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Energy Landscapes of Macromolecules with Unique 3D Structures

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Abstract—A paradox of the convergence of the potential energy surface (PES) presented as a sum of pairwise interactions was formulated. Regularities of the formation of a multidimensional PES in the space of torsion (dihedral) angles were considered for the case of the macromolecules that form unique 3D structures. The presence of a single global minimum on the PES was shown to be impossible for nonchiral macromolecules. The chirality of the spatial structure of a macromolecule creates conditions for the formation of a single global minimum on the PES. The structure was studied for a model PES such that components of the multidimensional Fourier series exponentially damped with the increasing number of harmonics. It was proposed to describe the interaction between conformational degrees of freedom in the space of torsion angles by assigning distribution functions to linear combinations of harmonic numbers. A mathematical tool was developed for this purpose. The structure of the PES was studied for the cases of the Gaussian and Lorentzian distribution functions for the linear combinations of the Fourier series harmonic numbers. It was shown that the properties of such a PES can be described by introducing two generalized variables. A feature of the PES is the existence of a central funnel, which leads to a global energy minimum, and satellite funnels, which acts as traps during the folding process. Relatively rapid folding events (the achievement of the global energy minimum) may take place in the configuration space region that corresponds to the central funnel. This structure of the PES makes it possible to identify the configuration space areas that are important for the folding and to understand the basic difference between reversible (in solution) and irreversible (using an atomic force microscope) unfolding of unique 3D structures of biopolymers.

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ACKNOWLEDGMENTS

I am grateful to M.S. Sidorenko for help in constructing function graphs; G.A. Armeev, F.Yu. Popelenskii, A.A. Tuzhilin, and other attendees of regular seminars on structural biology (the Biological Faculty and Mechanical and Mathematical Faculty of Moscow State University) for fruitful mathematical discussions; N.K. Balabaev and A.V. Finkel’shtein for discussing the results; and M.P. Kirpichnikov and A.T. Fomenko for attention and support of this work.

This work was supported by the Russian Science Foundation (project nos. 14-24-00031 (the second section of the article) and 14-50-00029 (the third section)). The basics of the dynamics of conformationally mobile systems (the first section of the article) were developed as part of the state program of the Federal Agency for Scientific Organizations of Russia (problem no. 0082-2014-0001, project no. AAAA-A17-117040610310-6).

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Authors and Affiliations

  1. Moscow State University, 119234, Moscow, Russia

    K. V. Shaitan

  2. Semenov Institute of Chemical Physics, Russian Academy of Sciences, 119991, Moscow, Russia

    K. V. Shaitan

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  1. K. V. Shaitan
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Correspondence to K. V. Shaitan.

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Translated by T. Tkacheva

1Abbreviations: PES, potential energy surface.

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Shaitan, K.V. Energy Landscapes of Macromolecules with Unique 3D Structures. BIOPHYSICS 63, 485–496 (2018). https://doi.org/10.1134/S0006350918040152

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