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

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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

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

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

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).

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 thehelix

Q. How is this different from the simple worm-like chain?

A. This model has internal degrees of freedom representing secondary

structure.

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)

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

The helix/coil worm-like chain: Pictorial

n-1 n n+1

Equivalent descriptions

Tangent vector:

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.

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:

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)

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:

For the HCWLC model:

The partition function as propagator:

with transfer matrix:

where

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

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:

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.

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

Bending the helix: Buckling

Torque required to hold a bend of :

Buckling instability!

(N = 15, > = 100, < = 1, h = 3.)

Buckling is related to coil nucleation

Fraction of the chain the

coil state

The buckling effect is associated with the appearance of coil regions.

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).

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:

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

+

k

+

Force extension curves: The mean length

Mean length at zero force, fixed angle

One step walks

m

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.

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

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

> = 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

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.

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

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

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.

The big picture?