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

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

3. 3 Simulations from the PhD Years

4. 4 Hardware from the PhD Years

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

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

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

8. 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. 9 Definition of Rotational Differential Operators Let X be an infinitesimal rigid-body rotation. Then X R 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. 10 Euclidean Group, SE(3) An element of SE(3): Basis for the Lie Algebra: Small Motions

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

12. 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. 13 Variational Calculus on Lie groups Given the functional and constraints one can get the Euler-Poincaré equation as: where

14. 14 Explicit Formulations 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

15. 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. 16 Inverse Kinematics – Graphical Explanation

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

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

19. 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. 20 Simulation Results - Conformations Red line: isotropic Black line: anisotropic Blue line: histone-binding part

21. 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. 22 Conformational statistics of Conformational statistics of semiflexible polymers semiflexible polymers

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

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

25. 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. 26 Model Formulation Get the Schrödinger-like equation corresponding to H and quantization, p i = -j X R i, Apply the classical Fourier inversion formula

27. 27 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 Initial condition: f(a,R,0)=  (a)  (R) Defining A General Semiflexible Polymer Model

28. 28 Differential operators for SE(3)

29. 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. 30 Propagating By Convolution

31. 31 Operational Properties of Fourier Transform

32. 32 Entries of  (X i, p) for i=1,2,3

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

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

35. 35 Numerical Examples 2 1 0.5 0.1

36. 36 Numerical Examples HW5 HW2 HW3 HW1 KP

37. 37 The Structure of a Bent Macromolecular Chain 1)A bent macromolecular chain consists of two intrinsically straight segments. 2)A bend or twist is a rotation at the separating point between the two segments with no translation. A General Algorithm for Bent or Twisted Macromolecular Chains

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

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

40. 40 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 A General Algorithm for Bent or Twisted Macromolecular Chains

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

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

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

44. 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. 45 1) J. S. Kim, G. S. Chirikjian, ``Conformational Analysis of Stiff Chiral Polymers with End-Constraints,’’ Molecular Simulation 32(14):1139-1154. 2006 2) Y. Zhou, G. S. Chirikjian, ``Conformational Statistics of Semiflexible Macromolecular Chains with Internal Joints,’’ Macromolecules. 39:1950-1960. 2006 3) Zhou, Y., Chirikjian, G.S., “Conformational Statistics of Bent Semi-flexible Polymers”, Journal of Chemical Physics, vol.119, no.9, pp.4962-4970, 2003. 4) 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 E. References

46. 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