Statistics and Minimal Energy Comformations of Semiflexible Chains

1. Statistics and Minimal Energy Comformations of Semiflexible Chains Gregory S. Chirikjian Department of Mechanical Engineering Johns Hopkins University

2. Overview of Topics • My Background • Kinematic analysis • Equilibrium conformations of chiral semi-flexible polymers with end constraints • Probabilistic analysis • Conformational statistics of semiflexible polymers

3. Simulations from the PhD Years

4. Hardware from the PhD Years

5. Equilibrium conformations of chiral semi-flexible polymers with end constraints

6. Inextensible Continuum Model • Elastic potential energy: • Inextensible constraint

7. A General Semiflexible Polymer Model The general representation of U KP model: c=0 Yamakawa model: MS model:

8. Definition of a Group • A group is a set together with a binary operation o satisfying: • Associative: a o (b o c) = (a o b) o c • Identity: e o a = a • Inverse: a-1 o a = e • Binary operation o: a o b  G whenever a,b  G • Examples: {R, +} where e=0; a-1 =-a; rotations; rigid-body motions

9. Definition of Rotational Differential Operators • Let X be an infinitesimal rigid-body rotation. Then • XR can be thought of as the right directional derivative of f in the direction X. In particular, infinitesimal rigid-body rotation in the plane are all combinations of:

10. Euclidean Group, SE(3) • An element of SE(3): • Basis for the Lie Algebra: Small Motions

11. Body-fixed frame Space-fixed frame Lie-group-theoretic Notation • Coordinates free  no singularities

12. Extensible Continuum Model • We can extend inextensible model by adding parameters such as stretching stiffness, shear stiffness, twist-stretch coupling factor, etc. • This model, and the inextensible one, do not include self-contact, which can be included by adding another potential function.  Note: no constraints

13. VariationalCalculus on Lie groups • Given the functional and constraints one can get the Euler-Poincaré equation as: where

14. Inextensible  Can be solved iteratively with I.C. (0) =  and given , together with  Position a(s) is determined by the constraint. Extensible where  Can be solved iteratively with I.C. (0),together with Explicit Formulations

15. How to get to the desired pose • To reach the desired position and orientation , we need an inverse kinematics. • Let be the vector of undetermined coefficients ((0) for extensible case), and denote the distal frame for a given  as • Let • Define an artificial path functions which satisfy • Use Jacobian-velocity relation and position correction term.

16. Inverse Kinematics – Graphical Explanation

17. Graphic Explanation – Cont’d Initial conformation Final conformation

18. Example – histone binding DNA • N: number of base pairs, varying from 351 to 366. • w: wrapping of DNA around the cylindrical histone molecule, 1.40 or 1.75. • hb: helical repeat length in bound section = 10.40 [bp/turn] • Pitch=2.7 nm, diameter=8.6 nm Swigon, et al., Biophysical Journal, 1998, Vol. 74, p.2515-2530. F. D. Lucia, et al. J. Mol. Biol. 289:1101, 1999.

19. Simulation Results • N: number of base pairs, w: number of wraps, Lk: linking number, Wr: Writhe, E: elastic energy of the loop. • Experimental data from F. D. Lucia, et al. J. Mol. Biol. 289:1101, 1999.

20. Simulation Results - Conformations • Red line: isotropic • Black line: anisotropic • Blue line: histone-binding part

21. Conclusions for Part I • A new method for obtaining the minimal energy conformations of semi-flexible polymers with end constraints is presented. • Our method includes variational calculus associated with Lie groups and Lie algebras. • We also present a new inverse kinematics procedure. • Numerical examples are in good agreement with the experimental results published. • Extensible model can be used to do the same if all parameters are known.

22. Conformational statistics of semiflexible polymers

23. A General Semiflexible Polymer Model Elastic Energy of an Inextensible Chiral Elastic Chain with L B b (s) Total arc length Stiffness matrix Chirality vector Spatial angular velocity

24. Model Formulation • Potential energies of bending and twisting of a stiff chain (e.g. see [Yamakawa]) • Path integral over the rotation group

25. Model Formulation • Apply the classical Fourier transform w.r.t. a • Treat the inner most integrand as j times a Lagrangian with • Calculates the momenta and Hamiltonian

26. Model Formulation • Get the Schrödinger-like equation corresponding to H and quantization, pi = -j XRi , • Apply the classical Fourier inversion formula

27. A General Semiflexible Polymer Model A diffusion equation describing the PDF of relative pose between the frame of reference at arc length s and that at the proximal end of the chain Defining Initial condition: f(a,R,0)= (a) (R)

28. Differential operators for SE(3)

29. Fourier Analysis of Motion • Fourier transform of a function of motion, f(g) • Inverse Fourier transform of a function of motion where g SE(N) , p is a frequency parameter, U(g,p) is a matrix representation of SE(N), and dg is a volume element at g.

30. Propagating By Convolution

31. Operational Properties of Fourier Transform

32. Entries of (Xi , p) for i=1,2,3

33. Entries of (Xi , p) for I = 4,5,6

34. A General Semiflexible Polymer Model Solving for the evolving PDF Applying Fourier transform for SE(3) where B is a constant matrix. Solving ODE Applying inverse transform

35. Numerical Examples 2 1 0.5 0.1

36. Numerical Examples HW5 HW2 KP HW1 HW3

37. A General Algorithm for Bent or Twisted Macromolecular Chains The Structure of a Bent Macromolecular Chain • A bent macromolecular chain consists of two intrinsically straight segments. • A bend or twist is a rotation at the separating point between the two segments with no translation.

38. A General Algorithm for Bent or Twisted Macromolecular Chains The PDF of the End-to-End Pose for a Bent Chain 1) A convolution of 3 PDFs • f1(a,R) and f3(a,R) are obtained by solving the differential equation for nonbent polymer. • f2(a,R)= (a)(Rb-1R), where Rb is the rotation made at the bend. 2) The convolution on SE(3)

39. A General Algorithm for Bent or Twisted Macromolecular Chains Computing the Convolution using Fourier Transform for SE(3) 1) An operational property 2) Fourier transform of the 3-convolution where

40. A General Algorithm for Bent or Twisted Macromolecular Chains Two Important Marginal PDFs 1) The PDF of end-to-end distance 2) The PDF of end-to-end distance and the angle between the end tangents

41. Examples 1. Variation of f(a) with respect to Bending Angle and Bending Location__KP Model

42. Examples 2. Variation of f(a) with respect to Bending Angle and Bending Location__Yamakawa Model

43. Examples 3. Variation of f(a) with respect to Bending Angle and Bending Location__MS Model

44. Conclusions for Part II • A method for finding the probability of reaching any relative end-to-end position and orientation has been developed • It uses the irreducible unitary representations of the Euclidean motion group and associated Fourier transform • The operational properties of this transform convert the Fokker-Planck equation into a linear system of ODEs in Fourier space. • The group Fourier transform can be used to `stitch together’ pdfs of segments joined by joints or at discrete angles.

45. E. References • J. S. Kim, G. S. Chirikjian, ``Conformational Analysis of Stiff Chiral Polymers with End-Constraints,’’ Molecular Simulation 32(14):1139-1154. 2006 • Y. Zhou, G. S. Chirikjian, ``Conformational Statistics of Semiflexible Macromolecular Chains with Internal Joints,’’ Macromolecules. 39:1950-1960. 2006 • Zhou, Y., Chirikjian, G.S., “Conformational Statistics of Bent Semi-flexible Polymers”, Journal of Chemical Physics, vol.119, no.9, pp.4962-4970, 2003. • G. S. Chirikjian, Y. Wang, ``Conformational Statistics of Stiff Macromolecules as Solutions to PDEs on the Rotation and Motion Groups,’’ Physical Review E. 62(1):880-892. 2000

46. Acknowledgements • This work was done mostly by my former students: Dr. Yunfeng Wang, Dr. Jin Seob Kim, and Dr. Yu Zhou • This work was partially supported by NSF and NIH