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

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).

Dec 09-11,2006Int.Symp. Recent Progress in Quantitative and Systems Biology 4 Collaborators  Beijing Zhong-Can Ou-Yang Jie Zhou Yang Zhang  Germany Reinhard Lipowsky  India Sanjay Kumar  Italy Martin Weigt  USA Yang Zhang

Dec 09-11,2006Int.Symp. Recent Progress in Quantitative and Systems Biology 5 outline 1.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 dimensions induced by temperaturesecond order? induced by external stretching second order?

Dec 09-11,2006Int.Symp. Recent Progress in Quantitative and Systems Biology 8 2-dimensional collapse transition: (first) an analytical approach Monomer-monomer contact (attractive) potential Bending stiffness External stretching Thermal energy ¢ ¡ ²

Dec 09-11,2006Int.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 At high temperature and/or high external stretching, the polymer prefers to be extended coil conformations to maximize structural entropy beta-sheetcoil

Dec 09-11,2006Int.Symp. Recent Progress in Quantitative and Systems Biology 10 Phase transition theory  partition function  free energy Z = X a ll con f : e ¡ E ( c )= T F = ¡ T l n Z = h E i ¡ T h S i

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

Dec 09-11,2006Int.Symp. Recent Progress in Quantitative and Systems Biology 12 energetics E ¯ = ¡ ² n ¯ ¡ 1 X j = 1 [ m i n ( l j ; l j + 1 ) ¡ 1 ] ¡ n ¯ f a ( n ¯ ¡ 1 ) ¢ n ¯ = 5

=0=0.01 =0.002

Dec 09-11,2006Int.Symp. Recent Progress in Quantitative and Systems Biology 15 energetics (continued) n c = 8 m c = 10 E co i l = ¡ n c f a 0 + m c ¢

Dec 09-11,2006Int.Symp. Recent Progress in Quantitative and Systems Biology 18 the total grand partition function G ( ³ ) = + 1 X N = 0 ¡ ³ = a ¢ N Z ( N ) = £ 1 + G ¯ ( ³ ) ¤ G co i l ( ³ ) 1 ¡ G ¯ ( ³ ) G co i l ( ³ ) free energy density of the system Z ( N ) = µ e ¡ g ( f ; T )= T ¶ N

Dec 09-11,2006Int.Symp. Recent Progress in Quantitative and Systems Biology 19 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 

Dec 09-11,2006Int.Symp. Recent Progress in Quantitative and Systems Biology 22 Scaling behaviors at fixed external force

The collapse transition of a 2D partially directed lattice 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!

Dec 09-11,2006Int.Symp. Recent Progress in Quantitative and Systems Biology 24 2D polymer collapse transition: Monte Carlo simulations

Dec 09-11,2006Int.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

Force=0 =0.3 Jie Zhou

Dec 09-11,2006Int.Symp. Recent Progress in Quantitative and Systems Biology 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 Jie Zhou N=800 monomers Temperature changes near 2.91

Dec 09-11,2006Int.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 Jie Zhou N=500 monomers T=2.5

Dec 09-11,2006Int.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