Conformational Analysis of Molecular Chains Using Nano-Kinematics

Home , Cyclohexane conformation

Conformational analysis of molecular chains using Nano-Kinematics

y z

Dinesh Mano cha Yunshan Zhu William Wright

Computer Science Department

UniversityofNorth Carolina

Chap el Hill, NC 27599-317 5

fmano cha,zhu,[email protected]

Fax: 919962-179 9

Abstract: We present algorithms for 3-D manipulation and conformational analysis of molecular

chains, when b ond length, b ond angles and related dihedral angles remain xed. These algorithms

are useful for lo cal deformations of linear molecules, exact ring closure in cyclic molecules and

molecular emb edding for short chains. Other p ossible applications include structure prediction,

protein folding, conformation energy analysis and 3D molecular matching and do cking. The algo-

rithms are applicable to all serial molecular chains and make no asssumptions ab out their geometry.

We make use of results on direct and inverse kinematics from rob otics and mechanics literature

and show the corresp ondence b etween kinematics and conformational analysis of molecules. In

particular, we p ose these problems algebraically and compute all the solutions making use of the

structure of these equations and matrix computations. The algorithms have b een implemented

and p erform well in practice. In particular, they take tens of milliseconds on currentworkstations

for lo cal deformations and chain closures on molecular chains consisting of six or fewer rotatable

dihedral angles.

Categories: 3D Protein Mo deling, 3D Molecular Matching and Do cking, Multiple alignmentof

genetic sequences

Supp orted in part by Junior FacultyAward, University ResearchAward, NSF Grant CCR-9319957, ARPA

Contract DABT63-93-C-0 048 and NSF/ARPA Science and Technology Center for Computer Graphics and Scienti c

Visualizatio n, NSF Prime Contract Numb er 8920219

Supp orted in part by NIH National Center for Researc h Resources grant RR-02170

Supp orted in part by NIH National Center for Research Resources grant RR-02170

1 Intro duction

Three dimensional manipulation of molecular chains is a fundamental problem in conformational

analysis and do cking applications. Conformational analysis deals with computation of minimal en-

ergy con gurations of deformable molecules and do cking involves matching one molecular structure

to a receptor site of a second molecule and computing the most energetically favorable 3-D con-

formation. These problems have b een extensively studied in bio chemistry and molecular mo deling

literature. Molecular chains are mo deled using b ond lengths, b ond angles and dihedral angles.

In the literature on conformational energy calculations, b ond lengths and b ond angles have

b een treated in two di erentways, as variables or as xed parameters[GoS70]. However, mo deling

the b ond-length and b ond-angle as variables signi cantly add to the complexity and even for small

chains consisting of six or seven b onds, minimal energy computation b ecomes a non-trivial problem.

In their classic pap er, Go and Scheraga highlight that the minimal energy conformations computed

by xing these parameters is a go o d guess to the actual minima and can b e used along with a

p erturbation treatment for the energy minimization [GoS70]. Therefore, in this pap er, we assume

that the b ond lengths and b ond angles are xed and compute the conformation of the molecular

chains bychanging the dihedral angles only.

Given the assumptions of xed b ond-lengths and b ond-angles, two fundamental problems arise

in conformational energy calculations: ring closure and local conformational deformations [GoS70].

The problem of ring closure arises when we deal with cyclic molecules, i.e. the calculation of

dihedral angles which corresp onds to exact ring closure. In conformational energy minimization

pro cedures, one selects a starting conformation and alters it step by step such that the corresp ond-

ing conformational energy decreases monotonically.However, a small change in a dihedral angle

lo cated near the middle of the long chain causes a drastic change in the overall conformation of

the molecule. It is therefore, desired to have a co op erativevariation of angles which con nes the

conformational changes to a local section of the chain. As a result we are interested in calculation of

changes in dihedral angles, which cause lo cal deformations of conformations in long p olymer chains

[GoS70, BA82, CMR89, BK85]. In this pap er, We also address the molecular embedding problem,

that is, to nd atom p ositions that satis es a set of geometric constraints. It is an imp ortant

problem in interpreting data from NMR exp eriment [Cri89].

Manyinteractive systems for molecular mo deling suchasSybyl [Tri88] and Insight [Bio91]

provide capabilities for building and changing molecular mo dels by rotating the dihedral angles.

The full conformation space of acyclic molecules is the Cartesian pro duct of all the dihedral angle 2

with one-dimensional cycles, known as the n-dimensional toroidal manifold, where n is the number

of rotatable b onds. There is a considerable amount of literature on exhaustive metho ds. A recent

survey of di erent metho ds is given in [HK88]. The complexity of the search metho ds is an

exp onential function of the numb er of rotatable joints and therefore limited to chains with low n.

For cyclic molecules having rotatable b onds in the rings, six degrees of freedom are lost due to the

closure constraint. Go and Scheraga have prop osed solutions to computing the ring conformations

[GoS70, GoS73]. The algorithm in [GoS70] is restricted to molecular chains where the rotation can

take place ab out all backb one b onds and the metho ds in [GoS73] are for symmetric chains only.

They reduce the problem to computing ro ots of a univariate p olynomial. Algorithms based on

distancegeometry to study the ring systems are prop osed in [We83, PD92]. They use a distance

matrix to generate a set of cartesian co ordinates that satis es the original set of distances as

much as p ossible. A similar approach based on successive in nitesimal rotations is presented by

[SLP86]. Although these approaches make no assumption ab out the geometry of molecules, they are

relatively slow in practice, may fail to converge and cannot b e used for analytic computation of the

conformation space for even small chains. More recently, Cripp en has intro duced the technique of

linear embedding and applied it to explore the conformation space of cycloalkanes [Cri89, Cri92]. It

is a variant on the usual distance geometry metho ds and makes use of the metric matrix as opp osed

to distance matrix. However, the application to ring structures has b een limited to molecular chains,

where all b ond lengths and b ond angles are equal. Other approaches for molecular conformations

are based on sto chastic optimization and Monte Carlo analysis [GoS78, Sau89, BK85]. These can

b e slow in practice and are not guaranteed to nd all the solutions.

In this pap er, we reduce the problem of p ose-computation to a system of algebraic equations

and showhow they relate to the problems of direct and inverse kinematics in mechanics and

rob otics [Cra89, SV89]. In particular, we show the corresp ondence b etween molecular conforma-

tional analysis and inverse kinematics of serial manipulators. We also showhowinverse kinematics

can b e used to solve the molecular emb edding problem for short chains. The problem of inverse

kinematics has b een extensively studied in the rob otics literature, but closed form solutions are

known for some sp ecial geometries only.We present techniques for computing the solutions of the

algebraic equations by making use of their sparse structure and reduce the problem to computing

the eigenvalues and eigenvectors of a matrix. Based on numerically stable and ecient algorithms

for computing the eigendecomp osition of a matrix, we present fast and accurate algorithms for

ring closure and lo op analysis. These algorithms have b een implemented and take ab out of tens 3

of milliseconds on currentworkstations for manipulation of molecular chains consisting of up to

six rotatable b onds. We demonstrate their p erformance on chain closures of cyclohexane and lo op

deformation on a p olyp eptide chain. These algorithms make no assumptions ab out the geometry of

the molecules and are applicable to all molecular chains. These techniques are based on kinematics

and applied to molecular b onds of the order of a few Angstroms. Therefore, we refer to them as

nano-kinematics

The rest of the pap er is organized in the following manner. In Section 2, we show the equivalence

between the kinematics problem and molecular mo deling problems. In Section 3, we showhow

the technique can also b e applied to molecualr emb edding problems. We present the algebraic

algorithms in Section 4 and show the reduction to an eigenvalue problem. Dep ending on the

geometry of the molecular chain, we obtain di erent matrix orders. We discuss extensions of the

algorithm to chains consisting of more than six dihedral angles in Section 5. The algorithms are

illustrated in Section 6 with several examples of applications. The details of the implementation

and p erformance are presented in Section 7.

2 Molecular Mo deling and Inverse Kinematics

The inverse kinematics problem for general serial manipulators is fundamental for computer con-

trolled rob ots. Given the p ose of the end e ector the p osition and orientation, the problem

corresp onds to computing the joint displacements for that p ose [SV89]. A rob ot manipulator may

consist of prismatic, revolute or spherical joints. A molecular chain with xed b ond lengths and

b ond angles corresp onds to a serial manipulator with revolute joints. Therefore, we only consider

the inverse kinematics problem for serial chains with revolute joints. Initially,we consider chains

with six rotatable joints denoted as 6R and later on extend the results to arbitrary chains.

We use the Denavit{Hartenb erg DH formalism, [DH55], to mo del a serial chain with n revolute

joints. Each b ond is represented by the line along its joint axis and the common normal to the next

joint axis. In the case of parallel joints, any of the common normals can b e chosen. A co ordinate

system is attached to each joint for describing the relative arrangements among the various b onds.

The z-axis of frame i, called z , is coincident with the joint axis i. Initially a base frame is chosen

such that the origin is on the z axis and the x , y axis are chosen conveniently to form a right

1 1 1

hand frame. The origin of frame i, o corresp onds to the p oint where the common normal to z

i i

and z intersection z .Ifz and z intersect, the origin z is their p ointofintersection. x

i1 i i i1 i i 4 X

Z Y

N Y Z H X

CN X

O C Ca Z

Figure 1: Co ordinate systems on a p olyp eptide units based on DH formalism

is along the common normal b etween z and z through o , or in the direction normal to the

i i1 i

z z plane if z and z intersect. Given x and z , y is chosen to complete the right hand

i1 i i1 i i i i

co ordinate system at o . This formulation is shown for a p eptide unit in Fig. 1. This is rep eated

for i =2;:::;n. Given a co ordinate system, we create the following parameters for the molecular

chain:

a = distance along x from o to the intersection of the x and z axes.

i i i i i1

d = distance along z from o to the intersection of the x and z axes.

i i1 i1 i i1

= the angle b etween z and z measured along x .

i i1 i i

= the angle b etween x and x measured ab out z .

i i i i1

These parameters for the p olyp eptide unit are shown in Fig. 2. The dihedral angles corresp ond to

the 's. Given this representation of the co ordinate systems for each b ond, the 4 4 transformation

matrix relating i 1 co ordinate system to i co ordinate system is: 5 P α1

d1 d2 Ca Ca H

C N α2 O Ca P

Figure 2: DH parameters for a p olyp eptide unit

0 1

c s s a c

i i i i i i i

B C

s c c a s

i i i i i i i

B C

A = ; 1

B C

0 d

i i i

@ A

0 0 0 1

where s = sin ; c = cos ; = sin ; = cos :

i i i i i i i i

For a given serial chain or a rob ot manipulator we are also given the con guration of the end

of the chain or the end-e ector. This con guration is describ ed with resp ect to the base link or

link 1. We represent this con guration as:

0 1

l m n q

x x x x

B C

l m n q

y y y y

B C

A = :

hand

B C

l m n q

z z z z

@ A

0 0 0 1

The problem of inverse kinematics corresp onds to computing the joint angles, ; ;:::; such

1 2 n

that

A A :::A = A : 2

1 2 n hand

This relationship can b e reduced to a system of algebraic equations by substituting x = tan .

i i

Therefore, sin = and cos = .Eventually we obtain a system of 6 algebraic equations

i 2 2 i

1+x 1+x

i i

in n unknowns. 6

The problems of ring closure and lo cal deformations turn out to b e a sp ecial case of inverse

kinematics formulation, as shown b elow.

2.1 Ring Closure

The ring closure problem turns out to b e a sp ecial case of the inverse kinematics problem highlighted

ab ove. In particular for ring closure, the right hand side matrix A corresp onds to the identity

hand

matrix:

0 1

1 0 0 0

B C

0 1 0 0

B C

A = :

hand

B C

0 0 1 0

@ A

0 0 0 1

As a result, all the solutions to the ring closure are obtained by substituting for A in 2.

hand

2.2 Lo cal Deformations

The problem of p erforming conformational variations on a lo cal p ortion of a chain corresp onds

to selecting the b onds formulating the chain, cho osing the subset of rotatable b onds and the

conformation of the end of the lo cal chain. The algorithm pro ceeds by assigning co ordinate systems

and computing the DH parameters for the lo cal chain. The rotatable b onds need not b e contiguous

as shown in Fig. 1. The resulting problem can b e p osed as shown in 2. A corresp onds

hand

to the conformation of the end of the chain and n corresp onds to the numb er of rotatable in the

selected chain.

2.3 Cases of Rotation ab out All Backb one Bonds

Go and Scheraga have considered a restricted version of the inverse kinematics problem, when the

rotation can take place ab out all backb one b onds as opp osed to a few selected dihedral angles

[GoS70]. This turns out to b e a sp ecial case of the inverse kinematics problem as well. In particular,

the Denavit-Hartenb erg parameters of suchachain corresp ond exactly to: a =0,d = b ond length

i i

and = b ond angle. Such a molecular chain has b een shown in Fig. 3.

i 7 F4 F6

F0 α 4 α6 α2

d5 d3 d4 d6 d1 d2

α 3 α5 α1

F5 F3

Figure 3: Parameters for Molecular Chain with All Bonds Rotatable

3 Molecular Emb edding and Inverse Kinematics

The molecular emb edding problem consists of nding one or more sets of atomic co ordinates such

that a given list of geometric constraints is satis ed. Such problems arise, when we are given data

derived from NMR exp eriments, where we are given some experimental constraints corresp onding

to the distance b etween some atoms. In these cases it is assumed that the molecule is under no great

strain, so that all b ond lengths and b ond angles are known from standard values taken from X-ray

crystallographic studies on small molecules [Cri89]. These constraints are referred to as holonomic

constraints. The goal of the emb edding algorithms is to nd all conformations which satisfy these

constraints. In this section, we show the equivalence b etween the emb edding problems and inverse

kinematics. Furthermore, fast algorithms for inverse kinematics can b e used for computing the

emb eddings.

There are many approaches to computing the molecular emb eddings and most general one

is based on EMBED [CH, Cri81]. It is frequently used for the determination of conformations

of small proteins in solution by NMR. Given the exp erimental and holonomic constraints, the

algorithm nds upp er and lower b ounds for all distances which satisfy the constraints and using

some random values in this range it computes the closest corresp onding three-dimensional metric

matrix. Cripp en has extended the algorithm using linearized emb edding and taking chiralityinto

account [Cri89]. The overall pro cess is iterative and may take considerable running time on some

cases. Furthermore, even for small chains it is not guaranteed to compute all the con gurations. 8

We show the equivalence b etween molecular emb edding and inverse kinematics for small chains.

It can b e combined with mo del building techniques for application to larger chains as shown in

[LS92].

3.1 Application to Small Chains

In this section we will consider emb eddings of molecular chains consisting of up to six rotatable

b onds. In the basic version of the problem we are given an initial b ond, sp eci ed by xed atom

p ositions, P and P , and a nal b ond sp eci ed by xed atom p ositions P and P . We are

1 2 6 7

interested in nding all the p ositions of all of the intermediate atoms P , P and P such that the

3 4 5

b ond lengths are all as required d ;d ;d and d and the angles b etween the b onds are all as

2 3 4 5

required ; ; ; and . The problem formulation is shown in Fig. 3. We can reduce this

2 3 4 5 6

problem to inverse kinematics in the following manner:

Assign a frame using the DH formulation at each atom shown as P in Fig. 3, such that the

z-axis is along the b onds. We know the p osition of P in the base co ordinate system assigned at

P . Let the orientation of the frame at P b e such that the z-axis is along the b ond P P and

1 7 6 7

the x and y axes are chosen to complete a right hand co ordinate system. A is therefore computed

appropriately. The Denavit-Hartenb erg parameters are computed using the d 's and 's. The

i i

inverse kinematics problem corresp onds to computing all the dihedral angles, given the p ose of the

end-e ector A . Given the dihedral angles, the p ositions of the intermediate atoms can b e easily

E computed.

P7 P3 P5 P1

α3 α5 d5 d3 d4 d6 d1 d2

α4 α6 α2

P4 P6

Figure 3: Molecular emb edding problem with unknown atom p ositions

Theorem 3.1 Assuming xedbond lengths and bond angles, the conformation of a molecular chain 9 F5 F7 F3 F1 α5 α3

d5 d3 d4 d6 d1 d2

α α2 4 α6

F4 F6

Figure 4: Inverse kinematics formulation of molecular emb edding

with up to 7 atoms can be determined its rst and last bond positions.

Pro of

We prove the theorem by showing that the molecular chains in Fig. 3 and Fig. 4 have equivalent

solution space. Therefore, the conformation of the molecular chain in Fig. 3 represented in terms

of unknown atom p ositions can b e computed using the algorithm for inverse kinematics highlighted

in the previous sections.

We initially show that for each solution to the emb edding problem, there exists a corresp onding

solution to the inverse kinematics problem. Given atom p ositions P ,P , P that satisfy the b ond

3 4 5

length and b ond angles constraints, the x-axis of lo cal frame at P can b e computed based on the

atom p ositions. Given the frames and their p ositions in the global co ordinate system, the s are

easily computed and they corresp ond to the solutions of the inverse kinematics problem.

Conversely,if s are the solutions for the inverse kinematics problem, using forward kinematics

the p ositions of p oints P , P , :::, P are computed. Given the p osition and orientation of the

2 3 6

frames, we know that P and P corresp ond to the given p ositions, and z is along P -P b ond, and

1 7 0 1 2

z is along P -P . Since all rotations are along the b onds, the p osition of P and P are invariant,

7 6 7 2 6

and they match the sp eci ed p ositions. The rotations along the b onds do not violate any b ond

length or b ond angle constraints.

Q.E.D.

Although wehave considered molecular chains with rotation along all backb one joints, the

relationship b etween emb edding problems and kinematics problems extends to all other geometries

like p eptide chains as well. 10

4 Inverse Kinematics and Matrix Formulation

Most of the literature in rob otics has concentrate on inverse kinematics of chains with six or fewer

joints. The complexityofinverse kinematics of a general six jointed chain is a function of the

geometry of the chain. While the solution can b e expressed in closed form for a variety of sp ecial

cases, such as when three consecutive joint axes intersect in a common p oint, no such formulation

is known for the general case. It is not clear whether the solutions for such a manipulator can

b e expressed in closed form. Iterative solutions based on numerical techniques like optimization

or Newton's metho d to the inverse kinematics for general 6R manipulators have b een known

for quite some time. However, they su er from two drawbacks. Firstly they are slow for practical

applications and secondly they are unable to compute all the solutions. As a result, most industrial

manipulators are designed suciently simply so that a closed from exists.

The inverse kinematics problem for six revolute joints has b een studied for more than two

decades in the rob otics literature. Given the multivariate p olynomial system, di erent algorithms

to reduce the problem to nding ro ots of a univariate p olynomial are given in [Pie68, RRS73 ,

AA79, DC80]. Given a 6R manipulator with a p ose of the end-e ector, the maximum number of

solutions assuming a nite numb er of solutions is 16 as shown in [Pri86, LL88, RR89 ]. Practical

metho ds to compute all the solutions are presented in [WM91] based on continuation metho ds and

in [MC92] based on matrix computations.

In this section, we consider the problem of inverse kinematics for a molecular chain with six

revolute joints. Chains with more than six joints or fewer than six joints are considered in next

section. We make use of the algebraic reduction of the problem to a univariate system presented

by Raghavan and Roth [RR89]. However the symb olic derivation highlighted by Raghavan and

Roth do es not take care of all the geometries e.g. the p olyp eptide con gurations. As opp osed

to reducing the problem to nding ro ots of a univariate p olynomial, we formulate it as an eigen-

value problem. Dep ending on the geometry of the molecular chain we obtain matrices of di erent

order, though the total numb er of solutions is still b ounded by 16 assuming a nite number of

con gurations. The main advantages of the matrix formulation are eciency and accuracy. The

reduction to a univariate p olynomial involves expanding a symb olic determinant, which is relatively

exp ensive. Moreover the problem of nding ro ots of p olynomial of degree 16 can b e numerically

ill-conditi oned [Wil59]. As a result, wemay not b e able to compute accurate results using IEEE

double precision arithmetic on currentworkstations. 11

4.1 Raghavan and Roth Formulation

Given the matrix equation for a six jointed manipulator, 2, Raghavan and Roth rearrange it as:

1 1 1

A A A = A A A A : 3

3 4 5 hand

2 1 6

As a result the entries of the left hand side matrix are functions of ; and and the entries

3 4 5

of the right hand side matrix are functions of , and . This lowers their degrees and reduces

1 2 6

the symb olic complexity of the resulting expressions. On equating the corresp onding entries of the

matrix equation, 3, and after simpli cation, these equations are expressed in a linear formulation

given as:

0 1

s s

4 5

B C

s s

1 2

B C

s c

4 5

B C

s c

1 2

B C

c s

B C

4 5

B C

c s

B C

1 2

B C

c c

B C

4 5

B C

c c

1 2

B C

=Q ; 4 R

B C

1 B C

B C

B s C

B C

@ A

B C

@ A

where R is a 14 8 matrix, whose entries are functions of the geometry of the chain and the

con guration of the end of the chain. Q isa149 matrix, whose entries are linear functions of

s and c and their co ecients are functions of the manipulator parameters and the p ose. The

3 3

relationship expressed in 4 helps us in eliminating four of the vevariables.

The rest of the algorithm in [RR89] pro ceeds by using 8 of the 14 equations to solve for the p ower

pro ducts on the right-hand-side, and then use these solutions to eliminate these p ower pro ducts

from the remaining 6 equations. Eventually,we obtain a linear system of the form:

Ss s s c c s c c s c s c 1 = 0; 5

4 5 4 5 4 5 4 5 4 4 5 5

where S is 6 9 matrix, whose entries are linear combinations of s ;c and 1.

3 3

4.2 Symb olic Representation

In our algorithm, we represent the entries of R and Q as symb olic functions of the geometry of the

molecular chain. Given a particular geometry,we substitute the corresp onding parameters. The

parameters for the p olyp eptide geometry are highlighted in Section 5. 12

It is p ossible that for some sp ecial geometries Q is rank de cient. As a result, we p erform the

rank computation after substitution using the singular-value decomp ositions SVD [GL89]. If the

rank, r , is less than 8 we only use r equations to eliminate the p ower pro ducts on the left hand

side of 4 and thereby obtain 14 r equations. Thus, the order of S in 5 is 14 r 9.

Typically, Q has rank 8 and therefore, S consists of 6 rows. In case S consists of more than 6

rows, we are given an overconstrained system. To solve such a system, we take 6 linear combinations

of the rows of S and let the matrix S now corresp ond to this new system of 6 combinations. Once

we obtain all the solutions to these new 6 equations, we back substitute them into the original

overconstrained system consisting of 14 r equations, to nd all the solutions of the overconstrained

system. In the rest of the pap er, wework with the assumption that S consists of 6 rows.

The resulting system is transformed into an algebraic system by replacing the variables s and

c corresp onding to the dihedral angle by:

1x 2x

;c = : s =

i i

2 2

1+ x 1+x

i i

The resulting system of equations are multiplied by1+x to transform them into p olynomial

equations. As a result, we get a system of p olynomial equations in three unknowns, x ;x ;x .

3 4 5

Furthermore it can b e represented as a linear system of the form

M x H x ;x = 0; 6

3 4 5

where M x isa69 matrix, whose entries are quadratic p olynomial in x and H x ;x isa

3 3 4 5

91vector consisting of entries, h x ;x ;:::;h x ;x . Each h x ;x is a monomial function

1 4 5 9 4 5 i 4 5

of degree at most four in x and x .

4 5

4.3 Formulating a Square System

The algebraic equations corresp onding to the inverse kinematics have b een represented in terms

of the matrix p olynomial, M x . In our case, the set of equations 6 corresp ond to a system

of non-linear equations in x ;x and x and we are interested in computing all their common

3 4 5

solutions.

Given the linear system consisting of 6 equations, wemultiply it with appropriate p ower pro d-

ucts to convert it into a square system. In particular, wemultiply each of the 6 equations by x

and obtain 6 new equations represented as.

M x x H x ;x = 0:

3 4 4 5 13

We combine these equations into 12 equations represented in the matrix form as:

M x H x ;x = 0; 7

3 4 5

where M x is the 12 12 matrix:

0 1

Mx O

@ A

M x = ;

O Mx

and O is a 6 3null matrix. Moreover, H x ;x isa121vector, whose entries are monomial

4 5

functions in x ;x of degree at most 5. H x ;x is constructed by combining the terms of H x ;x

4 5 4 5 4 5

and x H x ;x . As a result, M x represents a square system of order 12 [RR89].

4 4 5 3

4.4 Matrix Formulation

M x H x ;x , the problem of inverse kinematics corresp onds Given the matrix representation,

3 4 5

to nding all the solutions to the linear system 7.

M x isa1212 matrix p olynomial of degree 2. It can b e represented as:

M x = M +M x +M x ; 8

3 0 1 3 2

M are numeric matrices. The matrix p olynomial M x is regular if its symb olic determinant where

i 3

is non-zero. Otherwise it is a singular matrix p olynomial. We are interested in computing the zeros

of its determinant. For singular matrix p olynomials it corresp onds to computing the eigenvalues

of the matrix p olynomial [VDD83].

The following theorem shows how this problem of zero computation can b e linearized, and the

problem of inverse kinematics is reduced to computing appropriate zeros of a matrix p encil of the

form, A xB [MC92, VDD83].

Theorem 4.1 Given the matrix polynomial, M x the zeros of the matrix polynomial correspond

to the zeros of the matrix generalized system A xB , where

0 1 0 1

I 0 0 I

@ A @ A

B = A = : 9

0 M M M

2 0 1

O and I are nul l and identity matrices of order 12.

In case the matrix p olynomial is regular and M is well conditioned, this can b e further reduced

to computing the eigenvalues of [MC92]

1 0

0 I

A @

: 10 C =

1 1

M M M M

0 1

2 2 14

4.5 Solving for All Geometries

A generic molecular chain results in regular matrix p olynomials and the solutions are computed

using the eigendecomp osition of the matrices highlighted in the previous section. However, most

of the sp ecial geometries like the p olyp eptide unit or cyclohexanes result in singular matrix

p olynomials and matrix p encils. The current algorithms for eigendecomp osition of such p encils

can b e ill-condi tioned and relatively slow. In this section, we make use of the structure of the

matrix p olynomials corresp onding to the kinematics formulation and derive ecient and accurate

algorithms.

M x 's is singular is based up on substituting random values of A simple test to check whether

x into the matrix p olynomial and checking the rank of the resulting numeric matrix. Furthermore,

the kernel of singular M x mayormay not b e indep endentofx. The rest of the algorithm

3 3

involves checking the kernel and p erforming appropriate matrix computations.

4.5.1 Kernel of the Matrix Polynomial

Wecheck whether the matrix has a xed kernel based on the following lemma.

Lemma 4.2 Given the 12 12 matrix polynomial M x , its right kernel consists of r constant

T T T

linearly independent vectors, i the 12 36 matrix P = has r linearly inde-

M M M

0 1 2

pendent vectors in its left kernel.

In the ab ove lemma, M refers to the transp ose of M . The pro of of this lemma is simple. Based

on this lemma, it is trivial to check whether M x has a xed kernel or it is a function of x .

3 3

4.5.2 Matrix Polynomials with Constant Kernel

In this section, we consider matrix p olynomials, M x with a xed kernel. The algorithm for

inverse kinematics follows from the following theorem.

Theorem 4.3 Given the matrix polynomial, M x , such that

i The system of equations, 7, has a nite number of solutions.

M x has a xed right kernel. ii

0 0 0 0

For any solution, x ;x ;x , to the equations 7, x is a solution of DetMmx = 0, where

3 4 5 3

Mmx is a leading minor of M x .

3 3 15

We are omitting the pro of due to space requirements.

The theorem is constructive and is used to compute all the solutions of inverse kinematics. We

take a non-singular minor of M x . That corresp onds to a regular matrix p encil of order N r

and has degree 2. Using Theorem 4.1, all the ro ots of its determinant corresp ond to the eigenvalues

of the equivalent matrix, C as shown in 10. The eigenvalues corresp ond to x of a given solution.

0 0 0

Given x ,we compute x ;x in the following manner.

3 4 5

Takeanytwo linear combinations of the 6 equations in x and x :

4 5

M x H x ;x = 0: 11

4 5

Let the two equations b e l x ;x =0and l x ;x = 0. These are two equations of degree 4

1 4 5 2 4 5

each, however the highest degree of x or x in each of these equation is 2. We compute all the

4 5

solutions to these equations using Sylvester resultant formulation and reducing the problem to an

eigenvalue formulation. The resulting matrix is 8 8 and we get at most 8 real solutions of these

equations. Given the real solution, we back substitute them into the system 11 and obtain the

actual solution.

4.5.3 Matrix Polynomials with Dep endent Kernel

In this section we consider matrix p olynomials whose kernel is a function of x .We make use of

the 6 equations represented in 6 and use them to compute all the x ;x ;x corresp onding to

3 4 5

the solutions of the inverse kinematics problem.

Given the 6 equations b eing represented as M x H x ;x = 0. Mx is rank de cient.

3 4 5 3

Dep ending up on the rank of M x , we use the following approaches.

RankM x < 3 In this case, there are in nite solutions for the con guration of the end of the

chain. This follows from the fact that at most 2 of the 6 equations are indep endent. A system of one

or two algebraic equations, in three unknowns corresp onds to a two dimensional or one dimensional

algebraic set. As a result, there are in nite solutions of the system, M x H x ;x = 0, which

3 4 5

implies that there are in nite solutions of the matrix equation 2.

RankM x > 3Wecho ose four indep endent equations and represent them as

M x H x ;x = 0;

3 4 5 16

where M x isa49 minor corresp onding to the four equations. In order to solve this system,

we use dialytic elimination and linearize it. This is done bymultiplying the four equations by x ;x

4 5

and x x . As a result we obtain additional 12 equations of the form:

4 5

M x x Hx ;x = 0

3 4 4 5

M x x Hx ;x = 0

3 5 4 5

M x x x Hx ;x = 0:

3 4 4 4 5

These 16 equations are combined and put together in a linear form:

MM x HHx ;x = 0; 12

3 4 5

where MMx isa1616 matrix p olynomial in x of degree 2, and its entries consist of

3 3

HHx ;x is obtained by combining the p ower pro ducts in entries of M x and null matrices.

4 5 3

H x ;x ;xHx;x ;xHx;x ;xxHx;x . This system is solved by reducing it to a

4 5 4 4 5 5 4 5 4 5 4 5

32 32 eigenvalue problem. From the real solutions of this system, we pick the solutions of the 6

equations, M x H x ;x =0 by back substitution.

3 4 5

RankM x = 3 In this case, only 3 of the 6 equations of M x H x ;x = 0 are indep endent.

3 3 4 5

The resulting problem corresp onds to solving three equations in three unknowns. Each equation

2 2 2

has degree 6, however the highest degree term corresp onds to x x x . Such equations are solved

3 4 5

using Dixon's formulation of resultant [Dix08] and reducing the problem to eigendecomp osition of

a4848 matrix, followed by back substitution.

5 Extension to All Molecular Chains

In the previous section we had considered molecular chains with six dihedral chains and reduced

the problem to an eigendecomp osition problem. In this section, we highlight techniques for manip-

ulation of molecular chains with fewer than six dihedral angles or more than six dihedral angles.

Given a molecular chain with n dihedral angles, the problem of kinematic manipulation reduces to

n equations in 6 unknowns. In case, n<6, there may b e no solutions of this system. We solve for

all conformations of the chain in the following manner.

Any serial chain with less than 6 dihedral angles is treated as a 6-jointed system. We use

additional b onds with rotatable dihedral angles in the con guration. The additional b onds are 17

placed in such a manner that the dihedral angle corresp onding to their co ordinate systems are

zero. Moreover, this dihedral angle is chosen as .We treat the chain as a 6-jointed chain and

compute all the solutions for the given p ose. The original chain has any solutions, i the 6-jointed

chain has a solution with =0.

Chains with more than six dihedral angles are commonly used in molecular mo deling. Examples

include cyclo-heptanes, cyclo-nonanes, sugar molecules b esides the p olyp eptide units. Given a chain

with n>6 dihedral angles, it has n 6 dimensional solution space in the neighb orho o d of any real

solution. A simple strategy to solve for the solutions involves the use of n 6 dihedral angles as

indep endentvariables and the rest of the six are functions of the indep endentvariables computed

using the inverse kinematics algorithm. A simple exhaustive pro cedure assigns some discrete values

to the n 6 indep endentvariables and solves for the rest of the dihedral angles based on the

algorithm highlighted in the previous section. Values of the indep endentvariables are chosen using

exhaustive search metho ds or randomized techniques. In case there are no real solutions, all the

eigenvalues of the matrices formulated in the previous section have imaginary parts. We can use

the magnitude of the imaginary part of the eigenvalues in cho osing the indep endentvariables and

p erturbing them to guide to a real solution.

The resulting algorithm combines the analytic approach with exhaustive search or randomized

search metho ds. As compared to purely exhaustive metho ds, the complexity of the search space

go es down by six dimensions, due to the inverse kinematics pro cedure. For example, to compute all

the chain closure con gurations of cyclo-nonanes, we only use discrete values of three indep endent

variables as opp osed to all the nine indep endentvariables. For dihedral angle increments of 60 ,

we generate 216 con gurations of the three indep endent variables and apply the inverse kinematics

pro cedure. A purely exhaustive metho d would generate 216 con gurations. Similarly it can b e

combined with Monte Carlo metho ds, we only need to intro duce random changes to n 6 dihedral

angles as opp osed to all the n angles.

6 Applications

The algorithm presented in previous sections makes no assumptions ab out the geometry of molec-

ular chains, and it can b e applied to a variety of problems. In this section, we will apply the

algorithm to ring closure and lo cal deformation problem. For each case, we will compute its De-

navit Hartenb erg parameters, and discuss the results. 18

Number Link Length O set Distance Twist Angle

i a d

i i i

1 0.0 1.525 69.6

2 0.0 1.526 69.6

3 0.0 1.526 69.6

4 0.0 1.526 69.6

5 0.0 1.526 69.6

6 0.0 1.526 69.6

Table 1: Denavit-Hartenb erg Parameters of Cyclohexane Mo del

6.1 Cyclohexane Conformations

We studied cyclohexane conformation assuming rigid b ond geometry. In particular, all C-C b ond

lengths are xed at 1.526 Angstrom, and C-C-C b ond angles are xed at 110.4 degree. Since

we are only concerned with main chain conformation, the b ond lengths and b ond angles of C-H

are irrelevant to our analysis. It was proved in [GoS73] that cyclohexane has in nite geometrically

p ossible conformation due to its structural symmetry. In the following example, we slightly increase

the length of the last b ond. Since the symmetry no longer holds, we obtain a nite set of solutions.

Using the pro cedure in Section 2, the Devanit Hartenb erg parameters can b e computed for this set

of b ond geometry. They are listed in Table 1. The end e ector is an identity matrix b ecause of

the ring structure.

Four sets of solutions computed by the algorithm are listed in Table 2. The rst two sets of

solutions corresp ond to the dihedral angles of the chair conformation in Fig. 5a. The last two

sets corresp ond to those of the twistedboat conformation in Fig. 5b.

We should p oint out here that the four solutions are obtained based on the assumption that

b ond lengths and b ond angles are xed as listed in Table 1. If variation of b ond length or b ond

angle is allowed, other p ossible conformations exist.

In the data listed in Table 3, we p erturb the b ond lengths and b ond angles bytwo p ercent. A

di erent set of solutions are obtained in Table 4. The rst two solutions again corresp ond to the

chair conformation, while the last two corresp ond to the boat conformation, see Figure 5c. The

twistedboat conformation from the standard b ond geometry do es not show up here. 19 (a)

(b)

(c)

Figure 5: Cyclohexane conformations: a. top chair conformation, b. middle twistedboat

conformation, c. b ottom boat conformation 20

Through a series of small p erturbations of the b ond geometry,we noticed that the chair confor-

mation is more stable than the boat and twistedboat conformation. In all cases of small p erturbation

we tried, there are no more than four solutions for the ring closure problem. Two of these four

solutions constantly corresp ond to the chair conformation, while the other two corresp ond to either

the boat or the twistedboat conformation. Furthermore, the dihedral angles of the chair conforma-

tions stay almost constant, while those of either the boat or the twistedboat vary a lot for small

p erturbations on b ond geometry. Our result is in accord with previous exp erimental and theoretical

result that the chair conformation is the favored conformation of cyclohexane.

6.2 Cyclo o ctane Conformations

Wehave applied inverse kinematics analysis to study the conformations of ring structures of cy-

clo o ctanes. The Denavit-Hartenb erg parameters of a cyclo o ctane structure are shown in Fig. 6

o o

d =1:55A and = 115 . The p ose of end e ector in this case is an identity matrix b ecause of

the ring closure.

In the case of cyclo o ctane in Fig. 6, there are eight rotatable b onds. The conformation can

not b e determined solely based on ring closure prop erty which gives no more than six constraints.

In other words, the solution space is a two-dimensional space. In our study,we combined inverse

kinematics algorithm along with systematic search. Namely,we assign discrete values to the last

two dihedral angles and use inverse kinematics to compute the rest of the six dihedral angles. We

have used di erent increments of the angles. At 30 degree increments for and , the number

7 8

o o

of solutions are shown in Table 5. For example, if we x =60 and =30, there are

7 8

6 solutions. Due to the inherent symmetry in the problem, the resulting two-dimensional table

should b e symmetric. In general, the algorithm can compute the solutions to a go o d accuracy.

1 2 3 4 5 6

1 -57.69 57.67 -57.62 57.60 -57.62 57.67

2 57.69 -57.67 57.62 -57.60 57.62 -57.67

3 67.78 -32.04 -31.98 67.70 -31.98 -32.04

4 -67.78 32.04 31.98 -67.70 31.98 32.04

Table 2: The dihedral angles corresp onding to the Cyclohexane with standard b ond length and

b ond angles 21

Number Link Length O set Distance Twist Angle

i a d

i i i

1 0.0 1.525 69.5

2 0.0 1.525 69.6

3 0.0 1.526 69.8

4 0.0 1.527 69.7

5 0.0 1.526 69.5

6 0.0 1.526 69.6

Table 3: Denavit-Hartenb erg Parameters of p erturb ed Cyclohexane Mo del

However, near higher multiplicity ro ots say double ro ots, the problem can b e ill-conditi oned and

sp ecial pro cessing is needed to compute these high multiplicity solutions, as highlighted in [Man92].

The p erformance of the inverse kinematics algorithm for conformation search of a cyclo o ctane

has b een highlighted in Table 6. In particular, the running time is shown for di erent grid size

increments. It takes ab out 40 50 milliseconds for a single execution of the inverse kinematics

pro cedure and ab out 7 8 seconds to search the two dimensional space of and at 30 increments

8 7

to generate the results in Table 5. Our current implementation is not fully optimized and we b elieve

that the p erformance and accuracy can b e further improved. Our current implementation highlights

o o

that 20 to 30 degree grid increments generate a go o d approximation of the con guration space.

Furthermore, the p erformance of the overall algorithm is directly prop ortional to the grid size. It

is straightforward to predict that it will take less than 20 minutes to search the conformation of

1 2 3 4 5 6

1 -57.38 57.73 -57.95 57.93 -57.75 57.42

2 57.38 -57.73 57.95 -57.93 57.75 -57.42

3 -60.73 5.87 54.06 -61.25 6.42 53.50

4 60.73 -5.87 -54.06 61.25 -6.42 -53.50

Table 4: The dihedral angles corresp onding to the Cylcohexane with p erturb ed b ond length and

b ond angles 22

0.0 0 4 4 6 0 0 0 0 0 6 4 4

30.0 4 4 6 0 0 0 0 0 4 6 3 4

60.0 4 6 0 0 0 0 0 0 6 4 4 4

90.0 6 0 0 0 0 0 0 0 4 8 4 5

120.0 0 0 0 0 0 0 0 0 0 4 6 4

150.0 0 0 0 0 0 0 0 0 0 0 0 0

180.0 0 0 0 0 0 0 0 0 0 0 0 0

210.0 0 0 0 0 0 0 0 0 0 0 0 0

240.0 0 4 6 4 0 0 0 0 0 0 0 0

270.0 6 6 4 8 4 0 0 0 0 0 0 0

300.0 4 4 4 4 6 0 0 0 0 0 0 6

330.0 4 4 3 6 4 0 0 0 0 0 6 4

Table 5: Numb er of solutions for given values of and . The rows are for di erent values,

7 8 8

and the columns are for di erent values.

cyclo decane at 30 degree increment of the last four dihedral angles.

Our analysis has b een based on pure geometry,we are yet to study energy asp ects of our results.

We b elieve that these results can b e used as starting p oints for energy minimization pro cedures.

That involves variations in the dihedral angles, along with small variations in the b ond length and

b ond angles.

6.3 Lo cal Deformation

The algorithm is also applied to study lo cal deformation of protein segments. In the following

example, we compute the Denavit Hartenb erg parameters and the end e ector from protein data le

instead of using a standard set of b ond geometry. In a given data le from protein databank, the

Grid size increments 30 20 10 5

Time sec 7.81 18.12 73.4 295.2

Table 6: Running time of cyclo o ctane conformation searchatvarious grid sizes 23 θ3 θ4 α Y X Z θ2 θ5 d

θ1 θ6

θ8 θ7

Figure 6: Cyclo o ctane Conformation : systematic searchon and

7 8

b ond geometry can deviate from the standard geometry for as much as 10 p ercent. The error

from the exp erimental data is minimized by computing the b ond geometry directly from atom

co ordinates. This approach is useful for analyzing protein mo dels. In protein synthesis applications,

standard geometry for example Pauling-Corey geometry can b e used.

Given atom co ordinates, b ond lengths and b ond angles can b e easily computed. We assume

all covalent b onds are rigid, and the only freedom in a protein chain is the rotation along C-Ca

b ond and N-Ca b ond. Using the pro cedure in section 2, Denavit Hartenb erg parameters can b e

computed from the b ond geometry. The end e ector can b e either computed from the original le

or sp eci ed by the user, dep endent on the application.

In our example, wecho ose a segmentof -helix from protein Felix [Ric93]. Two orthogonal

views of this segment are shown in Figure 8a. The Denavit Hartenb erg parameters and the end

e ector are computed from the atom co ordinates.

The D-H parameters are listed in Table 7. The p ose of the end e ector is:

0 1

0:628 0:764 0:146 4:141

B C

0:552 0:569 0:609 0:152

B C

A = :

hand

B C

0:548 0:302 0:779 3:251

@ A

0 0 0 1

Two solutions for the given p ose are listed in Table 8. The rst set of solutions in Table 8 24

Figure 7: a. top frontview and sideview of an -helix, b. b ottom two p ossible conformations

obtained byinverse kinematics pro cedure. 25

Number Link Length O set Distance Twist Angle

i a d

i i i

1 0.0 -5.81 8.67

2 0.0 9.44 70.05

3 0.0 -5.86 8.61

4 0.0 9.49 70.11

5 0.0 -5.78 8.68

6 0.0 9.42 70.12

Table 7: Denavit-Hartenb erg Parameters of a Protein Segment

recovers the structure in the original data, it corresp onds to the dihedral angles of a -helix. The

second set corresp onds to an alternative conformation. These two solutions are showed in Fig.

7b.

This technique can b e used to exam the loops between secondary Structures. Since the lo ops

usually can have irregular structures, they represent a dicult part of the study of protein confor-

mation problem. Our algorithm computes all p ossible solutions to close the lo op. We b elieveitis

a useful to ol in analyzing and designing lo op structures.

6.4 Molecular Emb edding of Protein Chains

We applied the inverse kinematics pro cedure for computing the molecular emb eddings of small

protein chains. In this section, we demonstrate how to compute backb one atom p ositions of a

protein chain from two end b onds p ositions based on geometric constraints.

A p eptide unit is showed in Fig. 8a. Due to the the partial double b ond character of C-N

b ond, no rotation along C-N b ond is p ossible and all the atoms of a p eptide unit lie in a planar.

1 2 3 4 5 6

1 -51.19 124.30 -55.71 136.07 -50.14 124.48

2 -33.93 101.43 -38.57 115.47 -8.90 79.64

Table 8: The dihedral angles of the protein chain 26 P Ca P H 0.153

Ca 117 120 H 0.132 CN 121 Ca 120 CN 0.123 0.147 N C O O Ca Ca Ca

(a ) (b ) (c )

Figure 8: Geometry of a p eptide unit

The only degree of freedom is the rotation along C -C and N-C b onds. C -C and N-C intersect

at a p oint P in the same p eptide plane as showed in Fig. 8b. The angle b etween C -C and

N-C is not shown accurately. Obviously, rotation along C -C and N-C are the same as rotation

along C -P and P-C , resp ectively. Therefore, a p eptide unit is geometrically equivalenttotwo

consecutive rotatable b onds as showed in Fig. 8c. Based on standard b ond geometry in Fig 1,

all the distances and angles in Fig. 8c can b e easily computed. Based on this analysis, Theorem

3.1 can b e directly applied for protein segments, while the planar p eptide unit is still maintained.

If two adjacentC -C b ond and C-C b ond are given, the missing N atom p osition can b e

computed as shown in Fig. 8b. Extend C -C b ond to P, N is on the line of P-C and its

distance to Ca is 1.47 A . As the two given b onds move further apart, computing intermediate

atoms b ecomes more complicated. Based on Theorem 3.1, at most three intermediate p oints can

b e computed if all b onds are rotatable. For the case of protein, it means that there can b e no more

than four rotatable b onds b etween the two given end b onds. Otherwise , the intermediate atom

p ositions can not b e determined uniquely solely based on b ond length and b ond angle constraints

there is an entire family of solutions.

One example of molecular emb edding is showed in Fig. 9. In this protein segment, there are four

residues Glu, Asn, Phe and Gln. The C -C b ond of Glu is xed , and the N-C b ond of Gln is xed.

The four atoms p ositions of these two b onds are listed in Table 9. Four p ossible conformations of

Asn and Phe can b e derived based on the b ond lengths and b ond angles constraints.

Wehave shown examples of computing backb one atom p ositions of protein segments. The

same technique applies to side chain and other molecular structures. This is based on the fact 27 {n) (b}

(II)

Figure 9: Four sets of p ossible atom p ositions of residues Asn and Phe. The Ca-C b ond of Glu

and N-Ca b ond of Gln are xed. 28

X Y Z

Glu C 0.128 -2.805 -1.279

Glu C 0.769 -2.365 0.039

Gln N -1.171 0.389 4.312

Gln C -2.247 -0.494 5.359

Table 9: Atom p ositions of two end b onds

that we made no assumption ab out the molecule geometry in the pro of of Theorem 3.1 or in the

pro cedure of solving inverse kinematics. This technique is also applicable for inferring backb one

atom p ositions from the side chains.

7 Implementation and Performance

A system is implemented to p erform real time lo cal deformation. In this section, we will brie y

discuss its implementation and p erformance.

7.1 User Interface

A molecular chain sp eci ed by a user is displayed on a pro jector screen. A rob ot ARM with 6

degrees of freedom is used as an input device. One terminus of the chain is xed. The other

terminus is manipulated by the user with the rob ot ARM. When the inverse kinematics pro cedure

nds at least one solution to match the terminus controlled by the user, new atom p ositions are

computed and displayed. Alternative solutions can b e displayed by rotating a knob on the ARM.

7.2 System Description

The Denavit Hartenb erg parameters and lo cal frames for a given molecular chain are precomputed.

Atom p ositions with resp ect to the lo cal frames are also precomputed. The computation can

b e based on either atom co ordinates in a protein data le or standard protein geometry. The

parameters are loaded at the system initialization stage. This precomputaion pro cess takes very

little computing time.

At runtime, whenever the user moves the ARM, a new p ose of the terminus is computed. This 29

end p ose along with the Denavit Hartenb erg parameters of the molecular chain is passed to the

inverse kinematics pro cedure. If solutions are found, atom p ositions for the new set of dihedral

angles can b e calculated with forward kinematics. These new atom p ositions are used to up dated

the display, and the system is ready to read another end p ose from the ARM.

7.3 Performance

The system p erformance is determined mainly by the inverse kinematics pro cedure. The symb olic

computation for the inverse kinematics is done only once, and the result is co ded into the system.

The b ottleneck is the eigendecomp osition of a matrix, which takes more than 80 of the total time.

Both examples in section 4 cyclohexane conformation and lo cal deformation involve computing

eigenvalues for 32*32 matrices. The system takes 40-50ms on a SGI/ONYX workstation for each

case.

8 Conclusions

In this pap er, wehave highlighted techniques for 3D manipulation of molecular chains with as-

sumptions of rigid b ond length and b ond angles. We show the corresp ondence b etween the prob-

lems of lo cal deformations, ring closure and molecular emb edding with inverse kinematics of serial

manipulators in rob otics. Using results from kinematics, we present an algorithm for kinematic

manipulation of molecular chains. It makes no assumption on the geometry of the chain and works

very well in practice on chains with six or fewer dihedral angles. It can also b e used to reduce

the search space on chains with more dihedral angles. It has b een implemented and we illustrate

its real time p erformance on computing ring closure of cyclo-hexane and lo cal deformation and

emb edding of short p olyp eptide chains.

9 Acknowledgements

Wewould like to thank Fred Bro oks, Jane and David Richardson, Alex Tropsha for useful discus-

sions. 30

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