A hybrid method of molecular dynamics and harmonic dynamics for docking of flexible ligand to flexible receptor

A Hybrid Method of Molecular Dynamics and HarmonicDynamics for Docking of Flexible Ligand

to Flexible Receptor

RIE TATSUMI,1 YOSHIFUMI FUKUNISHI,2 HARUKI NAKAMURA2,3

1Japan Biological Information Research Center, Japan Biological Informatics Consortium, Aomi2-41-6, Koto-ku, Tokyo 135-0064, Japan

2Biological Information Research Center, National Institute of Advanced Industrial Science andTechnology, Aomi 2-41-6, Koto-ku, Tokyo 135-0064, Japan

3Institute for Protein Research, Osaka University, Yamadaoka 3-2, Suita,Osaka 565-0871, Japan

Received 18 June 2004; Accepted 9 August 2004DOI 10.1002/jcc.20133

Published online in Wiley InterScience (www.interscience.wiley.com).

Abstract: We have developed a new docking method to consider receptor flexibility, a hybrid method of moleculardynamics and harmonic dynamics. The global motions of the whole receptor were approximately introduced into thoseof the receptor in the docking simulation as harmonic dynamics. On the other hand, the local flexibility of the side chainswas also considered by conventional molecular dynamics. We confirmed that this new method can reproduce thefluctuations of the whole receptor by making a comparison of the directions and amplitudes of the global fluctuations.Then this method was applied to the docking of HIV-1 protease and its ligand. As a result, we observed a dockingprocess where the ligand enters into the binding pocket well, which implies that this method is effective enough toreproduce a molecular complex formation.

© 2004 Wiley Periodicals, Inc. J Comput Chem 25: 1995–2005, 2004

Key words: receptor–ligand docking; receptor flexibility; principal component analysis; harmonic dynamics;molecular dynamics

Introduction

The molecular recognition processes are the basis of all biologicalprocesses, and the docking process of a ligand to its receptor is alsoone of the recognition processes. Understanding the docking pro-cess is not only a great step toward the general grasp of thebiological system but also very important for drug design.

To predict ligand–receptor complex structures and estimatetheir binding affinity, many docking methods have been devel-oped. Based on the extent of flexibility, docking approaches aredivided into three types: (1) rigid ligand–rigid receptor, (2) flexibleligand–rigid receptor, (3) flexible ligand–flexible receptor. Earlystudies treated a ligand and its receptor as rigid bodies1 and thenonly ligand flexibility gradually came to be considered.2–5 Inrecent years, some methods to include receptor flexibility havebeen proposed, but none of these methods seems to suffice todescribe docking processes accompanied by large-scale conforma-tional changes of receptors.6–9

Jiang and Kim proposed a “soft” docking method that implic-itly allows small conformational changes of the surface of a

receptor and its ligand by representing them as cubes.10 Jones et al.regarded only atoms related to hydrogen bonds with a ligand asflexible.11 In some studies, only the side chains in the binding sitewere considered to be flexible.12,13 Totrov and Abagyan,14 andBastard et al.15 included not only the flexibility of the side chainsbut also that of the loops in the binding site.

Other studies treated all atoms in the binding site of a receptoras flexible. Luty et al. divided the receptor into three regions—theflexible binding site region, the rigid region, and the region be-tween them.16 All atoms of the ligand and the binding site weresimulated by molecular dynamics (MD) explicitly, while the rigidregion of the receptor was represented by grids. The atoms in theregion between the rigid and flexible regions of the receptor wereposition-restrained. Mangoni et al. used the same system as Luty et

Correspondence to: R. Tatsumi; e-mail: [email protected]

Contract/grant sponsors: New Energy and Industrial TechnologyDevelopment Organization of Japan, and the Ministry of Economy,Trade, and Industry of Japan.

© 2004 Wiley Periodicals, Inc.

al., except that the rigid region of the receptor was neglected.17 Toachieve a fast search of the receptor surface, they employed themolecular dynamics docking (MDD) method,18 where only thecenter of mass of the ligand was simulated at a higher temperature.Because all of these studies limit receptor flexibility to the bindingsite region, that is, the local region, they cannot describe large-scale conformational changes of receptors, which are global andcollective.

In contrast, although Sandak et al. considered the hinge-bend-ing motions of domains, subdomains, or substructural parts, theythemselves were held rigid.19 In other words, global motions wereallowed, but local flexibility was not.

Still other studies included receptor flexibility implicitly, usingthe information on the multiple complex structures obtained byexperiment. Verkhivker et al.20,21 and Murcia and Ortiz22 per-formed the docking of a ligand to each receptor crystal structure.Their results suggested that using a single rigid receptor structurefor docking is not necessarily sufficient. Knegtel et al. proposed acomposite grid method to consider the information on conforma-tional variety.23,24 The use of the composite grid improved theaccuracy of the identification of known ligands. Claussen et al.developed a new software FlexE to handle receptor flexibility.25

Based on the superimposed structures of some crystal structures,they regarded similar parts of them as the same and dissimilar partsas separate alternatives. Then they docked a flexible ligand intonew valid receptor structures created from various combinations ofthe discrete alternatives. Although all of these methods allowed asearch for the binding modes over a broader space, they also havelimits in that they depend on the availability of complex structuraldata.

Instead of using experimentally determined conformations,Cavasotto and Abagyan generated a discrete set of receptor con-formations using the ICM-flexible receptor docking algorithm(IFREDA).26 However, it is likely that global collective motions,such as hinge-bending motions, could not be captured becausebackbone movements are taken into account only through mini-mization. Pang and Kozikowski27 and Broughton28 performed MDsimulations to generate receptor conformations, but they were tooshort and were initiated from the receptor structure extracted fromone complex crystal structure. If only uncomplexed structure isobtained by experiment and the docking is accompanied by large-scale conformational change of the receptor, longer MD simula-tions would be required.

We introduce a new method that solves these problems, ahybrid method of molecular dynamics and harmonic dynamics(Hybrid MD/HD method). When the binding site is known, ingeneral, the docking simulation is performed in the limited localregion consisting of a ligand and the binding site of the receptorbecause the inclusion of the whole receptor requires a great deal oftime. The Hybrid MD/HD method has been constructed so that themotions of the whole receptor, that is, global and collective con-formational changes, are reproduced in such a local docking sim-ulation. We have assumed that the HD motions described by anensemble of modes of the entire protein molecule are relevant tothe functionally important motions, such as those produced by thedocking. In fact, this assumption has been proven to be correct inmany cases.29–37 Because of this assumption, docking problemsaccompanied by drastic conformational transition, such as local

unfolding, are beyond the scope of our method. The motions of thewhole receptor are approximated as HD and introduced into themotions of the C� atoms of the receptor in the docking. On theother hand, the motions of the other atoms are produced byconventional MD to allow local motions of the side-chain atoms.Therefore, the Hybrid MD/HD method is a multiscale simula-tion,38–40 which means a large-scale integrated simulation acrosswidely disparate spatial and temporal scales. Although Zachariaset al.41,42 also approximated the motion of a receptor as harmonicmodes or biquadratic modes, they ignored the local flexibility.Because conformational changes of the side chains have no cor-relation with those of the backbone atoms (or C� atoms),43 it ishighly desirable that global and local flexibility are treated sepa-rately.

Using the Hybrid MD/HD method, we confirmed first that itcan reproduce the fluctuations of the whole receptor. Then weapplied it to the preliminary docking of HIV-1 protease and itsligand, which is accompanied by flap motions of the receptor, anddemonstrated its effectiveness.

Method

In the Hybrid MD/HD method, we need to first perform a MDsimulation of the whole receptor, starting from a ligand-free con-formation to capture the characteristics of its motions. This simu-lation is computationally demanding, but once it is performed, insubsequent docking simulations of a large number of possibleligands to their receptor, the global conformational change of thereceptor is approximately taken into account without including thewhole receptor.

Second, to extract the collective motions of the whole receptor,we have to analyze the MD trajectory by principal componentanalysis (PCA) using the C� atoms.31,33,44 We can also use thebackbone atoms instead of the C� atoms. The covariance matrix Cis constructed as

Cij � �� xi � �xi�� xj � �xj��, (1)

where xi represents each x, y, z coordinate of the C� atoms and � �is the average over time, that is, the MD trajectory. The averagestructure is calculated over all structures superimposed on anyreference structure and then all structures are superimposed on theaverage structure. In this study, the initial structure of the MDsimulation was used as the reference structure. The eigenvaluesand their corresponding eigenvectors are obtained by diagonaliz-ing the covariance matrix and are arranged in decreasing order ofthe eigenvalues. Because the covariance matrix to be diagonalizedhas a dimension of 3N � 3N, where N is the number of the C�

atoms, our method can be applied to a large receptor with smallamounts of the computational time. The eigenvectors represent thedirections of the fluctuations and the eigenvalues, the mean-squarefluctuations (msf) along the eigenvectors. Most of the fluctuationsare occupied by the first M eigenvalues, where M is small com-pared to the total 3N � 6.

Third, we need to perform the docking simulation using allatoms of a ligand and the binding site of the receptor. Then the

1996 Tatsumi, Fukunishi, and Nakamura • Vol. 25, No. 16 • Journal of Computational Chemistry

collective motions of the whole receptor are introduced into themotions of the C� atoms as harmonic modes so that such globalfluctuations are reproduced even when only part of the receptor isused for the docking. Although anharmonic behavior may beobserved in the motions of the whole receptor, the characteristicsof the global motions are well captured by HD. We note that notonly the coordinates of the C� atoms in the binding site but alsothose of the whole receptor can be obtained by HD because theharmonic mode consists of those of the whole receptor. On theother hand, the motions of the other atoms are described byconventional MD to consider the local motions of the side-chainatoms. The potential energy in the Hybrid MD/HD method con-sists of three terms:

E � EAA � EA�B � EBB, (2)

where A denotes the C� atoms of the whole receptor, whosenumber is N; A�, the C� atoms only in the binding site out of A.B represents the other atoms, that is, all atoms except the C� atomsin the binding site of the receptor, all atoms of the ligand and watermolecules. The interaction between A atoms, EAA, is given by

EAA � ��1

m1

2k�c�

2, (3)

where k� is the spring constant of the �-th harmonic mode; c�, thedisplacement along the �-th eigenvector. The summation is takenover the number of harmonic modes, m, which is large enough todescribe the global fluctuations of the whole receptor. Althoughthe spring constant k� can be given arbitrarily, in the present study,unless it is mentioned which value was used, the following valuewas adopted so that the fluctuations in the MD simulation of thewhole receptor are reproduced:

k� �kBT

e�

, (4)

where e� is the �-th eigenvalue of the covariance matrix C, kB isthe Boltzmann constant and T is the absolute temperature. Theinteraction between A� and B atoms, EA�B, and the interactionbetween B atoms, EBB, are conventional MD potential functions,such as AMBER45 and CHARMM.46

When the above potential energy is used, it is easy for theequations of motion of the A atoms to be solved in the spaceconsisting of m eigenvectors and for those of the B atoms to besolved in the Cartesian coordinate space. The equations of motionof the A atoms are introduced as described below. We define theconnection between the Cartesian coordinate of an A atom, xi, andthe displacement along the �-th eigenvector, c�, as follows:

xi � xi0 � �

��1

m

ui�c�, (5)

where xi0 represents the coordinate of the vibrational center of an

A atom and ui� is the i-th element of the �-th eigenvector. We take

xi0 � �xi� so that the Hybrid MD/HD method can be applied even

to cases of the absence of complex structural data. Then thevelocity of each A atom in the Cartesian space vi is given by

vi � ��1

m

ui�c�, (6)

and their kinetic energy KA,

KA � �i�1

3N1

2mi� �

��1

m

ui�c�� 2

, (7)

where mi is the mass and the summation is taken over all A atoms.From eqs. (2), (3), and (7), the Lagrangian of the system is givenby

L � �i�1

3N1

2mi� �

��1

m

ui�c�� 2

� KB � ��1

m1

2k�c�

2 � EA�B � EBB,

(8)

where KB denotes the kinetic energy of the B atoms. The motionof the A atoms is determined by solving Lagrange’s equations ofmotion:

d

dt � �L

�c�� L

�c�� . (9)

Substituting eq. (8) for the above equation (9) gives:

��1

m �i�1

3N

miui�ui�c� � �k�c� � �i�1

3N� ��EA�B

�x�i�u�i�, (10)

where N� is the number of the A� atoms, x�i represents the Carte-sian coordinate of the A� atoms and u�i� also represents the elementcorresponding to the A� atoms out of all elements of the �-theigenvector (ui�, i � 1, . . . , 3N). Here, ¥i�1

3N miui�ui� of the leftside of eq. (10) can be taken as the (�, �) element of matrix S, s��.We multiply the above eq. (10) by inverse matrix W (�w��) of Sfrom the left and finally reach the following equation:

c� � ��1

m

w��k�c� � �i�1

3N� ��EA�B

�x�i �u�i��. (11)

After this equation of motion in the space consisting of m eigen-vectors is solved, the Cartesian coordinates and velocities of the A( A�) atoms can be obtained from eqs. (5) and (6).

A Hybrid Method of Molecular Dynamics and Harmonic Dynamics 1997

Results and Discussion

Reproduction of Global Motion

First of all, we need to verify whether the Hybrid MD/HD methodcan reproduce the fluctuations in the MD simulation of the wholereceptor because the motions of the whole receptor are approxi-mately introduced into those of the receptor in the docking as HD.For that purpose, HIV-1 protease was selected because its dockingis accompanied by flap motions of the arms, that is, global mo-tions.

First, we performed a MD simulation of the dimeric HIV-1protease in water, which is a natural form under physiologicalconditions and consists of two chains of 99 residues, that is, a totalof 198 residues. The system consists of 3126 atoms of HIV-1protease and 7509 TIP3P47 water molecules. The initial structurewas obtained from 3HVP in the PDB without a ligand,48 which isa synthetic protein, replacing both Cys67 and Cys95 to �-amino-N-butyric acid (Aba) residues. The MD simulation was performedby the program prestoX.49 The water molecules were confined ina sphere (diameter � 80 Å) with a harmonic potential. No potentialcutoff was used for the Lennard–Jones and the Coulomb interac-tions. The simulation was carried out at 310 K by the constraintmethod,50 and all equations of motion were integrated by theleapfrog method. We took a time step t � 1.5 fs by using theSHAKE algorithm.51 In all simulations of the present study, theAMBER parm96 force field52 was used and the Coulomb interac-tion was calculated with the dielectric constant 1.0. Because theatomic point charges of Aba residues of HIV-1 protease are notavailable in the AMBER database,52 they are determined by therestrained electrostatic potential (RESP) procedure.53,54 TheGAMESS55 and Gaussian 9856 were used at the HF/6-31G* levelto perform the quantum chemical calculations. The RESP charges

are summarized in Table 1. The simulation was done for 540 ps,after a 510-ps equilibrium run, so that the global motions could becaptured, which required 162 days of CPU time. In the presentstudy, all simulations were carried out using maximum 16 CPUs ofHP Alpha 21264 (833 MHz) in parallel.

Second, we constructed the covariance matrix using the MDtrajectory with a total of 3600 structures recorded at every 150 fsafter the equilibrium run. The covariance matrix has a dimensionof 594 � 594. By diagonalization of the covariance matrix, theeigenvectors and eigenvalues, that is, the directions and amplitudesof the fluctuations, were obtained. We also calculated the root-mean-square fluctuations (rmsf) of the C� atoms to find whichatom fluctuates greatly. Based on the data about the directions andamplitudes of the fluctuations, we need to confirm whether the flapmotions are captured by the MD simulation of the whole of HIV-1protease. If not, the Hybrid MD/HD method could not be appliedto the docking.

Figure 1 shows the rmsf of the C� atoms in the MD simulationof the whole of HIV-1 protease. The rmsf is larger around the 50-thresidues, which are the tips of the arms of HIV-1 protease. Fur-thermore, we show the direction of the third largest fluctuatingamplitude mode, that is, the direction of the eigenvector with thethird largest eigenvalue, in Figure 2a. The flap motions are iden-tified remarkably, which is consistent with the low-frequencycollective motions obtained by normal mode analysis in the studyof Cao et al. (Fig. 8 in ref. 57). These results imply that the flapmotions are the most dominant fluctuations; in other words, theyoccupy most of the total fluctuations. Because it is clear that theflap motions are functionally important by comparison between theunbound and bound form of HIV-1 protease, the assumption thatfluctuations and functionally important motions are closely con-nected has been proven to be correct also in HIV-1 protease.

Table 1. The RESP Charges of Aba and Nle Residues

Aba Nle

Atom name Charge Atom name Charge

N �0.4157 N �0.4157H 0.2719 H 0.2719CA �0.0498 CA 0.0168HA 0.0982 HA 0.0621CB 0.0689 CB �0.0225HB1 0.0070 HB1 0.0279HB2 0.0070 HB2 0.0279CG �0.1177 CG �0.0660HG1 0.0336 HG1 0.0316HG2 0.0336 HG2 0.0316HG3 0.0336 CD 0.0408C 0.5973 HD1 �0.0037O �0.5679 HD2 �0.0037

CE �0.0902HE1 0.0206HE2 0.0206HE3 0.0206C 0.5973O �0.5679

Figure 1. The rmsf of the C� atoms in the MD simulation of thewhole of HIV-1 protease. The black circles with the solid line showthat of the first chain; the open squares with the dotted line, that of thesecond one. The solid and dotted lines are merely guides for the eye.


Figure 2. The directions of the fluctuations in the MD simulation of the whole receptor and the HybridMD/HD simulations with a various number of modes: (a) the direction of the third largest fluctuatingamplitude mode in the MD simulation of the whole receptor, (b) that of the third mode in the HybridMD/HD simulation with all modes, 588 modes, (c) that of the second mode in the Hybrid MD/HD withthe first 91 largest amplitude modes (90% of the total fluctuation), (d) that of the second mode in theHybrid MD/HD with the first 38 modes (80% of the total), (e) that of the first mode in the Hybrid MD/HDwith the first 18 modes (70% of the total), and (f) that of the second mode in the Hybrid MD/HD with thefirst 3 modes (40% of the total). The structures of these drawings are taken from the average structure inthe MD simulation of the whole receptor, that is, that of the vibrational center. In addition, the length ofthe arrows is reasonably scaled so as to make it easier to find the directions of the fluctuations (drawn withMOLEKEL60).


Figure 3.


Third, we performed Hybrid MD/HD simulations in waterusing only atoms in the binding pocket of HIV-1 protease. Becauseour present aim is to confirm whether the global fluctuations ofHIV-1 protease are reproduced in the Hybrid MD/HD, there is noneed to include the ligand. The system includes the residuesaround the binding pocket, 8–9, 23–34, 45–56, and 76–88 of eachchain, that is, a total of 78 residues, which are covered with 2460TIP3P water molecules. Residues 45–56 represent the flappingarm. The number of harmonic modes to approximate the globalfluctuations is attempted in some values: (1) all modes, 588 modes,(2) the first 91 largest amplitude modes, which occupy 90% of thetotal fluctuation, (3) the first 38 modes (80% of the total), (4)the first 18 modes (70% of the total), and (5) the first 3 modes(40% of the total). The initial structure and the structure of thevibrational center were taken from the average one in the MDsimulation of the whole of HIV-1 protease. All simulationswere carried out at 310 K by the program prestoX with amodification for the Hybrid MD/HD. The water molecules wererestrained in a sphere with a diameter of 54 Å. The Lennard–Jones and the Coulomb interactions were cut off at a distance of12 Å and their list was updated every 50 steps. The simulationwas performed for 2 ns with a time step t � 0.5 fs. After therun, PCA was done again to find the directions and amplitudesof the global fluctuations. Then the covariance matrix was

constructed using the C� atoms of the whole receptor, whosecoordinates can be obtained even in the Hybrid MD/HD simu-lation. The rmsf of the C� atoms was calculated again. Bycomparing these data with those of the MD, we need to verifywhether the global fluctuations of the whole receptor are repro-duced in the Hybrid MD/HD.

Figure 2b–f shows the directions of the fluctuations in theHybrid MD/HD simulations with a various number of modes. Theflap motions can be found clearly at larger amplitude modes in allcases, although the inconsistency in the mode number (such as thethird mode in the case of all modes and the second mode in thecase of the first 91 modes) can be seen, which might originate fromthe fact that HD is not completely independent due to the termEA�B. This result strongly suggests that the directions of thefluctuations in the Hybrid MD/HD are qualitatively the same asthose of the MD (Fig. 2a). Figure 3 shows the correlation betweenthe rmsf, that is, the amplitude of the fluctuations, of the MD of thewhole receptor and that of the Hybrid MD/HD with a variousnumber of modes. Here the rmsf of the 78 residues used in theHybrid MD/HD simulation was employed. The strong correlationcan be found in all cases. Taken in connection with Figure 1,which shows the flap motions clearly, this result suggests that thefunctionally important flap motions occur also in the Hybrid MD/HD. We also found that the smaller the number of modes is, theweaker the correlation is and the smaller the rmsf itself is. Theyseem to be natural, considering that the number of modes deter-mines the extent of the approximation of the fluctuations and theoccupancy in the total fluctuation. The reason why the HybridMD/HD with all modes cannot entirely reproduce the rmsf of theMD itself might also be the imperfect independence of HD. Theseresults imply that the Hybrid MD/HD method can describe theglobal fluctuations of the whole receptor even if the number ofharmonic modes to approximate them is small.

Figure 4. The directions of the first largest fluctuating amplitude modes in the MD simulations limitedto the local region with the C� atoms (a) strongly (a � 37.0 kcal/[mol � Å2]) or (b) weakly (a � 0.370kcal/[mol � Å2]) position-restrained where the potential function is ¥i a(r�i � r�i0)2. The structures of thesedrawings are taken from the average structure in the MD simulations. In addition, the length of the arrowsis reasonably scaled so as to make it easier to find the directions of the fluctuations (drawn withMOLEKEL60).

Figure 3. The correlation between the rmsf of the MD of the wholereceptor and that of the Hybrid MD/HD with each mode: (a) all modes,588 modes, (b) the first 91 largest amplitude modes (90% of the totalfluctuation), (c) the first 38 modes (80% of the total), (d) the first 18modes (70% of the total), and (e) the first 3 modes (40% of the total).The solid line indicates the linear fit. The correlation coefficient is0.984 in (a), 0.989 in (b), 0.986 in (c), 0.946 in (d), and 0.903 in (e).


Furthermore, we examined whether ordinary MD docking sim-ulation, whose system is limited to the local region, can reproducethe motions of the whole receptor. In ordinary MD docking limitedto the local region, the C� atoms (or backbone atoms) of thereceptor are position-restrained13 because the local region cut offfrom the receptor is generally unstable. We carried out MD sim-ulations of the binding pocket in water with the C� atoms position-restrained. The system consists of the same 78 residues as thoseused in the Hybrid MD/HD and 2448 TIP3P water molecules. Theinitial structure was taken from the dimeric structure obtained byusing 3HVP in the PDB. The simulation was performed at 310 Kfor 2.1 ns with a time step t � 1.5 fs. The nonbonded interac-tions were cut off at a distance of 12 Å and the interaction list wasupdated every 20 steps. The C� atoms of the 78 residues wereposition-restrained and the water molecules were confined in asphere with a diameter of 54 Å. The subsequent PCA and calcu-lation of the rmsf were done using the C� atoms of the 78 residues.

As a result, in the MD simulations limited to the local regionwith the C� atoms position-restrained, each atom fluctuates indifferent directions (Fig. 4). In Figure 5, we also show the corre-lation between the rmsf of the MD of the whole receptor and thatof the MD with the C� atoms position-restrained. When thestrength of the position restraint is strong (Fig. 5a), there is almostno correlation between them and the rmsf itself is very small. Evenif a weak position restraint is imposed (Fig. 5b), the rmsf itselfbecomes larger, but the correlation is still weak. These resultsindicate that ordinary MD docking simulation with the C� atomsposition-restrained cannot reproduce the motions of the wholereceptor and therefore support the usefulness of the HybridMD/HD method.

Application to Receptor–Ligand Docking

After verification, the Hybrid MD/HD method was applied toreceptor–ligand docking. We tried to dock the modified inhibitorMVT101 (Acetyl-Thr-Ile-Nle-Nle-Gln-Arg-Amide)6 to HIV-1protease in water (Nle, norleucine). The system consists of the 78residues around the binding pocket of HIV-1 protease, which arethe same as those used in the verification of the Hybrid MD/HDmethod, 121 atoms of the ligand MVT101, and 2379 TIP3P watermolecules. The ligand used in the simulation is different from thereal MVT1016 in that the peptide bond between the two Nleresidues is not reduced, which has been assumed to have littleeffect on the docking. Because the atomic point charges of Nleresidues of the ligand are not obtained in the AMBER database,52

they are determined in the same way as those of Aba residue. TheRESP charges are listed in Table 1. The 91 harmonic modes (90%of the total fluctuation) with all spring constants 1/16 times thoseof eq. (4) were used so that the arms flap largely. The averagestructure in the MD of the whole receptor was adopted as the initialstructure of the receptor and the structure of the vibrational center.The initial conformation of the ligand is the same as that of thecomplex crystal structure, PDB code 4HVP.6 The initial positionof the ligand is determined by translating the ligand so as to put itout of the pocket after the complex crystal structure 4HVP issuperimposed on the average structure in the MD of the wholereceptor (Fig. 6a) (t � 0). The water molecules were confined ina sphere with a diameter of 54 Å. The cutoff distance of thenonbonded interactions was 12 Å and the interaction table wasupdated every 50 steps. The simulation was carried out at 700 Kfor 4 ns with a time step t � 0.5 fs, which took 40 days of CPU

Figure 5. The correlation between the rmsf of the MD of the whole receptor and that of the MD with theC� atoms (a) strongly (a � 37.0 kcal/[mol � Å2]) or (b) weakly (a � 0.370 kcal/[mol � Å2])position-restrained. The solid line indicates the linear fit. The correlation coefficient is �0.0237 in (a) and0.157 in (b).


Figure 6. Representative snapshots during one docking simulation where most of the ligandenters into the binding pocket of HIV-1 protease (drawn with Rasmol). In all, left: front, right:top. The ligand is shown as orange molecules, the water as red molecules, and the protein asthe molecules with other colors; (chain A) residue 8–9: blue, 23–34: sky blue, 45–56: lightblue, 76–88: turquoise; (chain B) 8–9: green, 23–34: light green, 45–56: yellow green,76–88: yellow.

time. If we perform a docking simulation for 4 ns using the wholereceptor, at least 617 days of CPU time would be needed, which ismore than 15 times the CPU time of the Hybrid MD/HD. Itrepresents that the computational time of the Hybrid MD/HDmethod is very reasonable.

One docking process where the ligand enters considerably intothe binding pocket is shown in Figure 6. The C-terminus of theligand comes to the entrance of the pocket at 200 ps (Fig. 6b).Then the root-mean-square deviation (RMSD) of all atoms of theligand is 16.5 Å, which was calculated against the ligand structureof 4HVP after the complex crystal structure 4HVP is superim-posed on the structure of the vibrational center, that is, the averagestructure in the MD of the whole receptor. Within dozens ofpicoseconds, almost half of the ligand goes into the pocket (Fig.6c, RMSD: 15.0 Å). The ligand enters a little more at 1230 ps(Fig. 6d, RMSD: 9.0 Å). At 1560 ps (Fig. 6e, RMSD: 8.0 Å),most of the ligand enters into the pocket, although the bindingmode is wrong, where the ligand has the shape of the letter “U”with the N-terminus on the right, the C-terminus on the left, andthe second Nle residue in the inmost part of the pocket. Theobserved complexed structure might correspond to one of theensemble of the stable docked conformations, which are composedof many energy minimum conformations. Then the ligand goes outof the pocket at 2070 ps. Moreover, observing how the arms ofthe receptor move in this docking simulation is highly meaningful.The time course of the distance between the C� atom of Gly27 andthat of Ile50 is shown in Figure 7. Because Gly27 is in the middleof the lower part of the pocket and Ile50 is at the tip of the arm, thechange in the distance between these two atoms represents the flapmotion, which is the important motion in the docking process. The

distance in the unbound open conformation 3HVP is 16.7 Å andthat in the bound closed conformation 4HVP is 11.5 Å. The largeopen–close motions of the arms can be found. On the other hand,at 310 K, such large motions could not be observed and there wasno docking process where the ligand entered into the pocket.Opening the arms larger might lead to a decrease of the free energybarrier and make it possible for the ligand to enter into the pocket.These results suggest that the Hybrid MD/HD method is veryuseful, although we could not succeed in reproducing the actualcrystal complexed structure.

Finally, we emphasize that the Hybrid MD/HD method isapplicable to the case of large receptors. In fact, normal modeanalysis based on the simplified elastic potential58 is able toexamine the entire motions of large biological systems such asribosome and viruses, when the method of the diagonalization ofa large matrix proposed by Tama et al. is adopted.36,37,59 Thosemodes can be used as the harmonic motions in the Hybrid MD/HDmethod.

Conclusion

We have proposed a new docking method to consider receptorflexibility as well as ligand flexibility, the hybrid method of mo-lecular dynamics and harmonic dynamics (the Hybrid MD/HDmethod), which is a multiscale simulation. The Hybrid MD/HDmethod satisfies four conditions: (1) only part of a receptor is usedfor docking so that the computational time is reasonable, (2) theglobal conformational changes of the whole receptor are repro-duced even in such a local docking simulation, (3) local flexibilityis also considered, (4) the method does not depend on the avail-ability of complex structural data. The motions of the wholereceptor are introduced into those of the receptor in the docking asa HD approximation. On the other hand, the local motions of theside-chain atoms are considered through conventional MD.

We have investigated whether this method can reproduce thefluctuations of the whole receptor by making a comparison of thedirections and amplitudes of the global fluctuations, using HIV-1protease. As a result, the flap motions have clearly been found atlarger amplitude modes. We have also found that the amplitudesare highly correlated. From these two points of view, we concludethat the Hybrid MD/HD method can describe the global fluctua-tions of the whole receptor even if the number of harmonic modesto approximate them is small. On the other hand, ordinary MDdocking simulation with the C� atoms position-restrained hasfailed to reproduce them. This result emphasizes the effectivenessof the Hybrid MD/HD method.

Then this method was applied to the docking of HIV-1 proteaseand its ligand. As a result, we obtained a case in which most of theligand enters into the binding pocket, which suggests that theHybrid MD/HD method is very promising.

Acknowledgments

We would like to thank Ikuo Fukuda for fruitful discussions. R.T.also thanks George Chikenji, Nobuya Maeshima, and Naoki Iharafor useful discussions and comments. We are also grateful to

Figure 7. Time course of the distance between the C� atom of Gly27and that of Ile50 in the Hybrid MD/HD docking simulation. The solidline shows that of the first chain; the dotted line, that of the second one.The upper and lower dashed lines correspond to the distances for theopen (3HVP) and closed (4HVP) crystal structures, respectively.


Yoshiaki Mikami, Takashi Kurosawa, and Satoru Kubota for tech-nical support.

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A hybrid method of molecular dynamics and harmonic dynamics for docking of flexible ligand to flexible receptor