Ультраметрическое случайное блуждание и динамика белковых молекул
This paper is a brief survey of applications of the p-adic equation of ultrametric random walk to the description of conformational dynamics of protein molecules. Two main experiments are considered that determine the properties of the fluctuation dynamic mobility of protein molecules from 300 to 4 K: the analysis of the kinetics of CO binding to myoglobin and spectral diffusion in proteins. It is shown that an ultrametric description allows one to build a unified picture of the conformational mobility of a protein molecule in the whole range of the indicated temperatures and realize the fact that it varies in a self-similar way. This feature of protein molecules, which has remained hidden to date, significantly expands the idea of the structure of nanoscale systems related to the family of molecular machines.
In this paper, the basic notions of ultrametric (p-adic) description of protein conformational dynamics and CO rebinding to myoglobin are presented. It is shown that one and the same model of the reaction — ultrametric diffusion type describes essentially different features of the rebinding kinetics at high-temperatures (300÷200 K) and low- temperatures (180÷60 K). We suggest this result indicates a special structural order in a protein molecule. Besides all the other structural features, it is organized by such a way that its conformational mobility changes self-similarly from room temperature up to the cryogenic temperatures.
Transitions between different conformational states are ubiquitous in proteins. A vast class of conformation-changing proteins includes evolutionary switches, which vary their conformation as an effect of few mutations or weak environmental variations. However, modeling those processes is extremely difficult due to the need of efficiently exploring a vast conformational space to look for the actual transition path. In this study, we report a strategy that simplifies this task attacking the complexity on several sides. We first apply a minimalist coarse-grained model to the protein, based on an empirical force field with a partial structural bias toward one or both the reference structures. We then explore the transition paths by means of stochastic molecular dynamics and select representative structures by means of a principal path-based clustering algorithm. We finally compare this trajectory with that produced by independent methods adopting a morphing-oriented approach. Our analysis indicates that the minimalist model returns trajectories capable of exploring intermediate states with physical meaning, retaining a very low computational cost, which can allow systematic and extensive exploration of the multistable proteins transition pathways.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.