CS273: Algorithms for Structure Handout # 2 and Motion in Biology Stanford University Thursday, 1 April 2004

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1 CS273: Algorithms for Structure Handout # 2 and Motion in Biology Stanford University Thursday, 1 April 2004 Lecture #2: 1 April 2004 Topics: Kinematics : Concepts and Results Kinematics of Ligands and Proteins Conformational Space and Sampling Molecular Motion Scribe: Gauhar Wadhera 1 Kinematics: The kinematics of a mechanical system specifies topology, degrees of freedom (DOF), velocities, and constraints, without explicit specification of externally applied forces/torques. In the context of molecule linkage models we shall only consider kinematic chains with two components - links and joints. Links are rigid bodies which are connected to each other by joints. We only consider revolute joints with rotation around a single axis. Figure 1 shows an example of such a linkage chain. 1.1 General Concepts and Results: Links, Joints and Degrees of Freedom: A critical concept for studying the kinematics of a system is the notion of degree of freedom. The degrees of freedom of a rigid body is defined as the number of independent movements it has. Figure 2 shows a rigid body in 3-D which can be translated along the three axes or rotated independently about each of them. Thus, it has 6 DOFs in 3-D space. An important concept that goes hand in hand with the degrees of freedom of a system is that of velocity space. The number of DOFs of a system is also the dimensionality of its velocity space General Results for Linkages: Grübler s Formula (1883): Grübler s Formula is used to calculate the number of DOFs as: N DOF = k*(n link -1) - (k-1)*n joint where, k = 3 for planar linkages and 6 for spatial linkages.

2 2 CS273: Handout # 2 Figure 1: Kinematic Chain: L denotes links and J denotes joints. The base is also a link with its coordinates in the world frame. Figure 2: Rigid Body in 3-Dimensional Space. The arrows indicate the 6 degrees of freedom of the body. - Simple Linear Linkage In Figure 1, N link = 4 and N joint = 3. Thus, N DOF = 3*(4-1) - (3-1)*3 = 3. - Simple Linear Linkage with Fixed End Point (a) Original Representation with coordinates of terminal point (b) Additional Link- Joint Interpretation of System Figure 3: Simple Linear Linkage with fixed position of the terminal point In Figure 3(a) the coordinates of the terminal point are fixed. One way to model the constraint is to rigidly attach this point to the base of the linkage as depicted in Figure 3(b). Therefore, N link = 4 and N joint = 4. Thus, N DOF = 3*(4-1) - (3-1)*4 = 1. - Simple Linear Linkage with Fixed Position and Orientation of End Point In Figure 4(a) the position and orientation of the terminal link are fixed. One way to model the constraint is to attach the last link to the linkage base as depicted in Figure 4(b).Therefore, N link = 3 and N joint = 3. Thus, N DOF = 3*(3-1) - (3-1)*3 = 0.

3 CS273: Handout # 2 3 (a) Original Representation with constraints at of terminal point (b) Additional Link- Joint Interpretation of System Figure 4: Simple Linear Linkage with fixed position/orientation of the terminal point 1.2 Forward Kinematics: Forward Kinematics is the problem of determining the positions of the individual links in the world coordinate frame given the rotation values of each joint in the linkage. Figure 5: Simple Linear Chain with 3 links and 2 joints. In Figure 5, given the data about the joint angles and link lengths, we can simple compute the world frame coordinates of (x,y) as: X = d 1 cosθ 1 + d 2 cos (θ 1 + θ 2 ) Y = d 1 sin θ 1 + d 2 sin (θ 1 + θ 2 ) 1.3 Inverse Kinematics: Inverse Kinematics of a serial linkage is the problem of determining the joint angles given the position and/or orientation of the last link. Figure 6 shows a simple example presented in class. In this case a closed form solution exists. In Figure 6 we can easily compute the values of θ 1 and θ 2 using the principles of geometry as:

4 4 CS273: Handout # 2 Figure 6: Inverse Kinematics problem: given (x,y) we need to find θ 1 and θ 2 Figure 7: Two solutions for θ 1 and θ 2 in the configuration given in Figure 6. ( (x θ 1 = cos y 2 d 2 1 d 2 ) 2 ) 2 d 1 d 2 θ 2 = x (d 2 sin θ 2 ) + y (d 1 + d 2 cos θ 2 ) y (d 2 sin θ 2 ) + x (d 1 + d 2 cosθ 2 ) However, a major point to be noted is that two solutions to the above equations exist due to the fact that cos 1 (x) has two values. These solutions are shown in Figure 7. As another example in Figure 8, the linkage with 3 joints and the coordinates for the end point fixed at (x,y), we have infinite solutions Issue: Number of Solutions (a) Original Representation of Inverse Kinematic Problem. (b) Diagram showing several solutions. The null space in this case is 1-D Figure 8: Visualization of the under-constrained system leading to infinte solutions In general, if the number of degrees of freedom is less than the number of constraints then no solutions exist. If the two are equal, there exist a finite number of solutions. In case of more degrees of freedom than constraints, we can have possibly infinite solutions. For example, a serial linkage with 6 joints and a fixed end point we can have upto 16 solutions.

5 CS273: Handout # Analytical IK Techniques: Analytical techniques exist only for systems with upto 6 degrees of freedom in 3D space. The methodoly is as follows: - Reduce the forward kinematic equations in polynomial form by epressing all trigonometric quantitites in terms of t = tanθ/2. - Use domain specific properties in order to simplify the types of configuration possible for the given problem. [1] apply such form of search space pruning in the context of protein structures. - Solve the equations analytically Incremental IK Techniques: The difficulty in solving IK problems analytically is the fact that they are non-linear problems due to the presence of trigonometric terms. Most of the numerical methods use linear approximations of the same. The Jacobian matrix is used in most of the incremental techniques. Let, - θ = n-vector of internal coordinates (usually the values θ i for all the joints). - X = m-vector for the end-point orientation (usually m=6 defining x,y,z,α,β,γ). The relationships can be given as : X = F(θ). dx = J dθ. n f i (θ) dx i = dθ j θ j j = 1 where F is the forward kinematics transform and J is the Jacobian. The Jacobian can be expressed formally as : J = f 1 (θ) θ 1 f 2 (θ) θ 1. f m(θ) θ 1 f 1 (θ) f 1 (θ) θ n f 2 (θ) θ n θ 2... f 2 (θ) θ f m(θ) θ 2... f m(θ) θ n Here, we assume that m is less than or equal to n as otherwise we would not have any solutions.the methods adopted totally depend on the relation between m and n. Jacobian Based Methods: 1. Case : m = n The method can be formulated as follows:

6 6 CS273: Handout # 2 (a) Problem: Given X, find θ such that X = F(θ). (b) Produce an initial guess (X 0, θ 0 ) such that X 0 = F(θ 0 ). (c) Iteration: i. Interpolate linearly between X 0 and X (X 1... X p ). ii. for j = 1 to p do A. θ i = θ i 1 + α J 1 (θ i 1 )(X i - X i 1 ) B. X i = F(θ i ) J is a mxn matrix and with m < n we can assume that the rank of J is at most m. Since the Jacobian is not square in this case we can only find a pseudo inverse following the relatioship J J + = I. We find a pseudo inverse of J and update : θ = J + dx + dθ 0. The computation of J + is done by performing singular value decomposition of J. We get: Figure 9: Result of SVD of J. J = U V T as shown in Figure 9. The p-inverse is then computed as: J + = V + U T where + = diag( 1 σ i ) with σ i = Eigen Values for J Optimization Based Methods: Cyclic Coordinate Descent (CCD) The basic idea of optimization based methods is to take a look at the primary equation θ = f 1 (X) as a minimization problem. Thus, the equation could be transformed into: E(θ) = (P X(θ)) 2. The CCD is based on minimization applied to each joint separately. The steps in one pass are ordered from the most distant segment to the base segment. A number of

7 CS273: Handout # 2 7 passes are made over the manipulator to find the global minimum of the above equation. Since only one joint variable is changing at any time, an analytic solution could be used significantly speeding up the minimization problem. 2 Kinematics of Proteins 2.1 Kinematic Models of Molecules Revisited - Atomistic model: The position of each atom is defined by its coordinates in 3-D space. Constraints on bond lengths/angles are encoded separately. - Linkage model: The kinematics is defined by internal parameters (bond lengths and angles, and torsional angles). Small local changes may have big global effects since even a small change in one torsion angle rotates the entire remaining molecule making it more prone to errors. Another difficulty with the method is that forces are difficult to express with most force models being distance based necessitating coordinate and distance computations. - Simplified Linkage Model: In this model bond lengths and angles are assumed constant and only torsional angles vary. This leaves less parameters to optimize. However, as a result of assuming all other parameters constant we lose the flexibility that exists in actual molecules wrt bond distances and angles. 2.2 Proteins: Basic Structural Features Proteins are polymer chains made up of monomeric units called amino acids. The key bonding component of the protein structure is the peptide bond between two amino acids which sets up the protein chain Amino Acids: Amino acids are the basic structural units of proteins. An alpha-amino acid consists of an amino group, a carboxyl group, a hydrogen atom, and a distinctive R group bonded to a carbon atom, which is called the alpha-carbon. An R group is referred to as a side chain. Figure 10 shows the above defined structure. The 20 amino acids that are found within proteins determine the biological activity of the protein as they contain the necessary information to determine how that protein will fold into a three dimensional structure, and the stability of the resulting structure.

8 8 CS273: Handout # 2 Figure 10: Generic Structure of Amino Acids Peptide Bonds: Even though amino acids that form the actual chain, peptide bonds are an important aspect of protein structure as well since they are the force behind the linking in the protein chain. The peptide bond is slightly shorter than a standard single bond due to partial delocalization of pi electrons from the carbonyl group into orbitals shared with the lone pair electrons of the amide nitrogen which inhibits rotation around the peptide bond; thus, the four atoms bound to the carbonyl carbon and amide nitrogen form a plane (Figure 11). Figure 11: Induced planarity in the peptide bond 2.3 Protein Linkage Model: The salient features of the linkage models of proteins are (given in Figure 12): Figure 12: Linkage Model for Proteins

9 CS273: Handout # The sequence of N-C α -C atoms form the backbone. This is analogous to the links in the simple linear linkage. - Rotatable bonds (torsion angles) along the backbone define the φ ψ torsional degrees of freedom. These bonds perform the role of joints with 1 degree of freedom. - We also have small side-chains corresponding to the amino acids which have a χ degree of freedom about the Cα-Cβ bond Example of Kinematic Chain in Proteins: In Figure 12 with 5 amino acids we have (assuming no role of the χ torsion) 10 links and 10 joints. Applying Grübler s Formula in 3-D we have: N DOF = 6*(n-1) - (6-1)*n = n - 6 = 4 (for n = 10) 3 Conformational Space: A conformation of a molecule or a system in general is a complete specification of the spatial placement of the entire system. In the case of molecules it could be the relative positions of all atoms wrt each other. The conformational space for a system is the set of all possible conformations. 3.1 Typical Representations: Typical Representations of the conformation space include: 1. Coordinate Representation: This representation describes the coordinates of each atom in the molecule and corresponds to the atomistic model of molecules. 2. Torsional Representation: This corresponds to representing each conformation in terms of the torsional angles of the rotatable bonds and assumes the bond lengths and bond angles as constant. 3. Intra-Molecular Distance Matrix: This representation stores the distances between all atoms of the molecule. Despites having a quadratic number of parameters it does capture some very interesting semantics especially in the case of complex folding systems such as proteins. As an example in Figure 13, we can clearly see the effect of folding in proteins. Apart from the diagonal distances to be small, we also have other distant C α atoms clearly indicating the presence of the molecule folding onto itself.

10 10 CS273: Handout # 2 Figure 13: Distances between Cα pairs of a protein with 142 residues. Darker squares represent shorter distances. 3.2 Conformational Space Metrics and the RMSD Measure: A good metric (with all the mathematical properties of a distance metric) should be able to measure how well the atoms in two conformations can be aligned. In this context the RMS Distance/Deviation measure seems to have a lot of relevance. The next few lines describe the basic RMSD measure while the following sections evalute metrics based on RMSD, namely crmsd and drmsd. Given two sets of points in <3 : A = {a1,...,an } and B = {b1,...,bn } The RMSD between A and B is: RMSD(A,B) = (1/n)[ ni=1 kai bi k2 ]1/2 where, kai bi k denotes the Euclidean distance between ai and bi in <3. RMSD(A,B) = 0 iff ai = bi i. P 3.3 crms Distance: Given a Molecule M with n atoms {a1,..., an } and two conformations c and c of M where ai (c) is position of ai when M is at c, we have the crmsd between c and c as the minimized RMSD between the two sets of atom centers: crmsd(c,c ) = mint [(1/n) Pn i=1 kai (c) T (ai (c0 ))k2 ]1/2. The minimization is over all possible rigid-body transforms T. Usually, crmsd is restricted to a subset of atoms, e.g., the Cα atoms on a protein backbone. However, this still remains a non-trivial task since it involves finding the aligning transform.

11 CS273: Handout # drms Distance: Given a Molecule M with n atoms {a 1,..., a n } and Two conformations c and c of M where d ij (c): n x n symmetrical intra-molecular distance matrix in M at c. The drmsd between c and c is : drmsd(c,c ) = [(1/n(n 1)) n i=1 nj=i+1 [d ij (c) d ij (c )] 2 ] 1/2 Usually d ij is restricted to a subset of atoms on order to reduce the load on the algorithm. Even though drmsd does not depend on finding an aligning transform it suffers from having to deal with a quadratic number of parameters. 3.5 Other Topics Covered: Conformational Space Search Mechanisms: In order to be able to simulate stable conformations of molecules (or for that matter any system) we need to develop two notions: 1. Energy Functions: What is a Good Conformation? Energy functions in the conformational space calculate the total amount of energy of the conformation and serve as measures of the quality of any given conformation. Various functions to be covered in later lectures include: - Potential Functions (eg. Electrostatic and Van Der Waal s Potential) - Heuristic Potential Functions (eg. Gō Model) - Explicit/Implicit Silvent Model 2. Conformational Sampling: How do we search in the conformational space? The answer to this question is extremely critical because of the vastness of the conformational space to be search. We need to decide on mechanisms which help us either formally or heuristically in pruning our search space. Certain Sampling Strategies to be discussed later include: - Generation of low-energy conformations - Random Selection and Minimization - Biased Sampling (eg. Ramachandran Plots with bias towards particular torsion angle values) - Monte Carlo Simulations Molecular Motion The lecture also covered briefly the types of molecular motion. These were divided into :

12 12 CS273: Handout # 2 Folding Motion This section briefly discussed the folding motion with two principle motivations: - To understand and build structure prediction models. - To study the folding pathway of the molecules (mostly proteins). The effects of protein folding were discussed by briefly describing the salient features of: (a) Alpha Helix Structure (b) Beta Sheet Structure Figure 14: Secondary Structures of Proteins - The Alpha Helix Structure (Protein Secondary Structure) The alpha helix structure is stabilized by H-bonds between the NH and CO groups which are at a distance of 4 in the main chain. The side chains protrude outwards from the helix and are only responsible for much of the functional bonding of the protein with the environment. - The Beta Sheet Structure (Protein Secondary Structure) These are antiparallel sheets formed from the peptide polymer chain. Adjacent strands run in opposite directions in the actual protein chain with H-bonds formed between the NH and CO groups of the strands. The side chains project out of the strand plane on either sides. - Protein Tertiary Structure The tertiary structure basically evolves from the secondary structure helices and sheets further folding onto each other resulting in complex structures containing varying fractions of helices and sheets. Binding Motion

13 CS273: Handout # 2 13 We shall not be discussing this topic in these lecture notes in detail and shall leave it for the future lectures to cover it in greater detail. The topics covered were: - Ligand-Protein Binding - Protein-Protein Binding References [1] E.A. Coutsias, C. Seok, M.P. Jacobson and K.A. Dill, A Kinematic View of Loop Closure, Jour. of Comutational Chemistry 2004.