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Register a new userTopology, Geometry, and Stability: Protein Folding and Evolution
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Authors
Walter Simmons
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The protein folding problem must ultimately be solved on all length scales from the atomic up through a hierarchy of complicated structures. By analyzing the stability of the folding process using physics and mathematics, this paper shows that features without length scales, i.e. topological features, are potentially of central importance. Topology is a natural mathematical tool for the study of shape and we avail ourselves of that tool to examine the relationship between the amino acid sequence and the shapes of protein molecules. We apply what we learn to conjectures about their biological evolution.
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We adopt the point of view that analysis of the stability of the protein folding process is central to understanding the underlying physics of folding. Stability of the folding process means that many perturbations do not disrupt the progress from the random coil to the native state. In this paper we explore the stability of folding using established methods from physics and mathematics. Our result is a preliminary theory of the physics of folding. We suggest some tests of these ideas using folding simulations. We begin by supposing that folding events are related in some way to mechanical waves on the molecule. We adopt an analytical approach to the physics which was pioneered by M.V. Berry, (in another context), based upon mathematics developed mainly by R. Thom and V.I. Arnold. We find that the stability of the folding process can be understood in terms of structures known as caustics, which occur in many kinds of wave phenomena. The picture that emerges is that natural selection has given us a set of protein molecules which have mechanical waves that propagate according to several mathematically specific restrictions. Successful simulations of folding can be used to test and constrain these wave motions. With some additional assumptions the theory explains or is consistent with a number of experimental facts about folding. We emphasize that this wave-based approach is fundamentally different from energy-based approaches.
In spite of decades of research, much remains to be discovered about folding: the detailed structure of the initial (unfolded) state, vestigial folding instructions remaining only in the unfolded state, the interaction of the molecule with the solvent, instantaneous power at each point within the molecule during folding, the fact that the process is stable in spite of myriad possible disturbances, potential stabilization of trajectory by chaos, and, of course, the exact physical mechanism (code or instructions) by which the folding process is specified in the amino acid sequence. Simulations based upon microscopic physics have had some spectacular successes and continue to improve, particularly as super-computer capabilities increase. The simulations, exciting as they are, are still too slow and expensive to deal with the enormous number of molecules of interest. In this paper, we introduce an approximate model based upon physics, empirics, and information science which is proposed for use in machine learning applications in which very large numbers of sub-simulations must be made. In particular, we focus upon machine learning applications in the learning phase and argue that our model is sufficiently close to the physics that, in spite of its approximate nature, can facilitate stepping through machine learning solutions to explore the mechanics of folding mentioned above. We particularly emphasize the exploration of energy flow (power) within the molecule during folding, the possibility of energy scale invariance (above a threshold), vestigial information in the unfolded state as attractive targets for such machine language analysis, and statistical analysis of an ensemble of folding micro-steps.
Processes that proceed reliably from a variety of initial conditions to a unique final form, regardless of moderately changing conditions, are of obvious importance in biophysics. Protein folding is a case in point. We show that the action principle can be applied directly to study the stability of biological processes. The action principle in classical physics starts with the first variation of the action and leads immediately to the equations of motion. The second variation of the action leads in a natural way to powerful theorems that provide quantitative treatment of stability and focusing and also explain how some very complex processes can behave as though some seemingly important forces drop out. We first apply these ideas to the non-equilibrium states involved in two-state folding. We treat torsional waves and use the action principle to talk about critical points in the dynamics. For some proteins the theory resembles TST. We reach several quantitative and qualitative conclusions. Besides giving an explanation of why TST often works in folding, we find that the apparent smoothness of the energy funnel is a natural consequence of the putative critical points in the dynamics. These ideas also explain why biological proteins fold to unique states and random polymers do not. The insensitivity to perturbations which follows from the presence of critical points explains how folding to a unique shape occurs in the presence of dilute denaturing agents in spite of the fact that those agents disrupt the folded structure of the native state. This paper contributes to the theoretical armamentarium by directing attention to the logical progression from first physical principles to the stability theorems related to catastrophe theory as applied to folding. This can potentially have the same success in biophysics as it has enjoyed in optics.
The protein folding problem is stated and a list of properties that do not depend upon specific molecules is compiled and analyzed. The relationship of this analysis to future simulations is emphasized. The choice of power and time as variables as opposed to energy and time is discussed. A wave motion model is reviewed and related to the action in classical mechanics. It is argued that the properties of the action support the idea that folding takes place in small steps. It is explained how catastrophe theory has been employed in wave motion models and how it can be used in examination of successful simulations
A model to describe the mechanism of conformational dynamics in secondary protein based on matter interactions is proposed. The approach deploys the lagrangian method by imposing certain symmetry breaking. The protein backbone is initially assumed to be nonlinear and represented by the Sine-Gordon equation, while the nonlinear external bosonic sources is represented by $phi^4$ interaction. It is argued that the nonlinear source induces the folding pathway in a different way than the previous work with initially linear backbone. Also, the nonlinearity of protein backbone decreases the folding speed.
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