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Biomolecular Modeling: Goals, Problems, Perspectives

Biomolecular Modeling: Goals, Problems, Perspectives
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  Molecular Dynamics  DOI: 10.1002/anie.200502655 Biomolecular Modeling: Goals, Problems, Perspectives Wilfred F.van Gunsteren,*DirkBakowies,RiccardoBaron,IndiraChandrasekhar,Markus Christen, Xavier Daura, Peter Gee, Daan P. Geerke, Alice Gl   ttli,Philippe H. H   nenberger, Mika A. Kastenholz, Chris Oostenbrink, Merijn Schenk,Daniel Trzesniak, Nico F. A. van der Vegt, and Haibo B. Yu  Angewandte Chemie Keywords: computer simulation · force-fieldtechniques · GROMOS · molecularmodeling · moleculardynamics W. F. van Gunsteren et al. Reviews 4064   2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim  Angew. Chem. Int. Ed.  2006 ,  45  , 4064–4092  1.  Introduction The role of computation in biology, biological chemistry,and biophysics has shown a steady increase over the past fewdecades. The continuing growth of computing power (inparticular in the context of personal computers) has made itpossible to analyze, compare, and characterize large andcomplex data sets that are obtained from experiments onbiomolecular systems. This has in turn led to the formulationof models for biomolecular processes that are amenable tosimulation or analysis on a computer. When undertaking abiomolecular modeling study of a particular system of interest, the level of modeling, that is, the spatial resolution,time scale, and degrees of freedom of interest, must beconsidered (Table 1).Which level of modeling is chosen to describe a particularbiomolecular process depends on the type of process. In thisReview we focus on three of the four biomolecular processesillustrated in Figure 1: 1) polypeptide folding, 2) molecularcomplexation (e.g. protein–ligand, DNA–ligand, protein–DNA, etc.), 3) partitioning of molecules between differentenvironments, such as lipid membranes, water, mixtures (e.g.water/urea, ionic solutions), and apolar solvents, and 4) theformation of lipid membranes or micelles out of mixtures of their components. These four processes play a fundamentalrole in the behavior of biomolecular systems and share thecommon feature that they are driven by weak, nonbondedinteratomic interactions. Such interactions govern the ther-modynamic properties of the condensed phase in which thefour processes occur. Therefore, these processes are mostpromisingly modeled at the atomic or molecular level (thirdrow in Table 1). Since the temperature range of interestbasically lies between room and physiological temperatures,andenergies involved in theseprocesses areon theorder of1–10  k B T   (which corresponds to tens of kJmol  1 ,  k B  is theBoltzmann constant), the processes are largely determined bythe laws of classical statistical mechanics. Although quantummechanics governs the interactions between the electrons of the atoms and molecules as well as the motions of lightparticles such as protons, the nonbonded interactions can bevery well described by a classical potential-energy function orforce field as part of a classical Hamiltonian of the system of interest. [1] Figure 2 shows the four choices to be made whenmodeling a biomolecular system: 1) which atomic or molec-ular degrees of freedom are explicitly considered in themodel, 2) which interaction function or force field is used todescribe the energy of the system as a function of the chosendegrees of freedom, 3) how these, generally many, degrees of freedom are to be sampled, and 4) how the spatial boundariesand external forces are modeled. As already mentioned, wemainly consider atomic and molecular degrees of freedomwith the corresponding classical force fields and classicalNewtonian dynamics to sample the degrees of freedom.System sizes that can be considered range up to 10 5 or [*] Prof. Dr. W. F. van Gunsteren, Dr. D. Bakowies, R. Baron,Dr. I. Chandrasekhar, M. Christen, Prof. Dr. X. Daura, Dr. P. Gee,D. P. Geerke, Dr. A. Gl  ttli, Dr. P. H. H  nenberger, M. A. Kastenholz,Dr. C. Oostenbrink, Dr. M. Schenk, D. Trzesniak,Dr. N. F. A. van der Vegt, Dr. H. B. YuLaboratory of Physical ChemistrySwiss Federal Institute of TechnologyETH8093 Zurich (Switzerland)Fax: (   41)44-632-1039E-mail: wfvgn@igc.phys.chem.ethz.chProf. Dr. X. DauraICREA, Institute of Biotechnology and BiomedicineUniversitat Aut  noma de Barcelona08193 Bellaterra (Barcelona) (Spain)Dr. C. OostenbrinkPharmaceutical Sciences/PharmacochemistryVrije UniversiteitDe Boelelaan 1083 P262, 1081 HV Amsterdam (The Netherlands)Dr. N. F. A. van der VegtMax-Planck-Institute for Polymer ResearchAckermannweg 10, 55128 Mainz (Germany)Dr. H. B. YuDepartment of ChemistryUniversity of Wisconsin1101 University Ave, Madison, WI 53706 (USA) C  omputation based on molecular models is playing an increasinglyimportant role in biology, biological chemistry, and biophysics. Sinceonly a very limited number of properties of biomolecular systems isactually accessible to measurement by experimental means, computer  simulation can complement experiment by providing not only aver-ages, but also distributions and time series of any definable quantity, for example, conformational distributions or interactions between parts of systems. Present day biomolecular modeling is limited in itsapplication by four main problems: 1) the force-field problem, 2) the search (sampling) problem, 3) the ensemble (sampling) problem, and4) the experimental problem. These four problems are discussed andillustrated by practical examples. Perspectives are also outlined for  pushing forward the limitations of biomolecular modeling. From the Contents 1.  Introduction  4065 2.  The Force-Field Problem  4067   3.  The Search Problem  4073  4.  The Ensemble Problem  4080   5.  The Experimental Problem  4083 6.  Perspectives in Biomolecular Modeling   4087  Biomolecular Modeling  Angewandte Chemie 4065  Angew. Chem. Int. Ed.  2006 ,  45  , 4064–4092   2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim  10 6 atoms or particles, which is still very small compared toAvogadro  s number, that is, macroscopic sizes. For such smallsystems, the modeling of the boundary or surface will have alarge effect on the calculated properties. Such surface effectscan be minimized by using periodic boun-dary conditions, where the box that containsthe molecular system is surrounded by aninfinite number of copies of itself (Figure 3).This avoids surface effects at the expense of introducing periodicity artefacts. [2–5] Present day biomolecular modeling islimited in its application by the four prob-lems highlighted in Table 2: 1) the force-field problem, 2) the search (sampling)problem, 3) the ensemble (sampling) prob-lem, and 4) the experimental problem.These four problems are the focus of thepresent Review and will be discussed andillustrated in Sections 2–5 by using exam-ples from our own work. We stress that theaim of this Review is not to review thecontributions of various research groups tothe field.The key reason why computer simula-tion is used in the study of biomolecularsystems in spite of the above-mentioned Wilfred F. van Gunsteren was born in 1947 in Wassenaar (The Netherlands). In 1968 hegained a BSc in physics at the Free Univer-sity of Amsterdam; in 1976 he was awarded a “Meester” in Law, and in 1976 a PhD innuclear physics. After postdoc research atthe University of Groningen and at Harvard University he was, 1980–1987, senior lec-turer and, until August 1990, Professor for Physical Chemistry at the University of Gro-ningen. In 1990 he became professor of  Computer Chemistry at the ETH Z   rich. Heis holder of a gold medal for research of theRoyal Netherlands Chemical Scoiety. His main interests center on thephysical fundamentals of the structure and function of biomolecules. Table 1:  Examples of levels of modeling in computational biochemistry and molecular biology.Methods Degrees of freedom Properties, processes Time scalequantum dynamics atoms, nuclei, electrons excited states, relaxation,reaction dynamicspicosecondsquantum mechanics(ab initio, density functional,semiempirical, valence bondmethods)atoms, nuclei, electrons ground and excited states,reaction mechanismsno time scaleclassical statistical mechanics(MD, MC, force fields)atoms, solvent ensembles, averages,system properties, foldingnanosecondsstatistical methods (databaseanalysis)groups of atoms, aminoacid residues, basesstructural homology andsimilarityno time scalecontinuum methods (hydro-dynamics and electrostatics)electrical continuum,velocity continuum etc.rheological properties supramolecularkinetic equations populations of species population dynamics,signal transductionmacroscopic FoldingMembrane or Micelle Formation   ComplexationPartitioning folded/nativedenaturedmicellemixtureboundunboundin membranein waterin mixtures Figure 1.  Four biomolecular processes that are governed by thermody-namic equilibria.  Boundary conditions MOLECULAR MODEL  Degrees of freedom: atoms are the elementary particlesForces or interactionsbetween atoms Methods to generate configurationsof atoms: Newton systemtemperaturepressurewallsexternal forcesForce field =physicochemicalknowledge Figure 2.  Four basic choices in the definition of a model for molecularsimulation. Vacuum Droplets Periodic:  system is surrounded by copies of itself  •Surface effects •No dielectric screening•Still surface effects•Only partial dielectric screening•Evaporationof the solventAdvantage:•No surface effectsDisadvantage:•Artificial periodicity•High effective concentration(surface tension)(at water – vacuum interface) Figure 3.  Three types of spatial boundary conditions used in molecularsimulation. W. F. van Gunsteren et al. Reviews 4066   2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim  Angew. Chem. Int. Ed.  2006 ,  45  , 4064–4092  limitations to its accuracy resides in the fourth of the fourreasons listed in Table 3: Only a very limited number of properties of a biomolecular system is actually accessible toexperimental measurement, whereas in a computer simula-tion notonly averages, but alsodistributions andtime series of any definable quantity can be determined. Thus, computersimulation represents a complement to experiment byproviding the detailed conformational and other distributionsthat determine the space and time averages obtainedexperimentally. As such, it is an indispensable tool tointerpret experimental data. Moreover, it can be used topredict properties under environmental conditions that aredifficult or expensive to realize. In the next four sections weillustrate the use, power, and limitations of biomolecularmodeling in conjunction with experimental efforts withregard to the four processes of interest (Figure 1). 2.  The Force-Field Problem A biomolecular force field generally consists of potential-energy terms representing covalent interactions betweenatoms (such as bond-stretching, bond-angle bending,improper and proper dihedral-angle torsion) on the onehand and nonbonded interactions on the other hand betweenatoms in different molecules and between atoms in amolecule that are separated by more than two or threecovalent bonds. [6,7] Since nonbonded interactions govern the thermodynamicequilibria and processes of interest depicted in Figure 1, wefocus on the formulation and parametrization of thesepotential-energy terms. Three problems dominate the topicof force-field development (Table 2, point 1, A–C).A first problem is that the (free) energy differencesdriving the processes of Figure 1 are of the order of 1–10  k B T  (which corresponds to tens ofkJmol  1 ). These relativelysmallenergies result from a summation over very many (10 6 –10 8 )atom pairs: A system of   N  = 1000 atoms has about  1 = 2 N  ( N   1) = 500000 pairs of atoms contributing to the non-bonded interaction. To reach the requested accuracy for thetotal nonbonded energy, the accuracy of the individual termsin the summation (the atom-pair energies) must be higher.This difficulty becomes increasingly severe when trying toderive a force field of high accuracy for larger systems, that is,for larger values of   N  .A second problem is to appropriately account for entropiceffects. Since we are not interested in biomolecular systems ata temperature of    273.15   C (0 K), we have to consider thecontribution of entropy  S  to the free energy  F  = U   TS  of thesystem of interest. It is well known that entropy plays anessential role in all four of the processes shown in Figure 1.Changes in free energy that drive processes may result fromchanges in internal energy ( U  ) or in entropy ( S ), which maywork together or against each other depending on the relativestrengths of the nonbonded interactions between the variouscomponents (atoms, molecules) of the system. [8,9] Figure 4illustrates the phenomenon of energy–entropy compensation:two conformations  x 1  and  x 2  of a molecule may have  U  (  x 1  ) ! U  (  x 2 ), while  F  (  x 1 ) > F  (  x 2 ) if at a given temperature  S (  x 1 ) ! S (  x 2 ). The entropy is a measure of the extent of conforma-tional space (  x ) accessible to the molecular system at a giventemperature  T  .Figure 4 also illustrates that searching for and finding theglobal energy minimum of a biomolecular system is mean-ingless when its entropy accounts for a sizeable fraction of itsfree energy. For example,  F  =  24 kJmol  1 ,  U  =  41 kJmol  1 , and  TS =  17 kJmol  1 for liquid water atroom temperature and pressure. The properties of water in Table 2:  Four basic problems of biomolecular modeling. 1. force-fieldproblem A) very small (free) energy differences, manyinteractionsB) entropic effectsC) variety of atoms and molecules 2. search problem  A) convergenceB) alleviating factorsC) aggravating factors 3. ensemble problem  A) entropyB) averagingC) nonlinear averaging 4. experimental problem  A) averagingB) insufficient number of dataC) insufficient accuracy of data Table 3:  Four reasons why computer simulation is used in science.Simulation can replace or complement an experiment:1. experimentis impossiblecollision of stars or galaxiesweather forcast2. experimentis dangerousflight simulationexplosion simulation3. experimentis expensivehigh pressure simulationwind channel simulation4. experimentis blindmany properties cannot be observed on veryshorttime scales and very smallspace scales Figure 4.  Energy–entropy compensation at finite temperatures. Biomolecular Modeling  Angewandte Chemie 4067  Angew. Chem. Int. Ed.  2006 ,  45  , 4064–4092   2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim  the condensed phase can therefore only be described througha conformational distribution, which in turn can be generatedby computer simulation. Similar considerations apply tobiomolecular systems: an energy-minimized structure of aprotein corresponds to a possible conformation at 0 K, andlacks information on the conformational distribution of theprotein at physiological temperatures. This state of affairs hasconsequences for the development of force fields: if a forcefield is to be used in computer simulations above 0 K, itsparameters should be derived or calibrated taking intoaccount entropic effects for it to be consistent. In otherwords, calibration of force-field parameters involves com-puter simulations to generate configurational ensembles,which makes it a more costly task than when only singleminimum-energy conformations or measured average struc-tures are used.A third problem in the development of a biomolecularforce field is the enormous variety of chemical compounds forwhich adequate force-field parameters should be derived. If the force-field parameters are (to some extent) transferablebetween atoms or groups of atoms in different molecules, thisproblem may be (at least partially) alleviated. In general,putting the force-field terms on a physical (instead of a purelystatistical) basis and keeping them simple and local willenhance the transferability of parameters from one com-pound to another. In addition, by keeping them computa-tionally simple, the efficiency of biomolecular simulation canbe enhanced, which facilitates the sampling of configurationalspace. 2.1.  Functional Form of the Force-Field Terms Most biomolecularforce fields are composedof termsthatpossess a rather simple functional form. [6] The GROMOSforce field, for example, consists of the following terms[Eqs. (1)–(8)]: [7,10] V   bond ð r  ; K  b , b 0 Þ¼ X N  b n ¼ 11 = 4 K  b n ½ b 2 n  b 20 n  2 ð 1 Þ V   angle ð r  ; K  q , q 0 Þ¼ X N  q n ¼ 11 = 2 K  q n ½ cos ð q n Þ cos ð q 0 n Þ 2 ð 2 Þ V   har ð r  ; K  x , x 0 Þ¼ X N  x n ¼ 11 = 2 K  x n ½ x n  x 0 n  2 ð 3 Þ V   trig ð r  ; K  f , d , m Þ¼ X N  f n ¼ 1 K  f n ½ 1 þ cos ð d n Þ cos ð m n  f n Þ ð 4 Þ V   LJ ð r  ;C 12 ,C 6 Þ¼ X pairs  i ,  j  C 12 ð i ,  j Þ r  12 ij  C 6 ð i ,  j Þ r  6 ij   ð 5 Þ V   C  ð r  ; q Þ¼ X pairs  i ,  j q i  q  j 4 p e 0 e 1 1 r  ij ð 6 Þ V   RF ð r  ; q Þ¼ X pairs  i ,  j q i  q  j 4 p e 0 e 1 ð 12 C  rf   r  2 ij Þ R 3 rf  ð 7 Þ V  RF c ð r  ; q Þ¼ X pairs  i ,  j q i  q  j 4 p e 0 e 1 ð 12 C  rf   1 Þ R rf  ð 8 Þ The first four equations describe the four types ofcovalent(bonded) interactions mentioned before, while the last fourspecify the nonbonded interactions: the van der Waals inter-action cast in the form of a Lennard–Jones term, theelectrostatic Coulomb interaction between (partial) atomiccharges  q i , the distance-dependent and distance-independent(constant) interactions arising from the dipolar reaction field(RF) induced by the charge distribution inside the cut-off sphere through the continuous dielectric medium outside thiscut-off sphere. Since this force field covers a variety of molecules (including polypeptides, polysaccharides, nucleicacids, lipids), it contains a large set of parameters: [7] 52 typesof bonds [Eq. (1)], 54 types of bond angles [Eq. (2)], 3 typesof improper (harmonic) dihedral angles [Eq. (3)], 41 types of proper torsional (trigonometric) dihedral angles [Eq. (4)],van der Waals interactions of 53 types of atoms [Eq. (5)], andmany different sets of atomic charges for the typical polar orcharged groups of atoms in the molecules mentioned above[Eqs. (6)–(8)]. [7,10] The functional forms are chosen such that they are easy tocompute. The nonbonded interactions only contain pairterms, and the more complex three- and four-body covalentterms [Eqs. (3) and (4)] are much fewer in number than thenonbonded pair terms. The solvent part of this biomolecularforce field only contains nonbonded terms, the intramoleculardegrees of freedom of solvent molecules are kept frozen. Themajor computational effort resides in evaluating the non-bonded interactions. 2.2.  Calibration of Force-Field Parameters Having specified the functional form of the interactionterms, the formidable task of finding appropriate, consistentvalues for the hundreds of force-field parameters remains tobe addressed. This task involves the choice of type of data,type of systems, thermodynamic phase, and properties to beused as the calibration set for specific force-field parameters.The choices made for the GROMOS force field are summar-ized in Table 4. Since biomolecular systems are generally inthe condensed phase, data for the condensed phase (exper-imental and theoretical) are used whenever possible. Fur-thermore, to maximize the transferability of parametersbetween groups of atoms in different molecules, only datafor small molecules are used. When using data from largemolecules such as proteins (e.g. from the protein data bank)properties of groups of atoms may be dependent on theirparticular environment in the folded molecule. Furthermore,the protein data bank contains structures measured at widelydifferent thermodynamic conditions (pH value, ionicstrength, etc.). Finally, certain properties will be stronglyrelated to specific force-field parameters and only weakly toothers. This situation offers the opportunity to reduce thecalibration effort by optimizing specific subsets of parametersseparately against a limited set of properties. W. F. van Gunsteren et al. Reviews 4068   2006 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim  Angew. Chem. Int. Ed.  2006 ,  45  , 4064–4092
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