1. The worm turns: The helix-coil transition on the worm-like chain
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, (1996).
The space-filling picture showing
all atoms.
The cartoon picture showing
the secondary structures.
Carbon
Oxygen
Nitrogen
The red is an -helix
The yellow is a  - sheet
The gray is random coil

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. 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
F-actin: Persistence length
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 n n+1
Equivalent descriptions
Tangent vector:

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

10. The Hamiltonian: Energy scales
Local thermodynamic
driving force to native structure.
Energy cost of a domain wall
between helical and random coil regions.
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. 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 So

15. The eigenvalues and the partition function
Where:
Angular momentum (Worm-like chain) eigenstates
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. The mean length and force-extension curves
Applied force
To include applied forces:
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 
One step walks k + k j
Two step walks + 1 + 1 +

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

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

28. Monte Carlo Simulations I: Denaturation
The radius of gyration by Monte Carlo
Theory 2
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
Force extension curves by Monte Carlo
 = 1.0 kB T
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?