# 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  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 & Continuous Dynamical Systems - B, 2020, 25 (7) : 2749-2774. doi: 10.3934/dcdsb.2020030
##### References:

show all references

##### References:
(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 $q_4$
 [1] Armando G. M. Neves. Eigenmodes and eigenfrequencies of vibrating elliptic membranes: a Klein oscillation theorem and numerical calculations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 611-624. doi: 10.3934/cpaa.2010.9.611 [2] Marcel Freitag. The fast signal diffusion limit in nonlinear chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1109-1128. doi: 10.3934/dcdsb.2019211 [3] Angelo B. Mingarelli. Nonlinear functionals in oscillation theory of matrix differential systems. Communications on Pure & Applied Analysis, 2004, 3 (1) : 75-84. doi: 10.3934/cpaa.2004.3.75 [4] John R. Graef, János Karsai. Oscillation and nonoscillation in nonlinear impulsive systems with increasing energy. Conference Publications, 2001, 2001 (Special) : 166-173. doi: 10.3934/proc.2001.2001.166 [5] Dong-Ho Tsai, Chia-Hsing Nien. On the oscillation behavior of solutions to the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4073-4089. doi: 10.3934/dcds.2019164 [6] Saroj Panigrahi, Rakhee Basu. Oscillation results for second order nonlinear neutral differential equations with delay. Conference Publications, 2015, 2015 (special) : 906-912. doi: 10.3934/proc.2015.0906 [7] Yuri V. Rogovchenko, Fatoş Tuncay. Interval oscillation of a second order nonlinear differential equation with a damping term. Conference Publications, 2007, 2007 (Special) : 883-891. doi: 10.3934/proc.2007.2007.883 [8] Jiying Ma, Dongmei Xiao. Nonlinear dynamics of a mathematical model on action potential duration and calcium transient in paced cardiac cells. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2377-2396. doi: 10.3934/dcdsb.2013.18.2377 [9] Petr Knobloch. Error estimates for a nonlinear local projection stabilization of transient convection--diffusion--reaction equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 901-911. doi: 10.3934/dcdss.2015.8.901 [10] Roberta Fabbri, Carmen Núñez, Ana M. Sanz. A perturbation theorem for linear Hamiltonian systems with bounded orbits. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 623-635. doi: 10.3934/dcds.2005.13.623 [11] Osama Moaaz, Omar Bazighifan. Oscillation criteria for second-order quasi-linear neutral functional differential equation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020136 [12] Hai-Yang Jin. Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3595-3616. doi: 10.3934/dcds.2018155 [13] Nguyen Dinh Cong, Doan Thai Son, Stefan Siegmund, Hoang The Tuan. An instability theorem for nonlinear fractional differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3079-3090. doi: 10.3934/dcdsb.2017164 [14] Shui-Hung Hou. On an application of fixed point theorem to nonlinear inclusions. Conference Publications, 2011, 2011 (Special) : 692-697. doi: 10.3934/proc.2011.2011.692 [15] Gábor Székelyhidi, Ben Weinkove. On a constant rank theorem for nonlinear elliptic PDEs. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6523-6532. doi: 10.3934/dcds.2016081 [16] Abdullah Özbekler, A. Zafer. Second order oscillation of mixed nonlinear dynamic equations with several positive and negative coefficients. Conference Publications, 2011, 2011 (Special) : 1167-1175. doi: 10.3934/proc.2011.2011.1167 [17] Jonathan P. Desi, Evelyn Sander, Thomas Wanner. Complex transient patterns on the disk. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1049-1078. doi: 10.3934/dcds.2006.15.1049 [18] C. D. Ahlbrandt, A. C. Peterson. A general reduction of order theorem for discrete linear symplectic systems. Conference Publications, 1998, 1998 (Special) : 7-18. doi: 10.3934/proc.1998.1998.7 [19] Roberta Fabbri, Russell Johnson, Carmen Núñez. On the Yakubovich frequency theorem for linear non-autonomous control processes. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 677-704. doi: 10.3934/dcds.2003.9.677 [20] José A. Cañizo, Chuqi Cao, Josephine Evans, Havva Yoldaş. Hypocoercivity of linear kinetic equations via Harris's Theorem. Kinetic & Related Models, 2020, 13 (1) : 97-128. doi: 10.3934/krm.2020004

2018 Impact Factor: 1.008

## Tools

Article outline

Figures and Tables