Free-Energy Landscapes for Macromolecules that Form a Unique 3D Structure
- 5 Downloads
Abstract—The entropic effects for the energy landscapes of macromolecules are considered. We use the ideas and methods of multidimensional geometry and topology, expansion into a multidimensional Fourier series for a potential-energy surface, and the Gaussian distribution function for the expansion coefficients to describe the interaction between conformational degrees of freedom to investigate the free energy surface in the space of dihedral (torsion) angles. We show that the free-energy surface for a macromolecule that forms a unique 3D structure in the framework of the proposed approach has the following features. The main properties of the free-energy surface can be represented in terms of a two-dimensional surface, which depends on two generalized variables. At low temperatures, the multidimensional space of torsion angles can be divided into region that belongs to the deepest central energy funnel and the region of satellite funnels. The funnels are separated by relatively high energy barriers, whose height decreases as the representative point moves up further away from the bottom of the funnel. The depths of satellite funnels decrease while the distances from the central funnel becomes larger. The funnels are surrounded by smooth entropic barriers. The height of entropic barrier is at a minimum for the central funnel, while the entropic barriers heights of satellite funnels increase as funnels move further away from the central funnel. As the temperature increases, the satellite funnels are destroyed, starting from the most distant central one. When the temperature is higher than the critical point (proportional to the specific gain in the potential energy during the formation of a unique 3D structure) the compact state becomes unstable and the global minimum of free energy disappears. As the number of interacting torsion angles (the length of the polymer chain) increases, the critical temperature becomes higher.
Keywords:3D structure of proteins energy funnel free-energy surface entropy barrier transition temperature folding structural biology energy landscapes
The author is grateful to M.S. Sidorenko for his help with plotting the graphs and G.A. Armeev, F.Yu. Popelenskii, and A.A. Tuzhilin, as well as other participants of the joint seminar of departments of biology and mechanics and mathematics of Moscow State University on structural biology for fruitful discussions. The author is grateful to A.V. Finkelstein for stimulating discussions and M.P. Kirpichnikov and A.T. Fomenko for attention to the work and general support.
This work was supported by the Russian Science Foundation (project no. 14-50-00029; sections Energy Surface… and Determination of Configurational Entropy…) and Russian Foundation for Basic Research (project no. 18-504-12045; section Free Energy Surface…). The bases of the dynamics of conformationally labile systems were developed in the framework of the state task of the Federal Agency of Scientific Organizations of Russia (theme 0082-2014-0001, no. AAAA-A17-117040610310-6).
- 4.S. O. Garbuzynskiy D. N. Ivankov, N. S. Bogatyreva, and A. V. Finkelstein, Proc. Natl. Acad. Sci. U. S. A. 110 (1), 147 (2013).Google Scholar
- 8.A. Zygmund, Trigonometric Series (Cambridge Univ. Press, Cambridge, 1960; Mir, Moscow, 1965), Vol. 1.Google Scholar
- 9.M. V. Fedoryuk, Saddlepoint Method (URSS, Moscow, 2009) [in Russian].Google Scholar
- 10.J. W. Gibbs, Elementary Principles in Statistical Mechanics (Charles Scribner’s Sons, New York, 1902; Nauka, Moscow, 1982).Google Scholar
- 11.Ya. I. Frenkel, Statistical Physics (USSR Acad. Sci., Moscow, 1948) [in Russian].Google Scholar
- 12.V. A. Zorich, Mathematical Analysis (NTsNMO, Moscow, 2001), Part 1 [in Russian].Google Scholar
- 13.A. Yu. Grosberg and A. R. Khokhlov, Giant Molecules: Here, There, and Everywhere, 2nd ed. (World Scientific, Singapore, 2011).Google Scholar
- 14.A. V. Finkelstein, Physics of Protein Molecules (Izhevsk Inst. Computer Research, Izhevsk–Moscow, 2014) [in Russian].Google Scholar