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Γ molecules P S ] and where p K lig- en- the Γ on S d S n . . 2 over ΓS to give a reduced Boltzmann distribution boundary effects). It is also independent of the precise defini- tion of Σ1, provided that Σ1 includes the protein-ligand bind- ing site and does not include a “macroscopic” volume beyond f(Γ) = f(Γ, ΓS) dΓS Z this. exp[ βE(Γ)] In this formulation, we have assumed that the system vol- = − , exp[ βE(Γ)] dΓ ume V0 is fixed. However, in most physiological systems, the − R pressure is held constant and the binding affinity is then re- where lated to the differences in the which intro- duces a correction term which is the product of the pressure 1 E(Γ) = ln exp[ βE(Γ, ΓS)] dΓS and the change in the volume caused by the formation of the −β Z −  LP complex [6]. We expect this correction to be small for is the energy of the system with ligand and protein configu- typical ligand-protein interactions in solution. rations specified by Γ and with the equilibrium effects of the We would like to cast Eq. (2) as the ratio of canonical av- solvent implicitly included as a solvation free energy. In this erages which can be computed using the canonical-ensemble paper, we will assume that E(Γ) is given. Monte Carlo method [7]. To achieve this, we combine the Typical molecular interactions have a short range. In view unbound and bound systems by extending phase space to in- clude a discrete index λ 0, 1 and consider a system in of this, let us define Σ0 to represent all accessible Γ space this extended space with energy∈ { E}∗(Γ) for which the canoni- (i.e., L and P somewhere in the system volume V0), and Σ1 λ cal average is defined by to represent that portion of Σ0 where there is an appreciable interaction between L and P which therefore form the com- dΓ exp[ βE∗(Γ)]X (Γ) X = λ − λ λ . plex LP. In the phase-space volume Σ1 we write the energy ∗ h i P R λ dΓ exp[ βEλ(Γ)] as E (Γ) which is just an alternate notation for the full energy − 1 ∗ P R We take E (Γ) to be infinite for Γ / Σλ and finite otherwise. E(Γ), while in the volume Σ0 Σ1 we may write the energy λ − Now Eq. (2) can be rewritten as ∈ as E0(Γ) which we define as the “unbound”energy,i.e., E(Γ) ∗ excluding the interaction between L and P. The dissociation 1 δλ0 exp β[E0(Γ) E (Γ)] K = − − 0 , (3) constant can then be written as d ∗  V0 δλ1 exp β[E1(Γ) E1 (Γ)] 2 − − −βE(Γ) ∗  e dΓ where δλµ is the Kronecker delta. If Eλ(Γ) Eλ(Γ), the 1 Σ0−Σ1 ≈   terms being averaged are O(1). Because the definition of K Kd = −RβE(Γ) −βE(Γ) d V0 e dΓ e dΓ Σ0 Σ1 is independent of V0, we can pick V0 1/Kd so that approx- R R 2 imately the same number of samples contribute∼ to each of the e−βE0(Γ) dΓ 1 Σ0−Σ1 =   canonical averages. We show later, Eq. 7, that this choice V R −βE1(Γ) 0 Σ1 e dΓ × minimizes the error in the estimate of Kd. ∗ R 1 We can evaluate Eλ(Γ) with short energy cutoffs allowing . e−βE0(Γ) dΓ+ e−βE1(Γ) dΓ it to be computed more quickly than Eλ(Γ) and the terms con- Σ0−Σ1 Σ1 tributing to the averages in Eq. (3) can be accumulated every R R hundred steps, for example. Since there is typically a high In the dilute limit V0 , this can be simplified by ignor- ing the second term in→ the ∞ denominator of the last factor and correlation between successive steps in a Monte Carlo sim- ulation, this method allows the averages to be computed to by extending the limits of the integrals over Σ0 Σ1 to in- − a given degree of accuracy in less time than if we had used clude Σ1. In extending the definition of E0(Γ) to Γ Σ1, we ∈ ∗ include the intramolecular energy and the solvation free en- Eλ(Γ) = Eλ(Γ). ergy but continue to omit the intermolecular (ligand-protein) The extension of phase space has been used in other free en- energy. This yields [1, 4, 5] ergy calculations, to combine, for example, systems at several different temperatures [8] or to treat the “reaction coordinate” controlling the transition between two chemical species as a 1 Σ0 exp[ βE0(Γ)] dΓ Kd = − . (2) dynamic variable [9, 10]. In our case, the use of the worm- V0 R exp[ βE1(Γ)] dΓ Σ1 − hole Monte Carlo (described in the next section) allows us to R The Helmholtz free energy of the unbound and bound systems include just the two systems of interest without the need to is [3] compute the properties of (possibly unphysical) intermediate systems. 1 Fλ = ln exp[ βEλ(Γ)] dΓ , −β ZΣλ −  Wormhole Monte Carlo for λ = 0 and 1, and Eq. (1) is obtained from Eq. (2) with ∆F = F1 F0. The definition of Kd, Eq. (2), is strictly in- We can compute the canonical averages in Eq. (3) using the − ′ ′ dependent of V0 because of translational symmetry (ignoring Monte Carlo method [7] to make steps from [Γ, λ] to [Γ , λ ] 3

(a) whose equilibrium distribution is proportional to g(Υ). The canonical average of a quantity X(Υ) is defined by ** ΓΓ ** ΓΓ EE00(( )) EE11(( )) dΥ g(Υ)X(Υ) X = . h i R dΥ g(Υ) R In our application, we make the identification Υ = [Γ, λ], dΥ= dΓ, and g(Υ) = exp[ βE (Γ)]. λ − λ R Let usP defineR a set of “portal functions,” w, w′, w′′,...,on Υ, with properties

0 w(Υ) 1/v < , ≤ ≤ ∞ ΓΓ ΓΓ

(b) dΥ w(Υ) = 1, ΓΓ; λλ = 0 ΓΓ; λλ = 1 Z where v is a representative phase-space volume of the portal function. A wormhole move consists of the following steps: ′ select a pair of portals (w, w ) with probability pww′ ; reject the move with probability 1 vw(Υ), where Υ is the current state; otherwise, with probability− vw(Υ), pick a configuration Υ′ with probability w′(Υ′); and accept the move to Υ′ with ′ probability Pww′ (Υ, Υ ). If the moveis rejected, Υ is retained ′ as the new state. We determine Pww′ (Υ, Υ ) by demanding ∗ ′ FIG. 1: (a) Schematic representation of E0 (Γ) (shallow and wide) that the rate of making transitions from Υ to Υ via portals ∗ ′ ′ and E1 (Γ) (deep and narrow). Conventional Monte Carlo moves (w, w ) is balanced by the reverserate from Υ to Υ via portals between λ = 0 and 1 (shown as dashed lines) are nearly always (w′, w), i.e., rejected because they lead to large increases in energy. (b) Schematic representation of typical portals for the wormhole moves for the case ′ ′ Rww′ (Υ, Υ )= Rw′w(Υ , Υ) illustrated in (a), The large ratio of the volume of the unbound (λ = 0) portal compared to the bound (λ = 1) portals compensates for the (the condition of detailed balance), where the rates are given higher energy of the unbound configurations. This results in accepted by wormhole moves (dashed lines) between all the portals.

′ g(Υ) R ′ (Υ, Υ ) = p ′ ww ww dΥ g(Υ) × with probability ′ ′ ′ vwR (Υ)w (Υ )Pww′ (Υ, Υ ). ∗ ′ ∗ min 1, exp β[E ′ (Γ ) E (Γ)] . − λ − λ   A possible solution for the acceptance probability is However, the estimate of the ratio in Eq. (3) will be very poor, ′ ′ ′ ′ pw w g(Υ ) v because transitions between λ = 0 and 1 will be extremely Pww′ (Υ, Υ ) = min 1, , ∗ ∗  pww′ g(Υ) v  rare—typically, E0 is shallow and wide, while E1 is deep and narrow; see Fig. 1(a). One possible way of remedying this is where we have made use of the identity α min(1, α−1) = to treat λ as a continuous variable [9, 10], providing a suitably min(1, α), which is valid for α> 0. interpolated definition of E (Γ), and allowing Monte Carlo λ In order to apply this move, let us use a specific rather sim- steps with small changes in λ. In practice, this approach only ple form for the portal functions, w(Υ), namely “works” if the two endpoints are sufficiently similar, thus lim- iting this method to the comparison of chemically close mol- 1/v, for Υ w, w(Υ) = ∈ ecules.  0, otherwise, Here we propose an alternative way of carrying out a Monte ∗ Carlo simulation of the Eλ(Γ) system. We restrict the stan- where we now denote a portal by w which defines an arbi- dard moves to changes in Γ only, and allow changes in λ trary subset of Υ space of volume v. In principle, there is via “wormhole moves” [11] which connect otherwise discon- no restriction on the choice of the portals; however, practical nected regions of configuration space. This obviates the need considerations, discussed below, dictate how they are chosen. to treat (possibly unphysical) intermediate values of λ, per- Furthermore, for simplicity, we will assume that the wormhole mitting us to compute the free energy differences needed to probabilities are all equal, pww′ = const. determine the absolute binding affinity. Let us now describe the wormhole move in [Γ, λ] space. Assume we have some system defined on a phase space Υ Starting with a configuration [Γ, λ], first pick a random portal 4 w. If [Γ, λ] / w, then reject the move. Otherwise, pick a separately for λ = 0 and 1. For each λ, we obtain a canon- random configuration∈ [Γ′, λ′] uniformly in a randomly chosen ical set of configurations Ξ (suppressing the λ subscripts portal w′, and accept the move to [Γ′, λ′] with probability for brevity), to which we fit{ n}-dimensional ellipsoids, as fol- lows. First we try to fit a single ellipsoid to Ξ . The center ∗ ′ ′ { } exp[ βE ′ (Γ )] v of the ellipsoid is given by the mean configuration Ξ . For min 1, − λ . (4)  exp[ βE∗(Γ)] v  each configuration in Ξ , we determine the deviationh i from − λ the mean, δΞ = Ξ {Ξ}, and compute a covariance matrix This differs from the standard Boltzmann acceptance proba- for the configurations− which h i can be diagonalized as bilty by the ratio of volumes, v′/v, which can be large enough to compensate for the difference in the mean energies in the δΞ δΞ = PΛPT, h i portals. Indeed, if the mean energy of configurations in w P scales as β−1 ln v+const., the acceptance probability, Eq. (4), where is the matrix of (column) eigenvectors and Λ is the is O(1), thereby allowing moves between shallow wide wells diagonal matrix of eigenvalues. Because of the properties of P P−1 PT and deep narrow ones; see Fig. 1(b). In practice, each por- the covariance, is real and orthogonal, = , and the ∗ eigenvalues are real and non-negative. If there are no hidden tal will occupy one of the energy wells of either E0 (Γ) or ∗ constraints on the motion, we additionally can assume that E1 (Γ). This implies that the λ = 0 portals will permit unre- stricted translation and rotation of the ligand and the protein the eigenvalues are strictly positive. We find it convenient to define subject to the V0 constraint, and the λ = 1 portals will allow unrestricted motion of one of the molecules. B = PΛ1/2, B−1 =Λ−1/2PT, The wormhole move embodies two concepts which have been used separately in other works: stretching or shrink- so that we can write ing Γ space when making the move compensates for possibly large energy differences between wells [12, 13]; and jump- δΞ δΞ = BBT. h i ing between disconnected regions of phase space enables the Markov chain to explore regions of phase space separated by We take the semi-axes of the ellipsoid to be the columns B large energy barriers [14, 15]. In the following sections, we of √n . The multiplier here, √n, is chosen to ensure that O(1) of the configurations in Ξ lie within the ellip- will show how these elements may be combined to permit the { } computation of protein binding affinities with realistic force soid. This choice is motivated by considering a symmetric fields. n-dimensional Gaussian exp( 1 r2) f(r)= − 2 , (2π)n/2 Finding the portals for which we have In orderfor the wormholemethod to be practical, we need a r2 = r2f(r) dnr = n. reliable way of choosing the portals. We describe this process h i Z first in the general case. Let us write Γ = [ΓL, ΓP], where ΓM = [XM, ΞM] represents the state of molecule M, XM rep- Ellipsoidsare anaturalchoiceto useto fit the set of configu- resents its position and orientation, and ΞM represents its con- rations for the following reasons: (1) The iso-density contours formation. of the distribution in a harmonic well are ellipsoids. (2) It is We assume that ΞM is expressedin such a way that anycon- easy to sample points randomly from an ellipsoid. (3) Con- straints on the positions of the atoms in M (e.g., bond lengths versely, it is easy to test that a point lies inside an ellipsoid. and bond angles) is implicitly accounted for, so that the di- (4) The volume of an n-dimensional ellipsoid is given by mensionality of ΞM reflects the number of degrees of confor- n πn/2 mational freedom, nM, for this molecule. v = a , n (n/2)! i Because of the translational and rotational symmetry of the iY=1 system, certain components of Γ are ignorable. We can there- ∗ ∗ where ai is the length of ith semi-axis. Note well the degener- fore write Eλ(Γ) = Eλ(Ξλ), with ate case of this result, v0 =1, which is used if the protein and ligand are both rigid (n = 0). The sampling and testing of Ξ0 = [ΞL, ΞP], 0 points, (2) and (3), can either be accomplished by transform- Ξ = [Y, Ξ , Ξ ], 1 L P ing with the matrix B. Alternatively, we can use the simpler Cholesky decomposition of the covariance matrix where Y = X X denotes the position and orientation of L − P the ligand relative to the protein. The dimensionality of Ξλ is δΞ δΞ = CCT, nλ with n0 = nL + nP and n1 = n0 +6. h i The strategy for determining the portals is to carry out con- where C is a lower triangular matrix. Both C and C−1 may be ∗ ventional canonical Monte Carlo simulations with Eλ(Ξλ) computed by direct (non-iterative) methods [16]. 5

Having defined an ellipsoid in this way, we test its suitabil- dihedral lies outside [ π, π] or the turn u lies outside the unit ity as a portal by demanding that O(1) of the configurations ball. In order to preserve− detailed balance, we reject the re- sampled uniformly from it have energies close to its mean en- sulting move. ergy E∗(Ξ) . If this test fails, we split Ξ into two sets There are four ways in which we can improve the qual- accordingh toi the sign of δΞ projected along{ the} largest semi- ity of the portals obtained by this method so as to make suc- axis of the ellipsoid and construct new trial portals from each cessful wormhole moves more likely. (1) The Monte Carlo of these sets. runs used to obtain the samples from which the portals are The ellipsoids that result from this process constitute our defined should begin with an “annealing” phase where the portals. When computing the volumes of the portals for use temperature is started at some high value and slowly reduced in Eq. (4), we need to account for the freedom to place 2 λ to T at which point we start gathering samples. This al- molecules at arbitrary positions and orientations in the volume− lows the Monte Carlo sampling to explore phase space more V0. In practice, this means we multiply the volume of the thoroughly. (2) Other methods of finding conformational en- λ = 0 portals by σV0 where V0 is the translational volume ergy minima [19] can be applied to provide additional starting and σ is the orientational volume (given below). points for the Monte Carlo runs. (3) The samples can be sup- In order to complete the specification of the portals, we plemented with those obtained by applying those symmetry need to describe how Ξ and δΞ are formed, since, as a operations which leave the molecules invariant. In the case consequence of workingh ini the subspace where the molecu- of non-chiral , we would also apply a reflection of the lar constraints are implicitly satisfied, Ξ is not simply a point ligand in the unbound case since this will leave the energy un- in Rn. We wish to represent each ellipsoid on a locally Carte- changed. In this way, the portal moves allow all the symmetric sian space Rn which lets us use familiar formulas for defining variants of the molecules to be explored so that symmetry is the ellipsoid. We also demand that the mapping to Rn have included in a systematic way in the computation of the bind- constant Jacobian in order to maintain detailed balance when ing affinity. (4) When forming δΞ0 δΞ0 , we should set the sampling from the ellipsoids. intermolecular cross terms to zeroh becausei the conformations We illustrate this by considering the case of the protein and of the two molecules are independent when λ =0. the ligand being made up of nM +1 rigid fragments simply This method of finding portals depends on the samples connected by nM bonds each of which allow rotation only “spanning” a volume of phase space. This requires that the (i.e., the bond lengths and bond angles are fixed). The relative dimensionality of phase space be sufficiently small and this, position and orientation of L with respect to P, Y , can be de- in turn, implies the use of an implicit solvation model. In ad- fined in terms of one of the rigid fragments of each molecule. dition the portals can be more reliably found if the “hard” de- Finally, we use unit quaternions [17] to represent orientation. grees of freedom in the molecules are replaced by constraints Since the quaternions q and q represent the same rotation, (e.g., by fixing the bond lengths and bond angles as described orientations are given by an axis− [18] of the sphere S3. above). The coordinates making up Ξ are then: (a) the position The method of successively subdividing the samples may component of Y , a point in R3; (b) the orientation component lead to suboptimal portals in some cases, for example, by di- of Y , an axis of the sphere S3; and (c) the dihedral angles viding a contiguous set of samples. We have recently exper- of the rotatable bonds of the ligand and protein, points on the imented with fitting a mixture of Gaussians to the samples circle S1. The definition of Ξ and δΞ is straightforward for using the expectation-maximization (EM) algorithm [20, 21]. (a), since we use the normalh arithmetici definitions. For (c), Since this optimizes the fit to all the samples, it usually results we define the mean for each angle [18] by the direction of the in fewer, better, portals. In this case, we are naturally lead to mean of the points on S1 embedded in R2. We form the devi- use Gaussians for the portal functions rather than the more re- ation in this case by subtraction modulo 2π so that the result strictive ellipsoids. By enforcing the symmetries of the ligand lies in [ π, π]. when making the fits, ligand symmetry can also be included To find− the mean of the orientations (b), we similarly em- in a rigorous way. These improvements will be described in a bed S3 in R4. We define the mean orientation [18] as the subsequent publication. axis in R4 about which the moment of inertia of the sample axes is minimum. To find the deviation of the orientation, we compute the differential rotation δq = q q ∗ which takes The free energy calculation the mean orientation to the sample orientationh i and we project this into a “turn” vector u in the unit ball in R3 so as to pre- Prior to the calculation of the free energy, we estimate a serve the metric. This is achieved by a generalization of the suitable value of V0, which enters into the definition of the Lambert azimuthal equal-area projection as described in the volumes of the λ = 0 portals, by assigning an estimated sta- Appendix. In this representation, the volume of orientational tistical weight of v exp( β E∗ ) to each portal and estimat- space (which contributes to the multiplier for the unboundvol- ing − h i 4 umes) is σ = 3 π. Occasionally, the point sampled from w′ correspondsto one v exp( β E∗ ) V λ=1 − h i , of the coordinates “wrapping around,” i.e., the change in the 0 ∼ P (v/V ) exp( β E∗ ) λ=0 0 − h i P 6

NH2 the primary goal of this exercise is to assess how well worm- hole moves allow the free energy calculation to converge. For 2NH + this purpose, we are not so interested in comparing the result- NH2 ing computed binding affinity to the experimental data since this will depend in large measure on the accuracy of the force FIG. 2: The structure of p-amino-benzamidine. field and of the protein structure. Nevertheless, since the con- vergence will depend on the complexity of the energy “land- where the sums in numerator (denominator) are over the scape,” we use this example to epitomize the binding of a small ligand to a protein. bound (unbound) portals and v/V0 for the unbound portals is the volume of the ellipsoid (multiplied by σ) and thus does The coordinates of the atoms in trypsin are taken from a trypsin-benzamidine complex, 1BTY [22]. At physiologi- not depend on V0. If this estimate for V0 results in the sam- cal pH, the amidine group is protonated (net charge of +1) pling being too heavily weighted toward λ = 0 or 1, V0 may and, in the complex, it is attracted to a negatively charged be changed and the current values of the sample sums for Kd can be adjusted to account for this change. aspartate residue in trypsin inhibiting its enzymatic action. The choice of the starting configuration for the free energy We employ the Amber 7 force field [23, 24, 25] and the calculation may introduce some bias in the results. We can GB/SA solvation model [26, 27, 28]. The protein is taken remove much of this bias by using the portals to select the to be rigid, nP = 0, and two bonds of the ligand are al- starting configuration: select a portal w with probability pro- lowed to rotate, namely, those connecting the amidine and portional to its estimated statistical weight; select a configura- the amino groups to the benzene ring, nL = 2. The pub- tion, [Γ, λ], uniformly from this portal; and accept the config- lished force field [23] does not provide a satisfactory tor- uration with probability sion for the bond between the benzene ring and the amidine group and this term was determined using Gaussian 98 [29] as min 1, exp β[E∗(Γ) E∗ ] , [ 14.2 cos(2φ)+3.3 cos(4φ)+0.5 cos(6φ)]kJ/mol, where − λ − h i −   φ is the dihedral angle. where E∗ is the canonical average energy over the portal. Five unbound and five bound canonical simulations of 1000 This procedureh i is repeated until a configuration is accepted. steps each were carried out to find the portals. The resulting The bias can be further reduced by running the Monte Carlo configurations were supplemented by those obtained by ap- calculation for several correlation times, defined by Eq. (6), plying the symmetry operations which leave the ligand invari- prior to gathering the data for Eq. (3). ant. This gave 16 unbound and 8 bound portals. The config- During the course of the free energy calculation, normal urations of the bound portals are all the same “pose” of the and wormhole Monte Carlo moves are mixed. With the nor- ligand on the protein and correspond to the symmetries of p- mal moves, we only attempt to change Γ and, for this reason, amino-benzamidinegiven by rotating the amino, benzene, and it is possible to have the Monte Carlo step size depend on amidine groups by 180◦. The unbound ligand also exhibits 8 λ (with, usually, the step size being larger with λ = 0). In symmetries given by rotating the amino and amidine groups practice, most ( 90%) of the attempted moves are worm- by 180◦ relative to the central benzene ring and by including hole moves because∼ frequently we have [Γ, λ] / w and the the mirror images. However since the bond parameters allow attempted wormhole move is inexpensively rejected.∈ partially free rotation of the amidine group, each symmetric The method is robust in the sense that it does not depend set of configurations is represented by two portals. −18 3 on the particular form of the energy function. In addition, a We estimated a suitable value for V0 of 0.39 10 m failure to make transitions between λ = 0 and 1 can be de- based on the volumes and the mean energies of× the com- tected. This may be because the test [Γ, λ] w never suc- puted portals. During the binding affinity calculation, worm- ceeds (i.e., the configuration has been trapped∈ in a new well), hole moves were attempted on 90% of the steps (the other or because the acceptance criterion is never met, which indi- 10% were standard Monte Carlo moves). Of these attempted cates that there is a deep well within the well of one of the wormhole moves, 97% failed (inexpensively) because the portals. In both cases, the problem can be corrected by adding configuration was not in the chosen w. Of the remaining 3%, a new portal based on recent configurations. about 60% lead to a successful move, of which about 40% in- volved switching from λ = 0 to 1 and vice versa. The major computational cost in the free energy calculation is the eval- Example uation of the bound energies. In this example, the wormhole moves required less than 3 bound energy evaluations, on av- The efficacy of the wormhole method depends on how fre- erage, to effect the transition from a bound configuration to quently a configuration lies within one of the portals and how an unbound one and back. This enables an accurate estimate often the jump to the new portal is accepted. In order to assess to be made of the ratio of the averages in Eq. (3) which after 6 these questions, we have computed the binding affinity of p- 5 10 steps yields pKd =7.99 0.01. The error estimate is amino-benzamidine,whose structure is shown in Fig. 2, to the derived× in the next section and represents± a 2% relative error digestive trypsin, at T = 290K. We emphasize that in Kd. 7

8.1

0.1

c (s) t d 8 C 0.01 b pK d

a 0.001 e 7.9 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 6 3 s × 10 t × 10

FIG. 3: Cumulative estimates pKd(s) obtained by sampling every FIG. 4: The λ-correlation function Ct. The curves show Ct for −18 3 1 1 100th step from the first s steps of 5 independent Monte Carlo runs. V0/(0.39 10 m )= (a) 10 , (b) 3 , (c) 1, (d) 3, and (e) 10. The × 1 100 The dashed lines shows convergence as 1/√s to the mean value of ratio q/p varies correspondingly (from approximately 8 to 8 ). The 7.99. correlation times are (a) 329, (b) 346, (c) 352, (d) 382, and (e) 415. In determining the correlation times, we limit the sum in Eq. (6) to 0 < t 2500 in order to avoid including the small but noisy terms This example illustrates that the method is effective at al- for large≤ t. lowing sufficient transitions between the bound and unbound states to enable the binding affinity of protein-ligand systems to be accurately computed. We note that our computed bind- verges depends, naturally, on the “correlation time” of the sys- ing affinity differs from the experimental results of 5.1 to 5.2 tem. If wormhole moves rarely cause the system to switch be- [30, 31]. This discrepancy may be accounted for by modest, tween bound and unbound states, the correlation time is large 20%, errors in the force field. and the convergence will be slow. ∼ In order to make these ideas quantitative, we define the λ- correlation function, Error analysis C = (λ p)(λ p) , t h s − s+t − is In order to assess the errors in the computation of pKd in where λ is the value of λ at simulation step s and ... de- more detail, we carried out 10 independent runs similar to the s h is one describedin previoussection. Each of these used the same notes an average over steps. Figure 4 shows Ct for several different values of V0. (From Eq. (5), we have q/p V0.) portals and the same value of V0. We computed cumulative In a Markov chain of length s, the expected number of∝ bound estimates for pKd based on the first s steps of the Markov states is ps, while, for s , the variance in the number of chains. When formingthe averagesin Eq. (3) we sample every → ∞ 100th step. (As we shall see, there is a high degree of correla- bound states is 2Ds, where tion within 100 steps; thus there would be little improvement 1 1 in the estimate of pK by sampling more frequently.) The re- D = C0 + Ct = C0τ. (6) d 2 2 sults for 5 of these runs are shown in Fig. 3. The convergence Xt>0 toward the mean is as 1/√s; after 5 106 steps, the error in This provides us with the definition of the correlation time, τ. pK has been reduced to about 0.01×. d From Fig. 4, we see that C decays approximately exponen- For the purposes of further analysis,± let us assume that the t tially so that the sum converges. computation is carried out with E∗(Γ) = E (Γ) so that all λ λ We can compare the Monte Carlo simulation to a simpler samples have the same statistical weight. The probability that Markov process of independent trials, e.g., tossing a coin the system is in the bound (resp. unbound) state is p = δ λ1 where the probability of heads is p. In this case, we have (resp. q =1 p = δ ) and Eq. (3) becomes h i − h λ0i C = 0 and D = 1 C = 1 pq. In the limit s , t>0 2 0 2 → ∞ 1 q the relative errors in estimating p for the Monte Carlo simu- Kd = . (5) lation will be the same as that for s/τ tosses of the coin. We V0 p can illustrate this by making 5 independent simulations of a The determination of Kd is then equivalent to estimating q/p coin tossing experiment to match the data in Fig. 3, for which by taking the ratio of the number of unbound and bound steps p =0.443 and τ = 352. The results are given in Fig. 5, where in the Monte Carlo simulation. How well this estimate con- we plot hn/tn against n where hn (resp. tn) is the number of 8

and bound systems to be treated as a single canonical system 1 and Kd to be expressed as the ratio of canonical averages, Eq. (3); (2) the wormhole move which allows transitions be- 0.9 tween the bound and unbound systems; and (3) the use of an implicit solvation model which reduces the number of degrees

n of freedom and so allows portals to be identified. /t n

h 0.8 A method similar to ours is “simulated mutational equili- bration,” [32] which employs an extended phase space, uses an implicit solvation model, and allows jumps between differ- 0.7 ent systems. However, in place of the wormhole move, this work employed a more restrictive “jumping between wells” move, which limited its applicability to computing the differ- 0 2 4 6 8 10 12 14 ence in the binding affinities for two enantiomers. In contrast, × 3 n 10 our wormhole method can be used to compute directly the binding affinity of a ligand with several rotatable bonds to a protein target. This allows us to study a wide range of inter- FIG. 5: Estimation of the bias of a coin with p = 0.443. Five esting drug-like ligands. independent runs are shown. In each run, the coin was tossed 5 106/τ = 14200 times, where τ is the correlation time for Fig. 4(c),× In the special case of binding rigid molecules (for which ∗ and the cumulative value of hn/tn is recorded. The expected value, we have E0 (Γ) = 0), wormhole Monte Carlo is isomorphic p/q, is shown as a dashed line. The axes have been adjusted to allow to a grand canonical Monte Carlo simulation [33]. Consider a the figure to be directly compared with Fig. 3. grand canonical system with a single protein molecule and with ligand molecules at a fixed chemical potential µ; we make the additional restriction that the ligand-ligand interac- heads (resp. tails) after n throws. As expected, Figs. 3 and 5 tion energy is infinite, so that the system can only accommo- both exhibit the same convergence behavior. date 0 or 1 ligand molecule. For the wormhole simulation, Since the distribution of outcomes h in the coin tossing n we specify the bound portal to include the full simulation vol- experiment is the binomial distribution, the mean and variance ume with arbitrary orientation; the unbound port is degenerate for l = ln(h /t ) can be calculated, yielding n n n and corresponds merely to specifying λ = 0. The wormhole p 1 p q moves from λ = 0 to 1 and vice versa correspond to particle l = ln + + O(n−2), h ni  q  2n q − p insertion and deletion in the grand canonical simulation; and we can verify the acceptance probabilities are the same. 2 1 −2 The application described here can be extended by allow- (ln ln ) = + O(n ). h − h i i npq ing a greater degree of flexibility for the protein and the lig-

The standard error in the estimate of pKd from a Monte Carlo and. This permits the treatment of side-chain rotation and run with s steps is found by substituting n = s/τ and scal- ligand-induced loop movement on the part of the protein and ing by ln 10 (since pKd is defined in terms of the common the treatment of flexible rings for the ligand. Many stan- logarithm) to give dard Monte Carlo techniques can be used with this method, if appropriate—preferential sampling, early rejection, force 1 τ bias, etc. Wormhole moves could also be used to treat other ∆pKd . (7) ≈ ln 10rspq discrete, or nearly discrete, transitions. Examples are: the In the case of the simulations shown in Fig. 3, we find treatment of molecules, such as cyclohexane, which can as- sume distinct conformations; discrete sets of side chain rota- ∆pKd 0.01 consistent with the data in the figure. From Fig. 4, we≈ see that τ is weakly dependent on p. Thus for a tions in the protein; protonation and tautomerization states for 1 either molecule. In each of these cases, the acceptance prob- given s, we minimize the error by choosing p = q = 2 . Al- ternatively, we may wish to minimize the error for a given ability for the transitions should account for the free energy amount of computational effort. If bound steps are f times difference between the discrete states. more expensive to carry out than unbound steps, the error is minimized by taking q/p = √f, i.e., p =1/(1 + √f).

Discussion Acknowledgment

The important aspects of this method that enable us to com- This work was supported, in part, by the U.S. Army Med- putethe binding affinity ofa ligandto a proteinare: (1)formu- ical Research and Materiel Command under Contract No. lation in the extended [Γ, λ] space, which allows the unbound DAMD17-03-C-0082. 9

Appendix: Generalized Lambert projection (The other hemisphere, corresponding to 1 π<θ π, maps 2 ≤ onto the shell 1

Hopkins University Press, 3rd edition, 1996. well-behaved electrostatic potential based method using charge [17] W. R. Hamilton. On quaternions. Proc. Royal Irish Acad., 1847, restraints for deriving atomic charges: The RESP method. J. 3, 1–16. URL http://www.maths.tcd.ie/pub/HistMath/People/ Phys. Chem., 1993, 97, 10269–10280. Hamilton/Quatern2/. [26] W. C. Still, A. Tempczyk, R. C. Hawley, and T. Hendrickson. [18] K. V. Mardia and P. E. Jupp. Directional Statistics. J. Wiley & Semianalytical treatment of solvation for molecular mechanics Sons, 1999. and dynamics. J. Am. Chem. Soc., 1990, 112, 6127–6129. [19] M. S. Head, J. A. Given, and M. K. Gilson. Mining minima: Di- [27] G. D. Hawkins, C. J. Cramer, and D. G. Truhlar. Pairwise solute rect computation of conformational free energy. J. Phys. Chem., descreening of solute charges from a dielectric medium. Chem. 1997, 101, 1609–1618. Phys. Lett, 1995, 246, 122–129. [20] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum like- [28] V. Tsui and D. A. Case. Molecular dynamics simulations of lihood from incomplete data via the EM algorithm. J. Royal nucleic acids with a generalized Born solvation model. J. Am. Stat. Soc., 1977, 39, 1–38. Chem. Soc., 2000, 122, 2489–2498. [21] J. J. Verbeek, N. Vlassis, and B. Kr¨ose. Efficient greedy learn- [29] M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuse- ing of Gaussian mixture models. Neural Computation, 2003, ria, M. A. Robb, J. R. Cheeseman, V. G. Zakrzewski, J. A. 15, 469–485. Montgomery, Jr., R. E. Stratmann, J. C. Burant, et al. Gaus- [22] B. A. Katz, J. Finer-Moore, R. Mortezaei, D. H. Rich, and R. M. sian 98, Revision A.9. Gaussian, Inc., Pittsburg, 1998. URL Stroud. Episelection: Novel Ki nanomolar inhibitors of http://www.gaussian.com. serine proteases selected by binding∼ or chemistry on an en- [30] M. Mares-Guia and E. Shaw. Studies of the active center of zyme surface. , 1995, 34, 8264–8280. URL trypsin. J. Biol. Chem., 1965, 240, 1579–1585. http://www.rcsb.org/pdb/cgi/explore.cgi?pdbId=1bty. [31] S. M. Schwarzl, T. B. Tschopp, J. C. Smith, and S. Fischer. [23] D. A. Case, D. A. Pearlman, J. W. Caldwell, T. E. Cheatham, Can the calculation of ligand binding free energies be improved III, J. Wang, W. S. Ross, C. L. Simmerling, T. A. Darden, K. M. with continuum solvent electrostatics and an ideal-gas entropy Merz, R. V. Stanton, et al. Amber 7. University of California, correction? J. Comp. Chem., 2002, 23, 1143–1149. San Francisco, 2002. URL http://amber.scripps.edu/doc7/. [32] H. Senderowitz, D. Q. McDonald, and W. C. Still. A prac- [24] W. D. Cornell, P. Cieplak, C. I. Bayly, I. R. Gould, K. M. Merz, tical method for calculating relative free energies of binding: Jr., D. M. Ferguson, D. C. Spellmeyer, T. Fox, J. W. Caldwell, Chiral recognition of peptidic ammonium ions by synthetic and P. A. Kollman. A second generation force field for the sim- ionophores. J. Org. Chem., 1997, 62, 9123–9127. ulation of proteins, nucleic acids and organic molecules. J. Am. [33] D. J. Adams. Grand canonical ensemble Monte Carlo for a Chem. Soc., 1995, 117, 5179–5197. Lennard-Jones fluids. Mol. Phys., 1975, 29, 307–311. [25] C. I. Bayley, P. Cieplak, W. D. Cornell, and P. A. Kollman. A