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The nonlinear behavior of metabolic systems can arise from at least two different sources. One comes from the nonlinear kinetics of chemical reactions in metabolism and the other from nonlinearity associated with regulatory processes. Consequently, organisms at a constant growth rate (as experienced in a chemostat) could display multiple metabolic states or display complex oscillatory behavior both with potentially serious implications to process operation. This paper explores the nonlinear behavior of a metabolic model of
Historically, microorganisms have been utilized for the production of valuable products in our daily life, e.g., bread, vinegar, wine and beer. With the advent of recombinant DNA technology several decades ago, it is common practice to make genetic modifications to microbes for the industrial production of food, energy, medicine and other valuable products. Towards ensuring the economic competitiveness of those commercial processes, maximizing productivity is one of the goals to achieve.
It is a challenge to manipulate cellular metabolism due to its complexity. Metabolic systems often exhibit intricate nonlinear behaviors, such as steadystate multiplicity and dynamic oscillations. It is necessary to understand what triggers this breadth of behavior and to predict when and under what conditions they would occur. Such a study is also practically important, as nonlinear behavior should be avoided if it prevents stable operations [
A basic source of nonlinearity in a metabolic system is the intrinsic kinetics of biochemical reactions. More importantly, however, nonlinear metabolic behavior becomes much more complex and diverse due to regulation that dynamically drives individual reactions in response to environmental changes. Dramatic shift between multitudes of metabolic pathways often arises in a dynamic environment as a consequence of metabolic regulation. For the nonlinear analysis of metabolic systems, therefore, it is essential to employ metabolic models that are able to appropriately account for dynamic regulation. Various modeling ideas have been developed for the analysis of metabolic systems, including metabolic pathway analysis [
A full kinetic description of metabolic regulation requires detailed knowledge of its molecular mechanism, which is incomplete in most cases. Alternatively, the cybernetic approach [
Cybernetic models have been successfully used to perform bifurcation analysis of metabolic systems, such as
In this article, Kim
This paper is organized as follows. In the subsequent sections, we provide a summarized description of the Kim
Dynamic mass balances of extracellular metabolites in a chemostat can be represented as follows:
Under the quasi steadystate approximation, the flux vector, r, can be represented as nonnegative (or convex) combinations of basic pathways, termed elementary modes (EMs) [
The cybernetic approach assumes a certain metabolic objective, such as the maximization of the carbon uptake rate (or growth rate) for which metabolic reactions are optimally regulated. The HCM framework views EMs as metabolic options to achieve such an objective and describe metabolic regulation in terms of their optimal combinations. Flux through the
Enzyme level
Kim
Bifurcation analysis of cybernetic models requires special treatment of the nonsmooth max function contained in the
Namjoshi and Ramkrishna [
Model equations and parameter values. EM, elementary mode.
Variables or parameters  Equations or parameter values 

Extracellular metabolites and biomass  Glucose: 
Enzymes  
Cybernetic variables  
Kinetics  
Parameters and stoichiometric coefficients  
Notations 
Four combinatorial cases for Kim
Case 





I  1  ≤1  ≤1  ≤1 
II  ≤1  1  ≤1  ≤1 
III  ≤1  ≤1  1  ≤1 
IV  ≤1  ≤1  ≤1  1 
In each case, we force
Hysteresis curves obtained from four cases considered in
Finally, we put together individual pieces of feasible branches of each case to obtain the hysteresis curve over the whole range of
The shape of the hysteresis curve becomes somewhat different at a higher fractional concentration of glucose,
Overall hysteresis curve generated by integrating individual pieces of feasible branches: (
While the combinatoric approach described above allows for rigorous bifurcation analysis in theory, it is ineffective in cases where the number of EMs is large. Alternatively, we may mollify the pain of handling nonsmooth functions by making smooth approximations.
The usefulness of the smooth approximation depends on THE cases in consideration [
Reproduction of the hysteresis curve of
Magnified views of two red windows around the catchup points in the lowerright panel of
If the approximate representation is acceptable as in our case, nonlinear analysis of piecewisesmooth functions is greatly facilitated by using an automated software, such as MATCONT, a standard continuation software package [
As a compromise, we may integrate combinatoric enumeration and smooth approximation. That is, we can sketch a bifurcation diagram conveniently using the smooth approximation and refine nonsmooth folds using rigorous computations based on the combinatoric approach, because they are only the regions where errors may occur. Catchup points are readily identified from the hysteresis curve using the approximate function. This combined approach is more accurate than the approximate function alone and more convenient than the full combinatoric enumeration.
Among three methods discussed in the previous section, we use the
To get a global bifurcation diagram, we explore the whole parameter space spanned by
Hysteresis curves of all components when
A global bifurcation diagram in the
To clarify the implication of this global bifurcation diagram, hysteresis curves drawn with nine different values of
Hysteresis of biomass concentration profiles with different
The effect of the total sugar concentration in the feed (
A global bifurcation diagram in the
Hysteresis curves of all components when
Kim
Through the comprehensive bifurcation analysis in this work, we could identify a new domain with seven steady states. Experimental verification would require precise control of conditions and concentration measurements, however.
Since the cybernetic variables for enzyme activity control are max functions and, therefore, nonsmooth, nonlinear analysis of cybernetic models has had to rely on a suitably convenient methodology to confront this issue. The
Using the approximate function, we could construct global bifurcation diagrams on the
Clearly, more detailed models comprising more EMs could produce a considerably greater number of steady states, which may be difficult to observe experimentally without accurate analytical measurements and precise control of experimental conditions. It is our premise that this is an area for extensive future exploration by researchers concerned with modeling metabolism.
The authors acknowledge support for this work from the Center for Science of Information (CSoI), a National Science Foundation (NSF) Science and Technology Center, under grant agreement CCF0939370.
The authors declare no conflict of interest.