The worm turns: The helix-coil transition on the worm-like chain

1. UCLA Department of Biomathematics February 2006 The worm turns:The helix-coil transition on the worm-like chain Alex J. Levine UCLA, Department of Chemistry & Biochemistry

2. References and collaborators • Alex J. Levine “The helix/coil transition on the worm-like chain” Submitted to Physical Letters PRL Submitted • Buddhapriya Chakrabarti and Alex J. Levine “The nonlinear elasticity of an alpha-helical polypeptide” PRE 2005 • Buddhapriya Chakrabarti and Alex J. Levine “Monte Carlo investigation of the nonlinear elasticity of an alpha-helical polypeptide.” PRE Submitted Collaborator: Buddapriya Chakrabarti

3. Protein mechanics: The appropriate level of description? Carboxypeptidase: data from x-ray diffraction. I. Massova et al. J. Am. Chem. Soc.118, 12479 (1996). The cartoon picture showing the secondary structures. The space-filling picture showing all atoms. • The red is an -helix • The yellow is a  - sheet • The gray is random coil • Carbon • Oxygen • Nitrogen

4. A first step toward protein mechanics Proteins often change conformational states in a manner related to their biological activity. Conformational change between apo and Calcium-loaded states of Calmodulin N-terminal domains. From: S. Meiyappan, R. Raghavan, R. Viswanathan, Y Yu, and W. Layton Preprint (2004).

5. Towards a lower-dimensional dynamical model Proposal: Take secondary structures as fundamental, nonlinear elastic elements Coarse-grained mechanics informed by multi-scale numerical modeling Coarse-graining thehelix Q. How is this different from the simple worm-like chain? A. This model has internal degrees of freedom representing secondary structure.

6. F-actin: Persistence length The worm-like chain and semiflexible polymers There is an energy cost associated with chain curvature – enhances the statistical weight of straight conformations on the length scale/T MacKintosh, Käs, Jamney (1995)

7. Bending modulus depends on secondary structure The bending stiffness of the alpha helix is enhanced by hydrogen bonding between helical turns. A model treating secondary structure as a two-state variable – Helix/coil A model for the conformational degrees of freedom of a semiflexible chain – Worm-like chain Couple them through the bending modulus: Helix/coil on the worm-like chain

8. The helix/coil worm-like chain: Pictorial n-1 n n+1 Equivalent descriptions Tangent vector:

9. n-1 n n+1 Energy cost of a domain wall between helical and random coil regions. Local thermodynamic driving force to native structure. The Hamiltonian: HCWLC Helical regions are stiffer than random coil regions.

10. Energy cost of a domain wall between helical and random coil regions. Local thermodynamic driving force to native structure. The Hamiltonian: Energy scales From experiment and simulation: H.S. Chan and K.A. Dill J. Chem. Phys. 101, 7007 (1994) A.-S. Yang and B. Honig J. Mol. Biol. 252, 351 (1995). From geometry and hydrogen bonding energies:

11. Exploring the model: Exploring the role of twist One polymer trajectory consistent with the boundary conditions. The Partition Function Fixing the ends (Two dimensional version)

12. Evaluating the partition function of the WLC Exploiting the analogy between the partition function and the quantum propagator of a particle on the unit circle We can write the partition function: We can write the partition function:

13. For the HCWLC model: The partition function as propagator: with transfer matrix: where

14. So Exploring the model: Exploring the role of bending In d = 2 using: We can now diagonalize the transfer matrix in angular momentum space (conjugate ) and in s-space: Where Is the exact wave function with angular momentum m In the diagonal representation

15. Angular momentum (Worm-like chain) eigenstates The eigenvalues and the partition function The Eigenvalues: Where: The fugacity of a coil segment at a given m Exponentiated Free Energy cost of a domain wall The Partition Function:

16. Making sense of Z: The  expansion Looking at the chain in the high cooperativity limit All helix to all coil transition One domain wall. Boltzmann weight associated with one domain wall Cost of changing one end to coil [left side + right side] Sum over lengths of the coil region.

17. Making sense of Z: Basic Phenomenology Start with an-helix: homogeneous nucleation of random coil Upon bending heterogeneous nucleation of random coil Complete melting of secondary structure

18. Bending the  helix: Buckling Torque required to hold a bend of : Buckling instability! (N = 15, > = 100, < = 1, h = 3.)

19. Buckling is related to coil nucleation Fraction of the chain the coil state The buckling effect is associated with the appearance of coil regions.

20. Applied force To include applied forces: The mean length and force-extension curves Where (si) is the length of the ith segment. Helical segments are shorter Exact answers are difficult since one cannot simultaneously diagonalize momentum and position operators. Numerical diagonalization and variational calculations e.g. J. F. Marko and E. Siggia Macromol. 28, 8759 (1995).

21. Force extension curves: Low force expansions I We can expand the partition function in powers of f: Where averages are computed with respect to the zero force Hamiltonian: We need to compute terms of the form:

22. Force extension curves: Low force expansions II Since: The length of the chain up to order Fn can be computed by examining all n+1 step random walks in momentum space Simplification: Average over  k One step walks + k j Two step walks + 1 + 1 +

23. k + Force extension curves: The mean length Mean length at zero force, fixed angle One step walks m

24. Denaturation experiments and mean length Mean length as function of h: N = 10, < = 10, > = 100 < = 1 > = 3 w = 6 Changing h is related to changes in solvent quality – i.e. denaturation experiments using urea.

25. Force-extension relations: small force limit Mean length vs. applied force for end-constrained chains with: > = 100, < = 1 N = 15. Stiff Helix F > = 10, < = 1, N = 15. F Flexible Helix

26. Force-extension curves: Mean-field analysis We write the free energy as the sum of the free energies of the left hand chain, the right hand chain and the junction. Remaining angular integral

27. > = 2, < = 1, w = 10, h = 1. > = 100, < = 1, w = 8, h = 2. Force extension curves: Mean field analysis II Coil WLC Denaturation Pseudo-plateau Helix WLC

28. 2 Monte Carlo Simulations I: Denaturation The radius of gyration by Monte Carlo Theory For parameter values: > = 100 < = 1 w = 10. h = 8.0 N=20.

29. Monte Carlo Simulations II: Force extension curves No applied torque Force extension curves by Monte Carlo For parameter values: > = 100 < = 1 w = 10. h = 8.0 N = 20. < = 1.0, > = 3.0 Mean Field Theory

30. Monte Carlo Simulations III: Force extension curves with applied torque  = 1.0 kB T Force extension curves by Monte Carlo For parameter values: > = 100 < = 1 w = 10. h = 8.0 N = 20. < = 1.0, > = 3.0

31. Summary We understand the nonlinear elasticity of the HCWLC under torques and forces • Under large enough applied torques the chain undergoes a buckling instability: • Does this bistability of the model underlie protein conformational change? • We have calculated the extension of the chain in response to small forces. • We have calculated the extensional compliance within a mean field approximation • and have explored non-mean field behavior via Monte Carlo simulations of the • model.

32. The big picture?