Reduced variable molecular dynamics - Semantic Scholar

Reduced Variable Molecular Dynamics JAMES TURNER and PAUL WEINER" Amdyn Systems lnc., 33 Sylvanus Wood Lane, Woburn Massachusetts 01801

BARRY ROBSON" Proteus Molecular Design Ltd., Proteus House, Lyme Green Business Park, Macclesfield, Cheshire S K I 1 OJl, England

RAVI VENUGOPAL, HARRY SCHUBELE 111, and RAMEN SINGH Dynacs Engineering Company, lnc., 28870 U.S. Highway 19 North, Suite 405, Clearwater, Florida 34621 Received 6 July 1994; accepted 3 February 1995

ABSTRACT This article describes an extension to previously developed constraint techniques. These enhanced constraint methods will enable the study of large computational chemistry problems that cannot be easily handled with current constrained molecular dynamics (MD) methods. These methods are based on an O( N solution to the constrained equations of motion. The benefits of this approach are that (1) the system constraints are solved exactly at each time step, (2) the solution algorithm is noniterative, (3) the algorithm is recursive and scales as O( N 1, (4)the algorithm is numerically stable, (5) the algorithm is highly amenable to parallel processing, and (6) potentially greater integration step sizes are possible. It is anticipated that application of this methodology will provide a 10- to 100-improvement in the speed of a large molecular trajectory as compared with the time required to run a conventional atomistic unconstrained simulation. It is, therefore, anticipated that this methodology will provide an enabling capacity for pursuing the drug discovery process for large molecular systems. 0 1995 by John Wiley & Sons, Inc.

Introduction

I

t is increasingly clear that computer sirnulations of molecular systems will have an enor-

*Authors to whom correspondence may be addressed.

mous impact on our understanding of the structure-function relationships pertinent to the discovery of new molecules with desired properties, such as new pharmaceutical drugs. The use of molecular simulations and the increasing performance of modem computers makes it possible to study the precise physicochemical nature of protein-ligand

Journal of Computational Chemistry, Vol. 16, No. 10, 1271-1290 (1995) 0 1995 by John Wiley & Sons, Inc.

CCCO192-8651/95/101271-20

TURNER ET AL.

interactions, protein engineering, and solvation phenomena and to characterize the thermodynamical properties of complex systems with many thousands of atoms.',' Molecular dynamics (MD) is one of the established simulation techniques in the prediction, analysis, and design of complex molecules.',' These techniques typically apply the Newtonian laws ( F = ma) to the motion of all of the atoms in the system once the forces have been defined as a function of variables such as atom type, bond type, dihedral type, and interatomic distances. There are many software packages that implement these techniques very efficiently on computers ranging from workstations to large parallel supercomputers. Unfortunately, the primary difficulties in modeling complex molecules are not overcome by attention to details of hardware and software implementation. There are several sources of computational complexity for MD simulations. The first difficulty is due to the N 2 nonbonded interactions (assuming no cutoff criteria), where N is the number of atoms that occur in the energy function because every atom can experience the forces of every other atom via long-range interaction effects. Even with the use of arbitrarily imposed cutoff distances, the evaluation of the nonbonded terms can take over 90% of the computer time involved in each energy function evaluation. The second difficulty relates to the step size limitation in the integration step of the dynamics algorithm. To integrate accurately the equations of motion in atomistic models, it is necessary to sample adequately the period of the highest frequency motion in the system. The high-frequency bond stretching motion that occurs in all molecules requires the use of a small step size (i.e., 0.5-1.0 fs) in the simulation. This limitation prevents very large simulations from being run, even on supercomputers, for periods much greater than hundreds of picoseconds. This is much too short a time scale to study most processes of biological interest or to study rapidly occurring but infrequent events, such as conformational barrier crossings. Even for the study of non-covalent protein-ligand interactions, the simulations may be too short to sample conformational space adequately and therefore to calculate accurately the free energy of interaction. The third difficulty relates to the presence of multiple minima in the 3 N - 6 dimensional conformational space that defines the potential energy

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surface. These minima can cause the simulations to become locally trapped or delayed. Even if one had enormous amounts of supercomputer time to carry out nanosecond trajectory simulations, it is difficult to ensure adequate sampling to estimate thermodynamic properties accurately, such as free energy, or even to ensure that the structures reached include all of the biologically relevant ones. For example, if each simulation takes thousands of supercomputer hours, it would be difficult to repeat the simulations for many other modified molecules. Additional simulations are important to gain an understanding of the specificity of the biological process being studied. One approach for solving this problem is to apply non-Newtonian dynamics. These methods are non-Newtonian in the sense that the Newtonian laws are highly modified or Newtonian motion is in some sense interr~pted.~ These methods include the use of increased dimensionality to avoid entrapment in three dimensional x , y, z Euclidean ~pace.4,~ By transforming the normal energy/force functions and extending the simulations into four dimensions, barriers, which exist due to movements in true Euclidean space, could be tunneled through and a lower minimum found. The appropriate direction of tunneling is controlled by "target functions," which express heuristic information concerning which direction the global minimum must lie in. The target function can also represent specific experimental data obtained from the protein of interest by experimental work? Other methods, such as Rush dynamics, modify the basic simulation equations in the familiar three-dimensional space. It has been shown, for at least small problems, that these methods can more efficiently search conformational space than the traditional MD methods.3~~ There is, however, still a need to reduce a very large problem to a simpler one even when applying these methods. A better approach is to reduce significantly the number of degrees of freedom in the Simulation (while retaining the quantitative aspects of the simulation). Then each simulation takes less time and one can run many different simulations. A class of dynamics algorithms, called reduced variable dynamics, makes this reduction in time possible. Since there are many fewer degrees of freedom, phase space can be more efficiently searched. The following sections will describe the currently available methods.

VOL. 16, NO. 10

REDUCED VARIABLE MD

Traditional Constrained Dynamics Approaches One approach for eliminating unnecessary degrees of freedom and high-frequency motions is through the use of defined constraints. The traditional approach consists of adding to Newton's equation of motion a forcelike term consisting of the gradient of a position-dependent constraint equation times an unknown Lagrange multiplier. The resulting atomistic constrained equation of motion follows as

subject to

V,Cr(R,,. . , , R N ) R j= 0;J = 1,.. ., NP;r = 1,. .., N, where Vj denotes the gradient with respect to the jth atom position coordinates, C,(*) denotes the functional form of the r th constraint function, and A, denotes the rth Lagrange multiplier. The solution for A, is obtained by differentiating the gradient of the constraint equation and introducing the equations of motion. Straightforward manipulations lead to a A, solution defined by a linear matrix-vector algebraic equation of dimension (N,X N,),where N, denotes the number of constraints. As N, becomes large, the computational burden associated with the inversion of the constraint matrix rapidly becomes prohibitive [e.g., O(N31. There are two alternative methods available for solving this problem. The first method involves the construction of a set of generalized coordinates, leading to

M(q)q = F(q, 4) where q denotes the (N X 1) generated coordinate vector, M ( q ) denotes the (N X N) mass matrix, and F(*) denotes the (N X 1) force vector. The method is completely general and has the added benefit of eliminating the constraint variables completely in the problem formulation. However, it requires the inversion of the (NX N)mass matrix M(q), which is an O ( N 3 ) operation. A further limitation is that the equations are cumbersome to implement and they are difficult to modify because of the strong interdependency of the mathematical relationships.

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Generalized coordinates have been used in MD for many years, though until recently the applications have been limited to short polymers.8 General-purpose formulations have not appeared frequently for several reasons. First, the classical analytical methods, such as Lagrange's, involve the formulation of kinetic and potential energy expressions, which require computing potentially thousands of time-varying first- and second-order partial derivatives with respect to the generalized coordinates.' This procedure is well defined but rapidly becomes unwieldy even for relatively low-order problems (i.e., 2 30 independent variables). In addition, there is still the problem of inverting an (N x N)mass matrix. Mazur and Abagyan"," have developed a modeling approach using a Lagrangian-based internal coordinate model for generating the equations of motion. The accelerations are computed by inverting the mass matrix for all the degrees of freedom at each time step in the integration process. An advantage of this approach is that the constraint equations are analytically eliminated from the problem. The major disadvantage of their approach is that the time-varying mass matrix must be computed and inverted at each time step. This approach is useful for small problems but does not scale up well for large problems because the computational effort required for inverting a large mass matrix [ O ( N 3 ) ]dominates the effort required for the force field calculations. If timevarying constraints or body flexibility are included, the mass matrix becomes larger to account for these degrees of freedom and the inversion problem becomes even more time consuming. In addition, the methodology has some limitations,12 including restrictions on intermolecular connectivity and torsion angle definitions. In a later article" these limitations were eliminated, but only a Monte Carlo technique was presented and there were no updates for the formulation of the equations of motion to account for the additional degrees of freedom. This update would involve a major modification of the original formulation and would still not remove the fundamental dynamics problem of requiring a large matrix inversion. Rudnicki and Harvey13 present an algorithm for computing the pseudorotation dynamics of a furanose ring. By retaining internal vibrational behaviors, their approach goes beyond the limitations of a rigid body model. However, the authors did not present a methodology for coupling their pseudorotation model to the underlying rigid body

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TURNER ET AL. motion of the ring, nor did they discuss how this model could be coupled to other bodies in a simulation. This must be carried out in such a way that the resulting equations of motion scale up efficiently for large systems. This method is most closely related to the generalized coordinate method, because the constraints are eliminated by a Lagrangian algorithmic approach. Their approach is useful for generating simplified dynamics models for subcomponents of more complex systems. Their results will be of interest to the current authors as they extend their methodology to embrance flexible dynamics. Other methods involve calculating approximate solutions to the constrained equations of motion. These methods often use an iterative approach to solve for the Lagrange multipliers and typically only need a few iterations if the corrections required are small. The most popular method of this type, SHAKE,'*,15 is easy to implement and scales as O ( N ) as the number of constraints increases. Therefore, the method is applicable to macromolecules. An alternative method, RATTLE,16 is based on the velocity version of the Verlet algorithm. Like SHAKE, RATTLE is an iterative algorithm. However, adding any other types of constraints, even bond angle constraints, can greatly slow the convergence of SHAKE and limit the maximum step size.17 Recent w ~ r k ~has , ' ~resulted in extensions of the SHAKE algorithm that allow for internal coordinate constraints. The newer methods no longer have the convergence problems associated with the original SHAKE algorithm. These methods are also iterative and straightforward to implement. However, the maximum time step size used in the articles describing these methods is still limited to 3 fs. The systems used to test the methods were small (< 50 atoms), and it is not known how the methods would perform for larger systems with many thousands of constraints. A new dynamics algorithm coupling implicit integration and normal mode techniques has recently been developed." This algorithm gains a factor of 10 X over other explicit integration schemes. The integration technique introduces no damping and is stable for step sizes as large as 50 fs. This new integration scheme is very interesting and could be useful even with the method described in this article. The method, however, requires a linearized model. As shown in the Appendix, dynamics equations between coupled bodies must have nonlinear terms to allow the simulation to be valid for arbitrarily large dis1274

placements and rapid motions. In very dense systems, such as explicit solvation studies, the linearized model could be a reasonable approximation. In large protein simulations with no solvent or an implicit solvent, such as might be used in folding studies, one wants large and rapid motions to occur so that interesting low-energy structures can be quickly located. These simulations would require the presence of the nonlinear dynamics terms. The shortcomings of these methods are one or more of the following: (1) They are not exact, (2) they are still limited to a relatively small step size, or (3) they do not scale up as O ( N ) . The proposed approach is based on a recursive generalized coordinate formulation for the equations of motion and can be generalized to handle both particles and rigid body constraints in a unified framework. The proposed algorithm maintains all internal constraints with no approximations. This new approach leads to a reduced variable molecular dynamic (RVMD) simulation technique. The RVMD will prove to be powerful because there are far too many variables, even with all bonds constrained, for macromolecular simulations to explore more than a small region of phase space. However, once the interesting events are located using RVMD, unconstrained simulations can be carried out to calculate the thermodynamic properties of interest. It is a practical necessity to be able to reduce greatly the number of variables in a macromolecular system so that qualitative aspects of its behavior, such as reaction pathways, can be studied and interesting events located. In two article^'^,^^ it has been noted that the use of constraints beyond bond length constraints can affect calculated properties, such as conformational interconversion rates. As a result, the effects of the constraints must be closely examined. The impact of imposing various constraints can be assessed by comparing the results of the reduced variable simulations with unconstrained simulations (when possible) as well as any available experimental data. This article describes an extension to previously developed constraint techniques applied by Turner et al.21-24and developed by Singh et al.25-27These enhanced constraint methods will enable the study of large computational chemistry problems that cannot be easily handled with current constrained MD methods. These methods are based on an O ( N ) solution to the constrained equations of motion. The benefits of this approach are that (1) the system constraints are solved exactly at each time step, (2) the solution algorithm is noniterative, (3) VOL. 16, NO. 10

REDUCED VARIABLE M D

the algorithm is recursive and scales as O( N ), (4) the algorithm is numerically stable, (5) the algorithm is highly amenable to parallel processing, and (6) greater integration step sizes are possible. It is anticipated that application of this methodology can potentially provide a 10- to 100-fold improvement in the speed of a large molecular trajectory as compared with the time required to run a conventional atomistic unconstrained simulation. It is anticipated that the RVMD methodology will provide an enabling capacity for pursuing the drug discovery process for large molecular problems.

General Constrained Dynamics Formulation PARTICLE CONSTRAINTS The mathematical models for constrained dynamics problems must account for the forces and torques required to maintain the kinematic constraint conditions defined by the researcher. Accordingly, in MD, one defines position-dependent constraints of the form

C j ( R ,,..., RN,)

=

cxj(t); j = 1,...,

N,

where Cj denotes the jth constraint, Ri for j = 1,.. . ,N p denotes the (3 X 1) position vector locating the jth atom relative to inertial space, aI denotes the constant or time-varying value of the constraint objective, and N,denotes the number of constraints. The functional form for the constraint permits arbitrary definitions for interrelationships between variables. Examples of potential types of constraints include bond and angle constraints. Because it is difficult to incorporate the position constraints directly into the Equation of Motion (EOM), the constraints are first differentiated with respect to time, yielding Np

r=l

ac. .

l R r = Gj 'Rr

where a( * ) / d R j denotes partial differentiation with respect to the jth particle position vector and ( ) denotes differentiation with respect to time. When 'Yj is nonzero the constraints are time varying and can be prescribed to satisfy arbitrary values (i.e., a non-holonomic rheonomic con~ t r a i n t ~ ~The , ~ ~equations ). of motion and con-

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OF COMPUTATIONAL CHEMISTRY

straint equation can be written as

miRj = Fj

+ bTA;

c buRu Nb

=

A;

u=1

bT

=

[ VTC1,. ..,VTCNc];A = [ A,,. . .,A N , ] T ; A

= [&l,...,'YNc] T

The analytic algebraic solution for A can be shown to bez8

which is valid for both fixed and rheonomic constraints. The evaluation for A is two time derivatives removed from the original definition for the position-dependent constraints. As a result, this formulation permits small numerical errors to accumulate during the integration of the equations of motion, because redundant degrees of freedom are retained in the physical model for the constrained MD system. Special algorithms are required to handle the induced constraint drift that arises in the integrated system response. The proposed RVMD method completely eliminates the potential for induced drift in the integrated system response.

RIGID BODY CONSTRAINTS For rigid bodies the constraint equations are generalized as follows:

where R j denotes the (3 X 1) jth body reference point position vector locating the jth body relative to inertial space and Oj denotes the vector of jth body kinematic variables used for describing the orientation of the jth body relative to the inertial frame. The constraint rates follow as

'

r= 1

acj .

-Rr (dR,

dCj .

+ -8,

4-

leading to constrained equations of motion of the form

M 191 . " = FI. + bTA; I

c bu(Ru&) Nb

T

. = cxj

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TURNER ET AL.

where

M j denotes the (6 X 6 ) jth rigid body mass matrix; Fi denotes the (6 X 1) jth body force, torque, and rotating frame kinematic effects vector; and di = d 2 ( R X jR , y j ,R Z j ,O x j , Oyj, 0 , , ) / d t 2 denotes the acceleration vector for the jth body translation and orientation. Other sets of orientation parameters can be used for characterizing the rotational motion of rigid bodies (e.g., Euler parameters, etc.). The (N,x 6 ) constraint matrix, b,, is generalized for both translational and rotational components. Typically, the constraints are defined at the interconnection hinges between contiguous bodies. For example, if only one rotational degree of freedom is allowed at a joint, then two rotational and three translational constraints must be defined and the foregoing constraint equation is a (5 x 1) vector. The solution for A can be expressed in a functional form that is identical to the form obtained for the particle formulation?’

Recursive Generalized Coordinate Methodology The modeling problem is naturally divided into two parts: (1) the formulation and solution process for the equations of motion, and (2) the development of mathematical models for the constraint functions to be supported by the RVMD algorithm. The generation of equations of motion for interconnected systems defined by generalized coordinates requires special consideration. Large matrices can occur either because of the need to solve the constraint equations, given above with a large Lagrange multiplier constraint or the need to solve the accelerations at the system level. The large matrices arise because conventional approaches attempt to obtain the solutions in a single operation, such as matrix inversion, for the desired unknowns. Recursive techniques reformulate the solution process to eliminate these large matrices by using body-level operations. The body-level operations lead to implicitly defined sets of equations that are noniteratively solved. Small matrices arise in the solution process because only local body-level models are considered at any one time. The recur1276

sive process leads to a multistep algorithm. An added benefit of recursive formulations is that they allow the user greater freedom in setting up new constraint models. The recursive algorithm greatly reduces the computational complexity involved in solving large-order linear equations of the form

AX = b Recursive algorithms work by using noninterative recursive equations, which involve many small matrices, to generate the solution x

=

A-’b

without forming A or A-’ explicitly. An example of a recursive a l g ~ r i t h m ~ha~s, been ~ ~ , run ~ ~ comparing two methods for solving the constraints between multiple rigid bodies. The O ( N 3 )generalized coordinate algorithm inverted a large matrix in a single step. The O ( N ) recursive algorithm used a series of steps that only require the inversion of (5 X 5) matrices. Each body has six degrees of freedom. There are five constraints between each interconnected body. For 14 bodies, the recursive algorithm is 1.9-fold faster than the generalized coordinate method. For 350 bodies, the recursive algorithm is 2523-fold faster. In this case, the generalized coordinate method had to invert a 1745 X 1745 matrix.

Current Modeling Approach The equations of motion are modeled using N generalized coordinates q = (ql, . . . , qN}. The equations of motion for a multibody system can be derived by Newton-Euler methods or by analytical mechanics methods (e.g., Langrange’s equations, Hamilton’s canonical equation, the BoltzmanHamel equation, etc.). The relative advantages and disadvantages of these various approaches depend on (1) the choice of dependent (kinematic) variables, (2) the geometrical organization and accounting procedure for a unique kinematic description, and (3) the systematic implementation. Several studies suggest that Kane’s method of generalized which is a generalization of Lagrange’s form of D’Alembert’s principle,34is most useful. Kane’s method is used to generate the rigid body equations of moti0n.3~Kane’s method combines the computational advantages of both Newton’ laws and Lagrangian formulation (i.e., the nonworking constraint forces and torques do not VOL. 16, NO. 10

REDUCED VARIABLE MD

appear and tedious differentiation of scaler energy functions are avoided). The resulting equation set is of minimum dimension?6 The multibody mathematical model consists of bodies and hinges. The bodies represent the models for the atoms that have been grouped together, where no internal motion is allowed. The hinges represent either the bonds, angles, or dihedrals that connect different atoms on bodies. The degrees of freedom for the multibody system are defined at the hinges (i.e., the allowed translations and rotations). For example, if the constraint is defined by a fixed length bond, then the second body is only free to rotate about the hinge axis connecting the atoms on different bodies. In this case, the hinge has one free rotational degree of freedom and two rotational and three translational constraints. A fictitious hinge is assigned to the reference body of the simulation, whose motion is measured relative to inertial space. As a result, for a systems consisting of a tree topology, the number of hinges equals the number of bodies in the system. A recursive problem formulation is presented that avoids the inversion of a large mass matrix.

the integral denotes three-dimensional integration over the volume occupied by the body, and a dot over a vector denotes time differentiation in inertial space. Rigid body motion is completely characterized by these equations. The equations remain valid if vector inner products are formed with both sides. A similar strategy is used in the classical Lagrangian approach, where the foregoing equations are expressed as

leading to

Kane extends this method by introducing generalized speeds (wj),defined as linear combinations of generalized velocities. The most commonly used generalized speeds, and the only ones used here, are the scalar components of angular velocity vectors. If w1 ...wN are chosen as generalized speeds, the critical translational and angular velocities are given by

KANE’S METHOD

The equations of motion are derived via Kane’s method extended to accommodate ”generalized speeds’’35(sometimes called derivatives of quasicoordinates) and further generalized to accommcdate nonholonomic constraints. Kane’s method begins with Newtonian equations in vector form (i.e., for a differential element of mass located inertially by the position vector R). df

=

Rdm

and for a material body with mass center located inertially by the position vector R ,

F

=

rnR,; M

=

where R locates a generic point in the kth body inertially and R,, locates its mass center. The basic equations of motion for the n-body system can be expressed as hl

fiL

where or F

=

/df; M

=

/ p x d f ; rn

=

/drn

R = R , + p, drn is the mass of the differential element, H , is the inertial angular momentum, These equations can be further written as

H,

=

/ p x p drn

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fj

+r

=

0; j

=

1, ..., N

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TURNER ET AL. where the generalized active forces can be shown to be

of the form

N

c

(FK*y,ck-Mck'Wkk)

k= 1

and the generalized inertia forces follows as N k= 1

The remaining details for formulating the equations of motion require lengthy presentations, which can be found

where the first equation has one Lagrange multiplier for the constraint between the jth and j lth bodies, and the second equation has two Lagrange multipliers for the constraints between the jth and j lth bodies and the j - lth and jth bodies. The constraint equation for the first body can be shown to be

+

+

+j&'

Recursive Formulation The recursive approach begins by using Kane's method to derive the individual body equations of motion as

where the constraint matrix C j selects a subset of the system-level Lagrange multipliers that act on the jth body. The recursive algorithm requires three major steps: (1) Eliminate the Lagrange multipliers acting on each body, leading to a set of implicit equations in which the accelerations of one body depend on the accelerations of another body; (2) recursively process the equations until a reference body is found; and (3) use the reference body solution to define the implicit parts of the previously defined implicit sets of equations. The success of the recursive strategy depends on the reference bodies motion being measured relative to an inertial frame. To motivate the algorithmic strategy, we consider systems consisting of bodies connected in a tree structure. We further assume that the number scheme for the bodies increases as bodies move away from the "trunk" of the structure. The structure can have "branches." The derivations presented here are limited to treelike topologies. Future articles will deal with the straightforward extensions required for handling ringlike topologies. The recursive algorithm begins with bodies at the end of branches of the tree structure. The body at the end of the branch is assumed to be numbered j 1 and the body that it is attached to is assumed to be numbered j . This leads to equations

+

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+ $jij

+pj

=

+ Cj, j + l q j

= Cj+1, j q j + l

If; $]Vj= 1,;

$T+. = 1 1

0

where f j denotes the free degrees of freedom at the joint connecting the jth and j lth bodies, ci denotes the constrained degrees of freedom at the joint connecting the jth and j + lth bodies (Note: i does not have to be zero, so that user-specified constrained rates can be handled with this formudenote matrices that select lation), and +j and the free and constrained degrees of freedom and maps them into a (6 X 1) vector. The factorization of the equations permits the equations to be manipulated into a form in which the Lagrange multipliers can be eliminated. The orthogonality of these matrices is very important for developing the recursive algorithms. The recursive algorithm begins by manipulating the constraint equation to define the following change of variables for cjj+ 1:

+

qj+l =

'~21, j ( + j f j

+ qjij

-

cj,j + l q i )

where the motion rate for the last body in the branch is now described in terms of the rates at the joint and the rate for the jth body. This equation is said to be implicit because of its dependence on the jth body rate. Before the change of variables substitution can be used in the equation of motion for the j + lth body, the transformation equation must be differentiated with respect to time, leading to

q j + l = c-' 1+1,1

.+.c+ hj+'(ijj) I I

where hi+ contains all derivative terms. Introducing the preceding equation into the equation of motion for the body at the end of the branch, and VOL. 16, NO. 10

REDUCED VARIABLE MD

multiplying the resulting equation by

The solution for hi represents an exact algebraic equation, where the dimension of the required matrix inverse is typically 3 to 6 for MD applications. At this stage everything that can be known about the j + lth body acceleration is complete until we have some means of evaluating the implicit dependence on the jth body acceleration. Next, the equation of motion for the jth body is processed. The first step is to introduce eq. (2) into the jth body equation of motion, leading to

4Tc-T I 1+1,1 leads to

~ T C ~jMj+ A, 1

4jjj + hj+1)

~ ~ 2j ( 1 ,

or

x ( - h j + I ( q j ) + 4Tc~T1, jFj+1) where term containing A j vanishes because of the following kinematic identity:

The body-level algorithm has replaced the degrees of freedom for the j + lth body entirely in terms of the free degrees of freedom available at the jth hinge. The great advantage of this approach is that the required inverse matrix only has the dimension of the free degrees of freedom at the joint (i.e., typically 1 to 3 degrees of freedom for MD applications). The reduced matrix size of the recursive algorithm provides the basis for its significant computational advantages over system-level approaches. Because the equation of motion for the jth body explicitly depends on A,, we must obtain a solution for this variable. The solution for A, is obtained by solving the constraint equation for the constraint rates, as follows:

6,

cl,l+141)

= iqc1+1,141+1 +

Differentiating this equation with respect to time leads to -

?]

=

4 T ( ~ , + l , 1 9+~Pl(9,)) +l

(1)

where p , is implicitly dependent on the jth body acceleration. To eliminate the acceleration for the j + lth body, one solves the original constrained equation of motion for the j + lth body as follows:

and introducing this equation into eq. solution for hi can be shown to be

(11, the

x ( z j - $ , Y ~ + ~ , ~ M ; : ~ F ~ + ~ )(2)

JOURNAL OF COMPUTATIONAL CHEMISTRY

where the mass matrix and the force vector have been modified. This equation has the same form as the first equation solved. Following the same procedure outlined earlier, we obtain equations of the form

and

The process is repeated until a reference body is reached, whose motion is referenced relative to the inertial frame. For topologies with multiple branches emanating from a single body, the same basic strategy works. This happens because the mass matrices and force vectors are modified to account for all effects of bodies beyond the current body branch position. When a reference body is reached in the algorithm, the transformed equation of motion can be shown to be -

-

M x q x = Fx where the solution follows as

This explicit solution is possible because the reference body motion is referenced to an inertial frame (i.e., a frame of reference with zero acceleration). By traversing the system topology in the reverse order used to generate the equations defined earlier, the implicit dependence in each of these equa-

1279

TURNER ET AL.

tions can now be eliminated, thereby completing the solution for the system accelerations.

Nonlinear Effects The preceding method leads to nonlinear equations of motion. The nonlinear effects play an essential part in allowing the rigid bodies to undergo arbitrarily large displacements and rapid motions and may be omitted or approximated only for simulations involving very limited motions. If large motions do occur in a system wherein these nonlinear terms are neglected, the predicted motions may not be physically meaningful (see the Appendix). The nonlinear effects arise from two sources. First, dynamic reference frames that move with each body are used to simplify the equations of motion. A significant advantage of this approach is that the rigid body mass properties are constant. Constraints are also easier to describe in terms of the dynamic reference frames. The use of dynamic reference frames leads to nonlinear w X ( * ) terms in the translational velocity and acceleration and the rotational angular momentum vector equations. These terms must be retained to account for rapid motions of the rigid bodies. Second, when large changes in the orientation of rigid bodies with respect to inertial space are possible, the direction cosine matrices describing the large rotational displacements must retain all the nonlinear products of sine and cosine terms. These effects are particularly important in tree structures, where many products of direction cosine matrices may be required to orient bodies with respect to a reference body's orientation. The nonlinear effects also influence the types of motion that can be predicted. For example, the coupling effects allow the energy to flow between axes, and linear models cannot predict these coupling effects. These differences can be seen even in the case of a single body with no forces or torques acting on it (see the Appendix). The nonlinear equations provide fundamentally different solutions, where the components of angular momentum are time varying, though they satisfy by the kinetic energy and angular momentum conservation laws. These solutions allow general tumbling motions of the body. The tumbling motions are directly the result of the w X ( * ) cross-axis coupling effects.

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Methods The initial conformations of alanine dipeptide were obtained by model building, using AMBER +37 using the Amber 3 force field.38,39 The phi and psi angles, shown in Figure 1, were set at the desired initial value and a constrained energy minimization was performed. This was followed by gradually heating (3-5 ps with a 1-fs step size) to the specified temperature (300 K or 600 K) using a dynamics simulation with SHAKE. The resulting coordinates, particle velocities, rigid group definitions, and a list of the allowed degrees of freedom were then passed to the RVMD program. This program, as shown in Figure 2, is completely independent from the molecular mechanics software. This enables the reduced variable method to be easily attached to any molecular mechanics or quantum mechanics code. The programs communicate via common blocks and an interface routine. The RVMD program takes care of all of the coordinate transformations, and the molecular mechanics program only needs to have routines that compute the forces in terms of Cartesian coordinates and does not have to be modified if the body definitions are changed. The RVMD program calculates equivalent rigid body masses, first mass moment vectors, and inertia tensors. From the particle position, velocity, and mass data, the linear and angular momentum are computed. After the rigid body mass data are computed, a model can be developed for the inertial linear and angular momentum, as follows: Nb

C mrvr r= 1 Nb

C mr Rr x V,

r=l

FIGURE 1. Alanine dipeptide.

VOL. 16, NO. 10

REDUCED VARIABLE MD PRE-PROCESSING Coordinates Velocities Constraints Group Definitions

MOLECULAR MECHANICS (AMBER)

INTERFACE x(t) Ei(0 PROGRAM A

5 POST-PROCESSING Analvsis

RIGID BODY

’ MODEL BUILDER

,

X(t+At)

;EB(t)3E#

&(to) W t o ) mi

X(t)

&(t+At)

%t) r

I

COORDINATE < CONVERSION

where J , denotes the rth bodies’ inertia tensor, S, denotes the first mass moment, (-) denotes that the first mass moment is expressed in terms of a (3 X 3) matrix equivalent of the vector cross product, M, denotes the mass of the rth body, and the R vector locates the rigid body reference point relative to the inertial frame. The only unknowns in this equation are wk and ok. The solution for these variables is obtained by inverting the matrices. The initial rates where further processed to account for the constraints in the system. The constraints are defined by specifying the degrees of freedom at each joint between the bodies. For example, if a body was attached with a single rotational degree of freedom, then three translational constraints and two rotational constraints where defined. This process consists of projecting the adjacent body rotational and translational rates along the specified joint axes. Constant energy and constant temperature simulations were then run. In addition, annealing runs in which the temperature was heated and then cooled were made in an attempt to locate the global minimum. For the later runs, it was necessary to implement a rigid body temperature scaling algorithm. The all-atom AMBER simulation used a one-point Verlet integrator, and the RVMD simulation used a four-point Runge-Kutta integrator. The accuracy of this integrator is extremely high. Constant energy simulations of alanine

MULTIBODY

’ DYNAMICS

X(t+At)

INTEGRATION

dipeptide with 100,000 points had less than a 10% fluctuation in the total energy with no restarts. Simulations are currently being undertaken with a two-point integrator.

TEMPERATURE SCALING The temperature was scaled by developing a model for the system kinetic energy, as follows:

where J, denotes the rth bodies’ inertia tensor, S, denotes the first mass moment, ( ) denotes that the first mass moment is expressed in terms of a (3 x 3) matrix equivalent of the vector cross product, and M , denotes the mass of the r th body. The angular velocities and linear velocities are linear functions of the degrees of freedom integrated by the equations of motion. Accordingly, the kinetic energy and the temperature can be adjusted by computing a scale factor

-

where K denotes the currently computed kinetic energy and K h s i r e d denotes the desired kinetic energy. This scale factor is used to modify the

TURNER ET AL.

give the results for the rigid body simulations with a total of 5 degrees of freedom (three methyl torsional angles and phi and psi dihedral angles with a total of six rigid bodies). The runs are plotted on phi/psi contours generated with rigid body rotations about the phi/psi torsional angles. These simulations are run at either constant energy or constant temperature. The ability of the multibody simulation to search conformational space and to locate the global minimum using annealing algorithms was investigated. The figure captions give the relevant details of each simulation, including starting point, time step, length of simulation, and type of simulation (all atom or rigid body). Runs made at 300 K with the AMBER SHAKE algorithm did not move out of the starting point

velocity terms in the equations of motion. Because the kinetic energy is linearly related to the temperature, scaling the kinetic energy is equivalent to temperature scaling.

Results Figures 3 and 4 show results from all-atom AMBER simulations of alanine dipeptide with SHAKE bond constraints (22 atoms with 39 degrees of freedom). These runs were carried out at constant energy and only varied according to their starting conformation, temperature, or length of simulation. All of these plots are made on phi/psi contours generated with all other degrees of freedom minimized at each point. Figures 5 through 9

-180

.I20

-60

0

MI

im

120

FIGURE 3. Constant energy AMBER simulation with SHAKE bond constraints. The system was heated to 600 K over 5 ps and then a 200-ps simulation was run with a 1-fs step size. Starting value of phi = - 60, psi = 60.

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-180

-120

do

0

60

im

I 0

FIGURE 4. Constant energy AMBER simulation with SHAKE bond constraints. The system was heated to 600 K over 5 ps and then a 1600-ps simulation was run with a 1-fs step size. The number of points was 1,600,000. Starting value of phi = -60, psi = 60.

minimum at phi = -60 and psi = 60. At 600 K, Figure 3 shows that the AMBER SHAKE simulation can explore a few local minima, but the structure still stays relatively near the starting conformation after 200 ps. Figure 4 explores the effect of a long simulation, 1.6 ns. This simulation explores all four low-energy minima and samples other reasonably low-energy portions of the surface. The molecule does remain trapped for long periods near the low-energy regions. This is a common phenomenon of molecular dynamics simulations. Figure 5 shows that the "normal temperature" (rigid body velocities are derived from the 300-K all-atom velocities) rigid body simulation appears to sample phase space as well as the 600-K all-atom simulation (Fig. 3). The contours for this rigid map are steeper than those of the flexible map in Figure

JOURNAL OF COMPUTATIONAL CHEMISTRY

3 and that it is more difficult for the simulation to move around the rigid map. By doubling the kinetic energy or the temperature of this simulation, Figures 6 and 7 show that the accessible regions of phase space are well sampled independent of the starting point or the step size (2 or 4 fs). The distribution of points is uniform (see the following discussion), and the simulation does not appear to get trapped in a local minimum for long periods of time. The molecule in Figure 7 had to surmount barriers of 5 to 10 kcal to go from the minimum at phi = 60 and psi = -120 to the portion of the surface, including the global minimum near phi = -60 and psi = 60. Every 100th point was saved during these runs. An analysis of the phi/psi values showed frequent changes of 60" or more between points saved. This

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-IRa

-120

do

0

60

im

180

FIGURE 5. Constant energy multibody simulation with 300 K AMBER velocities used as starting point to calculate rigid body velocities. The system was run for 400 ps with a 2-fs step size. Starting value of phi = - 60, psi = 60.

showed that energy is easily able to flow between the bodies. Figures 8 and 9 show the results of heating up and then cooling alanine dipeptide. All protocols were carried out over approximately a 300-ps time period. The simulations were stopped once they had been cooled sufficiently low that further movement on the energy surface was unlikely. Only the simulation carried out with the starting point in Figure 9 seemed to depend on the cooling protocol. If cooling began while the molecule was over the center strip, the simulation converged to the minimum near phi = 60, psi = -60. If the cooling began over the other regions, the simulation tended to converge to the global minimum near phi = -60, psi = 60. 1284

Two important issues that arise when comparing this method to more traditional all-atom methods are the speedup in the total simulation time and the amount of time required for convergence of the method. The first issue of speedup is only partially addressed in this study. The speedup in the use of the multibody technology results from (1) the use of a much simpler energy surface that can be searched more quickly because there are many fewer local minimum” (many fewer degrees of freedom), and (2) the increase in step size possible because of the reduction in high-frequency motion (it should also be noted that each step, even if taken with the same step size as the all-atom simulation, is taken using dihedral internal coordinates. This allows a much larger motion of the VOL. 16, NO. 10

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.in0

-120

60

0

60

im

180

FIGURE 6. Constant energy multibody simulation with each rigid body velocity component computed in Example 5 multiplied by the square root of 2, thus doubling the amount of KE. The system was run for 800 ps with a 4-fs step size. Starting value of phi = -60, psi = 60.

atoms than a step using Cartesian coordinates with all atoms moving independently.), and (3) the decrease in time for computing the energy function that arises because a multipole expansion can be used to compute the interactions between the rigid bodies. The reduction in the number of degrees of freedom in this problem is approximately 8 (39 degrees of freedom in the SHAKE all-atom simulation and 5 degrees of freedom in the reduced variable simulation). The speedup from step size because of reduction in high-frequency motion is a factor of 2 to 4 in the studies carried out (1 fs versus 4 fs). While step sizes as large as 2 fs may be carried out using the SHAKE algorithm, simulations at high temperatures and those for large molecules with starting points far from stable

JOURNAL OF COMPUTATIONAL CHEMISTRY

low-energy configurations may require a step size nearer 1 fs for stability of the simulation. This gives a speedup of 16 to 32. Larger molecules will have possibilities for greater reductions in the degrees of freedom because of possible groupings of larger numbers of atoms (i.e., all atoms in a helix). There is also the possibility of reduction in the time for the energy function evaluation through the use of multipole expansions (currently being implemented). Previous ~ o r k * ~has , * ~shown a time reduction of 44 for the time required to evaluate the electrostatic interaction energy between two helices using fourth-order multipole expansions. In an actual problem requiring a mixture of methods for calculating short- and long-range interaction energies, this factor will decrease but should be at least a factor of 2. Combining all of the aforemen-

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-1110

-120

-60

0

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im

FIGURE 7 . Constant energy multibody simulation with the same starting velocity algorithm as Example 6. The system was run for 800 ps with a 4-fs step size. Starting value of phi = 60, psi = 120.

tioned factors gives a conservative estimate of 10 to 100 times possible speedup in simulation time using this methodology. Of course, in a speedup comparison of simulation methods, one must also take into consideration the overhead for the multibody portion of the code. There is no measurable cost for communication between the molecular mechanics and multibody code since a single program is used and information is passed through common blocks. The proportion of the time required for the multibody software overhead rapidly decreases as the time for the energy function evaluation increases. For a small problem, such as alanine dipeptide, the total multibody overhead time compared to the total time for the simulation is 75 to 90% of the total

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time. This proportion decreases to 10 to 30% for simulations of moderate-size molecules (approximately 500 atoms and 100 rigid bodies). When one speaks of convergence of a simulation, there are two possible meanings of convergence: (1) Does the simulation give results in accordance with Boltzmann statistics and does it achieve global ergodic sampling so that free energy differences between the principal conformations can be accurately computed? or (2) Does the simulation adequately search phase space according to a given criterion? Within the number of iterations applied in the present study to alanine dipeptide, the accessible conformational space (i.e., within 10 kcal of the ground state) is well filled independent of the VOL. 16, NO. 10

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-I80

-120

40

0

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im

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FIGURE 8. Annealing multibody run with the same starting velocity algorithm as Example 6. The system was run for 67,400 points with a 4-fs step size. Starting value of phi = - 120, psi = 120.

starting point. Thus all of the low-energy regions are searched. This satisfies convergence according to the later criterion mentioned in the preceding paragraph. By direct sampling of populations in given volumes of phase space, preliminary studies show that the population density is consistent with the statistical distribution on energy according to the Boltzmann law that sample probability is proportional to exp( - E). However, the runs are too short (0.8 ns) to state that statistical "convergence" has occurred or that free energy differences can be computed accurately. Much longer runs will be carried out, and the issue of statistical convergence will be addressed in a future article. As one of the referees of this paper points out, this methodology appears to change the shape of the conformational energy surface. Clearly any po-

JOURNAL OF COMPUTATIONAL CHEMISTRY

tential surface will change with the potential functions (force field) used and, as a special instance of that, any united-atom representation (rigid bodies) or field of potential of mean force or multipole approximation. In addition, any given "slice" through an energy surface (in this case a two-dimensional representation of the potential energy of a system) with many degrees of freedom will appear to change as the method of handling the remaining variables changes. One should, therefore, reparameterize the force field to get a proper description of the potential surface. Strictly, the concept of self-consistencyand thereby of the consistent force field fostered by the Lifson group (Weizmann Institute of Science, Israel) requires that parameters should be redeveloped de nuvo for each different case if the surface is then put to some subsequent purpose. We have

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-180

-120

do

0

60

120

180

FIGURE 9. Annealing multibody run with the same starting velocity algorithm as Example 6. The system was run for 67,400 points with a 4-fs step size. Starting value of phi = 60, psi = - 120.

developed and refined potential functions specifically for the kind of consistent approach which is used here,40 and studies will be undertaken using these potential functions. However, the purpose of the present study is to introduce and demonstrate a new simulation technique, not to compare and contrast force fields. The methodology will make this task easier by creating a general, all-purpose simulation framework for carrying out this testing.

Conclusion The RVMD software appears to be stable for step sizes larger than those permitted in a conventional all-atom simulation and can be used to reduce easily the number of degrees of freedom in a problem. Since energy easily flows from one 1288

body to another without getting trapped in smallamplitude high-frequency motion, the molecule can more easily sample many low-energy regions of phase space. Larger problems are being run to test the scaling properties of the method, and integrators that require fewer energy evaluations are being investigated. The method appears to be a promising computational tool for drug design.

Acknowledgments Two of the authors Q. T. and P. W.) wish to thank Proteus for its generous hospitality and support for these studies, which were carried out at its headquarters in Macclesfield, England. All of the authors also wish to thank Jin Li, Chris Murray, and Eric Platt for helpful scientific discussions; VOL. 16, NO. 10

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Brian Petch and Richard Walker for computational assistance; and Peter Brady for general guidance. They are also grateful to John Pool for inviting them to England and for making this work possible. They thank one of the referees, who gave many useful suggestions for improving this article. In particular, the authors used several of the referee's suggestions provided in discussion of the speedup issues.

nonlinear models allow for coupling between axes that cannot exist in linear models. Without the nonlinear terms, one cannot be sure that the resulting motions are accurately predicted unless the rates are extremely small.

References 1. C. L. Brooks 111, M. Karplus, and B. M. Pettitt, Proteins: A

Appendix: Nonlinear Motion Models Nonlinear motion models allow simulations to be valid for arbitrarily large displacements and rapid motions. Unlike linear models, nonlinear models can lead to fundamentally different types of solutions. For example, when one considers the equations governing the angular momentum of a rotating body, the solution for even unforced models is radically different. In the following linear model, when no torques act, the angular momentum is constant, which implies a uniform rotation rate as seen in a body-fixed frame. On the other hand, the nonlinear model has time-varying momentum components. Indeed, the time-varying terms lead to nonuniform rotation rates. As a result, the orientation of the body in inertial space can be characterized as a general tumbling motion.

Linear Model

Nonlinear Model dP'/dt

+ o X P'= 8

The conservation laws are given by

Kinetic Energy

Angular Momentum

In both the linear and nonlinear cases, the kinetic energy and angular momentum are conserved. The resulting motions, however, are very different. The

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Theoretical Perspective of Dynumics, Structure, and Thermodynamics, John Wiley & Sons, New York, 1988. 2. W. F. V. Gunsteren, P. K. Weiner, and A. J. Wilkinson, Eds., Computer Simulations of Biomolecular Systems, Vol. 2, ESCOM Science Publishers, B.V., Leiden, The Netherlands, 1994. 3. D. Byme, J. Li, E. Platt, B. Robson, and P. Weiner, CAMD, 8, 67 (1994). 4. B. Robson, Cyber 205 Newsletter, p. 5, Winter 1985. 5. B. Robson, In Practical Protein Chemistry-A Handbook, A. Darbre, Ed., John Wiley & Sons, New York, 1986, p. 567. 6. B. Robson, In QSAR in Drug Design and Toxicology, D. Hadzi and B. Jerman-Blazic, Eds., Elsevier Science Publishers, B.V., Amsterdam, 1987, p. 239. 7. J. Li, E. Platt, B. Waskowycz, R. M. J. Cotterill, and B. Robson, Biophys. Chem., 43, 221 (1992). 8. M. R. Pear and J. J. Weiner, J. Chem. Phys., 71, 212 (1979). 9. J. I?. Rychaert, Mol. Phys., 55, 549 (1985). 10. A. K. Mazur and R. A. Abagyan, J. Biomol. Struct. Dyn., 6, 815 (1989). 11. R. A. Abagyan and A. K. Mazur, J. Biolmol. Struct. Dyn., 6, 833 (1989). 12. R. A. Abagyan, M. Totrov, and D. Kuznetsov, I. Comp. Chem., 15, 488 (1994). 13. W. R. Rudnicki, B. Lesyng, and S. C. Harvey, Biopolymers, 34, 383 (1994). 14. J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, I. Comp. Phys., 23, 327 (1977). 15. W. F. van Gunsteren and H. J. C. Berendsen, Mol. Phys., 34, 1311 (1977). 16. J. C. Andersen, J. Comp. Phys., 52, 24 (1983). 17. W. F. van Gunsteren and M. Karplus, Macromolecules, 15, 1528 (1982). 18. D. J. Tobias and C. L. Brooks 111, J. Chem. Phys., 89, 5115 (1988). 19. G. Zhang and T. Schlick, J. Comp. Chem., 14, 1212 (1993). 20. W. F. van Gunsteren, Mol. Phys., 40, 1015 (1980). 21. J. D. Turner, H. M. Chun, P. Weiner, S. Gallion, and C. Singh, "Order ( n ) Multibody Dynamics," Conference on Research Perspectives in Structural Biology and Chemistry, Hilton Head, SC, January 27-30, 1991. 22. J. D. Turner, H. M. Chun, P. Weiner, S. Gallion, and C. Singh, "Order ( n ) Multibody Dynamics," 7th International Congress of Quantum Chemistry, Menton, France, July 2-5, 1991. 23. J. D. Turner, H. Chun, V. Lupi, P. Weiner, S. Gallion, and C. Singh, Chem. Design Automation N m s , 7(12), 34 (1992). 24. J. D. Turner, H. Chun, V. Lupi, P. Weiner, S. Gallion, and C. Singh, Chem. Design Automation News, 8(1), 16 (1993).

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TURNER ET AL. 25. R. P. Singh, R. J. VanderVoort, and P. W. Likens, TREETOPS User’s Manual, Dynacs Eng. Co. Inc., 1990. 26. R. P. Singh and V. Ravi, Code Generator User‘s Document, Internal Report 920625-1, Dynacs Eng. Co., 1992. 27. R. Venugopal and M. Kumar, Proceedings of the Fifth N A S A / N S F / D O D Workshop on Computational Control, February 15, 1993. 28. E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, 4th ed., Cambridge University Press, Cambridge, UK, 1961. 29. L. Meirovitch, Methods of Analytical Dynamics, McGraw-Hill, New York, 1970. 30. H. P. Frisch, H. M. Chun, and J. D. Turner, NDISCOS-User‘s & Programmers’ Manual, Photon Research Technical Report, December 1992. 31. H. M. Chun, J. D. Turner, and H. P. Frisch, AAS/AIAA Astrodynamics Specialists Conference, Durango, Colorado, August 1991. 32. H. M. Chun, J. D. Turner, and H. P. Frisch, “Recursive Multibody Formulations for Robotic Applications with Har-

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monic Drives,” International Conference on Dynamics of Flexible Structures in Space, Cranfield, England, May 15-18, 1990. 33. R. R. Kane and C. R. Wang, J. Siam, 13,2 (1965). 34. T. R. Kane and D. A. Levinson, J. Guidance and Control, 3,2 (1980). 35. R. P. Sing and P. W. Likins, Automatic Control Conference, San Francisco, CA, June 1983. 36. T. R. Kane and D. A. Levinson, Dynamics: Theory and Applications, McGraw-Hill, New York, 1985. 37. AMBER + is a fully vectorized molecular mechanics code by U. C. Singh, K. Ramnarayan, P. K. Weiner, and P. Kollman. This code is distributed by Amber Systems, Inc. 38. S. J. Weiner, P. A. Kollman, D. A. Case, U. C. Singh, G. Ghio, G. Alagona, S. Profeta, and I?. K. Weiner, 1. Am. Chem. SOC.,106, 765 (1985). 39. S. J. Weiner, P. A. Kollman, D. T. Nguyan, and D. A. Case, 1. Comp. Chem., 7, 230 (1986). 40. B. Robson and E. Platt, J. Mol. Biol., 188, 259 (1986).

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