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

Alex J. LevineUCLA, Department of Chemistry & Biochemistry

UCLA Department of Biomathematics February 2006

Collaborator: Buddapriya Chakrabarti

1. Alex J. Levine “The helix/coil transition on the worm-like chain” Submitted to Physical Letters PRL Submitted

2. Buddhapriya Chakrabarti and Alex J. Levine “The nonlinear elasticity of an alpha-helical polypeptide” PRE 2005

3. Buddhapriya Chakrabarti and Alex J. Levine “Monte Carlo investigation of the nonlinear elasticity of an alpha-helical polypeptide.” PRE Submitted

References and collaborators

The space-filling picture showingall atoms.

• Carbon• Oxygen• Nitrogen

The cartoon picture showingthe secondary structures.

• The red is an -helix• The yellow is a - sheet• The gray is random coil

Carboxypeptidase: data from x-ray diffraction. I. Massova et al. J. Am. Chem. Soc. 118, 12479 (1996).

Protein mechanics: The appropriate level of description?

A first step toward protein mechanics

Proteins often change conformational states in a manner related to theirbiological activity.

Conformational change betweenapo and Calcium-loaded statesof 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

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

A. This model has internal degrees of freedom representing secondarystructure.

Coarse-graining the helix

The worm-like chain and semiflexible polymers

There is an energy cost associated with chain curvature – enhances thestatistical weight of straight conformations on the length scale /T

F-actin: Persistence length

MacKintosh, Käs, Jamney (1995)

Bending modulus depends on secondary structure

The bending stiffness of the alpha helix is enhanced by hydrogen bonding betweenhelical 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

1, HelixState

1, CoilStateis

Equivalent descriptions

n-1 n n+1

Tangent vector:

The Hamiltonian: HCWLC

1 1

1 10 0 0

ˆ ˆ(1 ) ( 1) ( ) 12 2

N N Nw

i i i i i ii i i

hH s s s s t t

Local thermodynamic driving force to native structure.

Energy cost of a domain wall between helical and random coil regions.

Helical regions are stifferthan random coil regions.

n-1 n n+1

The Hamiltonian: Energy scales

Local thermodynamic driving force to native structure.

Energy cost of a domain wall between helical and random coil regions.

From experimentand 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.

1

010

( ) exp ({ },{ })n

N N

n i i nsn

Z d H s

Fixing the ends

The Partition Function

(Two dimensional version)

Evaluating the partition function of the WLC

Exploiting the analogy between the partition function and the quantumpropagator of a particle on the unit circle

2

2exp cos2

mJ imJ

m

eJ e

J

2

11 ( )

2

0

( )2

nn n n

mN imn J

mn

dZ C e

We can writethe partition function:

2

2

are the angular momentum eigenstates.2

is the Hamiltonian where is the

angular momentum operator

im

L

J

em

H e L i

2 2

1( )2 2

1 1 1

nn n n

m Lim

J Jn n n n n ne m m e m m

2

20( ) 0

NL

JNZ C e

We can writethe partition function:

For the HCWLC model:

1

0 1

10 0 1 1

,

( ) , 0 ,N

N

NS N N

s s

Z s s

T

whereT TTTT T

, 1N N

hS s e

1 1

1 1

1 cos 1 cos

, 1 1 cos 1 cos

i i w i i

w i i i ii i

h h

e e

e e e e

T

The partition function as propagator:

with transfer matrix:

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:

,

( )w

w

m mh hs s

m m

em

e e

T

m mI e Where

Is the exact wave function with angular momentum m

In the diagonal representation

11

2

0,

0where

T Q D•Q D So

1N N T Q • D •Q

The eigenvalues and the partition function

Where:

m hm

m

z e

The fugacity of a coil segment at a given m

exp log 2 w Exponentiated Free Energy cost of a domain wall

m mI e Angular momentum (Worm-like chain) eigenstates

The Partition Function:

The Eigenvalues:

Making sense of Z: The expansion

0 1 22Z Z Z Z

Looking at the chain in the high cooperativity limit

0

AllHelix WLC AllCoil WLC

1im N h Nm m

m

Z e e z

All helix to all coil transition

1 1

1w

Nim N h m

m mm m

zZ e e z e

z

Boltzmann weight associated with one domain wall

Cost of changing one end to coil[left side + right side]

1

0

Npm

p

z

Sum over lengths of the coil region.

One domain wall.

Making sense of Z: Basic Phenomenology

Start with an -helix:

heterogeneous nucleation of random coil

homogeneous nucleation of random coil

Uponbending

Complete melting of secondary structure

Bending the helix: Buckling

log Z

Torque required to hold a bend of :

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

Buckling is related to coil nucleation

Fraction of the chain thecoil state

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

exp log 2 w

The mean length and force-extension curves

Applied force

To include applied forces:

Where (si) is the length of the ith segment.

Exact answers are difficult since one cannot simultaneously diagonalize momentumand position operators.

Helical segments areshorter

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 steprandom walks in momentum space

k

+

+

j

+

+

1

1

Two step walks

One step walks k

Simplification:Average over

Force extension curves: The mean length

Mean length at zero force, fixed angle

One step walks

+

k

m

( 1)N k m TΓ( )k mT

Denaturation experiments and mean length

Mean length as function of

h:

N = 10, < = 10, > = 100< = 1> = 3w = 6

Changing h is related to changes in solvent quality – i.e. denaturation experiments usingurea.

Force-extension relations: small force limit

Mean length vs. applied force for end-constrainedchains with:

> = 10, < = 1, N = 15.

F

F

> = 100, < = 1 N = 15.

Flexible Helix

Stiff 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 handchain and the junction.

Remaining angular integral

Force extension curves: Mean field analysis II

Helix WLCPseudo-plateau

Denaturation

Coil WLC

> = 2, < = 1, w = 10, h = 1.

> = 100, < = 1, w = 8, h = 2.

Monte Carlo Simulations I: Denaturation

For parameter values:

> = 100< = 1

w = 10.h = 8.0N=20.

The radius of gyration by Monte CarloTheory

2

Monte Carlo Simulations II: Force extension curves

For parameter values:

> = 100< = 1

w = 10.h = 8.0N = 20.

< = 1.0, > = 3.0

Force extension curves by Monte Carlo

Mean Field Theory

No applied torque

Monte Carlo Simulations III: Force extension curves with applied torque

For parameter values:

> = 100< = 1

w = 10.h = 8.0N = 20.

< = 1.0, > = 3.0

Force extension curves by Monte Carlo = 1.0 kB T

Summary

We understand the nonlinear elasticity of the HCWLC under torques and forces

2. We have calculated the extension of the chain in response to small forces.

3. We have calculated the extensional compliance within a mean field approximationand have explored non-mean field behavior via Monte Carlo simulations of the model.

1. Under large enough applied torques the chain undergoes a buckling instability: Does this bistability of the model underlie protein conformational change?

The big picture?