Stochastic biochemical systems

Stochastic biochemical systems and the chemical master equation... 

Part II of this book represents the bulk of the material on the analysis and modeling of biochemical systems. Concepts covered include biochemical reaction kinetics and kinetics of enzyme-mediated reactions simulation and analysis of biochemical systems including non-equilibrium open systems, metabolic networks, and phosphorylation cascades transport processes including membrane transport and electrophysiological systems. Part III covers the specialized topics of spatially distributed transport modeling and blood-tissue solute exchange, constraint-based analysis of large-scale biochemical networks, protein-protein interactions, and stochastic systems. 

HI F) 1978 Schulmeister, Th Sel kov, E. E. Folded Limit Cycles and Quasi-Stochastic Self-Oscillations in a Third Order Model of an Open Biochemical System, Studia Biophysica, vol. 72,111-112... 

Schulmeister, T. E.E. Sel kov. 1978. Folded limit cycles and quasi-stochastic self-osciUations in a third-order model of an open biochemical system. Studia Biophys. 72 111-12 and Microfiche 1/24-37. 

Goal. - Probability. - Informatioa -Chance. - Necessity. - Chance and Necessity. - Self-Organizatioa - Physical Systems. - Chemical and Biochemical Systems. - Applications to Biology. - Sociology A Stochastic Model for the Formation of Public Opinion. - Chaos. - Some Historical Remarks and Outlook. 

Indeed, semi-synthetic minimal cells often consist of a liposome enclosing defined biochemical pathways, specifically pathways capable to synthesize proteins that could, in principle, close the circle towards a complete self-sustainability of the minimal cell in other words, the different components of the minimal cells can cooperate to restore themselves. As of today, only a few papers have been published dealing with theoretical simulations, at different detail level, of PS behavior and time course [7,15 17], and only three deal with PS entrapped in liposomes [18 20]. This last condition is very interesting since the smaller the liposome, the lower the probability that a large amount of molecules of each constituent chemical species be entrapped in the volume of the liposome. This implies that a standard simulation of PS based on deterministic ODE formalism is no longer suitable to describe a biochemical system beyond the deterministic limit. If even a single chemical species is present with only a few molecules, the behavior of all the PS is turned to stochastic. 

When mechanistic information is available or obtainable for the components of a system, it is possible to develop detailed analyses and simulations of that system. Such analyses and simulations may be deterministic or stochastic in nature. (Stochastic systems are the subject of Chapter 11.) In either case, the overriding philosophy is to apply mechanistic rules to predict behavior. Often, however, the information required to develop mechanistic models accounting for details such as enzyme and transporter kinetics and precisely predicting biochemical states is not available. Instead, all that may be known reliably about certain large-scale systems is the stoichiometry of the participating reactions. As we shall see in this chapter, this stoichiometric information is sometimes enough to make certain concrete determinations about the feasible operation of biochemical networks. 

Figure 4.1). The realization that fluctuations can be important in such a system has led to a surge of interest in stochastic models of biochemical reactions in systems biology.

Deterministic dynamics of biochemical reaction systems can be visualized as the trajectory of (ci(t), c2(t), , c v(0) in a space of concentrations, where d(t) is the concentration of ith species changing with time. This mental picture of path traced out in the N-dimensional concentration space by deterministic systems may prove a useful reference when we deal with stochastic chemical dynamics. In stochastic systems, one no longer thinks in terms of definite concentrations at time t rather, one deals with the probability of the concentrations being xu x2, , Wy at time t ... 

It is widely appreciated that chemical and biochemical reactions in the condensed phase are stochastic. It has been more than 60 years since Delbriick studied a stochastic chemical reaction system in terms of the chemical master equation. Kramers theory, which connects the rate of a chemical reaction with the molecular structures and energies of the reactants, is established as a central component of theoretical chemistry [77], Yet study of the dynamics of chemical and biochemical reaction systems, in terms of either deterministic differential equations or the stochastic CME, is not the exclusive domain of chemists. Recent developments in the simulation of reaction systems are the work of many sorts of scientists, ranging from control engineers to microbiologists, all interested in the dynamic behavior of biochemical reaction systems [199, 210],... 

Stochastic models for biochemical reaction systems in terms of the CME are not an alternative to the differential equation approach, but a more general theoretical framework that deserves further investigation. In particular, the relation between the dynamic CME and the general theory of statistical thermodynamics of closed and... 

Special topics - explores spatially distributed systems, constraint-based analysis for large-scale networks, protein-protein interaction, and stochastic phenomena in biochemical... 

Additional complexity can be included through cell population balances that account for the distribution of cell generation present in the fermenter through use of stochastic models. In this section we limit the discussion to simple black box and unstructured models. For more details on bioreaction systems, see, e.g., Nielsen, Villadsen, and Liden, Bioreaction Engineering Principles, 2d ed., Kluwer, Academic/Plenum Press, 2003 Bailey and Ollis, Biochemical Engineering Fundamentals, 2d ed., McGraw-Hill, 1986 Blanch and Clark, Biochemical Engineering, Marcel Dekker, 1997 and Sec. 19. 

Stochastic modeling and simulation better approximates the biochemical reality, but its usage may not always be beneficial. It requires very detailed knowledge on the underlying reaction mechanisms. Even when such knowledge is available, the computational costs involved may become prohibitive for larger systems. 

In the post-genomic era, the development of kinetic models that allow simulation of complicated metabolic pathways and protein interactions is becoming increasingly important [189, 190]. Unfortunately, the difference between an in vivo biological system and homogeneous in vitro conditions is large, as shown by Schnell and Turner [191]. Mathematical treatments of biochemical kinetics have been developed from the law of mass action in vitro, but the modifications required to bring them in line with stochastic in vivo situations are stiU under development [192-194]. 

For biochemically driven reactions, embedding heat bath provides the source of stochastic dynamics. The stochastic model, in the form of the chemical master equation, is an infinite system of mathematically coupled ordinary differential equations (Vellela and Qian, 2009). Assuming that tia and b are the number of substrate molecules, which are fixed for a fixed volume, and p (f) is the probability of having nX molecule at time t. The stochastic model equations are... 

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