Ordinary differential equations , cellular

Nevertheless most of the biophysical methods do not observe single molecules but a huge amount of players in a concentration range between 1014 and 1020 molecules/1 (corresponding to 100 pmol/1 to 1 mmol/1, which are reasonable concentrations in cellular systems). While the dynamics of the single reactants is stochastic the macroscopic readout of the measuring system normally can be described with ordinary differential equations. 

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... 

No fewer than 11 variables appear in the reaction scheme (5.5). These are the concentrations of the metabolites and of the free and complexed forms of the receptor and adenylate cyclase. Because of the existence of conservation relations for the total amount of adenylate cyclase and of receptor, the number of independent variables can be reduced to nine. In the conditions of spatial homogeneity corresponding to the experiments in well-stirred cellular suspensions, the time evolution of the system is governed by the system of nine ordinary differential equations (Martiel Goldbeter, 1987a see Appendix, p. 234) ... 

Computational techniques are centrally important at every stage of investigation of nonlinear dynamical systems. We have reviewed the main theoretical and computational tools used in studying these problems among these are bifurcation and stability analysis, numerical techniques for the solution of ordinary differential equations and partial differential equations, continuation methods, coupled lattice and cellular automata methods for the simulation of spatiotemporal phenomena, geometric representations of phase space attractors, and the numerical analysis of experimental data through the reconstruction of phase portraits, including the calculation of correlation dimensions and Lyapunov exponents from the data. 

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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