Elasticity and structural phase transitions in single biopolymer systems

1. 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)

2. 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 Int.Symp. Recent Progress in Quantitative and Systems Biology

3. 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).

4. Beijing Zhong-Can Ou-Yang Jie Zhou Yang Zhang Germany Reinhard Lipowsky India Sanjay Kumar Italy Martin Weigt USA Yang Zhang Collaborators Int.Symp. Recent Progress in Quantitative and Systems Biology

5. outline • Collapse transition: A brief introduction • Collapse transition in 2D: An exactly solvable model and it’s predictions • Collapse transition in 2D: Monte Carlo simulations on a more general model • Conclusion Int.Symp. Recent Progress in Quantitative and Systems Biology

6. Collapse transition of a long polymer can be driven by changes in • temperature, • solvent conditions, • external force field, • …

7. 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 --------------------------------------------------------------------- 2 dimensions induced by temperature second order? induced by external stretching second order?

8. ¢ ¡ ² 2-dimensional collapse transition: (first) an analytical approach Monomer-monomer contact (attractive) potential Bending stiffness External stretching Thermal energy Int.Symp. Recent Progress in Quantitative and Systems Biology

9. Qualitative behavior of the toy model At low temperature and/or low external stretching, the polymer prefer to be in globule conformations to maximize contacting interaction beta-sheet coil At high temperature and/or high external stretching, the polymer prefers to be extended coil conformations to maximize structural entropy Int.Symp. Recent Progress in Quantitative and Systems Biology

10. X ( ) = E T ¡ c Z e = l l f l F T Z ¡ a c o n n = : h i h i E T S ¡ = Phase transition theory  partition function  free energy Int.Symp. Recent Progress in Quantitative and Systems Biology

11. The total partition function Int.Symp. Recent Progress in Quantitative and Systems Biology

12. 1 ¡ n ¯ X [ ( ) ] ( ) l l f E ¢ i 1 2 1 ¡ ¡ ¡ + ¡ ² m n n a n = ¯ ¯ ¯ j j 1 0 + ; j 1 = 5 n = ¯ energetics Int.Symp. Recent Progress in Quantitative and Systems Biology

14. =0.002 =0 =0.01

15. 1 0 m = c 8 n = c f E ¢ ¡ + n a m = l i 0 c o c c energetics (continued) Int.Symp. Recent Progress in Quantitative and Systems Biology

18. + £ ¤ 1 ( ) ( ) ³ ³ G G 1 + N ¯ l i X c o ¡ ¢ ( ) = ( ) ³ ³ G Z N a = = N ( ) ( ) ³ ³ µ G G ¶ 1 ¡ ¯ l i c o ( ) = f T T N 0 ¡ ( ) g = Z N ; e = the total grand partition function free energy density of the system Int.Symp. Recent Progress in Quantitative and Systems Biology

19. ( ) ³ ¢ ¢ G 0 0 > = the total free energy density The free energy density of the system can be obtained by analyzing the singular property of the function Zero bending stiffness: second-order phase transit. Positive bending stiffness first-order phase transit. Int.Symp. Recent Progress in Quantitative and Systems Biology

20. Fixed force f=0

21. Fixed temperature T=0.591 

22. Scaling behaviors at fixed external force Int.Symp. Recent Progress in Quantitative and Systems Biology

23. The collapse transition of a 2D partially directed lattice polymer: • is a second-order structural phase transition, if the polymer chain is flexible (with zero bending energy penalty) • is a first-order structural phase transition, if the polymer chain is semi-flexible (with a positive bending energy penalty). Bending stiffness matters!

24. 2D polymer collapse transition:Monte Carlo simulations Int.Symp. Recent Progress in Quantitative and Systems Biology

25. Temperature-induced collapse transition in the case of zero bending stiffness or very small bending stiffness: support the picture of a second-order continuous phase transition Force=0 =0 Jie Zhou Int.Symp. Recent Progress in Quantitative and Systems Biology

26. Force=0 =0.3 Jie Zhou

27. Temperature-induced collapse transition in the case of relatively large bending stiffness: support the picture of a first-order discontinuous phase transition Force=0 =5 N=800 monomers Temperature changes near 2.91 Jie Zhou Int.Symp. Recent Progress in Quantitative and Systems Biology

28. Force-induced collapse transition: support the picture of a first-order discontinuous phase transition for positive bending stiffness Force changes near 0.66 =5 N=500 monomers T=2.5 Jie Zhou Int.Symp. Recent Progress in Quantitative and Systems Biology

29. 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 Int.Symp. Recent Progress in Quantitative and Systems Biology