Heat Capacity and Conformation of Small Protein Masayuki Irisa

1. International Workshop on Continuum Modeling of BiomoleculesSeptember 14-16, 2009 in Beijing, China Heat Capacity and Conformation of Small Protein Masayuki Irisa Department of Bioscience and bioinformatics, Faculty of Computer science and systems engineering, Kyushu Institute of Technology, Japan

2. Outline • Background • Heat capacity change of protein • Two state model (native and denatured states) • Entropy convergence • Hydration entropy • Methods • Extended scaled particle theory • Results • Hydration entropy • Hydration heat capacity • Comparison with experimental results • Conclusions • Acknowledgements

3. Two state model of protein • Native state (N) and Denatured state (D) • Experimental heat capacity change ΔCpis constant at ordinary temperatures. • ΔCp has been believed to be caused by hydration of a protein. • ΔCp values are proportional to changes of accessible surface area of a protein between a native and an extended conformation.

4. Heat capacity • Heat capacities at constant volume (Cv) or pressure.(Cp) • Heat capacity gives curvature of free energy • Knowing a enthalpy （H0） and entropy （S0） at a temperature, T0, • When Cp is constant at ordinary temperatures. • Ideal gases and solids Deby app. Solid Liquid Gas “Physical Chemistry” Atkins

5. Entropy Convergence at ~ 120℃ Normalized Entropy Changesfor protein denaturation with respect to weight converge at ~ 120℃ Hydration entropyfor rare gases converge at ~ 120℃ Privalov, P. L. & Gill, S. J. (1988) Adv. Prot. Chem. 39, 191-234 Garde, S. et.al., P.R.L., 77,4967(1996)

6. The origin of temperature convergence Why does temperature convergence occur ? • Contribution from hydrophobic hydration is the origin of the temperature convergence (Experimental results) Litian Fu and Ernesto Freire, “On the origin of the enthalpy and entropy convergence temperatures in protein folding”, Proc. Natl. Acad. Sci. USA,Vol. 89, 9335-9338 (1992) Then, How does hydrophobic hydration make convergence temperature? ・Role of water ・Protein conformation (G. Graziano treats a protein as a single sphere: SPT or van der Waals eq.) Theory which can treat both contributions from the shape, conformation, of a protein and from temperature is required. .

7. Scaled particle theory (SPT) • Hard-Sphere System: • Reiss,H.,Fish，H.L.，andLebowitz,J.L，J.Chem.Phys.，31，369(1959) • Aquerous Solution for nonpolar spherical solutes: • Pierotti,R.A.，Chem.Rev.,76，717(1965) • Hard-Convex System: • Gibbons,R.M.,Molec.Phys.,17,8l(1969) • BouBLik,T.,Molec.Phys.,27,1415(1974) • Aquerous solution for arbitrary shaped solutes: “Extended Scaled Particle Theory (XSPT)” • Irisa，M., Nagayama,K.,and Hirata,F.,Chem. Phys.Lett.,207,430(1993) • Irisa，M., Takuya,T., Nagayama,K.,and Hirata,F.,Molec.Phys.,95,1227 (1985)

8. ↑ ● ● ↑

9. P is the the excluded volume of λ-folded solute molecule, ρ is the number density of bulk water. Water Excluded Region First and Second derivatives of excluded volume are required

11. α-shape Computational geometry methods

12. Analytical expression for volume and area of fused spheres • Richmond,T.,Molec.Biol.,178,63(1984) • Analytical expression with Cartesian coordinate system. • Too much subdivisions lose the connection with the weighted Voronoi diagrams. • Irisa,M.,Comp. Phys.Comm.,98,317(1996) • Analytical expression with sphere radii and distances without using Cartesian coodinates. • Tight connection with weighted Voronoi diagrams and alpha shape.

13. Extended scaled particle theory (XSPT) • Merit: • The expression of hydration free energy is analytic. This theory is not empirical one, but gives an intuitively understandable picture containing fundamental measures in geometry ( similar to morphologic approach by R. Roth). • A solute molecule at any conformation can be treated as fused spheres in shape. • Ensure the exact expressions of hydration free energy at two limiting cases in hypothetical scaling • when λ is small enough, statistical mechanics gives the exact expression • when λ is large in macroscopic size, thermodynamics gives the exact expression • Demerit: • XSPT cannot be applied to a protein having holes and cavities where water molecules can enter because of discontinuity of excluded regions of a protein.

14. Hydration free energies for n-alkanes Irisa，M., Takuya,T., Nagayama,K.,and Hirata,F.,Molec.Phys.,95,1227 (1985)

15. Hydration free energies for n-alcohols Irisa，M., Takuya,T., Nagayama,K.,and Hirata,F.,Molec.Phys.,95,1227 (1985)

16. Target protein

17. D.Goldenberg (2003): Calculated distribution of denatured conformations of BPTI Accessible surface area

18. Results: excluded volume function

19. Results: hydration entropy 120℃

20. Hydration Entropy Contributions from C2 term and others C2 term only Others

21. Results:hydration heat capacity

22. Results: heat capacity change experiments calculations Experiment: G.Makhatadze et al. (1993) Values in denatured state were obtained by using a chemically modified BPTI Calculated ΔCp values are constant in the range between 0-100℃.

23. Is there any simple explanation? Bilenar assumption for entropy change of protein (Bilenear function has a saddle-like shape) B. Lee ,Proc. Nati. Acad. Sci. USA,88, 5154-5158 (1991) ΔS = s0 + s1 ln T , ΔS = a0 + a1 X X: physical quantity of each protein Accessible surface area? Excluded volume? ….. Our answer : Condition 1 has been reproduced by XSPT Condition 2 corresponds to the contribution from C2 term in gc (hydration free energy) X is a geometric part of C2 term in XSPT. ・Has the same dimension as volume. ・Corresponds to integral of mean curvature over the surface of a protein ・roughly proportional to the accessible surface area of a protein. ΔS X Ln T Entropy Convergence f(x,y)=xy

24. Boltzmann factor （kBT） shifts the maximum of hydration energy to a higher temperature (~120℃) The maximum of hydration entropy gives zero value of hydration entropy －＞ convergence temperature.. Term C2 in gc from XSPT: main contribution to Cp • Term C2 Why hydration entropy converges at 400K? Answer: Temperature dependence of a term C2 is caused by thermal expansion of water. gc/T has a maximum at 277K, but gc and C2 have that at 400K. Packing fraction η of water

25. Conclusions • XSPT reproduces the entropy-convergence temperature and heat capacity changes, ΔCp, qualitatively. At ordinary temperatures, ΔCp is almost constant. • Hydration entropy and heat capacity are dominated by C2 term in XSPT that is roughly proportional to square of the size of a protein. • Excluded volume function of a scaling parameter λ can be approximated as a third polynomial in the case of a protein. This enables us to apply XSTP to a protein having holes and cavities where water molecules can enter.

26. Acknowledgements • Kyushu Institute of Technology Hitomi Kitazato Nobue Kurose Kana Yamato Kenta Matsuo Junji Yasuniwa • Institute of Molecular Science Professor Fumio Hirata

28. Iizuka Campus

29. 研究室のメンバー Nobue Kurose Kenta Matsuo

30. Junji Yasuniwa