Geometric and Kinematic Models of Proteins What is

Geometric and Kinematic Models of Proteins

What is Kinematics? Study of movement independent of the forces that cause them

Protein § Long sequence of amino-acids (dozens to thousands), also called residues from a dictionary of 20 amino-acids

Role of Geometric and Kinematic Models § Represent the possible shapes of a protein (compare/classify shapes, find motifs) § Answer proximity queries: Which atoms are close to a given atom? (computation of energy) § Compute surface area (interaction with solvent) § Find shape features, e. g. , cavities (ligand-protein interaction)

What are the issues? § Large number of atoms Combinatorial problems § Large number of degrees of freedom Large-dimensional conformation space § Need to efficiently update information during simulation (surface area, proximity among atoms): • What is the position of every atom in some given coordinate system? • Which atoms intersect a given atom? • What atoms are within some distance range from another one? § Complex metric in conformational space § Many shape matching issues

Geometric Models of Bio-Molecules § Hard-sphere model (van der Waals radii) Van der Waals surface

Van der Waals Potential Van der Waals interactions between two atoms result from induced polarization effect (formation of electric dipoles). They are weak, except at close range. The van der Waals force is the force to which the gecko's unique ability to cling to smooth surfaces is attributed! 12 -6 Lennard-Jones potential

Geometric Models of Bio-Molecules § Hard-sphere model (van der Waals radii) Van der Waals surface Van der Waals radii in Å H C N O F 1. 2 1. 7 1. 5 1. 4 1. 35 P S 1. 9 1. 85 Cl 1. 8

Geometric Models of Bio-Molecules § Hard-sphere model (van der Waals radii) Van der Waals surface Solvent- accessible surface Molecular surface

Computed Molecular Surfaces Probe of 1. 4Å Probe of 5Å

Is it art?

Computation of Hard-Sphere Surface (Grid method [Halperin and Shelton, 97]) § Each sphere intersects O(1) Why? spheres § Computing each atom’s contribution to molecular surface takes O(1) time § Computation of molecular surface takes Θ(n) time

Computation of Hard-Sphere Surface (Grid method [Halperin and Shelton, 97]) § Each sphere intersects O(1) Why? spheres § Computing each atom’s contribution to D. Halperin and M. H. Overmars Spheres, molecules, and hidden surface removal molecular surface Computational Geometry: Theory and Applications 11 (2), 1998, 83 -102. takes O(1) time § Computation of molecular surface takes Θ(n) time

Trapezoidal Decomposition

Trapezoidal Decomposition D. Halperin and C. R. Shelton A perturbation scheme for spherical arrangements with application to molecular modeling Computational Geometry: Theory and Applications 10 (4), 1998, 273 -288.

Possible project: Design software to update surface area during molecule motion Other approach: Alpha shapes http: //biogeometry. duke. edu/software/alphashapes/pubs. html

Simplified Geometric Models § United-atom model: non-polar H atoms are incorporated into the heavy atoms to which they are bonded § Lollipop model: the side-chains are approximated as single spheres with varying radii § Bead model: Each residue is modeled as a single sphere

Visualization Models § Stick (bond) model

Visualization Models

Visualization Models § Stick (bond) model § Small-sphere model

Kinematic Models of Bio-Molecules § Atomistic model: The position of each atom is defined by its coordinates in 3 -D space (x 5, y 5, z 5) (x 1, y 1, z 1) (x 4, y 4, z 4) (x 2, y 2, z 2) (x 3, y 3, z 3) (x 7, y 7, z 7) (x 6, y 6, z 6) (x 8, y 8, z 8) p atoms 3 p parameters Drawback: The bond structure is not taken into account

Peptide bonds make proteins into long kinematic chains The atomistic model does not encode this kinematic structure ( algorithms must maintain appropriate bond lengths)

Kinematic Models of Bio-Molecules § Atomistic model: The position of each atom is defined by its coordinates in 3 -D space § Linkage model: The kinematics is defined by internal coordinates (bond lengths and angles, and torsional angles around bonds)

Linkage Model T?

Issues with Linkage Model § Update the position of each atom in world coordinate system § Determine which pairs of atoms are within some given distance (topological proximity along chain spatial proximity but the reverse is not true)

Rigid-Body Transform z T(x) y T x x

2 -D Case y x

2 -D Case yy xx

2 -D Case y y x x

y 2 -D Case y x x

2 -D Case y y x x

x 2 -D Case y y x

x 2 -D Case y y Rotation matrix: j i cos q -sin q cos q q ty tx x

x 2 -D Case y y Rotation matrix: j i i 1 i 2 q ty tx x j 1 j 2

x 2 -D Case y y Rotation matrix: a j b’ ty b q a’ i q i 1 a’ b’ = i 2 j 1 j 2 v a a tx x Transform of a point? a b

x Homogeneous Coordinate Matrix i 1 i 2 0 y y y’ q ty x’ y’ 1 tx x’ = cos q -sin q tx sin q cos q ty 0 0 1 tx ty 1 § T = (t, R) § T(x) = t + Rx y x j 1 j 2 0 x x y 1 = tx + x cos q – y sin q ty + x sin q + y cos q 1

3 -D Case q 2 ? q 1

Homogeneous Coordinate Matrix in 3 -D R z y x i z k y j i 1 i 2 i 3 0 j 1 j 2 j 3 0 x with: – i 12 + i 2 2 + i 3 2 = 1 – i 1 j 1 + i 2 j 2 + i 3 j 3 = 0 – det(R) = +1 – R-1 = RT k 1 k 2 k 3 0 tx ty tz 1

Example z cos q 0 -sin q 0 y q x 0 1 0 0 sin q 0 cos q 0 tx ty tz 1

Rotation Matrix R(k, q) = kxkxvq+ cq kxkyvq+ kzsq kxkzvq- kysq where: • k = (kx ky kz)T • sq = sinq • cq = cosq • vq = 1 -cosq kxkyvq- kzsq kykyvq+ cq kykzvq+ kxsq kxkzvq+ kysq kykzvq- kxsq kzkzvq+ cq k q

Homogeneous Coordinate Matrix in 3 -D z (x, y, z) y z (x’, y’, z’) k x i y j x’ y’ z’ = 1 i 2 i 3 0 j 1 j 2 j 3 0 k 1 k 2 k 3 0 x Composition of two transforms represented by matrices T 1 and T 2 : T 2 T 1 tx ty tz 1 x y z 1

Questions? What is the potential problem with homogeneous coordinate matrix?

Building a Serial Linkage Model Rigid bodies are: • atoms (spheres), or • groups of atoms

Building a Serial Linkage Model 1. Build the assembly of the first 3 atoms: a. Place 1 st atom anywhere in space b. Place 2 nd atom anywhere at bond length

Bond Length

Building a Serial Linkage Model 1. Build the assembly of the first 3 atoms: a. Place 1 st atom anywhere in space b. Place 2 nd atom anywhere at bond length c. Place 3 rd atom anywhere at bond length with bond angle

Bond angle

Coordinate Frame z x y

Building a Serial Linkage Model 1. Build the assembly of the first 3 atoms: a. Place 1 st atom anywhere in space b. Place 2 nd atom anywhere at bond length c. Place 3 rd atom anywhere at bond length with bond angle 2. Introduce each additional atom in the sequence one at a time

Bond Length z x y 1 Ti+1 = 0 0 0 ct -st 0 st ct 0 0 0 1 cb -sb 0 sb cb 0 0 0 1 0 0 d 0 0 1 0 0 0 1

Bond angle z x y 1 Ti+1 = 0 0 0 ct -st 0 st ct 0 0 0 1 cb -sb 0 sb cb 0 0 0 1 0 0 d 0 0 1 0 0 0 1

Torsional (Dihedral) angle z x y 1 Ti+1 = 0 0 0 ct -st 0 st ct 0 0 0 1 cb -sb 0 sb cb 0 0 0 1 0 0 d 0 0 1 0 0 0 1

Transform Ti+1 y i+1 z Ti+1 z x t i-1 x b i-2 d i y 1 Ti+1 = 0 0 0 ct -st 0 st ct 0 0 0 1 cb -sb 0 sb cb 0 0 0 1 0 0 d 0 0 1 0 0 0 1

Transform Ti+1 y Readings: i+1 Ti+1 z J. J. Craig. Introduction to Robotics. Addison Wesley, reading, MA, z 1989. x t for Fast and Accurate Zhang, M. and Kavraki, i-1 L. E. . A New Method Derivation of Molecular Conformations. Journal of Chemical x d Information and Computer Sciences, 42(1): 64– 70, 2002. i http: //www. cs. rice. edu/CS/Robotics/papers/zhang 2002 fast-compb mole-conform. pdf i-2 y 1 Ti+1 = 0 0 0 ct -st 0 st ct 0 0 0 1 cb -sb 0 sb cb 0 0 0 1 0 0 d 0 0 1 0 0 0 1

Serial Linkage Model T 1 0 -2 -1 1 T 2

Relative Position of Two Atoms Ti+2 Ti+1 k-1 Tk k i Tk(i) = Tk … Ti+2 Ti+1 position of atom k in frame of atom i

Update § § Tk(i) = Tk … Ti+2 Ti+1 Atom j between i and k Tk(i) = Tj(i) Tj+1 Tk(j+1) A parameter between j and j+1 is changed § Tj+1 § Tk(i) = Tj(i) Tj+1 Tk(j+1)

Tree-Shaped Linkage Root group of 3 atoms p atoms 3 p -6 parameters Why?

Tree-Shaped Linkage T 0 Root group of 3 atoms world coordinate system p atoms 3 p -6 parameters

Simplified Linkage Model In physiological conditions: § Bond lengths are assumed constant [depend on “type” of bond, e. g. , single: C-C or double C=C; vary from 1. 0 Å (CH) to 1. 5 Å (C-C)] § Bond angles are assumed constant [~120 dg] Only some torsional (dihedral) angles may vary Fewer parameters: 3 p-6 p-3

Bond Lengths and Angles in a Protein C C N Ca f w: Ca f: C C y: N N w=p w 3. 8Å

f-y Linkage Model peptide group side-chain group

Convention for f-y Angles § f is defined as the dihedral angle composed of atoms Ci-1–Ni–Cai–Ci § If all atoms are coplanar: C C N Ca f=0 C Ca N C f=p § Sign of f: Use right-hand rule. With right thumb pointing along central bond (N-Ca), a rotation along curled fingers is positive § Same convention for y

Ramachandran Maps They assign probabilities to φ-ψ pairs based on frequencies in known folded structures ψ φ

f-y-c Linkage Model of Protein Cb c c c Ca c c § The sequence of N-Ca-C-… atoms is the backbone (or main chain) § Rotatable bonds along the backbone define the f-y torsional degrees of freedom § Small side-chains with c degree of freedom

Side Chains with Multiple Torsional Degrees of Freedom (c angles) 0 to 4 c angles: c 1, . . . , c 4

Kinematic Models of Bio-Molecules § Atomistic model: The position of each atom is defined by its coordinates in 3 -D space Drawback: Fixed bond lengths/angles are encoded as additional constraints. More parameters § Linkage model: The kinematics is defined by internal parameters (bond lengths and angles, and torsional angles around bonds) Drawback: Small local changes may have big global effects. Errors accumulate. Forces are more difficult to express § Simplified (f-y-c) linkage model: Fixed bond lengths, bond angles and torsional angles are directly embedded in the representation. Drawback: Fine tuning is difficult

In linkage model a small local change may have big global effect Computational errors may accumulate

Drawback of Homogeneous Coordinate Matrix x’ y’ z’ = 1 i 2 i 3 0 j 1 j 2 j 3 0 k 1 k 2 k 3 0 tx ty tz 1 x y z 1 Too many rotation parameters Accumulation of computing errors along a protein backbone and repeated computation Non-redundant 3 -parameter representations of rotations have many problems: singularities, no simple algebra A useful, less redundant representation of rotation is the unit quaternion

Unit Quaternion R(r, q) = (cos q/2, r 1 sin q/2, r 2 sin q/2, r 3 sin q/2) = cos q/2 + r sin q/2 R(r, q) R(r, q+2 p) Space of unit quaternions: Unit 3 -sphere in 4 -D space with antipodal points identified

Operations on Quaternions P = p 0 + p Q = q 0 + q Product R = r 0 + r = PQ r 0 = p 0 q 0 – p. q (“. ” denotes inner product) r = p 0 q + q 0 p + p q (“ ” denotes outer product) Conjugate of P: P* = p 0 - p

Transformation of a Point x = (x, y, z) quaternion 0 + x Transform of translation t = (tx, ty, tz) and rotation (n, q) Transform of x is x’ 0 + x’ = R(n, q) (0 + x) R*(n, q) + (0 + t)