### Article

## Wave packet spreading and localization in electron-nuclear scattering

The wave packet molecular dynamics (WPMD) method provides a variational approximation to the solution of the time-dependent Schr¨odinger equation. Its application in the field of high-temperature dense plasmas has yielded diverging electron width (spreading), which results in diminishing electron-nuclear interactions. Electron spreading has previously been ascribed to a shortcoming of the WPMD method and has been counteracted by various heuristic additions to the models used. We employ more accurate methods to determine if spreading continues to be predicted by them and how WPMD can be improved. A scattering process involving a single dynamic electron interacting with a periodic array of statically screened protons is used as a model problem for comparison. We compare the numerically exact split operator Fourier transform method, the Wigner trajectory method, and the time-dependent variational principle (TDVP). Within the framework of the TDVP, we use the standard variational form of WPMD, the single Gaussian wave packet (WP), as well as a sum of Gaussian WPs, as in the split WP method. Wave packet spreading is predicted by all methods, so it is not the source of the unphysical diminishing of electron-nuclear interactions in WPMD at high temperatures. Instead, the Gaussian WP’s inability to correctly reproduce breakup of the electron’s probability density into localized density near the protons is responsible for the deviation from more accurate predictions. Extensions of WPMD must include a mechanism for breakup to occur in order to yield dynamics that lead to accurate electron densities.

The method ofWave Packet Molecular Dynamics Method (WPMD) is a promising replacement of the classical molecular dynamics for the simulations of many-electron systems including nonideal plasmas. In this contribution we report on a packet splitting technique where an electron is represented by multiple Gaussians, with mixing coefficients playing the role of additional dynamic variables. It provides larger flexibility and better accuracy than the original WPMD with a single Gaussian per electron. As a test case we consider ionization of hydrogen atom in a short laser pulse, where the split packets provide a basis for quantum branching.

Helical segments are common structural elements of membrane proteins. Dimerization and oligomerization of transmembrane (TM) α-helices provides the framework for spatial structure formation and protein-protein interactions. The membrane itself also takes part in the protein functioning. There are some examples of the mutual influence of the lipid bilayer properties and embedded membrane proteins. This work aims at the detail investigation of protein-lipid interactions using model systems: TM peptides corresponding to native protein segments. Three peptides were considered corresponding to TM domains of human glycophorin A (GpA), epidermal growth factor receptor (EGFR) and proposed TM-segment of human neuraminidase-1 (Neu1). A computational analysis of structural and dynamical properties was performed using molecular dynamics method. Monomers of peptides were considered incorporated into hydrated lipid bilayers. It was confirmed, that all these TM peptides have stable helical conformation in lipid environment, and the mutual adaptation of peptides and membrane was observed. It was shown that incorporation of the peptide into membrane results in the modulation of local and mean structural properties of the bilayer. Each peptide interacts with lipid acyl chains having special binding sites on the surface of central part of α-helix that exist for at least 200 ns. However, lipid acyl chains substitute each other faster occupying the same site. The formation of a special pattern of protein-lipid interactions may modulate the association of TM domains of membrane proteins, so membrane environment should be considered when proposing new substances targeting cell receptors.

Within the framework of the Lagrangian approach a method for describing a wave packet on the surface of an infinitely deep, viscous fluid is developed. The case, in which the inverse Reynolds number is of the order of the wave steepness squared is analyzed. The expressions for fluid particle trajectories are determined, accurate to the third power of the steepness. The conditions, under which the packet envelope evolution is described by the nonlinear Schrödinger equation with a dissipative term linear in the amplitude, are determined. The rule, in accordance with which the term of this type can be correctly added in the evolutionary equation of an arbitrary order is formulated.

Plasmatic membranes contain high amount of membrane proteins. They perform vital functions of life, so any disruptions in their structure result in pathologies and diseases. Studies of these proteins with experimental methods are very complicated and expensive, as they require the membrane environment. Despite considerable progress achieved so far in methods of structure determination and property analysis, many computational methods are developing to predict the structural and dynamical parameters of proteins in membranes. Among the algorithms of modeling are the homology analysis, de novo structure prediction, molecular dynamics simulations and other. With growing computational capabilities, sophisticated techniques are developed taking into account more environmental factors. Combined approaches with different levels of approximation of intermolecular interactions are widely used. The major interest in studies of membrane proteins is focused on their transmembrane domains that are fundamental structural elements and are constituted by α-helices or helical bundles incorporated into lipid bilayer in most cases. Therefore, the fundamental problem of interaction of a pair of helices in membrane arises: the exact mechanism of this process is still not so clear. In place of the prevailing concept of dimerization motifs that states the importance of protein-protein contacts, a new model of the membrane as an adaptable lipid matrix is proposed. It states that biological membrane can adjust its properties around proteins and also modulates their activity. This mechanism of the mutual influence of two components is challenging modern computational methods of membrane model- ing because these systems are quite large and include many components to be treated accurately. Nowadays, investigations of the complex multi-component model systems become possible with modern methods of computational experiment.

Transmembrane α-helical domains are common structural elements in membrane proteins structure. They are involved into functioning of receptors and ion channels. Protein-protein interactions in lipid environment underlie the function of the most membrane systems. The properties of lipid environment can modulate the activity of membrane proteins, such as receptor tyrosine kinases. Glycophorin A is a glycoprotein that forms a very stable dimer. Its transmembrane domain is known as a good model system to study dimerization of α-helices. The major mechanism of the disturbance of a dimer by point mutations is thought to be a change of protein-protein contacts, but the role of the membrane is not well understood. In present work we study the behavior of transmembrane segment of human glycophorin A and two mutant forms G83A and T87V using molecular dynamics simulations in lipid environment. The free energy of dimerization has been estimated and the analysis of lipid properties was done. We propose different mechanisms for each mutation: T87V strongly changes protein-protein contacts. For G83A we demonstrate with the decomposition approach the major contribution of non-favorable protein-lipid contacts coupled with the redistribution interfacial protein-protein interactions. For monomers and dimers of all three forms of glycophorin A we found lipid binding sites near the interface of dimerization in the hydrophobic region of the bilayer. Surprisingly, in the case of monomers lipid acyl chains bind to the interfacial residues. Thus, the membrane plays an active role in dimer formation.

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.

Event logs collected by modern information and technical systems usually contain enough data for automated process models discovery. A variety of algorithms was developed for process models discovery, conformance checking, log to model alignment, comparison of process models, etc., nevertheless a quick analysis of ad-hoc selected parts of a journal still have not get a full-fledged implementation. This paper describes an ROLAP-based method of multidimensional event logs storage for process mining. The result of the analysis of the journal is visualized as directed graph representing the union of all possible event sequences, ranked by their occurrence probability. Our implementation allows the analyst to discover process models for sublogs defined by ad-hoc selection of criteria and value of occurrence probability

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

Let G be a semisimple algebraic group whose decomposition into the product of simple components does not contain simple groups of type A, and P⊆G be a parabolic subgroup. Extending the results of Popov [7], we enumerate all triples (G, P, n) such that (a) there exists an open G-orbit on the multiple flag variety G/P × G/P × . . . × G/P (n factors), (b) the number of G-orbits on the multiple flag variety is finite.

I give the explicit formula for the (set-theoretical) system of Resultants of m+1 homogeneous polynomials in n+1 variables