Calculations of Protein-Ligand Binding Affinities

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Calculation ofProtein-Ligand BindingAffinities

Michael K. Gilson1 and Huan-Xiang Zhou2

1Center for Advanced Research in Biotechnology, University of MarylandBiotechnology Institute, Rockville, Maryland 20850; email: [email protected]

2Department of Physics and Institute of Molecular Biophysics and School ofComputational Science, Florida State University, Tallahassee, Florida 32306;email: [email protected]

Annu. Rev. Biophys. Biomol. Struct 2007. 36:2142

First published online as a Review in Advance onJanuary 2, 2007

The Annual Review of Biophysics and BiomolecularStructure is online at biophys.annualreviews.org

This articles doi:10.1146/annurev.biophys.36.040306.132550

Copyright c 2007 by Annual Reviews.All rights reserved

1056-8700/07/0609-0021$20.00

MKG dedicates this article to the memory of hisfather, Saul Bernard Gilson, MD: scholar, poet,and physician, 19222006.

Key Words

computation, drug-design, entropy, free energy, modeling

Abstract

Accurate methods of computing the affinity of a small molecule with

a protein are needed to speed the discovery of new medications andbiological probes. This paper reviews physics-based models of bind-

ing, beginning with a summary of the changes in potential energysolvation energy, and configurational entropy that influence affinity

and a theoretical overview to frame the discussion of specific computational approaches. Important advances are reported in modeling

protein-ligand energetics, such as the incorporation of electronicpolarization and the use of quantum mechanical methods. Recen

calculations suggest that changes in configurational entropy stronglyoppose binding and must be included if accurate affinities are to be

obtained. The linear interaction energy (LIE) and molecular me-

chanics Poisson-Boltzmann surface area (MM-PBSA) methods areanalyzed, as are free energy pathway methods, which show promise

and may be ready for more extensive testing. Ultimately, major im-provements in modeling accuracy will likely require advances on

multiple fronts, as well as continued validation against experiment.

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Ligand: in general,

any molecule thatbinds another; usedhere for a drug-likeorganic molecule,typically ofmolecular weightless than 600 Da

Contents

INTRODUCTION . . . . . . . . . . . . . . . . . 22MODELING THE PHYSICAL

CHEMISTRY OF BINDING:FORCES AND FLEXIBILITY . . . 23

Potential Energy . . . . . . . . . . . . . . . . . 23

Effects of Water, DissolvedElectrolytes, and pH. . . . . . . . . . . 24

Theoretical Framework for

Calculating Affinities . . . . . . . . . . 26Strain, Configurational Entropy,

and Preorganization . . . . . . . . . . . 27

APPROACHES TOCALCULATING AFFINITIES . . 28

Docking and Scoring . . . . . . . . . . . . . 28Free Energy Methods . . . . . . . . . . . . 30

PROSPECTS AND

CHALLENGES . . . . . . . . . . . . . . . . . 33

INTRODUCTION

Molecular recognition is central to biology,

forming the foundation for precise hormonalcontrol of distant organs, immune targeting

of nonself proteins, and the specificity of en-zymatic catalysis, for example. It also offers a

powerful mechanism for exogenous control ofbiological systems, and many medications and

biological probes act by binding and usually

inhibiting a specific macromolecular target.Indeed, the discovery of a ligand that binds

a targeted protein with high affinity is a ma-jor preoccupation of early-stage drug discov-

ery and also of chemical genomics projectsseeking small-molecule inhibitors to eluci-

date gene function. However, discovering asmall molecule that binds a specific protein

tightly, while retaining favorable pharmaco-logical properties, can be a major and costly

challenge.Computation can speed this process (62)

by artificial intelligenceand machine-learning

technologies and through simulation andmodeling. In particular, structure-based mod-

eling uses the three-dimensional atomic co-

ordinates of the targeted protein to calculatethe binding free energy of a proposed lig-

and, or at least to rank candidate ligands ac-cording to their predicted affinities for the

target. This information is then available toguide molecular design and synthesis. Ide-

ally, computational methods would be as ac-curate as experiment, with binding affini-ties correct to within 1 kcal mol1, but this

goal has not been achieved. Indeed, cal-culating protein-ligand affinities is difficult

One reason is that the free energy of bind-ing is typically a small difference between

large numbers, namely the interactions of theligand and protein with each other versus

their interactions with water and counteri-ons in their unbound state. Also, the inter-

atomic forces are strong and short ranged,making for an energy function sharply de-pendent on the details of molecular confor-

mation. Finally, the protein and the ligandare flexible and have many degrees of free-

dom; therefore exploring all potentially rele-vant conformations is a large computationa

task.Despite these difficulties, computer mod-

eling has enormous potential to make accu-rate predictions and provide insights that can

guide molecular design. In this paper, we re-view recent progress and discuss prospects for

further advances. We begin with a discussionof the forces involved in molecular recogni-

tion and how they are modeled. In partic-

ular, many approaches to calculating bind-ing affinities can be analyzed by identifying

how they treat the potential energy of theprotein and ligand as a function of confor-

mation, the influence of the aqueous solventand the conformational flexibility of the two

molecules. This discussion provides the basisfor a review of approaches to modeling bind-

ing, which are broadly categorized as dock-ing and scoring and free energy methods

A short conclusion touches on the prospectsfor further progress. Space limitations pre-

cludeaddressing manyinterestingtopics, such

22 Gilson Zhou


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as the treatment of protonation equilibria,

the possible consequences of macromolecularcrowding on binding, the automated design

of ligands, and the special challenges posedby nucleic acid targets. For the same rea-

son, the literature is summarized with a broadbrush, using representative, rather than ex-

haustive citations, in hopes of conveying ina few pages the structure of the field and pro-viding a framework that helps the reader make

sense of a range of papers.

MODELING THE PHYSICALCHEMISTRY OF BINDING:FORCES AND FLEXIBILITY

Perhaps someday there will be computers fastenough to compute binding affinities from

first-principles quantum mechanical simula-tions of all the nuclei and electrons in a fem-toliter solvent droplet, with dissolved protein

and ligand molecules. For now, however, onemust rely on a theoretical framework that

yields the affinity from calculations on justone protein and one ligand, and on approx-

imations that allow energies and forces to becomputed more rapidly than does an ab ini-

tio quantum mechanical treatment. In gen-eral, such approximations entail reducing the

number of degrees of freedom that are treatedexplicitly and replacing them with new forces

that account for the implicit degrees of free-

dom. For example, one can model the wa-ter around a protein by explicitly simulating

the motions of thousands of water molecules,allowing the properties of water to emerge

automatically from the simulation. Alterna-tively, one can discard the explicit water

molecules and instead use an implicit watermodel consisting of new energy terms for the

hydrophobic effect and dielectric screening.Such models can provide a useful intuitive

understanding of the system, but it is alsoimportant to remain alert to their potential

weaknesses.

Below we outline current thinking on keyenergycomponents that drive or opposebind-

Strain: the energycost of adopting thebound conformation

ing, breaking them into potential energy (as

if in vacuo) and the effects of solvent. Thena theoretical framework for computing affini-

ties based on these forces is described, fol-lowed by some of the implications of the

theory for binding thermodynamics, such asstrain and entropy.

Potential Energy

The potential energy of an isolated moleculeor complex in a given conformation can be

computed by quantum mechanical methodsthat explicitlyaccount for the electronicstruc-

ture. Such quantum calculations can be donein various ways, but a central division is

that between more time-consuming ab ini-tio methods and faster semiempirical meth-

ods, which substitute fitted parameters forquantities that would otherwise need to becalculated. The details of these approaches

are beyond the scope of this review. Quan-tum methods in principle are broadly ap-

plicable, but they are time-consuming rel-ative to the force-field models discussed in

the next paragraph, typically requiring min-utes to months of computer time, depending

upon the size of the system and the level ofthe calculation. Therefore, it is difficult to

use them in methods that calculate the en-ergy of many different conformations. It is

also important to be aware that many ab initio

methods markedly underestimate the attrac-tive forces due to electron-electron correla-

tions between atoms, i.e., dispersion forces,the attractive part of the van der Waals inter-

action.Empirical force fields (8) represent a

computationally fast alternative to quantummechanical calculations of the potential en-

ergy, because they encapsulate the chief con-sequences of electronic structure in a few

energy terms that are easily calculated, of-ten in well under 1 s for one conforma-

tion of a system. Current examples includeCHARMM (15), AMBER (30), and GRO-

MOS (121). A typical functional form is given

www.annualreviews.org Calculation of Protein-Ligand Binding Affinities 23


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Electronicpolarization:inducedredistribution ofelectronic charge inan atom or molecule

by Equation 1:

U =Nbonds

i=1

kb,i(b i b0,i)2

+

Nanglesi=1

k,i(i 0,i)2

+

Ndihedralsi=1

k,i(cos(nii 0,i))

+

Nimprpdhdi=1

k,i(i 0,i)2

+

Natomsi

Natomsj=i

Ai j

r12i j

Bi j

r6i j

+

Natomsi

Natomsj>i

qiqjDri j

. 1.

The first two terms treat bond stretch-

ing and angle bending by harmonic springswith large force constants. The third termaccounts for bond torsions (dihedrals), and

the fourth, improper dihedral, term is usedto keep planar groups planar. The fifth term

accounts for the van der Waals interactionsamong atoms with a short-ranged hard re-

pulsion that prevents atoms from overlap-ping and a slightly longer-ranged weak at-

traction attributable to correlated motionsof the electrons (dispersion forces). The fi-

nal term accounts for Coulombic interactionsbetween atoms, which result chiefly from thefact that electrons tend to concentrate around

atoms with large electronegativities and be-come depleted elsewhere, leading to negative

and positive partial atomic charges; those withcharges of large magnitude are polar atoms.

It is evident from Equation 1 that an empir-ical energy function contains many parame-

ters, such as the spring force constants k, thevan der Waals parameters A and B, and the

atomic charges q. These parameters can of-ten be obtained by fitting to high-level quan-

tum calculations, but it is best if they canbe tested and adjusted by computing physical

properties, such as solvation energies of model

compounds,and comparing them with experi-mental data. When suitable experimental data

are not available, it can be difficult to generate

trustworthy parameters.Newer force fields are becoming more

complex in order to achieve greater accuracyMore detailed models of the charge distri-

bution can be employed (87, 102, 115), astouched on below, andthere is alsogreatinter-

est in accounting for electronic polarization,the redistribution of atomic charges induced

by changes in the electrical fields to which the

atoms are subjected (66, 77).

Effects of Water, DissolvedElectrolytes, and pH

Protein-ligand binding normally occurs in a

salt-water environment, which has a strongeffect on energetics. Water has a dielectric

constant (D in Equation 1) of about 80whereas the dielectric constant of vacuum is

1. This leads to a new, one-body solvation en-

ergy for each atomiccharge, which arises fromthe favorable interaction between the charge

andthe high-dielectric environment (12). As aconsequence,therecanbeasubstantialenergy

penalty for moving the polar part of a ligandout of water and into the binding site. In ad-

dition, water screens the charge-charge inter-actions of fully hydrated atoms by80-fold

as indicated in the Coulombic term in Equa-tion 1. However, atoms in a protein-ligand in-

terface are sequestered from the solvent andtherefore interact with an effective dielectric

constant less than 80. In general, atoms that

are further apart are more likely to interactthrough solvent, and this idea led to the intro-

duction of a crude screening model consistingof a distance-dependent dielectric. For atoms

i and j in Equation 1, Dij = Crij, where C isa constant often set to 4 and rij is the inter-

atomic distance. This model allows one of thechief effects of the solvent to be accounted for

in a computationally efficient manner, andit isusedinanumberofligand-proteindockingal-

gorithms. However, it fails to account for theone-body solvation energy of each atom. In

addition, the electrostatic interaction of two

24 Gilson Zhou


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atoms actually depends not only on their mu-

tual distance, but also on the positions of allthe other protein and ligand atoms, because

these determine wherethe high-dielectricsol-vent can penetrate.

These additional complexities can be cap-tured by continuum electrostatics models,

which treat the protein and ligand as low-dielectric regions with embedded atomiccharges, surrounded by the high-dielectric

aqueous solvent (55). In particular, numeri-cal solutions of the Poisson-Boltzmann (PB)

equation offer a reasonably efficient methodof accounting for the subtleties of molec-

ular electrostatics, including the additionalelectrostatic screening produced by dissolved

salts. The PB equation usually can be solvedfor a protein in seconds to minutes of com-

puter time for a biomolecular system, and ityields the interaction energy of the protein,

ligand, or complex with the water in which

it is immersed. This electrostatic part of thesolvation energy can be added to the poten-

tial energy obtained by a force field or othermethod to provide a more complete account-

ing of conformational energetics. Whenmanyconformations need to be processed, faster

generalized Born (GB) models are valuable asapproximations to the full PB model (34, 51,

58, 59, 80, 97). These are based on a recog-nition that dielectric screening of a charge-

charge interaction correlates with the degreeto which each charge interacts with surround-

ing water (43).

Water also gives rise to the hydropho-bic effect, the tendency of water molecules

to drive nonpolar solutes together (68). Thispromotes the association of nonpolar sur-

faces of the ligand and the protein. The hy-drophobic effect is often accounted for by an

additional solvation energy term that is pro-portional to molecular surface area, with a

positive coefficient. The effect is to add a pos-itive (unfavorable) solvation energy to con-

formations with more surface area and thusto favor binding, which reduces surface area.

Combining the PB or GB electrostatics mod-

Poisson-Boltzmann(PB) equation: apartial differentialequation solved toobtain the molecularelectrostatic

potential, whichdepends on thelocations and valuesof atomic charges,the dielectricconstant as afunction of position,and the ionicstrength

Generalized Born(GB) model: a fastapproximation to thePoisson-Boltzmannequation

PBSA:Poisson-Boltzmannsurface area

GBSA: generalizedBorn surface area

Implicit solventmodel: a model thaaccounts for solventwithout explicitlymodeling individualwater molecules

MD: moleculardynamics

MC: Monte Carlo

els with such a surface area term yields the

PBSA (112) and GBSA (97) solvation mod-els, respectively. These are called implicit sol-

vent models because they do not treat anywater molecules explicitly. Once parameter-

ized, the PBSA and GBSA models providerather good agreement with experimental sol-

vation energies of small model compounds(10, 112), but they may be less accurate formore complex molecules, such as proteins,

that can bind or completely sequester individ-ual water molecules. Unfortunately, it is not

straightforward to generate experimental datathat directly address this issue, but computa-

tional studies are beginning to be applied to it(50, 83, 91, 118).

The influence of solvent on binding canalso be treated with molecular dynamics (MD)

or Monte Carlo (MC) simulations that in-clude thousands of explicit water molecules

modeled with an empirical force field (6, 64,

102). Dielectric screening, the solvation ofpolar groups, and the hydrophobic effect all

emerge automatically with this approach. Inaddition, it should provide a better treatment

of bound and sequestered water molecules, atleast in principle. However, an explicit treat-

ment of solvent is substantially more costlycomputationally than an implicit model, by

perhaps an order of magnitude, depending onthe specifics of the comparison.

Physiological solutions also buffer protonsat a specific pH and thus control the elec-

trical charges of acidic and basic groups of

both the protein and the ligand. As a re-sult, the pH can strongly influence confor-

mational preferences and binding affinities.Indeed, protonation and binding are thermo-

dynamically linked, so that the pH affectsthe apparent binding affinity and binding in

turn affects protonation states. A number ofresearch groups are working on methods of

predicting protonation states, which will un-doubtedly be important for computing accu-

rate affinities. The potential for many drug-like molecules to exist in multiple tautomeric

states poses similar challenges.



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Theoretical Framework forCalculating Affinities

Given a method of computing the potential

and solvation energies of a protein, a ligand,and their complex, a theory of binding is still

needed to compute their affinity. One widelyused approach (42) is based on recognition

that the free energy of binding is the changein free energy when one protein and one lig-

and react to form one complex. The change infree energy of a solution when one complex is

added to the system is defined as the complexschemical potential, P L; similarly, the changein free energy when one protein and one lig-

and are removed is P L. Thus, the freeenergy of binding is G = P L P L.

Statistical thermodynamics provides an ex-

pression for the chemical potential of the pro-

tein, for example, as

P = RTln

QNP+1,NS

QNP,NS

= RTln

82

CP

e (U(rP)+W(rP))/RTdrp

,

2.

where R is the gas constant; T is the abso-

lute temperature; QNP+1,NS and QNP,NS arethe canonical partition functions of solutions

ofNP + 1 proteins and NS solvent moleculesandNP proteins andNS solvent molecules, re-

spectively; the factor of 82 results from thefact that overall rotation of the protein does

not affect the integral over the other coordi-

nates of the system; CP is the concentration

of the protein; U(rP) is the potential energy of

the protein as a function of its internal coor-dinates rP and thus of its conformation; and

W(rP) is the corresponding solvation energy.

(A mass-dependent prefactor that cancels in

the final expression for the binding affinity hasbeen omitted, as has a small pressure-volume

term.) The Boltzmann factor in the integralis larger for more stable conformations of the

protein, so that the more low-energy confor-mations the protein can adopt, the greater the

integral and the more negative its standardchemical potential, implying greater stabilityand less reactivity. Also, the higher the con-

centration CP, the more positive the standardchemical potential, implying greater reactiv-

ity, consistent with the law of mass action.Supplying analogous formulae for the

chemical potentials of the ligand and thecomplex allows the free energy of binding to

be written as

G = RTln

182

CPCLCP L

e (U(rP L )+W(rP L ))/RTdrP L

e (U(rP)+W(rP))/RTdrP

e (U(rL )+W(rL ))/RTdrL . 3

For the most part, the integrals in this expres-

sion are considered to range over all confor-mations, but only conformations that are low

in energy actually require attention; for ex-ample, conformations with highly stretched

bonds can be neglected. However, a sub-tlety arises for the protein-ligand complex

because the integral should include only con-

figurations in which the ligand is consideredboundtotheprotein.Fortightcomplexes,any

integration domain that includes the lowest-energy configurations and not too much more

suffices (42); for weak binding, a more in-volved definition is needed (88).

Replacing each concentration in Equation3 by the standard concentration C (1 M)

yields the standard free energy of binding:

Go

= RTln Co

82e (U(rP L )+W(rP L ))/RTdrP L

e (U(rP)+W(rP))/RTdrP

e (U(rL )+W(rL ))/RTdrL

. 4

The factor of C makes it apparent that the

higher the standard concentration, the morenegative the standard free energy of binding

again consistent with the law of mass actionThe relationship ofGo to the equilibrium

26 Gilson Zhou


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constantK is established by recognizing thatG= 0 when the system is at equilibrium.Equation 3 then gives the equilibrium ratio

of concentrations as:

CP LCP CL

eq

=1

82e

(U(rP L )+W(rP L ))/RTdrP Le (U(rP)+W(rP))/RTdrP

e (U(rL )+W(rL ))/RTdrL

5.

Multiplying both sides byCo and using Equa-tion 4 yields that

K = e Go/RT =

CP LCo

CPCL

eq

6.

Equation 4 can be used to derive variousmethods of computing binding affinities, in-

cluding Double Decoupling andMining Min-ima (23, 52). Furthermore, Equation 4 can be

manipulated into the form

Go = UP L UP UL + WPL

WP WL TSoconfig, 7.

where UP L and WP L, for example, arethe Boltzmann-averaged potential and sol-

vation energies of the complex; and Soconfig,the change in configurational entropy, is the

change in the entropy associated with ligandand protein motions, including translational

motion, for binding at standard concentra-tion. The solvation terms are actually solva-

tion free energies and include both energetic

and entropic components. Equation 7 is rec-ognizable as the basis of most physics-based

scoring functions as well as the molecularmechanics Poisson-Boltzmann surface area

(MM-PBSA) method (46, 114).Some methods of computing binding

affinities yield predicted values ofGo thatdo not depend on the standard concentra-

tion. This is problematic because the stan-dard concentration is arbitrary. Indeed, if it

were changed from 1 M to 1 nM, for example,then every experimental standard free energy

of binding would be reported as RTln109 =

12kcalmol1 more positive. Therefore, a casecan be made that the success of a model is

fortuitous if the models predictions do not

Configurationalentropy: theentropy associatedwith motions of theligand and/or protein

depend on standard concentration. On the

other hand, a model that omits standard con-centration can legitimately provide the differ-

ence between two binding affinities, because

the standard concentration cancels in suchdifferences.

Strain, Configurational Entropy,and Preorganization

Equations 4 and 7 not only provide a basis forcomputational models, but also have impor-

tant implications that can be understood intu-itively. In particular, they bear directly on theconcepts of strain, configurational entropy,

and preorganization. If the ligand adopts astrained conformation in the bound state, the

rise of its internal energy on binding will makea positive contribution to UP L UP UL

and thus oppose binding. Alternatively, if theligand accesses a wide range of equally sta-

ble conformations in the free state, but only anarrow range of conformations in the bound

state, its configurational entropy will drop andthus oppose binding. Finally, if the bound

conformation of the ligand corresponds to an

especially stable conformation of the free lig-and, then the ligand is preorganized for bind-

ing, and both the strain and entropic penaltieswill be low. The same considerations apply to

the protein.Changes in configurational entropy upon

binding are the subject of a literature too com-plex and extensive to be summarized here; a

key problem has been the lack of a methodto compute these quantities. The Mining

Minima technique does allow such calcula-tions, and applications to host-guest model

systems (23, 26) and a protein-ligand sys-

tem (22) indicate that the entropic penalty ofEquation 7 is large, opposing binding nearly

as strongly as the energy changes favor it.



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Thus, the loss in configurational entropy can-

cels most of the binding energy, and the netbinding free energy is a small difference be-

tween larger numbers. Importantly, leavingout the configurational entropy, and thus fo-

cusing only on energy, leads to significantlyworse correlation with experimental free en-

ergies of binding. Nonetheless, there is signif-icant entropy-energy correlation, and scalingthe energy terms by a factor of perhaps 0.1 to

0.3 yields numbers in the experimental range.Similar results should hold for protein-ligand

binding.

APPROACHES TOCALCULATING AFFINITIES

Docking and Scoring

Docking methods try to identify the moststable conformation of the ligand-protein

complex using a simplified energy model, fre-quently an empirical force field with a sim-

ple solvent model for the sake of computa-tionalspeed.Theconformationisthenusedto

assign a binding energy or score to the ligand,sometimes using the same energy model that

was used during docking and sometimes witha more sophisticated and time-consuming

model. Reliance on a single bound conforma-tion in effect assumes that this conformation

is the only one that is significantly occupied,

so that it dominates the averages in Equation7. Pioneering initial approaches (32, 45, 76)

have been followed by further advances fromthe same groups and many others, as recently

reviewed (14, 113), and a number of meth-ods are actively used for computer-aided drug

discovery (110).The proliferation of docking and scor-

ing methods also has inspired evaluations andcomparisons (25, 69, 125, 127), which are of

particular interest to drug companies mak-ing purchasing decisions. Some of the most

commonly used criteria are a methods abil-

ity to reproduce the correct bound conforma-tion of a ligand-protein complex, its ability to

assign better energy scores to known high-

affinity ligands than to a large set of decoy

compounds (enrichment), and the ability togenerate scores that correlate with measured

binding affinities of known ligands. Over-all, existing methods frequently yield accurate

bound complexes and can usually provide sig-nificant enrichment. However, the results are

system dependent, and different methods per-form better on different systems. In addition,

scores are at best weakly predictive of affini-

ties across a series of known ligands. Carryingout a fair evaluation poses its own challenges

(29); for example, enrichment studies pro-vide different results when the known ligands

have nanomolar versus micromolar affinitiesDocking results also depend on procedural

details outside the actual docking protocolsuch as the choice of protonation sites on the

ligands (75, 120). Results are sometimes im-proved when multiple scoring functions are

combined in a single consensus score, withthe idea that a ligand that ranks high on mul-

tiple scoring functions is more likely to bind

tightly (7, 24, 90). Nevertheless, the gains ofthis approach are limited by the simplicity of

typical scoring functions, which are designedfor computational throughput.

As a consequence, there is great interest inmore sophisticated and hence more accurate

scoring functions. If these prove to be slow, itwill still be possible to prescreen compounds

with a fast scoring function and apply themore accurate methods only to compounds

that pass the prescreen. The fast method alsocan provide initial docked conformations to

initiate calculations with the slower one. The

following subsections review efforts to in-crease the sophistication of docking and scor-

ing methods.

Improved potential energy models. Oneimportant direction involves using potential

energy models that are more advanced than atypical empirical force field. A particularly in-

teresting study provides evidence that dock-ing accuracy is substantially better when the

energy model accounts for the redistributionof ligand charges due to the electric field of

28 Gilson Zhou


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the binding site, i.e., for electronic polariza-

tion (28). The procedure involves iterativelyadjusting partial atomic charges via quantum

mechanicalcalculations during a docking pro-cedure. Rather large shifts in charge were ob-

served, up to an average of 0.4 e across allatoms for some ligands. A number of other

studies also explore force fields with more de-tailed treatments of electrostatics (31, 37, 87),and a different style of force field, designed for

use with metalloenzymes, has now been usedto model protein-ligand systems (48).

Another approach has been to use a quan-tum mechanical treatment of the ligand, and

eventheprotein,ratherthanrelyonanempir-ical force field (47, 71, 98, 122). Because such

methods capture dispersion forces poorly, avan der Waals dispersion term has been com-

bined with the quantum approach, as done inan intriguing trial application to 165 protein-

ligand systems (99). A recent review provides

a useful perspective on the use of quantumcalculations in structure-based drug discovery

(95).

Improved solvent models. Many scoringfunctions have relied on simple models of the

solvent, such as the distance-dependent di-electric. These fail to capture the sensitivity

of the interactions between atomic charges tothe details of their locations relative to the

solvent, as well as the fact that polar groupshave strong, favorable interactions with the

solvent that may be lost upon binding. This

issue has been addressed by incorporatingPBSA and GBSA solvation models into dock-

ing and scoring, and a number of papers re-port improvements in accuracy (56, 84, 86,87,

111, 135, 136). However, increased accuracyin some systems requires careful assignment

of protonation and tautomer states (38, 56).It seems that, in these cases, removing one

source of error exposes another one.Several studies also address the formation

ofwaterbindingsitesduringdocking(41,100,123, and references therein), but the methods

tried so far do not seem to markedly improve

accuracy. In addition, it is not yet clear how to

adjust the score of a ligand if it is predicted to

bind with a water molecule.

Accounting for conformational changes of

the receptor. The first docking algorithms

typically treated the protein as rigid, althoughbinding sites in reality adjust to accommodate

different ligands. This simplification can beproblematic, because even a small steric over-

lap can make a good ligand look like a misfit

if it is docked into a single conformation ofthe receptor that was solved with a different

ligand bound. However, because the recep-tor possesses many internal degrees of free-

dom, treating it as flexible can vastly expandthe problem of discovering the most stable

conformation of the protein-ligand complex.Moreover, as the number of candidate energy

minima rises, the energy model must becomeincreasingly accurate in order to identify the

global minimum correctly. A wide variety ofapproaches have been devised to address this

problem, as previously reviewed (18). Meth-

ods range from using a single conformation ofthe receptor, but softening it to approximate

the consequences of flexibility (60); to dock-ing into a single representation of the binding

site obtained by averaging across multiple re-ceptor structures (16, 74, 93); to docking into

multiple discrete receptor conformations (1,20, 39, 130, 131); to reoptimizing the confor-

mation of the binding site during the dockingcalculation itself (65, 70, 108, 133, 134). The

relative merits of the various approaches arenot yet clear, though one study provides ev-

idence that soft docking yields worse enrich-

ment of known ligands than does docking intomultiple structures of the receptor (39).

Configurational entropy. Even if a docking

algorithm always identifies the global energyminimum of a protein-ligand complex with

a perfectly accurate potential and solvationenergy function, it will not yield an accurate

binding free energy. One reason is that thebinding free energy includes a loss in con-

figurational entropy due to the decrease infreedom of the ligand and the protein upon



10/24

binding. As noted above, the loss of configu-

rational entropy is nearly as large as the gainin favorable binding energy and correlates

roughly with the energy change. Therefore,it is expected that a model that accounts for

changes in potential and solvation energy, butneglects the loss of configurational entropy,

will tend to overestimate binding affinities. Infact, the energy terms in a physics-based scor-ing function had to be scaled down roughly

10-fold to bring the results into range withexperiment (89), as previously noted (26); a

similar result is observed in calculations withthe MM-PBSA method when configurational

entropy is neglected (17).A number of scoring methods use the

rigid-rotor/harmonic-oscillator formalism toseparate the ligands loss of entropy into a

rotational and translational part that resultsfrom trapping in the binding site and another

part associated with the ligands loss of flex-

ibility. It is generally assumed that any en-tropy change associated with the hard degrees

of freedom (bond stretches and angle bends)is negligible. This is reasonable because bond

lengths andbond angles areunlikely to be per-turbed much by the noncovalent forces that

drive protein-ligand binding. It is also widelyassumed that changes in rotational and trans-

lational entropy are either negligible or con-stant across all binding reactions, but this view

is not well founded. In fact, the translationaland rotational motions of a bound ligand de-

pend on exactly how tightly it is held (23, 26),

and this can vary not only from one complexto another, but also among alternative bound

conformations of a single complex (26). Thistopic poses nontrivial theoretical issues and is

now the subject of increasing computationalscrutiny (2, 33, 78, 85, 117, 138). For example,

a recent study reports improved docking ac-curacy with a novel method of estimating the

entropy of a bound ligand from the rangesof rotational, translational, and torsional co-

ordinates associated with multiple dockedconformations (107).

Changes in entropy due to the decreased

flexibility of the ligand upon binding are com-

monly regarded as resulting from a reduction

in the number of energetically accessible lig-and rotamers upon binding and are often ac-

counted for with empirical termsproportionalto the number of rotatable bonds of the ligand

affected by binding (9, 36, 44, 101, 124). Thecorresponding free energy penalties are typi-

cally on the order of 0.4 kcal mol1

torsion1

but run at least to 1 kcal mol1 torsion1 (99)However, a simple argument suggests that

such estimates markedly overestimate the lossof entropy due to the drop in accessible ro-

tamers. For a ligand with 10 rotatable bondsthe torsional penalty would be 4 kcal mol1

corresponding to a drop of e4/RT = 800-fold in the number of equally stable ligand

rotamers upon binding. This implies a mini-mum of 800 equally stable, rotamerically dis-

tinct ligand conformations in the free stateThis number seems improbably large. It isnot clear whether the literature has directly

addressed this issue, but programs designedto generate a large number of diverse ligand

conformations typically yield far fewer con-formations (13, 72). Moreover, of these con-

formations, only a few are likely to be within1 kcal mol1 of the global energy minimum

Therefore, it is unlikely that rotamer count-ing accounts for much of a ligands entropy

loss upon binding. Recent studies of modelsystems do reveal substantial losses in con-

figurational entropy upon binding, but they

attribute these losses to the fact that the en-ergy wells of a bound complex are narrower

than those of the free species, rather than toa reduction in the number of stable rotamers

(22, 26). Overall, it appears that a deeper un-derstanding of changes in configurational en-

tropy upon binding could lead to improvedaccuracy in docking and scoring.

Free Energy Methods

Free energy methods use conformationa

sampling to generate thermodynamic av-

erages, in contrast with docking methodswhich focus on a single bound conforma-

tion. The use of conformational averaging is

30 Gilson Zhou


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advantageous because it removes sensitivity to

the details of the single representative confor-mation relied on by docking methods. On the

other hand, free energy methods need morecomputer time in order to generate unam-

biguous, converged results. Also, the confor-mational searches tend not to be wide ranging

and thus may need to be seeded with severalplausible docked conformations. Two generalapproaches are considered here: end-point

methods, which generate conformations ofonly the free and bound species and compute

the binding free energy by taking a differ-ence, and pathway methods, which compute

the sum of small changes along a multisteppathway connecting the initialand final states.

End-Point methods. The linear interaction

energy (LIE) method (3) involves runningtwo MD simulations with an empirical forcefield: one for the ligand in solution, and the

other for the ligand in the protein bind-ing site. Snapshots saved from the simula-

tions represent Boltzmann ensembles of con-

formations and are used to compute theBoltzmann-averaged electrostatic (Uelec) and

van der Waals (Uvdw) interaction energies ofthe ligand with its environment in the bound

and free states. The binding free energy is

then estimated as

G o (Uelecbound Uelecfree)

+(Uvdwbound Uvdwfree), 8.

where the angle brackets indicate Boltzmannaverages. Changes in the internal energy of

the solvent and the protein are considered tobe accounted for by the factors and , on

the basis of their reorganization energy in re-sponse to the ligand (3). As recently summa-

rized (19), good correlations between calcu-lation and experiment have been reported in

a number of publications, both with the stan-dard coefficients = 0.18 and = 0.33 (19)

and with system-specific values of these co-

efficients (126). An additional surface-area-dependent term has also been incorporated

(61), and more recent variants speed the MD

LIE: linearinteraction energy

calculations by using implicit treatments of

the solvent (19, 135). In general, it does notseem that a single set of coefficients can be

applied to all systems; therefore some experi-mental binding data for the system of interest

may be needed before this method can be usedeffectively.

Nonetheless, the quality of the results issomewhat surprising, because LIE does notaccount explicitly for standard concentration

or for changes in the configurational entropyor the internal energy of the ligand. The

method may be successful in part because itis generally used to compare ligands within a

single chemical series. Entropy-energy com-pensation (23, 26), and cancellation of the

neglected drop in solute entropy by the ne-glected rise in the solvation entropy, may also

help.A second end-point approach, the

MM-PBSA method, along with its GB vari-

ant (MM-GBSA), uses MD simulations of thefree ligand, free protein, and their complex

as a basis for calculating the average potentialand solvation energies in Equation 7 (46,

114). The MD runs typically use an empiricalforce field and an explicit solvent model.

The resulting snapshots i are postprocessedby stripping them of their explicit solvent

molecules and computing their potentialenergies Ui with the empirical force field

and their solvation energies Wi with either

the PBSA or GBSA implicit solvent model.Averaging over each trajectory allows the

changes in mean potential and solvationenergy to be calculated, as per Equation 7.

The change in configurational entropy isthen estimated by energy-minimizing a few

snapshots of the free and bound molecules,computing their entropy with the rigid-

rotor/harmonic-oscillator approximation,and obtaining the average entropy change

over these snapshots. This approach neglectsany contribution to the entropy change that

may result from a change in the number

of thermodynamically accessible energyminima. The quasiharmonic approximation

(67) also has been used to estimate the change



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in configurational entropy from the MD sim-

ulations (57, 79), despite concerns regardingits accuracy (21). It is not always clear from

the literature whether or how the standardconcentration has been incorporated into

MM-PBSA or MM-GBSA calculations, buta correct approach is available (equation 3

in Reference 96); therefore the MM-PBSAmethod allows a clear connection with thetheory of binding.

Although early applications of MM-PBSAappeared promising, it apparently is diffi-

cult to converge the energy averages reli-ably. One reason for this difficulty may be

that they encompass energy fluctuations notonly of the ligand and the binding site, but

also of parts of the protein remote from thebinding site, which are less relevant to bind-

ing. Convergence has been accelerated by asingle-trajectory approach (17, 79), in which

only one MD simulation of the protein-ligand

complex is carried out. Conformations of thenominally free ligand are then derived sim-

ply by deleting the protein from the result-ing snapshots, and likewise for the free pro-

tein. In this single-trajectory approach, onlyligand-protein distances contribute to the

computed change in potential energy, becauseall other interatomic distances stay exactly the

same. The single-trajectory approach resem-bles LIE because it similarly neglects changes

in internal energy and configurational en-tropy of both ligand and protein. However,

it differs from LIE in that it assumes the same

conformational distribution for the free andbound ligand, whereas these change in LIE.

Free energy pathway methods. The

change in free energy between two states ofa system, such as before and after binding,

can be written formally as RTln e U/RT,where U is the change in the energy

function between the initial and final states,and the angle brackets indicate a Boltzmann

average taken in the initial state (137). For

binding, U would reflect the introductionof interactions between the protein and

ligand, and the average could, in principle,

be obtained by MD or MC simulations. In

reality, such a simulation is extraordinarilydifficult to converge unless the initial and

final states are similar. The free energyperturbation method solves this problem

by breaking the change into N small stepsU (perturbations) and running a separate

simulation for each resulting energy functionUi to obtain the stepwise free energy changesRTln

e U/RT

i

associated with each step

where i indicates a Boltzmann average withthe energy function Ui. Thus, the initial and

final states are linked by a pathway of smallsteps for which the small free energy differ-

ences can be computed. Another pathwayapproach, thermodynamic integration (TI)

(73, 116), uses MD or MC to compute thefirst derivative of the free energy with respect

to the distance along the path and thenestimates the total change in free energy via anumerical integral of the derivative along the

path. Because the first derivative of the freeenergy is effectively a force, TI effectively

involves a work integral. Either free energyperturbation or TI can be applied to a given

free energy calculation, and both methodsare widely used.

The first applications of such methods tobinding did not seek to compute the stan-

dard free energy of binding, but rather tocompute the difference between the binding

free energies of two similar ligands, Go

by a method termed computational alchemy(119). This involves using pathways to com-

pute the change in free energy when ligandA is changed to ligand B within the bind-

ing site, as well as in solution. A free en-ergy cycle shows that the difference between

these two free energy changes equals the dif-ference between the binding free energies

of the two ligands. The calculations typi-cally employ lengthy MD or MC simulations

(hours to many days, depending on the cal-culation) with an empirical force field and

an explicit treatment of solvent. Relative to

computing full-fledged standard free energiesof binding, in which a ligand must be re-

moved from the binding site, computationa

32 Gilson Zhou


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alchemy has the merit of focusing on a rela-

tively small perturbation, in which one ligandis changed to another similar one, and thus of

converging relatively rapidly. Computationalalchemy also avoids the problem of account-

ing for standard concentration, because it canbe shown from Equation 4 that Co cancels

in Go

. The method is theoretically soundand is supported by a literature reporting rela-tively accurate results, often within 1 to 2 kcal

mol1 of experimental data. Still, computa-tional alchemy has not yet been subjected to

extensive validation studies like those used toevaluate docking and scoring methods.

The use of computational alchemy hasbeen limited in part by the amount of com-

puter time it requires, especially when ligandA and ligand B are markedly different. How-

ever, advances in computer technology andsimulation algorithms allow practical applica-tion of this method to drug discovery (4, 27,

49, 104). Also, a clever extension of computa-tional alchemy by TI yields the relative affini-

ties of multiple similar ligands from a singlesimulation run, at least for rigid ligands (92).

At least two studies suggest that the methodis more accurate than MM-PBSA (46, 94), as

noted above. Its convergence properties alsomay be better than that of the original imple-

mentation of MM-PBSA because unlike MM-PBSA the work terms it computes are not di-

rectly influenced by fluctuations far from the

binding site.Pathway methods can also be used to com-

putethestandardbindingfreeenergyofapro-tein and ligand. The Double Decoupling ap-

proach (42), which was in effect first used tocompute the affinity of xenon for myoglobin

(54), draws on related approaches (53, 106),notably the DoubleAnnihilationmethod (63).

It involves using a pathway technique to com-pute the work of gradually decoupling the lig-

and from the binding site and then effectivelycoupling it with an energy well or trap of de-

fined size in bulk solution. The work of allow-ing the ligand to escape from the trap into a

volume equal to 1/Co can be computed ana-

lytically, so that the simulation results can be

connectedwith the appropriatestandard state.

This approach has been further developed andapplied to several protein-ligand systems (11,

35). A computationally massive application ofthe similar but nonrigorous (42) Double An-

nihilation method to the association of eightligands with the protein FKBP is also of con-

siderable interest (40).It has been argued that Double De-coupling is difficult to converge for highly

charged ligands because the final result mustaccurately balance strong interactions be-

tween the ligand and the solvent against sim-ilarly strong interactions between the ligand

and the protein (57). Alternative approachescompute the free energy of binding via a

pathway in which the ligand is gradually ex-tracted from the binding site. Preliminary

results from techniques of this type appearpromising (57, 79), though their applicability

to ligands that bind in a closed cavity within

the protein is not yet defined.Because the pathway methods are so

promising and yet so difficult to run to con-vergence, there is great interest in accelerat-

ing them or developing useful shortcuts, asrecently reviewed (105). Approaches include

improved methods of extracting free energiesfrom the simulations (109, 132), novel sam-

plingmethodstospeedconvergence(82,103),and hybrid implicit/explicit solvent models

that reduce the computational load (5, 81).

PROSPECTS AND CHALLENGES

Accurate calculation of protein-ligand bind-

ing affinities is an important and still unsolvedproblem. Nonetheless, significant progress is

being made through a deepening understand-ing of the physical chemistry and the im-

plementation of tractable algorithms basedon this understanding. Advances in com-

puter hardware also continue to extend thelevel of detail that can be modeled. There

is room for improvement in many aspects

of existing methods, including models of thepotential energy and of solvent, treatment

of protonation and tautomer equilibria, and



14/24

Solvent entropy:entropy associatedwith motions of thesolvent molecules

algorithms and heuristics for conformational

sampling. Indeed, one should not expectmarked improvement in affinity calculations

from progress against any single part of theproblem because theremainingparts still gen-

erate large errors. Rather, success likely re-quires a concerted effort on all fronts. Un-

derstanding the nature of the approximationsused is essential in this process, so that suc-cesses or failures will be interpretable and a

path of improvement can be discerned. Oth-erwise, model development risks devolving to

trial and error.Comparison with experimental data re-

mains indispensable. Validation studies areeasier for computationally faster approaches,

such as docking and scoring, but are alsoneeded for the more time-consuming meth-

ods. Simple model systems can play animportant role in validation. For example, ar-

tificial protein cavities that bind small, rigid

ligands allow a focus on intermolecular inter-actions rather than on issues of ligand flexibil-

ity (128, 129). Similarly, studies of host-guestmodel systems (23, 26) have provided insights

into configurational entropy that bear on the

development of improved scoring functionsExperimental data also are necessary as a basis

for improving specific components of bind-ing models, and expansion of certain data sets

would be welcome and valuable. For exam-ple, solvation models cannot be applied con-

fidently to many candidate ligands becausethe existing database of solvation free ener-gies does not include many of the chemical

moieties that frequently appear in drug-likecompounds.

Finally, todays understanding of the phys-ical chemistry of molecular recognition may

prove to be incomplete. Specific issues likelyto be of continuing interest include changes

in ligand, protein, and solvent entropy uponbinding; the puzzling and apparently erro-

neous tendency of many binding models toyield affinities that correlate strongly with the

molecular weight of the ligand; electronic po-

larization; the modeling of metal-containingbinding sites; and the thermodynamic impli-cations of water binding sites at the ligand-protein interface.

SUMMARY POINTS

1. Recent evaluations show that docking methods have considerable success in correctly

identifying bound conformations and in enriching known ligands against a back-ground of decoy compounds, but yield less impressive results for ranking known

ligand according to affinity.

2. Important advances are being made in calculating the potential energy as a function

of conformation, through more sophisticated force fields and the incorporation ofquantum mechanical treatments in ligand-protein modeling.

3. Recent calculations suggest that changes in configurational entropy on binding arelarge, cancel much of the energy change that drives binding, and must be accounted

for to obtain good correlations with measured affinities.

4. Free energy pathway calculations with explicit solvent appear to yield good agreement

with experiment, and it may be time to undertake more comprehensive validationstudies.

5. Achieving accurate affinity calculations will likely require advances on multiple fronts:potential energy functions, solvation models, and the treatment of conformational

flexibility.

34 Gilson Zhou


15/24

DISCLOSURE STATEMENT

MKG is a member of the company VeraChem LLC.

ACKNOWLEDGMENTS

The authors thank Dr. Hillary Gilson for valuable comments on the manuscript. This publica-tion was made possible by grant no. GM061300 to MKG and grant no. GM058187 to HXZ,

from the National Institute of General Medical Sciences of the National Institutes of Health.Its contents are solely the responsibility of the authors and do not necessarily represent the

official views of the National Institute of General Medical Sciences.

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Calculations of Protein-Ligand Binding Affinities

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