Compartmental systems

The reaction exothermicities ( —AG°) for forward and back ET in polar media were approximately estimated to be 1.39 and 2.18 eV, respectively [120], Since the back ET is highly exothermic, the relatively small kb-1 values for the compartmentalized system may be ascribed to the combined effect of the inverted region [97-99] and the loose ion-pair state. 

Although the electrostatic potential on the surface of the polyelectrolyte effectively prevents the diffusional back electron transfer, it is unable to retard the very fast charge recombination of a geminate ion pair formed in the primary process within the photochemical cage. Compartmentalization of a photoactive chromophore in the microphase structure of the amphiphilic polyelectrolyte provides a separated donor-acceptor system, in which the charge recombination is effectively suppressed. Thus, with a compartmentalized system, it is possible to achieve efficient charge separation. 

Fig. 10. Mechanisms of steady-slqte kinetics of sugar phosphorylation catalyzed by E-IIs in a non-compartmentalized system. (A) The R. sphaeroides 11 model. The model is based on the kinetic data discussed in the text. Only one kinetic route leads to phosphorylation of fructose. (B) The E. coli ll " model. The model in Fig. 8 was translated into a kinetic scheme that would describe mannitol phosphorylation catalyzed by Il solubilized in detergent. Two kinetic routes lead to phosphorylation of mannitol. Mannitol can bind either to state EPcy, or EPpe,. E represents the complex of SF (soluble factor) and 11 and II in A and B, respectively. EP represents the phosphorylated states of the E-IIs. Subscripts cyt and per denote the orientation of the sugar binding site to the cytoplasm and periplasm, respectively. PEP, phosphoenolpyruvate.

Usually, the buffer compartment is not accessible and, consequently, the absolute amount of X cannot be determined experimentally. For this reason, we will only focus our discussion on the plasma concentration Cp. It is important to know, however, that the time course of the contents in the two compartments is the sum of two exponentials, which have the same positive hybrid transfer constants a and p. The coefficients A and B, however, depend on the particular compartment. This statement can be generalized to mammillary systems with a large number of compartments that exchange with a central compartment. The solutions for each of n compartments in a mammillary model are sums of n exponential functions, having the same n positive hybrid transfer constants, but with n different coefficients for each particular compartment. (We will return to this property of linear compartmental systems during the discussion of multi-compartment models in Section 39.1.7.)... 

Extensive literature is available on general mathematical treatments of compartmental models [2], The compartmental system based on a set of differential equations may be solved by Laplace transform or integral calculus techniques. By far... 

We assume that compartment i is occupied at time 0 by qto amount of material and we denote by q, (t) the amount in the compartment i at time t. We also assume that no material enters in the compartments from the outside of the compartmental system and we denote by Rj,0 (t) the rate of elimination from compartment i to the exterior of the system. Let also Rji (t) be the transfer rate of material from the jth to ith compartment. Because the material... 

Mathematics is now called upon to describe the compartmental configurations and then to simulate their dynamic behavior. To build up mathematical equations expressing compartmental systems, one has to express the mass balance equations for each compartment i ... 

This law may be applied to the transfer rates Rji (t) of the previous equation for all pairs j and i of compartments corresponding to l and r and for the elimination rate R (t), where the concentration is assumed nearly zero in the region outside the compartmental system. One has for the compartment i,... 

The process was considered as continuous and compartmental models were used to approximate the continuous systems [335]. For such applications, there is no specific compartmental model that is the best the approximation improves as the number of compartments is increased. It order to put compartmental models of continuous processes in perspective it may help to recall that the first step in obtaining the partial differential equation, descriptive of a process continuous in the space variables, is to discretize the space variables so as to give many microcompartments, each uniform in properties internally. The differential equation is then obtained as the limit of the equation for a microcompartment as its spatial dimensions go to zero. In approximation of continuous processes with compartmental models one does not go to the limit but approximates the process with a finite compartmental system. In that case, the partial differential equation... 

The real world of compartmental systems has a strong stochastic component, so we will present a stochastic approach to compartmental modeling. In deterministic theory developed in Chapter 8, each compartment is treated as being both homogeneous and a continuum. But ... 

Erlang- and phase-type distributions provide a versatile class of distributions, and are shown to fit naturally into a Markovian compartmental system, where particles move between a series of compartments, so that phase-type compartmental retention-time distributions can be incorporated simply by increasing the size of the system. This class of distributions is sufficiently rich to allow for a wide range of behaviors, and at the same time offers computational convenience for data analysis. Such distributions have been used extensively in theoretical studies (e.g., [366]), because of their range of behavior, as well as in experimental work (e.g., [367]). Especially for compartmental models, the phase-type distributions were used by Faddy [364] and Matis [301,306] as examples of long-tailed distributions with high coefficients of variation. 

If the hazard rate of any single particle out of a compartment depends on the state of the system, the equations of the probabilistic transfer model are still linear, but we have nonlinear rate laws for the transfer processes involved and such systems are the stochastic analogues of nonlinear compartmental systems. For such systems, the solutions for the deterministic model are not the same as the solutions for the mean values of the stochastic model. 

As previously, initial conditions for the compartmental model and the enzymatic reaction were set to tiq = [100 50], and so = 100, eo = 50, and cq = 0, respectively. Figures 9.31 and 9.32 show the deterministic prediction, a typical run, and the average and confidence corridor for 100 runs from the stochastic simulation algorithm for the compartmental system and the enzyme reaction, respectively. Figures 9.33 and 9.34 show the coefficient of variation for the number of particles in compartment 1 and for the substrate particles, respectively. 

Agrafiotis, G., On the stochastic theory of compartments A semi-Markov approach for the analysis of the k-compartmental systems, Bulletin of Mathematical Biology, Vol. 44, No. 6, 1982, pp. 809-817. 

Matis, J. and Wehrly, T., On the use of residence time moments in the statistical analysis of age-department stochastic compartmental systems, Mathematics in Biology and Medicine, edited by V. Capasso, E. Grosso, and S. Paveri-Fontana, Springer-Verlag, New York, 1985, pp. 386-398. 

In compartmentation systems (or precursor activation systems), precursors and activating enzymes work synergistically to exert plant defense however, other kinds of synergism were observed. For example, chemicals that enhance the toxicity of furanocoumarin by inhibiting detoxifying enzyme of insects (i.e., cytochrome P-450 monooxygenase) coexist with furanocoumarin in plants.5... 

Jacquez JA, Simon CP. Qualitative theory of compartmental systems. SIAM Rev 1993 35 43-79. 

We now show how to evaluate the MWD in a monodisperse compartmentalized system. It will be seen that the problem may be solved with complete generality if chain-branching reactions do not occur moreover, analytical solutions can be obtained for the steady-state regime. 

It will be recalled that the polydispersity ratio P. defined in Eq. (35), gives a compact description of the MWD. It is instructive to compare values of P computed from bulk and solution MWD theory with equivalent values of P for a compartmentalized system. i.e., from the theory developed in Sections and 2. 

For the combination-dominated compartmentalized system shown in Fig. 9, the polydispersity ratio is found to ha 1.9. Here it can be seen that compartmentalization has a significant effect, this value for P being appreciably greater than the bulk value of 1.5. Thus, for compartmentalized combination-dominated systems the MWD is significantly broader than in the equivalent bulk system. 

Il is particularly informative to compute P as a function of H for combination-dominated and disproportionation-dominated stems (note incidentally that 0.5 is the lower limit of n in such a system). It will be seen that when S 2, the compartmentalized system is indeed a minibulk system, with compartmentalization effects becoming more pronounced as n becomes smaller. The P versus n results shown in Figs. 5 and 6 were obtained by fixing p and systematically varying either (Fig. 12) or (Fig. I3I. The values of P were obtained from tbe MWD expressions given in Sections I1.D.I (valid for n < 0.7) and II D,2 (valid for any ii). 

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