t ion is correct . Good agreement may be due to any of thefollowing reasons:1. The model is correct , that is , any ot her assumption used

to der ive the model , or any other choice of parametervalues would give bad agreement with experiment.

2. The proper ty that is compared is insensi t ive to the as-sumptions or parameter values of the model , that is ,whatever parameter values are used in the m odel calcula-t ion, the ag reement with exper iment will be good.

3. Compensation of er rors occurs, ei ther by chance or byf i t ting of the model param eters to the desired proper t ies .A numbe r of examples of the last case, viz. good agree-

ment for the wrong reason, are given in Ref . [28]. Here, wewould only like to remind th e reader tha t it is relatively easy,when modeling high-dimensional systems with manyparameters , to choose or f i t parameters such that goodagreement is obtained for a limited number of observablequanti t ies . O n the oth er hand, disagreement between simula-t ion and exper iment may also have dif ferent causes: 1 ) th esimulat ion is not correct , the exper iment is, 2) the s imulat ionis correct , the exper iment is not.

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With these cau t ionary r ema rks in mind we would l ike totu rn to a n evaluat ion o f the accuracy o f cur ren t s imula tions tud ies by com par ing s imula ted wi th exper imenta l quan t i -ties. Table 3 co n t a i n s a scheme of the a tom ic quan t it i es o fmolecular systems for which com par i son between simulatedand exper imental values is feasible. F ou r phases are dist in-guished. Th e most interest ing phase f rom a chemical point ofview is t h a t o f a molecule in solut ion. For this phase only fewaccurately measured atomic proper t ies are avai lab le fo r com -par ison, leaving therm ody nam ic system proper t ies l ike thefree energy of solvation a s a t es t g round fo r the s imula t ion .Cons iderably m ore a tomic da t a ar e avai lable f rom crysta llo -graphic dif f ract ion exper iments : atomic pos i t ions and mo-bili ty , al thou gh the accuracy of the lat ter is much lower thanthat o f the fo rmer , due to the s impli fy ing approx im at ions(harm onic isotropic motion ) used in the crystal lographic re-f inement process.

Table 3. Possible comparison of simulated properties with experimental onesfor complex molecules.Atomic properties Experimental Phase

method gas solu- mem- crystaltion braneStructuri,~ positions- distance- orientationsMohili l i '

X-ray diffractionneutron diffractionN M RN M RX-ray diffraction[y ron diffraction

ray diffractionneutron diffraction

- B-factors~ occupancy factorsDynuniic p r o p r r w y- vibrational infrared

frequencies spectroscopy~ relaxation rates various N M R- diffusion N O spin labelT h r r m o d i n u m i r proper l i es- density~ free energy~ viscosity.

optical techniques

conductance

XX

XX

X

X

X X

X XX

X

In the fol lowing subsections examples will be given of acomp ar i son o f var ious a tomic a nd sys tem proper t ies fo r d i f-f eren t compoun ds . The examples are taken f rom ou r ownwork. and so are a l l based on the G R O M O S force f ield. ['2'They on ly serve as an ind ica tion o f the degree o f accuracythat can be obtained for dif ferent proper t ies using currentmodel ing methods .2.3.1. Atomic Properties: Positions and Mobilities in Crystals

From a s imula tion the average molecu lar s t ruc tu re can beeasi ly calculated and compared with exper iment. Table 4contains the deviat ion between simulated and measuredatomic pos i t ions fo r a number o f molecu lar c rys ta l s . Thestructural deviat ion depen ds on th e s ize of the molecule andranges f rom 0.2 A t o 1.2A.T h e n u m b er s a r e an av e r ag e ov e rp a r t or o v e r all ato ms in the molecules. Th is means that par t sof the molecule will deviate more a nd othe r par ts will deviate

Table4. Comparison of M D time-averaged structures with experimental X-rayor neutron diffraction structures.Molecule Size (number Root mean square ( R M S ) differ-

of glucoseunits or amino C, or C , all atoms excl.acid residues) atoms H atoms

ence in atomic positions [A]

cyclodextrin (a/P)1291 6 0.1310.35 0.2510.51cyclosporin A [30] 11 0.3 0.6trypsin inhibitor [31] 58 1 o 1.5subtilisin [32] 27s 1 0 !2

less f rom the exper imental s tructure. This is illustrated inFigure 4, which shows the deviat ion for the backbone C,a tom s as a funct ion o f r es idue num ber averaged over all fourBPTI (bovine pancreat ic t rypsin inhibitor) molecules in thecrystal unit cel l . Most of the atoms deviate less than 1 A.Figure 5 shows the roo t mean square ( rms) a tomic pos it ional

4 02630 k r[ nml OI L -

010 -006-002I . . . . . , . . , I u .10 20 30 LO 50 60N

Fig. 4. Root mean square difference between the trypsin inhibitor simulatedtime- and molecule-averaged C. atomic positions and the X-ray positions ac-cording to [31]. A =root mean square deviation, N =aminoacid residue num-ber.

f luc tua t ions ca lcula ted f rom the M D s imula t ion and f romthe crystal lographic B-factors . Th e largest discrepancy is s t i llr a ther smal l , abou t 0.3 A .

012

",:I , , , , , , , , , , ,10 20 30 LO 50 60NFig. 5. Root mean square positional flucta tions for the trypsin inhibitor atomsaveraged over the backbone atoms of each residue according to 1311. Solid line:simulated results. Broken line: fluctuations derived from a set of X-ray temper-ature factors. rms =root mean square fluctuation, N =amino acid residuenumber.

A mo re str ingent test of the s imulat ion is to check whetherpar t ially occupied a tom ic sites are reproduce d. Th e neutrondif f ract ion work on p-cyclodextrin shows tha t 16 hydrogenatom s occupy tw o al ternative s ites. In the crystal s imulat iona non-zero occupancy is observed for 84% of the hydrogensites , and for 62YO f the hydrogen a tom s the r e la t ive occu-

Angrw. C 'hem. Inr. Ed . Engl. 29 (1990) 992-1023 997


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pancy of the two alternative sites is qualitatively repro-duced

The molecular structure can also be described in terms ofhydrogen bonds. Generally the experimentally observed hy-drogen bonds are also observed in the simulation. Againcyclodextrin crystals form a sensitive test case for the simula-tions, since peculiar geometries such as three-center hydro-gen bonds, and dynamic processes such as flip-flop hydrogenbonds have been experimentally observed. Almost all exper-imentally observed three-center hydrogen bonds in crystalsof CL- and P-cyclodextrin are reproduced in M D simulation,even as far as the detailed asymmetric geometry is con-~ e r n e d . ' ~ ~ ]n a so-called flip-flop hydrogen bond the direc-tionality is inverted dynamically. In a MD simulation ofB-cyclodextrin, 16 of the 18 experimentally detected flip-flopbonds are

2.3.2. Atomic Properties: Distances in SolutionStructural data of solutions can be obta ined by two-

dimensional nuclear magnetic resonance spectroscopy (2D-NMR) with exploitation of the nuclear Overhauser (NOE)experiments.[351 he data come in the form of a set of upperbounds or constraints to specific proton-proton distances.This affords the possibility of comparing these proton-pro-ton distances as predicted in a simulation with the experi-mentally measured NOE bounds. In the literature,[251 uch acomparison is given for a set of 174 NOE's in an eight base-pair D NA fragment in aqueous solution: 80% of the NOEdistances are satisfied by the simulation within experimentalerror, the mean deviation is 0.22 A, and the maximum devi-ation amounts to 2.9 A.

2.3.3. Atomic Properties: Orientation of MolecularFragments in MembranesThe degree of order in a membrane or lipid bilayer can be

measured selectively along the aliphatic chain by deuteriumNM R spectroscopy.[361 n Figure 6[j71 both experimentaland M D order parameters are displayed for all CH, units ina sodium ion/water/decanoate/decanol bilayer system as afunction of the position of the carbon atom in the aliphaticchain. The experimentally observed plateau in the orderparameters is well reproduced.

O P 0 4 I OP 0 4" 1 ; 1 ; I 0 000 2 4 6 8 1 0N NFig. 6 . Carbon-deuterium order parameters as a function of carbonatom num-be r along th e aliphatic chain of decanol (upper panel) and decanoate (lowerpanel) [37]. The head group has number zero. Solid l ine: simulated result.Squares:experimental values. O.P. =order parameter, N =carbon atom n u m -ber.

2.3.4. Atomic Properties: Diffusion in Solutions and inMembranes

The dynamic properties of simulations of molecular sys-tems are hard to test by comparison with experiment. Anexception is the diffusion constant. Its calculation and com-parison with experiment is a standard test for models ofliquid water. The diffusion constant of the simple three-pointcharge (SPC) water model is 4.3 x cm2 s- ' comparedwith an experimental value of 2 . 7 ~ 1 0 - ~ c m ~ s - ~t305 K.[381 nclusion of a correction term for self-polarizationin the SPC model led to a reparametrization, to the SPC/E(extended simple point charge) model, which yields a consid-erably smaller diffusion constant of 2.5 x lo-' cm2 s-1.[391

In the bilayered sodium ion/water/decanoate/decanol sys-tem mentioned in the previous section the simulated diffu-sion constants are: 2.7 x cm2 s- ' (decanoate) and5.2 x cm2 s - (decanol), which values are to be com-pared with the value of 1.5 x cm2s- ' measured withnitroxide spin labels.

2.3.5. System Properties: Thermodynamic QuantitiesInstead of atomic properties, thermodynamic system

properties like the density or free energy of solvation can becalculated from a simulation and compared to experimentalvalues.

For a crystal of a cytidine derivative X-ray diffractiondata and M D simulation data were compared at two differ-ent temperatures.[401Upon raising the temperature from113 K to 289 K the volume of the unit cell increased by 2.5%in the simulation at constant pressure, which value is to becompared with an experimentally determined increase of3.7%. At 289 K the simulated density was only 1.3% toolarge. A slightly larger discrepancy was found in a MD sim-ulation of crystalline cyclosporin A: he simulated density of1.080 g cm-3 was 3.6% larger than the experimental valueof 1.042 g cm-3.[301 omparable deviations have been foundfor other systems: the SPC/E model yields 0.998 g cm -3at 306K, compared with an experimental value of0.995 g cm

Finally, we quote an example of a free energy of hydra-tion. For the solvation of methanol in water the standardGROMOS parameters yield a free enthalpy of 7 kJ mol-'compared with an experimental value of 5 kJ mol- *. [2s]However, when charged moieties are involved the accuracyby which free energy of solvation can be calculated is n obetter than about 10-20 kJ mol-'.[28.411

at 305 K.[391

2.4. Is Computer Simulation Useful?The prediction of properties by computer simulation of

complex molecular systems is certainly not accurate enoughto justify abandoning the measurement of properties. If ameasurement is not too difficult it is always to be preferredover a prediction by simulation. The utility of computersimulation studies does not lie in the (still remote) possibilityof replacing experimental measurement, but rather in its abil-ity to complement experiments. Quantities that are inaccessi-ble to experiment can be monitored in computer simulations.

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We consider the computer simulation of complex molecu-lar systems to be useful for the following reasons.1 .

2.

3.

4.

3.

It provides an understanding of the relation between mi-croscopic properties and macroscopicbehavior. In a com-puter a microscopic molecular model and force field canbe changed at will and the consequences for the macro-scopic behavior of the molecular system can be evaluated.During the last few years computer simulation has be-come a standard tool in the determination of spatialmolecular structure on the basis of X-ray diffraction or2D-NMR data.Under favorable conditions computer simulation can beused to obtain quantitative estimates of quantities likebinding constants of ligands to receptors. This is especial-ly useful where the creation of the ligand or the measure-ment of its binding constant is costly or time-consuming.Finally we mention the possibility of carrying out simula-tion under extreme (unobservable) conditions of temper-ature and pressure.

MethodologyHere we briefly describe the methodology of classical com-

puter simulation as it is applied to complex molecular sys-tems.

3.1. Choice of Molecular Model and Force Field3.1.1. Current Force Fields for Molecular Systems

A typical molecular force field or effective potential for asystem of N atoms with masses m, (i =1,2,. ..,N)nd carte-sian position vectors r; has the form ( 1 ) .

The first term represents the covalent bond stretching inter-action along bond b. It is a harmonic potential in which theminimum energy bond length b , and the force constant K bvary with the particular type of bond. The second term de-scribes the bond angle bending (three-body) interaction insimilar form. Two forms are used fo r the (four-body) dihe-dral angle interactions: a harmonic term for dihedral angles5 that are not allowed to make transitions, e.g. dihedralangles within aromatic rings, and a sinusoidal term for theother dihedral angles cp, which may make 360 degree turns .The last term is a sum over all pairs of atoms and representsthe effective nonbonded interaction, composed of the vander Waals and the Coulomb interaction between atoms i andj with charges qi nd q j at a distance r i j .

There exists a large number of variants of expression(I)."1 3 . 4 2 - 531 Some force fields contain mixed terms like

K,,[b - ,][O- ,], which directly couple bond-length andbond-angle vibrations.[421 thers use more complex dihedralinteractions terms.[43*41The choice of the relative dielectricconstant E, is also a matter of dispute. Values ranging fromc, =1 [ I ', 2] to E , =814'] have been used, while others take c,proportional to the distance r, , I 2 2 , 46,471 Sometimes theCoulomb term is completely ignored.[44]Although hydrogenbonding can be appropriately modeled using expression(1),["* 1 2 * 4 8 s 4 9 1 in some force fields special hydrogen bond-ing potential terms are used to ensure proper hydrogenb ~ n d i n g . [ ~ ~ - ~ ~ ]n other way to refine expression ( 1 ) is toallow for non-atomic interaction centers or virtual sites, thatis , interactions between points (e.g. lone pairs) not located onatoms.[501 or solvents, especially water, a variety of molec-ular models is a ~ a i l a b l e , [ ' ~ * ~ * ~ ~ ~ ~f which a few have beendeveloped explicitly for use in mixed solute-water sys-tems." 1 3 4 9 1 Models for nonpolar solvents like carbon tetra-chloride are also available.[5z1

When determining the parameters of the interaction func-tion (1 ) there are essentially two routes to take. The mostelegant procedure is to fit them to results (potential or field)of ab-initio quantum calculations on small molecular clus-ters. However, due to various serious approximations thathave to be made in this type of procedure, the resulting forcefields are in general not very satisfactory. The alternative isto fi t the force field parameters to experimental data (crystalstructure, energy and lattice dynamics, infrared, X-ray dataon small molecules, liquid properties like density and en-thalpy of vaporization, free energies of solvation, nuclearmagnetic resonance data, etc.). In our opinion the best re-sults have been obtained by this semi-empirical ap-proach." ' 7 5 31

We wish to stress that one should fit force field parametersto properties of small molecules, which may be considered asbuilding blocks of larger molecules such as proteins andDNA, and subsequently apply them to these largermolecules as a test without any further adaptations to im-prove the test results.

The choice of a particular force field should depend on thetype of system for which it has been designed. The MM2force field1131s based on gas phase structures of small or-ganic compounds. The AMBER[471and CHARMM1461force fields are aimed at a description of isolated polypep-tides and polynucieotides, in which the absence of a solvent(aqueous) environment is compensated by the use of a dis-tance-dependent dielectric constant c,. The ECEPP[43,41and UNICEPP[551orce fields use E, =4, whereas the GRO-MOS I 2 %91 force field uses E, = 1 , since it has been set up forsimulation of biomolecules in aqueous environment. Thisalso holds for the OPLS["] force field which is aimed at aproper description of solvation properties. Force fields thatare applicable to a more restricted range of compounds likeions,[s61 iquid metals,[571 or carbohydrate~,[~*]remanifold. The quality of the various force fields should bejudged from the literature concerning their application tomolecular systems.

3.1.2. Inclusion of PolarizabilityThe term in the interaction function ( 1 ) representing the

nonbonded interactions consists only of a summation overAngew. Che m . I n [ . Ed . Engl. 29 (1990) 992-1023 999


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all pair interactions in the system. Nonbonded many-bodyinteractions are neglected. Yet, inclusion of polarizability ofatoms or bonds will be inevitable if one would like to simu-late, e.g., the binding of a charged hgdnd, which will polarizethe part of the receptor to which i t is binding (Table 1) .About 10 % (4 kJ mol -') of the energy of liquid water ispolarization energy. The infinite frequency dielectric con-stant of water is E , =5.3, its effective dipole momentchanges from 1.85 D in the gas phase to 2.4 D in the liquid.An analysis of the contribution of polarizability to localfields in proteins is given in Ref. [59].

Inclusion of polarizability in the molecular model is nottoo difficult, as can be observed from the following simpleoutline.[601We consider a system of N point dipoles withCartesian position vectors r i, dipole moments p , and polariz-abilities a, (constant, isotropic). The induced dipoles Ap tobey a field equation (2), where E, denotes the electric field

NApt =a i E i=ai C K j ; , [ p j+Apj]j = 1

* i

at position r i and the field tensor Tij s given by Equation ( 3 ) .The field equation can be solved for Ap i either by the inver-( 3 )

sion of a matrix of size 3N x 3 N , or by iteration. The formermethod is impractical for a molecular system containingthousands of atoms. The latter method is well suited for usein MD simuiations, since the induced dipoIes at the previousMD integration time step, Api(t - At), will be an excellentstarting point for the iterative solution of the field equation(2) at time t , which yields Api(t). In this way inclusion ofpolarizability may increase simulation times by only 20 -100%.

Computer simulations that include polarizability haveonly been performed for a limited number of molecular sys-tems, such as ionic liquids,[10,61] and a singleprotein.[63. 41 Many aspects of the treatment of polarizabil-ity in MD simulations are still under investigation. What isthe most practical way to model polarizability?( I ) By inducing point dipoles, as sketched above,(2) by changing the magnitudes of (atomic) charges,( 3 ) by changing the positions of (atomic) charges.How should the size of the atomic polarizabilities ai e cho-sen, etc?

3.1.3. Treatmentof Long Range Coulomb ForcesThe summation of the last term in the interaction function

(3 ) covering the nonbonded interaction runs over all atompairs in the molecular system. It is proportional to N 2 , hesquare of the number of atoms in the system. Since the otherparts of the calculation are proportional to N , computation-al efficiency can be much improved by a reduction of thissummation.The simplest procedure is to apply a cut-offcriterion for the nonbonded interaction and to use a list ofneighbor atoms lying within the cut-off, which is only updat-ed every so many simulation steps. The cut-off radius R,usually has a value between 6 8, nd 9 8, and the neighbor listis updated about every 10 or 20 M D time steps. This proce-

dure does not introduce any errors as long as the range of thenonbonded interaction is smaller than R,. However, theCoulomb term in (1) is propor tional to r- ', which makes itlong-ranged. If the molecular model does not involve bare(partial) charges on atoms, but only dipoles or higher multi-poles, the electrostatic interaction term becomes proportion-al to F3, hich makes it of much shorter range. However,when dipoles are correlated over larger distances, as is thecase for secondary structure elements like a-helices inproteins, their interactions again become l~ n g -r a ng e d. [ ~~ JIn the following subsections we briefly discuss a variety ofmethods for the treatment of long-range electrostatic inter-actions in molecular systems.[66.671

3.1.3.1. Distance Dependent Dielectric ConstantA simple way to reduce the range of the Coulomb interac-

tion is to introduce a reIative dielectric constant proportionalto r, viz. E , =r (in 8,).The interaction becomes proportionalto r - ' . It is difficult to find a physical argument in favor ofthis approximation. Its overall effect is to effectively reduceall types of long-ranged interactions. Due to its simplicity ithas been incorporated into a number of current forcefields.[46. 71Nevertheless, we think this approximation is toocrude for practical applications.

3.1.3.2. Cut-off Radius and Neutral Groups of AtomsWhen applying a cut-off radius &, he discontinuity of

the interaction at a distance r =R, will act as a noise sourcein a M D simulation, and this will artificially increase thekinetic energy of the atoms and thus the temperature of thesystem. A possible way to reduce the noise is to multiply thenonbonded interaction term in (1)with a so-called switchingfunction (4), which satisfies the conditions S(R,) = 1,

r R,- )2(R, +2 r - R,)/(R, ~ Rs)3 R,


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3.1.3.3. Cut-offRadius plus Multipole Expansion plicitly taken into account as degrees of freedom in the sys-tem that is simulated. If parts of the system are homoge-neous, like the bulk solvent surrounding a solute, the numberof atoms or degrees of freedom can be reduced considerablyby modeling of the homogeneous par t as a continuous medi-um, e.g. a continuous dielectric. In this type of approxima-tion the system is divided into two parts : ( 1 ) an inner regionwhere the atomic charges q, are explicitty treated (dielectric

The technique of using neutral groups of atoms is basedon the more general fact that a charge distribution of a finitegroup of atoms can be approximated by a multipole expan-sion (monopole, dipole, quadrupole, etc.). The electric inter-action between two groups of atoms can be formulated asthe product of the two multipole expansions. The terms inthe resulting expression can be grouped according to theirdistance dependence: monopole-monopole (r- ), mono-poleedipole (r- *),monopoleq uadru pole and dipole-dipole( Y 3 ) , tc. At long distance only the leading terms in theseries need to be taken into account. Application of thismultipole expansion approximation in MD simulations wassuggested by Ladd.[] It is also used in combination with thecut-off plus atom pair list technique in Ref. [46].3.1.3.4. Twin Range Method

The twin range method[661s illustrated in Figure 7. Twocut-off radii are used. The atomsj lying within a distance Rffrom atom i are stored in a neighbor list of atom i. Theinteraction of the at om sj or which RA


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t h a t $ , ( r ) is also a solu tion of (7). The potential in the outerregion is deno ted by $ z ( r ) . f the ou te r reg ion con t inuum ha sa finite ionic streng th I , the potential $ , ( r ) must be a solut ionof the Poisson-Boltzmann equation (9) with inverse Debye

length IC =21F2/(&,Rr)n which Fis Faradays constant , Ris the gas cons tan t , and T the tempera ture . For zero ionicstrength, (9) reduces to (8). The bounda ry cond i tion a t in fin-i ty is given by ( lo) , an d a t the bo und ary between the innerlim i / iZ ( r )=01 - 0 0

and ou ter r eg ions the po ten tia l $ ( r ) [Eq. (1 ) ] and the dielec-tr ic displacement & o $ ( r ) Eq. (12)] must be co ntinu ous :

where the componen t o f the g rad ien t normal to thebou nda ry is denoted by V,,.The exact fo rm of $ ( r ) will depend on the shape of th eboundary , the va lues of c Z a n d K , an d t h e co m p u t a t io n a lmethod tha t is used to solve the field equations (7)-(12).3.1.3.6. Continuum Methods:Image Charge Approximation to the Reaction Field

When the boundary i s a sphere of radius R the react ionf ield due to a charge qi located at posi t ion riwith respect tothe or igin of the sphere can be approximated (dipolar ap-proximation, c Z % e l ) by the field that is generated by a so-called image charge 41 =- E, - , ) ( E ~ +E 1 ) q i R/r, that islocated a t posi tion rim=(R/r i )2ri[711see Fig. 8) . T h e n am eof this approximation is due t o the mi rror type of behaviorof the spher ical bou nda ry: i f qi approaches the boundary, 4:will do likewise, c ausing a pole in the potential at theboundary . Th is means tha t the image approx imat ion b reaksdown fo r charges close to the boun dary . The o ther l imi ta-t ions of the method are the requirement of a spher icalboundary , an d the cond i t ions tha t I =0 and c 2 +8, .3.1.3.7. Continuum Methods:Series Expansion of the Reaction Field

When th e boundary i s of i r regular s hap e the react ion f ieldpotential cannot be writ ten in a closed form, but must beapproximated numerical ly . I f we assume that I =0 a n dE~ =00 , the fo llowing approa ch is possible. Since the reac-tion field potential $&) must satisfy the Laplace equation(8) i t may be expand ed in Legendre polynomials , which alsosatisfy Equ ation (8) [Eq . (13)]. If c Z = 00,we have $ z ( r )=0

a t the bounda ry , o r us ing (11) and (5) Equ ation (14), a t thebounda ry , w i th $,-(r)fixed by (6). Th e expansion coeff icients

C;can be determined numerical ly by performing a leastsquares f i t of $&) to the given -I)&) a t a chosen numberof po in ts on the boundary .

The m ethod works (Berendsen and Zwinderman , p r iva tecommunication) , but is rather expensive due to the matr ixinversion inheren t in the least squares fit procedure. Close tothe boundary the approximation is poor . The other l imit-a t ions are tha t I =0 an d cZ =cc.

3.1.3.8. Continuum Methods:Three-dimensional Finite Difference Techniques

Another numerical approach is to compute the electr icpotential on a three-dimensional (3D) grid using finite diffe-rence m ethods (Fig. 8). The a tomic charges qi are distributedover gr id points . To each grid point a position-dependentdielectric constant c 1 is assigned. Then the field equations(7)-(12) ca n be numerically ~olved .[~ his involves the so-lut ion of a set of l inear equations for the potential at gr idpoints . The method is too t ime-consuming to be used in M Dsimulations, since the density of grid points must be suffi-cient to adequa tely represent the ato mic charge distr ibution.The use o f a position-dependent dielectric constant introdu-ces an arbitrary element into the method.

3.1.3.9. Continuum Methods:Two-dimensional Boundary Element Techniques

A recent development is the applicat ion of G reens func-tion techn iques to th e electric field problem .[731 he Greens

function (15) for the inner region satisfies Equation (16) (aPoisson equation with charge a t ro). The Greens func-

t ion for the o ute r region [Eq. (17)] satisfies Equa tio n (18) (aPoisson-Boltzmann equation with charge co a t yo).

Th e next s tep is to integrate the fol lowing integran d over the[G , xexpression (7) for $1 $, xexpression (16) for G,]

(19)inner region and to conver t i t by Greens theorem into anintegral over the surface of the boundary region. This yieldsE q u a t i o n (20), consist ing of a surface term and a sourceterm.$ l ( r ) = J [G1(rIr)Vn$l(r) - l ( r O V n G l ( r I r ) d ~surface

1002 4 n g e r . Chem. Int Ed. Engf.29 (1990) 992--1023


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By integrating ( 2 1 ) and conversion we obtain equation(22) .

Application of the boundary conditions (1 1) and ( 1 2 )yieldstwo linear integral equations in I c / l and V J I ~ on the boun-dary surface. These equations can be solved numerically bydiscretization on a surface net, and subsequently solved bymatrix inversion. The (large) matrix that is to be invertedonly depends on the shape and position of the boundarysurface, not on the positions of the charges in the inner re-gion. This means that in a M D simulation the time-consu-ming matrix inversion can be avoided as long as the boun-dary remains fixed.

The promise of this method lies in the reduction of a 3Dproblem to a 2 D problem. On the boundary surface a chargedensity is generated which produces the proper reactionfield. Also the methods of Shaw,t741nd of Zauhar and Mor-

transform the problem from a three-dimensional oneinto a two-dimensional one.3.1.3.10. Langevin Dipole Mo del

The Langevin dipole model of W a r ~ h e Z I ~ ~ . ~ ~ ]ses a 3 D -grid, but it is not a continuum method. The medium in theouter region is mimicked by polarizable point dipoles p i onthe grid points i of a 3 D grid. The dipoles may be thought ofas representing the molecular dipoles of the solvent mole-cules forming the outer region, so the spacing of the gridpoints corresponds to the size of the solvent molecules. Thesize and direction of a dipole p, s determined by the electricfield Ei at the grid point according to the Langevin formula[Eq. ( 2 3 ) ] .Here, pa is the magnitude of the dipole moment

of a solvent molecule, and Cis a parameter representing themolecular resistance to reorientation. The model has beenapplied to protein simulations.t641Questionable features ofthis model are the representation of the field in the outerregion and the discreteness near the boundary.

3.1 3.11. Periodic Lattice Summation MethodsIn computer simulations of liquids or solutions, periodicboundary condit ions are often used t o minimize the boun-dary effects. The computational box containing the molecu-lar system is surrounded by an infinite number of copies ofitself (see Fig. 9 ) . In this way an infinite periodic system issimulated.

The electric interaction in a periodic system is obtained bya summation over all atom pairs in the central box (Fig. 9 )and over all atom pairs of which one atom lies in the centralbox and the other is a periodic image [Eq. ( 2 4 ) ] .

i # j if n = O

I II II-- - ---_ _I I+ I + - 1

I II-t----- +I+- + II L F I + -I I I I II 1 I 1 II I I I I

Fig. 9. Periodic cubic system. Central computational box (solid line) with twoof the infinite number of periodic image boxes.

The sum over n is a summation over all simple cubic latticepoints n = (n ,L , nyL ,n,L) with n,, ny, , integers. The infi-nite sum of point charges must be converted into a formwhich converges faster than Equation (24) .3.1.3.12. Periodic Me tho ds: the Ewald Sum

The Ewald sum is a technique for computing the interac-tion of a charge and all its periodic images.16*76-781hecharge distribution e(r) in the system is an infinite set ofpoint charges, mathematically represented as delta functions

Each point charge q, is now surrounded by an isotropicGaussian charge distribution of equal magnitude and oppo-site sign [Eq. ( 2 6 ) ] .ec (r) =- i(a/&)3e-~21r-r~12 ( 2 6 )This smeared charge screens the interaction between thepoint charges, so that the interaction calculated using thescreened charge distribution ( 2 7 ) becomes short-ranged[Eq. (28)]due to the appearance of the complementary errorfunction (29) .

i t j f n = O

merfc(x) =2x- ' / ' e-y2dy (29)Thus, E" can be well approximated using a finite summationin (28) . Of course a purely Gaussian charge distribution-&(r) must be added to &(r) in order to recover theoriginal charge distribution ei( r). The interaction of theseGaussian distributions is expressed as a lattice sum inreciprocal space minus a self term [Eq. ( 3 0 ) ] , where

N- 4 n s o ) - ' x - " Z a q; ( 3 0 )j = 1AnKen Chczm lnr Ed Engl 29 ( 1 9 9 0 ) 992-1023 1003


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k =2xL- ' ( l , , l y , 1,) a n d l,, l v , 1, are in teger s. Du e to thepresence of the exponential factor the inf ini te lat t ice sum canbe well approximated by a f ini te one. The parameter CYshould be chosen such a s to optimize the convergence prop-er t ies of both sums (28) and (30) which contr ibute to

Th e Ewald sum technique is routinely used in s imulat ionsof ionic systems. When applied to non-crystalline systemssuch as l iquids or so lut ions i t ha s the disadvantage of enh an-cing the ar t i fact of the applicat ion of per iodic boundarycondit ions.

E = E " ~ .

3.1.3.13. Periodic Methods: Fourier TechniquesThe Poisson equation (31) for the potential $ ( r ) is a sec-

ond-order par t ial dif ferential equation in r -space. However ,when transformed into reciprocal space (k-space) i t becomesa simple algebraic equation (32) with the solution (33).- 2$(k) =- 0 Q ( k ) (32)$(k) = (Eok2) -1h(k) (33)Here, the 3D Fourier t ransformation is def ined according toE q u a t i o n (34), and l ikewise for th e charge densi ty e(r).T h einverse transform of Equation (33) yields (35).

$ ( r ) =F-' { ( ~ ~ k ~ ) - ' F { ~ ( r ) } } (35)Du e to the availabil i ty of fast Four ier t ransform techniquesE q u a t i o n (35) is a fast method to solve the Poisson equationo n a periodic three-dimensional grid. For a practical imple-mentat ion of the method we refer to Ref . [l]. I t h as beenemployed in the s imulat ion of sal ts .[ ' ]

3.1.4. SummaryTh e choice of molecular m odel a nd force field is essential

to a proper predict ion of the proper t ies of a system. There-fore, i t is of great importance to be aware of the fundam entalassumptions, s implif ications and approxima tions that a reimplicit in the variou s types of models used in the literature.When treat ing molecular systems in which Coulomb forcesplay a role one sho uld be aw are of the way these long-rangedforces are handled . S ince these forces play a domin an t ro le inmany molecular systems we have tr ied t o give an overview ofthe different methods-and so the dif ferent approxima-tions-that ar e in use.

F r o m T ab le I and the discussions in the previous sect ionsit should be clear tha t a "best" force field does nor exist.What will be the best choice of model and force field willdepend on the type of molecular system and the type ofproper ty o ne is interested in . This means tha t the molecularmodeler must have a picture of the s trengths and weaknessesof the var iety of force f ields that are available in order tomak e a p roper cho ice.

3.2. Searching Configuration Spaceand Generating an Ensemble

Once the molecular model an d force f ield V ( r ) have beenchosen, a metho d to search conf igurat ion space for conf igu-rations with low energy V ( r ) has to be selected. Var iousmethods are available, each with their par t icular s trengthand weakness, which depend on- the form a nd type of the interact ion energy function V( r ) ,- the number of degrees of freedom (size of the system),- he type of degrees of freedom, viz. Cartesian coordinates

versus othe r coordinates (e.g . bond lengths, bond angles,torsional angles plus center of m ass coordinates of a mol-ecule).

3.2.1. Systematic Search MethodsIf the molecular system contains only a small number of

degrees of f reedom (coordinates) and if V ( r )does no t havetoo m any ( relevant) minima up on var iat ion of the degrees off reedom, i t is possible to systematical ly scan the com pleteconf igurat ion space of the system. For example, one candescr ibe conformations of n-alkanes in terms of C-C-C-Ctorsional angles, each of which has three conformationalminima, t rans ( O O ) , gauche' (120") and gauche- ( - 20"). Tofind the lowest energy confo rma tion of n-decane (seven tor-sional angles) one would have to co mp ute V ( r ) o r 3' =2187combinations of torsional angles. The relative weight of thedif ferent conformations in the ensemble representing thismolecule at tem perature T is given by the Boltzmann factor(36) .

The computing effor t required by a systematic search ofthe degrees of f reedom of a system grows exponentially withtheir number. Only very small molecular systems can betreated by systematic search methods.[79* The number ofdegrees of freedom th at still can be handled within a rea son-able computing t ime strongly depends on the complexity ofthe function V ( r ) , hat is , the t ime required to com pute V(v)for each conf igurat ion. A possibility for speeding up thecalculation is to split the selection of low V ( r ) onf igurat ionsint o different stages. In a first stage a simplified low comput-ing cost energy fun ction VSimple(r)s used to quickly discardhigh Vsimple(i) onf igurations. In a second stage the com pleteV ( r ) s only to be evaluated for the remaining conf igurat ions.This type of filtering proc edure ha s been used t o predict theloop structure in proteins[* '] and to predict the s table con-form atio n of small pepti des (five am in o acid residues)!**] fo rexample. The basic problem of filtering using simplifiedforms of V ( r ) s to en sure that the s implified Vsimple(r)s acorrect project ion of the complete function V ( r ) : whenVs,mp,e(r)s large, V ( v ) hould also be large, otherwise a con -figuration with low energy V ( r ) might be discarded in thefirst stage due to a high energy Vsimpl,(r).3.2.2. Random Search Methods

I f a system contains too many degrees of f reedom,straightforward scanning of the complete conf igurat ionspace is impossible. In tha t case a collect ion of conf igura-

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t ions can be genera ted by r an dom sampl ing . Such a collec-t ion becomes an ensemb le of conf igurat ions when each con -f igurat ion is given i ts Boltzmann factor [cf . Eq . (36)] asweight factor in the collect ion. Two types of ran dom searchmethods can be dist inguished :- Mo nte Ca r lo methods , in which a sequence of conf igura-

t ions is generated by an algor i thm which ensures that theoccurrence of conf igurat ions is propor t ional to the i rBoltzma nn factors [Eq. (36)].

- Other methods which p roduce an arb i t r ary (non-Bol tz-ma nn) collect ion of conf igurat ions, which ca n be trans-fo rmed in to a Boltzma nn ensemble by applying the weightfactors f rom Equation (36) .

3.2.2.1. Monte Carlo SirnulalionThe M onte Car lo (M C) s imula tion p rocedure by which a

(canonical) ensemble is produced consists of the fol lowingsteps.1 . Given a star t ing conf igurat ion r , a new conf igurat ion

r s + = , +Ar is generated by random displacement ofone (o r more) a toms . The r andom d isp lacements Arshould be such that in the l imit of a large number ofsuccessive displacements the available Cartesian spaceof a l l a tom s is un i fo rmly sampled . Th is does no t meanthat the actual sampling mus t be carr ied out in Car tesianspace. I t can be done, e.g . , in internal coordinate space(r , 0 , cp) , but s ince the equivalent volume element isr2sin 0 drdodcp, the sampling in internal c oord inate spacemus t be non-un i fo rm in o rder to p roduce a uni fo rm sam-pling in terms of Cartesian coordinates.

2 . Th e newly generated conf igurat ion r ,+ is either acceptedor rejected on t he basis of a n energy cr i ter ion involvingt h e ch an g e A E =V ( r , + - V(r , ) of the potential energywith respect to th e previous Conf igurat ion. Th e new con -f igurat ion is accepted when A E 0 whene-A"'kBT>R , where R is a r an d o m n u m b er t aken f r o m auniform distr ibution over the interval (0,l).Up on accep tance , the new conf igura t ion is counted andused a s a s ta r t ing po in t fo r the next r ando m d isp lacement .I f the cr i ter ia are not met, the new conf igurat ion r ,, isrejected. This implies that the p revious conf igurat ion r , iscounted again and used as a s tar t ing po in t fo r ano th errando m d isplacement .

It is relat ively easy to und erstand tha t this procedure willgenerate a Boltzmann ensemble. We consider two conf igu-rat ions r1 a n d r 2 with energies El =V ( r , )


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3.2.2.3. Other Random Search TechniquesThere exist infinitely many ways for generating a set of

molecular configurations in a random manner. Whether theset of generated configurations may be considered to forman ensemble that can be used for statistical mechanical eval-uation of quantities will depend on the sampling propertiesof the method that is used. It should sample the important(low energy) regions of configuration space and the configu-rations should occur according to their Boltzmann weightfactors. Otherwise the set of configurations can only beviewed as such, not as an ensemble representative of the stateof the system that is considered.

3.2.3. Dynamic Simulation MethodsAnother way to generate an ensemble of configurations is

to apply Nature's laws of motion for the atoms of a molecu-lar system. This has the additional advantage that dynamicalinformation about the system is obtained as well. The twomajor simulation techniques of this type are Molecular Dy-namics (MD), in which Newton's equations of motion areintegrated over time, and Stochastic Dynamics (SD), inwhich the Langevin equation of motion for Brownian mo-tion is integrated over time.3.2.3.1. Molecular Dynamics Simulation

In the Molecular Dynamics (MD) method a trajectory(configurations as a function of time) of the molecular sys-tem is generated by simultaneous integration of Newton'sequations of motion (39) and (40) for all the atoms in the

d2ri(t)/dt2=m;' F, (39)6 =- V(r,, . . .rN)/ari (40)system. The force on atom i is denoted by 6 and time isdenoted by t . MD simulation requires calculation of thegradient of the potential energy V(r), which therefore mustbe a differentiable function of the atomic coordinates r i. Theintegration of Equation (39) is performed in small time stepsA t, typically 1-10 fs for molecular systems. Static equilibri-um quantities can be obtained by averaging over the trajec-tory , which must be of sufficient length to form a representa-tive ensemble of the state of the system. In addition, dynamicinformation can be extracted. Another asset of M D simula-tion is that non-equilibrium properties can be efficientlystudied by keeping the system in a steady non-equilibriumstate, as is discussed in Section 5.4. Viewed as a technique tosearch configuration space, the power of M D lies in the factthat the kinetic energy present in the system allows it tosurmount energy barriers that are of order of k , Tper degreeof freedom. By raising the temperature T a larger part ofconformation space can be searched, as has been shown byDiNola et a1.,'881 who generated a series of different confor-mations of the hormone somatostatin by applying MD atT =600 K and at 1200 K. At the elevated temperature thetotal energy and potential energy are monitored for conspic-uous fluctuations which may signal a possibly significantconformational change. When minima in the total energyoccur, the system is cooled down and equilibrated at normal

temperature (300 K) . In this way different conformationswith comparable free energy were obtained. We note how-ever that the search at elevated temperature favors selectionof higher ent ropy conformations.

Searching conformation space by M D is expected to beefficient for molecules up to about 100 atoms. For largermolecules, which may and are likely to show a particulartopological fold, MD methods will not be able to generatemajor topological rearrangements. Even when the barriersseparating two topologically different low energy regions ofconformation space are of the order of k,T, the time neededfor traversing them may be much too long to be covered ina MD simulation of 10-100 ps.3.2.3.2. Stochastic Dynamics Simulation

The method of stochastic dynamics (SD) is an extension ofMD. A trajectory of the molecular system is generated byintegration of the stochastic Langevin equation of motion(411.d2r,(t)/dt2=m,-' , +m ; ' R , - ,dr,(t)/dt (41)

Two terms are added to Equation (39), a stochastic forceR i and a frictional force proportional to a friction coefficienty , . The stochastic term introduces energy, the frictional termremoves (kinetic) energy from the system, the condition forzero energy loss being that given in Equation (42), where

qers the reference temperature of the system.SD can be applied to establish a coupling of the individual

atom motion to a heat bath,[891or to mimic a solvent ef-fect.lgO. In the latter case the stochastic term representscollisions of solute atoms with solvent molecules and thefrictional term represents the drag exerted by the solvent onthe solute atom motion. An introduction to S D simulationtechniques is given in Refs. [90, 921.

3.2.4. Other Search MethodsThere exists a variety of other methods for searching con-

figuration space which are variants of the basic types dis-cussed in the previous subsections.

The method of simulated annealing[931s a MC or a MDsimulation in which the temperature is gradually lowered toOK. In this way the system generally ends up in a lowerenergy state than when an ordinary gradient energy minimiz-er is used.

In a M D simulation the atomic velocities are large whenthe system explores the low energy regions of the potentiaIenergy function V(r), and small when it crosses barriers athigher energy. For an efficient searching of configurationspace one would like to invert this behavior: the atomsshould move slowly in the valleys of V(r). A search algorithmdeveloped along this line seems to perform well for certainsystems.1941Within the framework of MD the ability ofatoms to cross barriers can be greatly enhanced by keepingtheir kinetic energy nearly constant by a tight coupling to aheat bath.

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Another possibility is the combination of different tech-niques; Monte Carlo simulation, gradient minimization, etc.in one computa tional scheme.195,9613.2.5. Summary

The various methods for searching configuration spacecan be classified as follows:A. Systematic search methods, which scan the complete

configuration space of the molecular system.B. Methods which aim a t generating a representative set ofconfigurations. These may be divided into two types.1. Non-step methods, such as the DG method, which

generate a (at least in principle) uncorrelated series ofrandom configurations.

2. Step methods, such as MC, M D and SD, which gener-ate a new configuration from the previous one.

Most search techniques fall in this class. They can be dis-tinguished by the way the step direction and the step size arechosen :- according to the gradient - o V ( r ) ,- according to a memory of the path followed so far,or- at random.

For example, in the MC method, the step direction ischosen a t random, the actual step size is determined by thechange in energy A E (gradient) and a random element in caseA E >0. In a MD simulation both step size and direction aredetermined by the force (gradient) and the velocity (memo-ry). In general a search algorithm will make a step which isa linear combination of the gradient , previous steps (memo-ry) and a random contribution. It will depend on the energysurface V(r )which is the optimal linear combination of thesethree ingredients.Another classification of algorithms which generate a setof configurations is whether they produce an ensemble ornot. Of the schemes discussed above only MC, M D and SDgenerate a Boltzmann weighted ensemble.

3.3. Boundary ConditionsWhen simulating a system of finite size, some thought

must be given to the way the boundary of the system will betreated. The simplest choice is the vacuum boundary condi-tion. When simulating a liquid, solution or solid rather thana molecule in the gas phase, it is common practice to mini-mize edge or wall effects by the application of periodicboundary conditions. If the irregularity of the system is in-compatible with periodicity, edge or wall effects may be re-duced by treating part of the system as an extended wallregion in which the motion of the atoms is partially restrict-ed.3.3.1. Vacuum Boundary Condition

Simulation of a molecuIar system in vacuo, that is, with-out any wall or boundary, corresponds to the gas phase atzero pressure. When the vacuum boundary is used for a solidor a molecule in solution, properties of atoms near or at thesurface of the system will be distorted.Ig7] The vacuumboundary condition may also distort the shape of a (non-

spherical) molecule, since it generally tends to minimize thesurface area. Moreover, the shielding effect of a solvent withhigh dielectric permittivity, like water, on the electric interac-tion between charges or dipoles in a molecule is lacking invacuo. We note that the water molecules do no t need to bepositioned between the charges or dipoles in order to pro-duce a screening of the interaction between these. Therefore,simulation of a charged extended molecule like DNA in vac-uo is a precarious undertaking. Solvation of DNA in asphere with solvent molecules will shift the boundary effectsfrom the DNA-vacuum interface to the water-vacuum in-terface and so improve the treatment of the DNA.Ig8Thebest results in vacuo are obtained for relatively large globu-lar molecular systems.

3.3.2. Periodic Boundary ConditionsThe classical way to minimize edge effects in a finite sys-

tem is to use periodic boundary conditions. The atoms of thesystem that is to be simulated are put into a cubic, or moregenerally into any periodically space-filling shaped box,which is treated as if it is surrounded by 26 (=33- 13)identical translated (over distances kRboxn the x- , - , z-di-rections) images of itself. The next layer of neighbor imagesof the central computational box contains 53 - 3- 3 =98 boxes, and so on. When an infinite lattice sum of theatomic interactions is to be performed (Sections 3.1.3.1 1-13), the interactions of an a tom in the central computationalbox with a11 its periodic images are computed. In most casesthis is not desirable. Then only interactions with nearestneighbors are taken into account. The black atom in thecentral computational box in Figure 10a will only interactwith atoms or images of atoms tha t lie within the dashed line(nearest image, NI, or minimum image, MI, approximation).The anisotropy of the interaction due to the cubic shape ofthe nearest image box can be avoided by the application ofa spherical cut-off (radius Rc) .The periodic boundary condi-tion affects not only the computation of the forces, but alsothe positions of the atoms. It is common practice (though notnecessary) to keep the atoms together, that is, in the centralcomputa tional box: when an a tom leaves the central box onone side, it enters it with identical velocity at the oppositeside at the translated image position.

Application of periodic boundary conditions means thatin fact a crystal is simulated. For a liquid or solution theperiodicity is an artifact of the computation, so the effectsshould be minimized. An atom should not simultaneouslyinteract with another atom and a periodic image of thatatom. Consequently the length Rbox f the edge of the period-ic box should exceed twice the cut-off radius R,. Possibledistorting effects of the periodic boundary l oz 1may be traced by simulation of a system in differently shapedboxes (see below) of different size.

When simulating a spherical solute, use of a more spheri-cally shaped computational box instead of a cubic or rectan-gular one may considerably reduce the number of solventmolecules that is needed to fill the remaining (after insertionof the solute) empty space in the box. A more sphericallyshaped space-filling periodic box is a truncated octahedron,(Fig. 10b).1103]t is obtained by cutting off the corners of acube in such a way that the symmetry of the cube is main-

A n g e n Chem. I n [ . Ed . Engl . 29 (1990) 992-1023 1007


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a

C

b

d

ful l motion +periodicful l motion

vocuum vacuumFig. 10. Types of boundary conditions: a) cubic periodic, b) troncated octahe-dron periodic, c) spherical extended wall region, d) flat extended wall region fora gas-solid simulation.

tained, and such that the distance between opposite hexago-nal planes equals$&imes the distance between oppositesquare planes. In this way half the volume of the originalcube is cut away. The rat io of the inscribed-sphere volume tototal volume of the truncated octahedron is 0.68, whereas i tis only 0.52 for a cube. Use of a truncated octahedron insteadof a rectangular periodic box may yield a sizeable reductionof the system to be simulated. For example, the small proteinbovine pancreatic trypsin inhibitor (BPTI) (58 amino acidresidues) consists of 568 atoms (hydrogen atoms bound tocarbons excluded), and its dimensions are 28 A by 28 A by40 8,. If we approximate it by a sphere of radius 16 A and putit into a cube with a minimum solvent layer thickness of 6 8,to the walfs, about 2300 water molecules (6900 atoms) areneeded to fill the non-protein part of the cube of volume[2 x (16 +6)13A3 .To fill the non-protein par t of a truncatedoctahedron of volume t [ 4 x (16 +6)/&13 A3 takes about1600 water molecules (4800 atoms). This means a reductionof the system size by at least one quarter of the number ofatoms.3.3.3. Extended Wall Region Boundary Condition

If a molecular system is irregular, i.e. not nearly transla-tionally periodic, periodic boundary conditions cannot beapplied. In this case the distorting effect of the vacuum ou t -side the molecular system may be reduced by designating alayer of atoms of the system to be an extended wall region,in which the motion of the atoms is restrained in order toavoid the deforming influence of the nearby vacuum(Fig. lOc, d). The atoms in the extended wall region can bekept fixed Is) or harmonically restrained to stationary posi-tions. Their motion may be coupled to a heat bath, e.g. by

applying stochastic dynamics[sg' in order to account for ex-change of energy with the surroundings. In any case the typeof force applied to these atoms should be chosen such thattheir motion in the finite system resembles as closely as pos-sible the true motion in an infinite system. The extended wallregion forms a buffer between the fully unrestrained part ofthe system and the (unrealistic) vacuum surrounding it.

This extended wall region technique has been applied inthe simulation of solid~,[ '~~l 'O6I and pro-tein~.[' '~.08] Although the technique yields considerablesavings in computing time, it should be carefully analyzedhow far the restraining of the motion of the atoms in the wallregion will affect the motion of the freely simulated atoms inthe system.

3.4. Different Types of Molecular DynamicsWhen Newton's equations of motion (39) and (40) are

integrated the total energy is conserved (adiabatic system)and if the volume is held constant the simulation will gener-ate a microcanonical ensemble. For various reasons this isnot very convenient and a variety of approaches has ap-peared in the literature to yield a type of dynamics in whichtemperature and pressure are independent variables ratherthan derived properties. When MD is performed in non-equilibrium situations in order to study irreversible process-es, catalytic events or transport properties, the need to im-press external constraints or restraints on the system isapparent. In such cases the temperature should be controlledas well in order to absorb the dissipative heat produced bythe irreversible process. But also in equilibrium simulationsthe automatic control of temperature and pressure as inde-pendent variables is very convenient. Slow temperaturedrifts that are an unavoidable result of force truncation er-rors are corrected, while also rapid transitions to new desiredconditions of temperature and pressure are more easily ac-complished.3.4.1. Constant- Temperature Molecular Dynamics

Several methods for performing MD at constant tempera-ture have been proposed, ranging from ad-hoc rescaling ofatomic velocities in order to adjust the temperature, t o con-sistent formulation in terms of modified Lagrangian equa-tions of motion that force the dynamics to follow the desiredtemperature constraint. Different types of methods can bedistinguished.

1. Constraint methods, in which the current temperatureT(t)at time point t is exactly equal to the desired referencetemperature To .This can be achieved by rescaling the veloc-ities at each MD time stepIZolby a factor [To/T(t)]"2,herethe temperature T(t) s defined in terms of the kinetic energythrough equipartition IEq. (43)]. The number of degrees offreedom in the system is Ndf.

(43)N 1 1Ekin(t)=C -mi$(t) =- N d f k , T ( t )i=12 2The same result can be obtained in a more elegant way by

suchmodification of the equations of motion (39);"'- '1008 Angen. Chem. I n t . Ed. Enfi. 9 (f990)992-fO23


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that the kinetic energy instead of the total energy becomes aconstant of motion. This method has two disadvantages.First, numerical inaccuracy of the algorithm may produce adrift in temperature that is not stabilized, since the referencetemperature To does not appear in the equations to be inte-grated. Second, although the system is modeled by a Hamil-tonian, this Hamiltonian does not represent a physical sys-tem, for which fluctuations in the kinetic energy arecharacteristic. In practical applications which are aimed atrealistically simulating a physical system, the use of a non-physical Hamiltonian is questionable, even though mathe-matically consistent equations of motion are obtained.

2. Extended system methods,[' I in which an extra degreeof freedom s epresenting a heat bath is added to the atomicdegrees of freedom of the molecular system. A kinetic and apotential energy term invoking this extra degree of freedomare added to the Hamiltonian and the simulation is carriedout for the Ndf+ degrees of freedom of this extended sys-tem. Due to the presence of a kinetic term +,(ds/dt)2 in theHamiltonian, energy is flowing dynamically from the heatbath to the system and back, the speed being controlled bythe inertia parameter m,. A disadvantage of this type ofsecond-order coupling to a heat bath is the occurrence ofspurious energy oscillations, the period of which depends onthe value of the adjustable coupling parameter m,.3. Weak coupling methods," 131 in which the atomic equa-tions of motion are modified such that the net result o

1990, computer simulation of molecular dynamics- methodology, applications, and perspectives in...

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