The mitogen-activated protein kinase (MAPK) cascades are ubiquitous in eukaryotic signal transduction, and these pathways are conserved in cells from yeast to mammals [1]. They relay extracellular stimuli from the plasma membrane to targets in the cytoplasm and nucleus, initiating diverse responses involving cell growth, mitogenesis, differentiation and stress responses in mammalian cells.

Detailed kinetics models of MAPK cascades have been constructed in recent years that are comprised of mixed sets of differential and algebraic equations (DAEs) [2,3]. Such models typically involve many parameters, such as the kinetic rate constants and the concentration ratios between various kinases and phosphatases, the values of which are not directly accessible in vivo and are subject to large uncertainty. Dynamic optimization has proved to be a very useful tool to help relate the model parameters to functions in MAPK networks [4,5]. Large-scale, nonlinear DAE models can be handled within this framework, as well as a large variety of objective functions and constraints.

In a recent work [5], the response of an interconvertible monocyclic cascade (phosphorylation-dephosphorylation cycle) has been studied. It was shown, using dynamic optimization, that values of the kinetic parameters that confer, at the same time, (i) a short response time, (ii) a large amplification capability, and (iii) a steep response profile to a graded input (ultrasensitivity) can be found. However, it was also found that, in a monocyclic cascade, these properties are not robust towards variations in the ratio between signaling enzyme and substrate kinase concentrations as well as the ratio between phosphatase and substrate kinase concentrations.

In this presentation, we extend the analysis to the general case of multiple levels of cascades, with emphasis on a linear three-kinase model as described in [2]. The same response properties as in the monocyclic case are considered, and dynamic optimization is employed to identify parameter values that optimize these response properties. Special emphasis is placed on the robustness of the resulting tricyclic cascades in the face of variations in kinase and phosphatase concentration ratios. Comparisons with the monocyclic cascade case are also presented.

References

[1] E. Klipp, R. Herwig, A. Kowald, C. Wierling, and H. Lehrach. “Systems Biology in Practice. Concepts, Implementations and Applications,” Wiley-VCH, Weinheim (Germany), 2005.

[2] C. Y. Huang, and J. E. Ferrel Jr. “Ultrasensitivity in the mitogen-activated protein kinase cascade,” Proc. Natl. Acad. Sci. U.S.A., 93:10078–10083, 1996.

[3] R. Heinrich, B. G. Neel, and T. A. Rapoport. “Mathematical models of protein kinase signal transduction,” Mol. Cell, 9:957–970, 2002.

[4] B. S. Adiwijaya, P. I. Barton, and B. Tidor. “Biological network design strategies: Discovery through dynamic optimization”, Mol. BioSyst., 2:650–659, 2006.

[5] B. Chachuat, D. Bonvin, and V. Hatzimanikatis. “Analysis of signal transduction pathways through dynamic optimization – Application to MAP kinase cascades,” AIChE Annual Meeting, Salt Lake City (UT), 2007.