# American Institute of Mathematical Sciences

July  2020, 25(7): 2749-2774. doi: 10.3934/dcdsb.2020030

## No-oscillation theorem for the transient dynamics of the linear signal transduction pathway and beyond

 1 LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China 2 Department of Bioengineering and Institute of Engineering in Medicine, University of California, San Diego, San Diego, CA 92093-0021, USA

Received  May 2019 Revised  September 2019 Published  July 2020 Early access  April 2020

Understanding the connection between the topology of a biochemical reaction network and its dynamical behavior is an important topic in systems biology. We proved a no-oscillation theorem for the transient dynamics of the linear signal transduction pathway, that is, there are no dynamical oscillations for each species if the considered system is a simple linear transduction chain equipped with an initial stimulation. In the nonlinear case, we showed that the no-oscillation property still holds for the starting and ending species, but oscillations generally exist in the dynamics of intermediate species. We also discussed different generalizations on the system setup. The established theorem will provide insights on the understanding of network motifs and the choice of mathematical models when dealing with biological data.

Citation: Tiejun Li, Tongkai Li, Shaoying Lu. No-oscillation theorem for the transient dynamics of the linear signal transduction pathway and beyond. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2749-2774. doi: 10.3934/dcdsb.2020030
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##### References:
 [1] U. Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman and Hall/CRC, Boca Raton, FL, 2007. [2] F. Capuani, A. Conte, E. Argenzio, L. Marchetti, C. Priami, S. Polo, P. {Di Fiore}, S. Sigismund and A. Ciliberto, Quantitative analysis reveals how EGFR activation and downregulation are coupled in normal but not in cancer cells, Nature Comm., 6 (2015), Article number, 7999. doi: 10.1038/ncomms8999. [3] M. Feiberg, Chemical reaction network structure and the stability of complex isothermal reactors Ⅰ. the deficiency zero and deficiency one theorems, Chem. Eng. Sci., 42 (1987), 2229-2268. [4] M. Feiberg, Chemical reaction network structure and the stability of complex isothermal reactors Ⅱ. multiple steady states for networks of deficiency one, Chem. Eng. Sci., 43 (1988), 1-25. [5] M. Feinberg, Foundations of Chemical Reaction Network Theory, Applied Mathematical Sciences, 202. Springer, Cham, 2019. [6] J. E. Ferrell Jr, T. Y.-C. Tsai and Q. Yang, Modeling the cell cycle: Why do certain circuits oscillate?, Cell, 144 (2011), 874-885. [7] D. Gillespie, Markov Processes: An Introduction for Physical Scientists, Academic Press, London, 1992. [8] F. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Arch. Rat. Mech. Anal., 49 (1972), 172-186.  doi: 10.1007/BF00255664. [9] R. Horn and C. Johnson, Matrix Analysis, 1st edition, Cambridge University Press, Cambridge, 1990. [10] C. Jia, M.-P. Qian and D.-Q. Jiang, Overshoot in biological systems modeled by markovchains: A nonequilibrium dynamic phenomenon, IET Syst. Biol., 8 (2014), 138-145. [11] J. Keener and J. Sneyd, Mathematical Physiology I: Cellular Physiology, 2nd edition, Springer Science+Business Media, New York, 2009. doi: 10.1007/978-0-387-79388-7. [12] E. Klipp and W. Liebermeister, Mathematical modeling of intracellular signaling pathways, BMC Neurosci., 7 (2006), S10. doi: 10.1186/1471-2202-7-S1-S10. [13] W. Ma, A. Trusina, H. El-Samad, W. A. Lim and C. Tang, Defining network topologies that can achieve biochemical adaptation, Cell, 138 (2009), 760-773.  doi: 10.1016/j.cell.2009.06.013. [14] J. Norris, Markov Chains, Cambridge Series in Statistical and Probabilistic Mathematics, 2. Cambridge University Press, Cambridge, 1998. [15] S.-Y. Shin, A.-K. Müller, N. Verma, S. Lev and L. K. Nguyen, Systems modelling of the EGFR-PYK2-c-Met interaction network predicts and prioritizes synergistic drug combinations for triple-negative breast cancer, PLoS Comp. Biol., 14 (2018), e1006192. doi: 10.1371/journal.pcbi.1006192. [16] S. Strogatz, Nonlinear Dynamics and Chaos, Westview Press, Boulder, CO, 2015. [17] G. Teschl, Ordinary Differential Equations and Dynamical Systems, American Mathmatical Society, Rode Island, 2012. doi: 10.1090/gsm/140. [18] J. Tóth, A. Nagy and D. Papp, Reaction Kinetics: Exercises, Programs and Theorems, Springer Science+Business Media, New York, 2018. doi: 10.1007/978-1-4939-8643-9.
(a). Illustration of a simplified MAPK signaling cascade. Here ${\rm Raf}^{\rm p}$, ${\rm MEK}^{\rm p}$ and ${\rm ERK}^{\rm p}$ represent the phosphorylated protein kinase. (b). Part of planar cell polarity WNT signaling pathway
The oscillations can appear for the middle species when the system is nonlinear. Shown here is the solution of $q_2$ for a three-node example with quadratic reaction rates
The oscillations can appear for the middle species when the system is nonlinear. Shown here is the solution of $q_2$ for a three-node example with Hill function type reaction rates. The inset figure shows the amplified detail of $q_2$ for $t\in [0, 12]$
Illustration of the linear signal transduction pathway with two branches, where we assume each species has decay but not been plotted here
Counter example which shows the oscillatory behavior for the middle speices in the sub-branches. Left panel: the network topology and reaction rates. Right panel: the history of $q_2$ which shows oscillation. The inset figure shows the amplified detail of $q_2$
Two kinds of more complicate tree-structured networks. Left panel: two levels but with more sub-branches. Right panel: trees with more than two levels
Left panel: matrix A. Right panel: oscillatory behavior of $q_2$. The inset figure shows the amplified detail of $q_2$
Left panel: matrix A. Right panel: oscillatory behavior of $q_4$
Left panel: network topology. Right panel: oscillatory behavior of $q_4$. The inset figure shows the amplified detail of $q_4$
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