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A geometric analysis of protein folding, which complements many of the models in the literature, is presented. We examine the process from unfolded strand to the point where the strand becomes self-interacting. A central question is how it is possible that so many initial configurations proceed to fold to a unique final configuration. We put energy and dynamical considerations temporarily aside and focus upon the geometry alone. We parameterize the structure of an idealized protein using the concept of a ribbon from differential geometry. The deformation of the ribbon is described by introducing a generic twisting Ansatz. The folding process in this picture entails a change in shape guided by the local amino acid geometry. The theory is reparamaterization invariant from the start, so the final shape is independent of folding time. We develop differential equations for the changing shape. For some parameter ranges, a sine-Gordon torsion soliton is found. This purely geometric waveform has properties similar to dynamical solitons. Namely: A threshold distortion of the molecule is required to initiate the soliton, after which, small additional distortions do not change the waveform. In this analysis, the soliton twists the molecule until bonds form. The analysis reveals a quantitative relationship between the geometry of the amino acids and the folded form.
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The intricate three-dimensional geometries of protein tertiary structures underlie protein function and emerge through a folding process from one-dimensional chains of amino acids. The exact spatial sequence and configuration of amino acids, the biochemical environment and the temporal sequence of distinct interactions yield a complex folding process that cannot yet be easily tracked for all proteins. To gain qualitative insights into the fundamental mechanisms behind the folding dynamics and generic features of the folded structure, we propose a simple model of structure formation that takes into account only fundamental geometric constraints and otherwise assumes randomly paired connections. We find that despite its simplicity, the model results in a network ensemble consistent with key overall features of the ensemble of Protein Residue Networks we obtained from more than 1000 biological protein geometries as available through the Protein Data Base. Specifically, the distribution of the number of interaction neighbors a unit (amino acid) has, the scaling of the structures spatial extent with chain length, the eigenvalue spectrum and the scaling of the smallest relaxation time with chain length are all consistent between model and real proteins. These results indicate that geometric constraints alone may already account for a number of generic features of protein tertiary structures.
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.
The energy for protein folding arises from multiple sources and is not large in total. In spite of the many specific successes of energy landscape and other approaches, there still seems to be some missing guiding factor that explains how energy from diverse small sources can drive a complex molecule to a unique state. We explore the possibility that the missing factor is in the geometry. A comparison of folding with other physical phenomena, together with analytic modeling of a molecule, led us to analyze the physics of optical caustic formation and of folding behavior side-by-side. The physics of folding and caustics is ostensibly very different but there are several strong parallels. This comparison emphasizes the mathematical similarity and also identifies differences. Since the 1970s, the physics of optical caustics has been developed to a very high degree of mathematical sophistication using catastrophe theory. That kind of quantitative application of catastrophe theory has not previously been applied to folding nor have the points of similarity with optics been identified or exploited. A putative underlying physical link between caustics and folding is a torsion wave of non-constant wave speed, propagating on the dihedral angles and $Psi$ found in an analytical model of the molecule. Regardless of whether we have correctly identified an underlying link, the analogy between caustic formation and folding is strong and the parallels (and differences) in the physics are useful.
The microcanonical analysis is shown to be a powerful tool to characterize the protein folding transition and to neatly distinguish between good and bad folders. An off-lattice model with parameter chosen to represent polymers of these two types is used to illustrate this approach. Both canonical and microcanonical ensembles are employed. The required calculations were performed using parallel tempering Monte Carlo simulations. The most revealing features of the folding transition are related to its first-order-like character, namely, the S-bend pattern in the caloric curve, which gives rise to negative microcanonical specific heats, and the bimodality of the energy distribution function at the transition temperatures. Models for a good folder are shown to be quite robust against perturbations in the interaction potential parameters.
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
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