Ordinary differential equations , cellular models

Odell, G. M. (1980) Qualitative theory of systems of ordinary differential equations, including phase plane analysis and the use of the Hopf bifurcation theorem. Appendix A.3. In L. A. Segel, ed.. Mathematical Models in Molecular and Cellular Biology (Cambridge University Press, Cambridge, England). 

Work has been done to infer differential equation models of cellular networks from time-series data. As we explained in the previous section, the general form of the differential equation model is deceit = f(Cj, c2,. cN), where J] describes how each element of the network affects the concentration rate of the network element. If the functions f are known, that is, the individual reaction and interaction mechanisms in the network are available, a wealth of techniques can be used to fit the model to experimental data and estimate the unknown parameters [Mendes 2002]. In many cases, however, the functions f are unknown, nonlinear functions. A common approach for reverse engineering ordinary differential equations is to linearize the functions f around the equilibrium [Stark, Callard, and Hubank 2003] and obtain... 

In the first chapter several traditional types of physical models were discussed. These models rely on the physical concepts of energies and forces to guide the actions of molecules or other species, and are customarily expressed mathematically in terms of coupled sets of ordinary or partial differential equations. Most traditional models are deterministic in nature— that is, the results of simulations based on these models are completely determined by the force fields employed and the initial conditions of the simulations. In this chapter a very different approach is introduced, one in which the behaviors of the species under investigation are governed not by forces and energies, but by rules. The rules, as we shall see, can be either deterministic or probabilistic, the latter leading to important new insights and possibilities. This new approach relies on the use of cellular automata. 

Each term from the right side of this representative equation of the model has a particular meaning. The first term shows that the number of the reactant species molecules in the k cell decreases as a result of the consumption of species by the chemical reaction and the output of species from the cell. The second term describes the reduction of the number of molecules as a result of the transport to other compartments. The last term gives the increase in the number of the species in the k compartment because of the inputs from the other cells of the assembly. With reference to the mathematical formalism, our model is described by an ordinary system of differential equations. Indeed, for calculations we must specify the initial state of the probabilities. So, the vector P] (0), k = 1, N must be a known vector. The frequencies Oj wUl be established by means of the cellular... 

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