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CellX : A Self-Organization Cell Simulator CellX simulates the multi-dimensional dynamics of a cell. Reaction-transport equations are solved along fibrils (1-D), on or within membranes (2-D) and in bulk media (3-D). A cell is divided into compartments separated by membranes. In all three domains (1,2,3-D) partial differential equations are solved via finite element techniques. Molecular exchange between these domains is accounted for by boundary conditions. CellX is an advanced cell simulator for understanding the self-organizational and other autonomous behavior of a cell accompanying cell transformation, transition to abnormality and other dramatic phenomena. A cell involves structure on a variety of scales from the molecular to that of overall intra-cellular organization. Cellular processes occur on timescales from nanoseconds to days or longer. These multiple scales present challenges for the development of a predictive model of cell dynamics. Self-organization has been proposed since Turing (1952) as an explanation for the autonomous behavior of a living system and its ability to reorganize its inner and outer structure during development. Prigogine and coworkers of the Brussels school discovered that a necessary condition for self-organization to occur is that the system be maintained sufficiently far-from-equilibrium. We suggest that this necessary condition implies the need for multiple scales in the network of transport processes. Thus key species fueling the cellular metabolic network should be particularly mobile so that much of the cells interior, and not just a narrow boundary layer near the outer membrane, be rich in these molecules. CellX is designed to meet the challenges brought about by this multiple timescale dynamic. In understanding cellular dynamics one must also account for self-assembly driven by the Second Law of thermodynamics and the implied tendency to minimize free energy. Templated polymerization (i.e. replication, transcription and translation) operates at the mesoscopic scale (i.e. between the microscopic and macroscopic). Simultaneously addressing these multiple length and time scales, and these autonomy-supporting processes is the objective of CellX and our ongoing research. CellX addresses a key element of this multi-scale character, including the separation of scales in the network of biochemical processes and those associated with rapidly and slowly transporting species. The presence of multiple scales in the reaction network has been shown to underlie many of the observed chemical wave and patterning phenomena in biological and other systems. It was shown that the fast versus slow reaction dichotomy can lead to folded, cusped and other hyper-surface topologies in the space of the descriptive variables (e.g. concentrations) that explain much of reaction-transport patterning and wave behavior. CellX is based on a reformulation of the intra-cellular reaction-transport dynamics that takes into account the separation of timescales between the transport of small molecules and larger entities (e.g. proteins, ribosomes) and its interplay with fast and slow reactions. Our approach is to introduce a timescale ratio following from the rate and transport coefficients. We then analyze the behavior of the system in the asymptotic limit of this ratio to arrive at a set of equations for the rigorous limiting behavior of reaction-transport dynamics that can be computed efficiently. A well-known limit to the numerical simulation of reaction-transport systems is placed on the time step. The latter must be smaller than any characteristic time for reaction. However, another restriction on most algorithms is the time associated with a spatial grid size and a diffusion coefficient. But if the latter is very large for some species then the time step must be small. In CellX we overcome this difficulty so that the time step is determined by the diffusion coefficient of the slower migrating species. CellX has been applied to T. brucei and E. coli. |
