Numerical study for multi-strain tuberculosis (TB) model of variable-order fractional derivatives.

In this paper, we presented a novel multi-strain TB model of variable-order fractional derivatives, which incorporates three strains: drug-sensitive, emerging multi-drug resistant (MDR) and extensively drug-resistant (XDR), as an extension for multi-strain TB model of nonlinear ordinary differential equations which developed in 2014 by Arino and Soliman [1]. Numerical simulations for this variable-order fractional model are the main aim of this work, where the variable-order fractional derivative is defined in the sense of Grünwald-Letnikov definition. Two numerical methods are presented for this model, the standard finite difference method (SFDM) and nonstandard finite difference method (NSFDM). Numerical comparison between SFDM and NSFDM is presented. It is concluded that, NSFDM preserves the positivity of the solutions and numerically stable in large regions than SFDM.

Legendre spectral-collocation method for solving some types of fractional optimal control problems.

In this paper, the Legendre spectral-collocation method was applied to obtain approximate solutions for some types of fractional optimal control problems (FOCPs). The fractional derivative was described in the Caputo sense. Two different approaches were presented, in the first approach, necessary optimality conditions in terms of the associated Hamiltonian were approximated. In the second approach, the state equation was discretized first using the trapezoidal rule for the numerical integration followed by the Rayleigh-Ritz method to evaluate both the state and control variables. Illustrative examples were included to demonstrate the validity and applicability of the proposed techniques.

A quantitative model of the major pathways for radiation-induced DNA double-strand break repair.

We have developed a model approach to simulate the major pathways of DNA double-strand break (DSB) repair in mammalian and human cells. The proposed model shows a possible mechanistic explanation of the basic regularities of DSB processing through the non-homologous end-joining (NHEJ), homologous recombination (HR), single-strand annealing (SSA) and two alternative end-joining pathways. It reconstructs the time-courses of radiation-induced foci specific to particular repair processes including the major intermediate stages. The model is validated for ionizing radiations of a wide range of linear energy transfer (0.2-236 keV/µm) including a relatively broad spectrum of heavy ions. The appropriate set of reaction rate constants was suggested to satisfy the kinetics of DSB rejoining for the considered types of exposure. The simultaneous assessment of several repair pathways allows to describe their possible biological relations in response to irradiation. With the help of the proposed approach, we reproduce several experimental data sets on γ-H2AX foci remaining in different types of cells including those defective in NHEJ, HR, or SSA functions. The results produced confirm the hypothesis suggesting existence of at least two alternative Ku-independent end-joining pathways.

The role of the bacterial mismatch repair system in SOS-induced mutagenesis: a theoretical background.

A theoretical study is performed of the possible role of the methyl-directed mismatch repair system in the ultraviolet-induced mutagenesis of Escherichia coli bacterial cells. For this purpose, mathematical models of the SOS network, translesion synthesis and mismatch repair are developed. Within the proposed models, the key pathways of these repair systems were simulated on the basis of modern experimental data related to their mechanisms. Our model approach shows a possible mechanistic explanation of the hypothesis that the bacterial mismatch repair system is responsible for attenuation of mutation frequency during ultraviolet-induced SOS response via removal of the nucleotides misincorporated by DNA polymerase V (the UmuD'2C complex).