Spatially Well-mixed Systems

The need for multiscale modeling of biological networks in zero-dimensional (well mixed) systems has been emphasized in Rao et al. (2002). The multiscale nature of stochastic simulation for well-mixed systems arises from separation of time scales, either disparity in rate constants or population sizes. In particular, the disparity in species concentrations is commonplace in biological networks. The disparity in population sizes of biological systems was in fact recognized early on by Stephanopoulos and Fredrickson (1981). This disparity in time scales creates slow and fast events. Conventional KMC samples only fast events and cannot reach long times. 

As another example of hybrid simulation touched upon above, Haseltine and Rawlings (2002) treated fast reactions either deterministically or with Langevin equations and slow reactions as stochastic events. Vasudeva and Bhalla (2004) presented an adaptive, hybrid, deterministic-stochastic simulation scheme of fixed time step. This scheme automatically switches reactions from one type to the other based on population size and magnitude of transition probability. 

Two prototype reaction examples (reversible first-order and irreversible second-order kinetics) were discussed to address issues of rounding when switching from deterministic variables to stochastic (i.e., conversion of real numbers to integers), as well as the thresholds of population sizes and transition probabilities to control accuracy in the first two moments of the population (mean and variance). Other more complex examples were also mentioned. The 

As temporal upscaling methods for acceleration of KMC simulation become mature and more robust, I expect that they will have a significant impact on the modeling of biological reaction networks. 

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This reaction can oscillate in a well-mixed system. In a quiescent system, diffusion-limited spatial patterns can develop, but these violate the assumption of perfect mixing that is made in this chapter. A well-known chemical oscillator that also develops complex spatial patterns is the Belousov-Zhabotinsky or BZ reaction. Flame fronts and detonations are other batch reactions that violate the assumption of perfect mixing. Their analysis requires treatment of mass or thermal diffusion or the propagation of shock waves. Such reactions are briefly touched upon in Chapter 11 but, by and large, are beyond the scope of this book. 

Illustration Short-time behavior in well mixed systems. Consider the initial evolution of the size distribution of an aggregation process for small deviations from monodisperse initial conditions. Assume, as well, that the system is well-mixed so that spatial inhomogeneities may be ignored. Of particular interest is the growth rate of the average cluster size and how the polydispersity scales with the average cluster size. 

Often, to simplify the analysis, the gas is assumed to be well mixed or to flow as a plug with no diffusion. In the well-mixed system, no gradients exist, and the set of coupled partial differential equations becomes sets of coupled algebraic equations, which is an enormous simplification. In general, however, spatial variations must be considered. 

The different behavior of the spatially distributed and of the homogeneously well mixed systems was also confirmed by the experiments of Kerr et al. (2002), who studied cyclic competition of three strains of E. coli bacteria in a well mixed liquid environment and on a solid agar surface. In these experiments a killer strain (R), with a relatively low fitness, i.e. low reproduction rate, can eliminate the sensitive type cells (S) by producing a toxic substance. A third resistant strain (R), however, is immune to the toxin and has higher fitness than the killer cells, but is less fit than the sensitive ones, due to the metabolic cost of producing proteins for protection from the toxin. 

A stream leaving a well-mixed region, such as a well-stirred tank, has the identical properties as in the system, since for perfect mixing the contents of the tank will have spatially uniform properties, which must then be identical to the properties of the fluid leaving at the outlet. Thus, the concentrations of component i both within the tank and in the tank effluent are the same and equal to Cj], as shown in Fig. 1.11. 

The simplest and often most suitable modeling tool is the one-box model. One-box models describe the system as a single spatially homogeneous entity. Homogeneous means that no further spatial variation is considered. However, one-box models can have one or several state variables, for instance, the mean concentration of one or several compounds i which are influenced both by external forces (or inputs) and by internal processes (removal or transformation). A particular example, the model of the well-mixed reactor with one state variable, has been discussed in Section 12.4 (see Fig. 12.7). The mathematical solution of the model has been given for the special case that the model equation is linear (Box 12.1). It will be the starting point for our discussion on box models. 

Hulburt and Katz (HI7) developed a framework for the analysis of particulate systems with the population balance equation for a multivariate particle number density. This number density is defined over phase space which is characterized by a vector of the least number of independent coordinates attached to a particle distribution that allow complete description of the properties of the distribution. Phase space is composed of three external particle coordinates x and m internal particle coordinates Xj. The former (Xei, x 2, A es) refer to the spatial distribution of particles. The latter coordinate properties Ocu,Xa,. . , Xt ) give a quantitative description of the state of an individual particle, such as its mass, concentration, temperature, age, etc. In the case of a homogeneous dispersion such as in a well-mixed vessel the external coordinates are unnecessary whereas for a nonideal stirred vessel or tubular configuration they may be needed. Thus (x t)d represents the number of particles per unit volume of dispersion at time t in the incremental range x, x -I- d, where x represents both coordinate sets. The number density continuity equation in particle phase space is shown to be (HI 8, R6)... 

A phase is a restricted part of a system with distinct physical and chemical properties (Wood and Fraser 1976). A phase can also be defined as a physically and chemically homogeneous portion of a system with definite boundaries (Brownlow 1979). These attributes mean that a phase should be mechanically separable from a system. Example phases are minerals and well-mixed gases and liquids. Not true phases, because they are comprised of more than one mineral, are rocks such as granite or minerals such as the feldspars when they are chemically zoned and have spatially variable compositions. 

There may be several chemical reactions occurring simultaneously, some of which generate A while others consume it. The net rate0 A will be positive if there is net production of component A and negative if there is net consumption. Unless the system is very well mixed, concentrations and reaction rates will vary from point to point within the control volume. The component balance applies to the entire control volume so that d, and0 A denote spatial averages. 

Piston flow reactors and most other flow reactors have spatial variations in concentration such as fl = a z). Such systems are called distributed. Their behavior is governed by an ODE when there is only one spatial variable and by a partial differential equation (PDE) when there are two or three spatial variables or when the system has a spatial variation and also varies with time. We turn now to a special type of flow reactor where the entire reactor volume is well mixed and has the same concentration, temperature, pressure, and so forth. There are no spatial variations in... 

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