Linear and structural stability of a cell division process model

The paper investigates the linear stability of mammalian physiology time-delayed flow for three distinct cases (normal cell cycle, a neoplasmic cell cycle, and multiple cell arrest states), for the Dirac, uniform, and exponential distributions. For the Dirac distribution case, it is shown that the model exhibits a Hopf bifurcation for certain values of the parameters involved in the system. As well, for these values, the structural stability of the SODE is studied, using the five KCC-invariants of the second-order canonical extension of the SODE, and all the cases prove to be Jacobi unstable

Pressure and temperature dependence of growth and morphology of Escherichia coli: Experiments and Stochastic Model

We have investigated the growth of Escherichia coli E.coli, a mesophilic bacterium, as a function of pressure $P$ and temperature $T$. E.coli can grow and divide in a wide range of pressure (1-400atm) and temperature ($23-40^{\circ}$C). For $T>30^{\circ}$ C, the division time of E.coli increases exponentially with pressure and exhibit a departure from exponential behavior at pressures between 250-400 atm for all the temperatures studied in our experiments. For $T<30^{\circ}$ C, the division time shows an anomalous dependence on pressure -- first decreases with increasing pressure and then increases upon further increase of pressure. The sharp change in division time is followed by a sharp change in phenotypic transition of E. Coli at high pressures where bacterial cells switch to an elongating cell type. We propose a model that this phenotypic changes in bacteria at high pressures is an irreversible stochastic process whereas the switching probability to elongating cell type increases with increasing pressure. The model fits well the experimental data. We discuss our experimental results in the light of structural and thus functional changes in proteins and membranes.Comment: 28 pages, 12 figure

Mathematical modeling of cell population dynamics in the colonic crypt and in colorectal cancer

Colorectal cancer is initiated in colonic crypts. A succession of genetic mutations or epigenetic changes can lead to homeostasis in the crypt being overcome, and subsequent unbounded growth. We consider the dynamics of a single colorectal crypt by using a compartmental approach [Tomlinson IPM, Bodmer WF (1995) Proc Natl Acad Sci USA 92: 11130-11134], which accounts for populations of stem cells, differential cells, and transit cells. That original model made the simplifying assumptions that each cell popuation divides synchronously, but we relax these assumptions by adopting an age-structured approach that models asynchronous cell division, and by using a continuum model. We discuss two mechanims that could regulate the growth of cell numbers and maintain the equilibrium that is normally observed in the crypt. The first will always maintain an equilibrium for all parameter values, whereas the second can allow unbounded proliferation if the net per capita growth rates are large enough. Results show that an increase in cell renewal, which is equivalent to a failure of programmed cell death or of differentiation, can lead to the growth of cancers. The second model can be used to explain the long lag phases in tumor growth, during which news, higher equilibria are reached, before unlimited growth in cell number ensues

A Stochastic model for dynamics of FtsZ filaments and the formation of Z-ring

Understanding the mechanisms responsible for the formation and growth of FtsZ polymers and their subsequent formation of the $Z$-ring is important for gaining insight into the cell division in prokaryotic cells. In this work, we present a minimal stochastic model that qualitatively reproduces {\it in vitro} observations of polymerization, formation of dynamic contractile ring that is stable for a long time and depolymerization shown by FtsZ polymer filaments. In this stochastic model, we explore different mechanisms for ring breaking and hydrolysis. In addition to hydrolysis, which is known to regulate the dynamics of other tubulin polymers like microtubules, we find that the presence of the ring allows for an additional mechanism for regulating the dynamics of FtsZ polymers. Ring breaking dynamics in the presence of hydrolysis naturally induce rescue and catastrophe events in this model irrespective of the mechanism of hydrolysis.Comment: Replaced with published versio

Computational Evolutionary Embryogeny

Evolutionary and developmental processes are used to evolve the configurations of 3-D structures in silico to achieve desired performances. Natural systems utilize the combination of both evolution and development processes to produce remarkable performance and diversity. However, this approach has not yet been applied extensively to the design of continuous 3-D load-supporting structures. Beginning with a single artificial cell containing information analogous to a DNA sequence, a structure is grown according to the rules encoded in the sequence. Each artificial cell in the structure contains the same sequence of growth and development rules, and each artificial cell is an element in a finite element mesh representing the structure of the mature individual. Rule sequences are evolved over many generations through selection and survival of individuals in a population. Modularity and symmetry are visible in nearly every natural and engineered structure. An understanding of the evolution and expression of symmetry and modularity is emerging from recent biological research. Initial evidence of these attributes is present in the phenotypes that are developed from the artificial evolution, although neither characteristic is imposed nor selected-for directly. The computational evolutionary development approach presented here shows promise for synthesizing novel configurations of high-performance systems. The approach may advance the system design to a new paradigm, where current design strategies have difficulty producing useful solutions

Modelling the spatial organization of cell proliferation in the developing central nervous system

How far is neuroepithelial cell proliferation in the developing central nervous system a deterministic process? Or, to put it in a more precise way, how accurately can it be described by a deterministic mathematical model? To provide tracks to answer this question, a deterministic system of transport and diffusion partial differential equations, both physiologically and spatially structured, is introduced as a model to describe the spatially organized process of cell proliferation during the development of the central nervous system. As an initial step towards dealing with the three-dimensional case, a unidimensional version of the model is presented. Numerical analysis and numerical tests are performed. In this work we also achieve a first experimental validation of the proposed model, by using cell proliferation data recorded from histological sections obtained during the development of the optic tectum in the chick embryo

Examples of mathematical modeling tales from the crypt

Mathematical modeling is being increasingly recognized within the biomedical sciences as an important tool that can aid the understanding of biological systems. The heavily regulated cell renewal cycle in the colonic crypt provides a good example of how modeling can be used to find out key features of the system kinetics, and help to explain both the breakdown of homeostasis and the initiation of tumorigenesis. We use the cell population model by Johnston et al. (2007) Proc. Natl. Acad. Sci. USA 104, 4008-4013, to illustrate the power of mathematical modeling by considering two key questions about the cell population dynamics in the colonic crypt. We ask: how can a model describe both homeostasis and unregulated growth in tumorigenesis; and to which parameters in the system is the model most sensitive? In order to address these questions, we discuss what type of modeling approach is most appropriate in the crypt. We use the model to argue why tumorigenesis is observed to occur in stages with long lag phases between periods of rapid growth, and we identify the key parameters

Conformational mechanism for the stability of microtubule-kinetochore attachments

Regulating the stability of microtubule(MT)-kinetochore attachments is fundamental to avoiding mitotic errors and ensure proper chromosome segregation during cell division. While biochemical factors involved in this process have been identified, its mechanics still needs to be better understood. Here we introduce and simulate a mechanical model of MT-kinetochore interactions in which the stability of the attachment is ruled by the geometrical conformations of curling MT-protofilaments entangled in kinetochore fibrils. The model allows us to reproduce with good accuracy in vitro experimental measurements of the detachment times of yeast kinetochores from MTs under external pulling forces. Numerical simulations suggest that geometrical features of MT-protofilaments may play an important role in the switch between stable and unstable attachments

Open and cut: allosteric motion and membrane fission by dynamin superfamily proteins.

Cells have evolved diverse protein-based machinery to reshape, cut, or fuse their membrane-delimited compartments. Dynamin superfamily proteins are principal components of this machinery and use their ability to hydrolyze GTP and to polymerize into helices and rings to achieve these goals. Nucleotide-binding, hydrolysis, and exchange reactions drive significant conformational changes across the dynamin family, and these changes alter the shape and stability of supramolecular dynamin oligomers, as well as the ability of dynamins to bind receptors and membranes. Mutations that interfere with the conformational repertoire of these enzymes, and hence with membrane fission, exist in several inherited human diseases. Here, we discuss insights from new x-ray crystal structures and cryo-EM reconstructions that have enabled us to infer some of the allosteric dynamics for these proteins. Together, these studies help us to understand how dynamins perform mechanical work, as well as how specific mutants of dynamin family proteins exhibit pathogenic properties