elasticity and structural phase transitions in single biopolymer systems haijun zhou ( 周海军 )...

Elasticity and structural phase transitions in single biopolymer systems

Haijun Zhou ( 周海军 )

Institute of Theoretical Physics, the Chinese Academy of Sciences, Beijing( 中国科学院理论物理研究所，北京 )

Dec 09-11,2006 Int.Symp. Recent Progress in Quantitative and Systems Biology

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Application of statistical physics ideas to complex systems

Bio-polymers: elasticity and structural transitions of DNA, RNA, and

proteins

Bio-mimetic networks:topology, dynamics, and topology evolutions

Systems with quenched disorders:spin-glasses, hard combinatorial optimizations

problems

some publications

“Bending and base-stacking interactions in double-stranded DNA”, (1999).

“Stretching Single-Stranded DNA: Interplay of Electrostatic, Base-Pairing, and Base-Pair Stacking Interactions”, (2001).

“Hierarchical chain model of spider capture silk elasticity”, (2005).

“Long-Range Frustration in a Spin-Glass Model of the Vertex-Cover Problem”, (2005).

“Message passing for vertex covers”, (2006).

“Distance, dissimilarity index, and network community structure”, (2003).

“Dynamic pattern evolution on scale-free networks”, (2005).


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Collaborators Beijing

Zhong-Can Ou-Yang Jie Zhou Yang Zhang

Germany Reinhard Lipowsky

India Sanjay Kumar

Italy Martin Weigt

USA Yang Zhang


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outline1. Collapse transition: A brief introduction

2. Collapse transition in 2D: An exactly solvable model and it’s predictions

3. Collapse transition in 2D: Monte Carlo simulations on a more general model

4. Conclusion

Collapse transition of a long polymer can be driven by changes in (1) temperature, (2) solvent conditions, (3) external force field, (4) …

The order of the collapse transition has been an issue of debate for many years.

3 dimensions

induced by temperature second order ?

induced by external stretching first order

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2 dimensions

induced by temperature second order?

induced by external stretching second order?


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2-dimensional collapse transition: (first) an analytical approach

Monomer-monomercontact (attractive)potential

Bending stiffness

External stretching

Thermal energy

¢¡ ²


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Qualitative behavior of the toy model

At low temperature and/or low external stretching, the polymerprefer to be in globule conformations to maximize contactinginteraction

At high temperature and/or high external stretching, the polymerprefers to be extended coil conformations to maximize structuralentropy

beta-sheet coil


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Phase transition theory

partition function

free energy

Z =X

all conf:

e¡ E (c)=T

F = ¡ T lnZ

= hE i ¡ ThSi


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The total partition function


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energetics

E ¯ = ¡ ²n¯ ¡ 1X

j =1

[min(lj ; lj +1) ¡ 1]¡ n¯ f a0 + 2(n¯ ¡ 1)¢

n¯ = 5

=0 =0.01

=0.002


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energetics (continued)

nc = 8

mc = 10

Ecoil = ¡ ncf a0 + mc¢


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the total grand partition function

G(³) =+1X

N =0

¡³=a

¢N Z(N ) =

£1+ G¯ (³)

¤Gcoil(³)

1¡ G¯ (³)Gcoil(³)

free energy density of the system

Z(N ) =

µe¡ g(f ;T )=T

¶N


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the total free energy density

The free energy density of the system can be obtained by analyzing the singular property of the function G(³)

Zero bending stiffness:second-order phase transit.

Positive bending stiffness

first-order phase transit.

¢ = 0

¢ > 0

Fixed force f=0

Fixed temperature T=0.591


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Scaling behaviors at fixed external force

The collapse transition of a 2D partially directedlattice polymer:

(a) is a second-order structural phase transition, if the polymer chain is flexible (with zero bending energy penalty)

(b) is a first-order structural phase transition, if the polymer chain is semi-flexible (with a positive bending energy penalty).

Bending stiffness matters!


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2D polymer collapse transition:Monte Carlo simulations


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Temperature-induced collapse transition in the case ofzero bending stiffness or very small bending stiffness: support the picture of a second-order continuous phase transition

Force=0

=0

Jie Zhou

Force=0

=0.3

Jie Zhou


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Temperature-induced collapse transition in the case ofrelatively large bending stiffness: support the picture of a first-order discontinuous phase transition

Force=0

=5

Jie Zhou

N=800 monomers

Temperaturechanges near 2.91


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Force-induced collapse transition: support the picture of a first-order discontinuous phase transitionfor positive bending stiffness

Force changesnear 0.66

=5

Jie Zhou

N=500 monomers

T=2.5


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Conclusion

Bending stiffness can qualitatively influence the co-operability of the globule-coil structural phase transition of 2D polymers

The collapse transition of 2D semi-flexible polymers can be first-order

elasticity and structural phase transitions in single biopolymer systems haijun zhou ( 周海军 )...

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