This is the first report, to our knowledge, on a systematic method for constructing a large scale kinetic metabolic model with incomplete information on kinetic parametersr, and its initial application to the modeling of central metabolism of Methylobacterium extorquens AM1, a methylotrophic and environmental important bacterium, with all necessary constraints. Through a systematic and consistent procedure of finding a set of parameters in the physiological range we overcome an outstanding difficulty in large scale kinetic modeling: the requirement for a massive number of enzymatic reaction parameters. We are able to construct the kinetic model based on general biological considerations and incomplete experimental kinetic parameters. The success of our approach with incompletely input information is guaranteed by two known principles in biology, the robustness of the system and the cooperation among its various parts. (Will be pleased to be informed on other methodologies dealing with same type of problems: aoping@u.washington.edu)
In the post genomic era, many of our current global concerns, such as health, energy, and environment, need perspectives from bioengineering. In response integrated and large scale wet and dry lab approaches associated with systems biology have been developing rapidly [1][2][3] . Microbiology has played and will continue to play an important role, because bacteria have several billion years of experience in exploring various living conditions. This approach may give us better ways to produce important medical agents, to increase the efficiency of using carbon sources, and to turn harmful materials into environmentally friendly ones. A key component in such endeavors is the mathematical modeling of large biological systems. Among them is metabolic modeling whose goal is to create a comprehensive multidimensional representation of all of the biosynthetic reactions in an organism. For this, we require mathematical models [4] to perform two ma jor functions. First, they should be able to reliably describe experimental observations [5] , for instance, the models should indicate the metabolic fluxes and concentration of metabolites in each organelle or cell type at a given moment under specified conditions. Secondly, it is capable of generating experimentally [6,7] testable hypotheses leading to new experiments and new results which can then be used to refine the model. In the process of constructing such a model, new insights may often be gained as well. Modeling from the metabolic perspectives has been presented in many recent excellent reviews [8] . Metabolic networks have been discussed parallel to signaling networks [9] , and the effect of noise in metabolic networks have also been discussed from various perspectives [10,11] .
A survey of the literature suggests that models for metabolic networks are primarily (i) stoichiometric models [12] which display-in many cases on a genomewide level-an organism’s metabolic capabilities and (ii) kinetic models [13][14][15] , which describe at the enzymatic level, the rate at which reactions proceed. Flux Balance Analysis (FBA) [16] , in which mass conservation and other constraints are imposed on the metabolic network to determine a feasible solution space, is often used to evaluate the stoichiometric models. These constraints can be, for example, thermodynamic [17] and transcription regulatory [18] . Stoichiometric models can also be characterized by network-based pathway definitions [19] . While useful, important temporal behaviors are beyond the reach of FBA, such as the transient accumulation of toxic intermediate metabolites and the dynamical deficiency of an important nutrient. These time-dependent behaviors can drastically affect the life process of an organism. Nevertheless, kinetic models have not been studied as extensively as they should be. The common reasons are: a mechanistic formulation of even single enzyme kinetics is complicated with many parameters [20] and the experimental data for such parameters are scarce [7] . However, they are important for modeling behaviors such as oscillations [21,22] or bi-stability [23,24] that often occur in biological networks. Realistic kinetic modeling of large metabolic networks has been difficult. A major challenge is how to achieve biologically meaningful predictions from the mathematical model in the face of sparse experimental kinetic parameters and other necessary inputs.
In this paper, we present a systematic methodology to solve this important parameter issue. The first problem to be addressed is how to effectively and accurately represent an enzymatic reaction which may easily contain tens or more molecular parameters. Our solution is based on an observation that a complicated and exact (in the quasi-steady state sense) enzymatic rate equation can be rigorously cast into a generic form with the smallest set of kinetic parameters, which have transparent biochemical interpretations and can be directly related to experimental values.
The second problem to be addressed is that given the set of reactions how can a plausible set of fluxes be identified. We have at least two ways to solve this problem. One is to use a method related to flux balance analysis. Another is to set up a robust kinetic model which can generate various fluxes, although the kinetic parameters used may not be related to a realistic situation. The reason that realistic steady state fluxes can be obtained by crude kinetic models is that such fluxes are not sensitive to most kinetic parameters.
The third problem is how to obtain a consistent set of all parameters needed with given fluxes. Our solution is based on two considerations: to make use of as much available experimental data as we can, and to use a matching rule. Once a reasonable set of parameters is obtained, various predictions can be made, and, can be discussed in the biological contexts. In addition, various mathematical analyses can be carried to further test the consistency of
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