(142a) Finding Steady State Solutions of Cybernetic Models of Biological Systems

Authors: 
Mantravadi, V. K., Indian Institute of Technology Madras
Kienle, A., Otto von Guericke University Magdeburg


Biological reactors, unlike simple chemical reactors, have additional degrees of complexity due to the regulatory features of the microorganisms or bacteria. The nonlinear behavior of bioreactors can be due to the regulatory features which drive organisms to achieve an optimal goal. The regulatory mechanism of the microorganisms acts as a positive feedback as well as a feedforward controller which enables the microorganisms to adapt to the continuously changing environment consisting of multiple substrates. Cybernetic models for biological systems have elegantly described these regulatory features of the microorganism/bacteria. The regulatory features are captured in these cybernetic models by incorporation of control/cybernetic variables. These cybernetic variables control the synthesis (by induction/repression) and activity (by inhibition/activation) of various key enzymes catalyzing the synthesis of key growth precursors. In this contribution, we deal with a cybernetic model for the uptake of multiple substrates which includes maintenance effects. The expression for the cybernetic variable which controls the activity of the enzymes has a ?max? function in the denominator. This renders the model intractable for classical bifurcation analysis to understand the steady state behavior of the system. A combinatorial strategy is used to enable classical continuation methods. This strategy involves finding out all possible combinations of the cybernetic variables controlling the activity of the key enzymes and applying each case to the model equations. The right hand sides of the ODEs describing the model consist of reaction rates which are linear combination of rational functions of variables and parameters. This work depicts usage of POLSYS_PLP, a globally convergent probability one homotopy method, for determining the steady state solutions for primary branches of bifurcation diagrams. This methodology ensures the detection of all feasible steady states. A detailed procedure is figured out for finding out the steady states. All possible steady states can be determined of which some may be spurious, due to conversion of ODEs to polynomials, and a few may be physically infeasible.

References

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Wise, S. M., Sommese, A. J., and Watson, L. T. (2000) "Algorithm 801: POLSYS_PLP: A partitioned linear product homotopy code for solving polynomial systems of equations", ACM Transactions on Mathematical Software, 26, 176 - 200.


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