A cybernetic modeling framework for analysis of metabolic systems
Introduction
Biological systems are extremely complex not only due to the large number of reactions that take place in a single cell, but also due to the extremely non-linear interactions between different metabolites and enzymes. Such interactions stem from the genetic make-up of the cells wherein the response to external stimulus is encoded in the genome of the cells. Thus, perturbations in the extracellular environment can lead to variations in the levels of the mRNA of the cells, and correspondingly in the enzymes. The enzymes themselves act on different intracellular metabolites, thereby changing their levels. The fluxes in different pathways therefore change and such changes can lead to different substrate uptake patterns as well as differences in the growth of cells as well as the syntheses of secondary products. The metabolic engineer must therefore deal with a large number of reactions in order to predict cellular behavior in order to optimize cells as well as cell-based processes. The cybernetic approach has been extremely successful in describing bioreactor behavior (Kompala et al., 1986Kompala, Ramkrishna, Jansen, & Tsao, 1986; Namjoshi, Kienle, & Ramkrishna, 2003; Ramkrishna, 1982; Ramakrishna, Ramkrishna, & Konopka, 1996; Ramkrishna, 1982, Straight and Ramkrishna, 1990; Varner and Ramkrishna, 1999a, Varner and Ramkrishna, 1999b) based on the postulate that cells are objective-oriented organisms which change the internal enzyme levels in response to environmental changes.
The cybernetic modeling framework relies on proposing objective functions for intracellular reactions.1
Once an objective function is proposed, the matching law and proportional law can be employed to derive cybernetic “u” and “” control variables which modify enzyme synthesis as well as enzyme activity, respectively. As the number of reactions that need to be modeled expand, the given network can be broken down into elementary reaction units in non-unique ways. The existing strategies to guide this include the use of network fluxes calculated from metabolic flux analysis (MFA) (Namjoshi, Hu, & Ramkrishna, 2003) which is based on the measurement of extracellular substrates or intracellular metabolites. Fluxes can help the modeler decide the elementary units to be modeled. This approach has been demonstrated in the past to formulate a model for the growth of hybridoma cells. However, as detailed measurement of intracellular metabolites, enzymes and mRNAs become available, the strategy needs to be modified to take advantage of these measurements in order to reduce the complicated network into a small set of candidate models. The modeling framework and strategy are therefore continuously evolving to accommodate new measurements as well as theoretical or computational advances. In this paper, we report on strategy devised so far for the sake of completeness, but treat some key aspects in more detail. Subsequent publications will present more examples as well as further modifications. We propose here a framework for the analysis of large-scale metabolic systems. Within the framework, we will expound on the methodical decomposition of the detailed regulatory network into four simple systems, viz. the convergent, divergent, linear and cyclic pathways (Straight & Ramkrishna, 1990), which can then be modeled by choosing suitable objective functions. This identification part of the strategy is demonstrated for a sample regulatory network. Next, we focus on a strategy to identify cybernetic models based on large-scale enzyme and metabolite measurements. The model of Ramakrishna et al. (1996) is chosen as an example to demonstrate our identification strategy by using the model of Namjoshi and Ramkrishna, 2001 to generate data. Finally, we lay down some preliminary thoughts on the use of genetic algorithms for identification of model parameters.
Section snippets
A general cybernetic modeling framework for biosystems
The modeling framework consists of several state variables:
- (1)
Biomass component vector: We denote by X the mass concentration (mass per unit volume of culture) of biomass which consists of n master components Mj with concentration cj and intracellular level mj. In vector form, the biomass components (m) can be related to the concentrations (c) as c = Xm.
- (2)
Enzyme vector: Enzymes are treated separately from the biomass vector, since they serve as catalysts for reactions between species in the biomass
Strategy for model formulation using the general modeling framework
This paper presents a cybernetic modeling strategy which modifies considerably one proposed earlier (Namjoshi, Hu, et al., 2003) to include measurements of metabolites as well as enzymes. We present this strategy in this section. Metabolic pathways may comprise numerous reactions, and in many situations, it is possible and sufficient to model a few key reaction steps that are relevant to the phenomenon being explained. The first step in the cybernetic modeling process is therefore to identify
Power law formulation for determining cybernetic competitions
In the preceding sections, we have discussed how cybernetic models can be formulated, once the elementary units are identified. However, since this can be done in a multiple number of ways, we propose a diagnostic tool for identifying the most likely underlying competition. Such an analysis assumes that measurements of enzymes or mRNAs along with metabolite concentrations are available and is based on past work (Gupta, Varner, & Maranas, 2005; Yeung, Tegner, & Collins, 2002). Yeung et al. (2002)
Future recommendations
With progressive understanding gained from more detailed experimental observations, the cybernetic framework is continuing to evolve in its formulation that will appear in future publications. The cybernetic modeling identification strategy presented in this paper may lead to a large number of possible cybernetic structures, some of which overlap. In that case, this might present a formidable optimization challenge which may be tackled from either a rigorous mathematical programming approach (
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