Diauxic Growth at the Mesoscopic Scale.

In the present paper, we study a diauxic growth that can be generated by a class of model at the mesoscopic scale. Although the diauxic growth can be related to the macroscopic scale, similarly to the logistic scale, one may ask whether models on mesoscopic or microscopic scales may lead to such a behavior. The present paper is the first step towards the developing of the mesoscopic models that lead to a diauxic growth at the macroscopic scale. We propose various nonlinear mesoscopic models conservative or not that lead directly to some diauxic growths.

Mathematical modeling of the intracellular protein dynamics: the importance of active transport along microtubules.

In this paper we propose a mathematical model of protein and mRNA transport inside a cell. The spatio-temporal model takes into account the active transport along microtubules in the cytoplasm as well as diffusion and is able to reproduce the oscillatory changes in protein concentration observed in many experimental data. In the model the protein and the mRNA interact with each other that allows us to classify the model as a simple gene regulatory network. The proposed model is generic and may be adapted to specific signaling pathways. On the basis of numerical simulations, we formulate a new hypothesis that the oscillatory dynamics is allowed by the mRNA active transport along microtubules from the nucleus to distant locations.

Mathematical Model Explaining the Role of CDC6 in the Diauxic Growth of CDK1 Activity during the M-Phase of the Cell Cycle.

In this paper we propose a role for the CDC 6 protein in the entry of cells into mitosis. This has not been considered in the literature so far. Recent experiments suggest that CDC 6 , upon entry into mitosis, inhibits the appearance of active CDK 1 and cyclin B complexes. This paper proposes a mathematical model which incorporates the dynamics of kinase CDK 1 , its regulatory protein cyclin B, the regulatory phosphatase CDC 25 and the inhibitor CDC 6 known to be involved in the regulation of active CDK 1 and cyclin B complexes. The experimental data lead us to formulate a new hypothesis that CDC 6 slows down the activation of inactive complexes of CDK 1 and cyclin B upon mitotic entry. Our mathematical model, based on mass action kinetics, provides a possible explanation for the experimental data. We claim that the dynamics of active complexes CDK 1 and cyclin B have a similar nature to diauxic dynamics introduced by Monod in 1949. In mathematical terms we state it as the existence of more than one inflection point of the curve defining the dynamics of the complexes.

Chaotic behavior of semigroups related to the process of gene amplification-deamplification with cell proliferation.

In the last few years there has been a renewed interest in infinite systems of differential equations, similar to the classical birth-and-death system of population dynamics, due to their rôle in modelling the evolution of drug resistance in cancer cells. In [J. Banasiak, M. Lachowicz, Topological chaos for birth-and-death models with proliferation, Math. Models Methods Appl. Sci. 12 (6) (2002) 755] such systems were shown to generate a chaotic dynamics under, however, very restrictive assumptions on the growth of coefficients. In this paper, using recently developed concept of subspace chaos [J. Banasiak, M. Moszynski, A generalization of Desch-Schappacher-Webb criteria for topological chaos with applications, Discrete Contin. Dyn. Syst. - A 12 (5) (2005) 959], we show that for a linear growth of the coefficients, which are more acceptable from biological point of view, the dynamics of these systems is chaotic in some subspaces of the original state space.

Preface.

This volume was inspired by the topics presented at the international conference "Micro and Macro Systems in Life Sciences" which was held on Jun 8-12, 2015 in Bedlewo, Poland. System biology is an approach which tries to understand how micro systems, at the molecular and cellular levels, affect macro systems such as organs, tissue and populations. Thus it is not surprising that a major theme of this volume evolves around cancer and its treatment. Articles on this topic include models for tumor induced angiogenesis, without and with delays, metastatic niche of the bone marrow, drug resistance and metronomic chemotherapy, and virotherapy of glioma. Methods range from dynamical systems to optimal control. Another well represented topic of this volume is mathematical modeling in epidemiology. Mathematical approaches to modeling and control of more specific diseases like malaria, Ebola or human papillomavirus are discussed as well as a more general approaches to the SEIR, and even more general class of models in epidemiology, by using the tools of optimal control and optimization. The volume also brings up challenges in mathematical modeling of other diseases such as tuberculosis. Partial differential equations combined with numerical approaches are becoming important tools in modeling not only tumor growth and treatment, but also other diseases, such as fibrosis of the liver, and atherosclerosis and its associated blood flow dynamics, and our volume presents a state of the art approach on these topics. Understanding mathematics behind the cell motion, appearance of the special patterns in various cell populations, and age structured mutations are among topics addressed inour volume. A spatio-temporal models of synthetic genetic oscillators brings the analysis to the gene level which is the focus of much of current biological research. Mathematics can help biologists to explain the collective behavior of bacterial, a topic that is also presented here. Finally some more across the discipline topics are being addresses, which can appear as a challenge in studying problems in systems biology on all, macro, meso and micro levels. They include numerical approaches to stochastic wave equation arising in modeling Brownian motion, discrete velocity models, many particle approximations as well as very important aspect on the connection between discrete measurement and the construction of the models for various phenomena, particularly the one involving delays. With the variety of biological topics and their mathematical approaches we very much hope that the reader of the Mathematical Biosciences and Engineering will find this volume interesting and inspirational for their own research.

Flexibility vs. robustness in cell cycle regulation of timing of M-phase entry in Xenopus laevis embryo cell-free extract.

During the cell cycle, cyclin dependent kinase 1 (CDK1) and protein phosphatase 2A (PP2A) play major roles in the regulation of mitosis. CDK1 phosphorylates a series of substrates triggering M-phase entry. Most of these substrates are dephosphorylated by PP2A. To allow phosphorylation of CDK1 substrates, PP2A is progressively inactivated upon M-phase entry. We have shown previously that the interplay between these two activities determines the timing of M-phase entry. Slight diminution of CDK1 activity by the RO3306 inhibitor delays M-phase entry in a dose-dependent manner in Xenopus embryo cell-free extract, while reduction of PP2A activity by OA inhibitor accelerates this process also in a dose-dependent manner. However, when a mixture of RO3306 and OA is added to the extract, an intermediate timing of M-phase entry is observed. Here we use a mathematical model to describe and understand this interplay. Simulations showing acceleration and delay in M-phase entry match previously described experimental data. CDC25 phosphatase is a major activator of CDK1 and acts through CDK1 Tyr15 and Thr14 dephosphorylation. Addition of CDC25 activity to our mathematical model was also consistent with our experimental results. To verify whether our assumption that the dynamics of CDC25 activation used in this model are the same in all experimental variants, we analyzed the dynamics of CDC25 phosphorylation, which reflect its activation. We confirm that these dynamics are indeed very similar in control extracts and when RO3306 and OA are present separately. However, when RO3306 and OA are added simultaneously to the extract, activation of CDC25 is slightly delayed. Integration of this parameter allowed us to improve our model. Furthermore, the pattern of CDK1 dephosphorylation on Tyr15 showed that the real dynamics of CDK1 activation are very similar in all experimental variants. The model presented here accurately describes, in mathematical terms, how the interplay between CDK1, PP2A and CDC25 controls the flexible timing of M-phase entry.

Equilibrium solutions for microscopic stochastic systems in population dynamics.

The present paper deals with the problem of existence of equilibrium solutions of equations describing the general population dynamics at the microscopic level of modified Liouville equation (individually--based model) corresponding to a Markov jump process. We show the existence of factorized equilibrium solutions and discuss uniqueness. The conditions guaranteeing uniqueness or non-uniqueness are proposed under the assumption of periodic structures.

A singularly perturbed SIS model with age structure.

We present a preliminary study of an SIS model with a basic age structure and we focus on a disease with quick turnover, such as influenza or common cold. In such a case the difference between the characteristic demographic and epidemiological times naturally introduces two time scales in the model which makes it singularly perturbed. Using the Tikhonov theorem we prove that for certain classes of initial conditions the nonlinear structured SIS model can be approximated with very good accuracy by lower dimensional linear models.