^{1}and Hong Qian

^{1}

### Abstract

We establish a mathematical model for a cellular biochemical signaling module in terms of a planar differential equation system. The signaling process is carried out by two phosphorylation-dephosphorylation reaction steps that share common kinase and phosphatase with saturated enzyme kinetics. The pair of equations is particularly simple in the present mathematical formulation, but they are singular. A complete mathematical analysis is developed based on an elementary perturbation theory. The dynamics exhibits the canonical competition behavior in addition to bistability. Although widely understood in ecological context, we are not aware of a full range of biochemical competition in a simple signaling network. The competition dynamics has broad implications to cellular processes such as cell differentiation and cancer immunoediting. The concepts of homogeneous and heterogeneous multisite phosphorylation are introduced and their corresponding dynamics are compared: there is no bistability in a heterogeneous dual phosphorylation system. A stochastic interpretation is also provided that further gives intuitive understanding of the bistable behavior inside the cells.

A large fraction of intracellular biochemical reactions in high-level biological organisms are for “information processing” known as signal transduction in biomedical literature. One of the most important biochemical reaction networks involved in signal transduction consists of only three kinds of proteins: a substrate which has an active state and an inactive state and two enzymes that catalyze the activation and deactivation reactions. The discoveries of two such systems were awarded Nobel prizes in physiology or medicine: in 1992 to Edmond Fischer and Edwin Krebs for reversible protein phosphorylation and in 1994 to Alfred Gilman and Martin Rodbell for G proteins. A nonlinear mathematical model can be developed for such a small biochemical network. For the case of two-step regulation, it is known that the system is bistable in a certain parameter range. The present work is an in-depth study of the mathematical model. We show that the molecular system exhibits a full range of competition behavior depending upon the absolute and relative strengths of the two activation steps: both weak but balanced, both strong but balanced, or imbalanced with one weak and one strong. Such canonical behavior is textbook material in mathematical ecology, but there has been no simple mathematical example in the past for cellular biochemistry. Such nonlinear behavior can have implications for a wide range of cellular processes. The transitions from imbalanced competition to balanced strong competition, for example, might be relevant to cancer immunoediting.

We thank Jian-Dong Ding, Jeremy Gunawardena, Boris Kholodenko, Wenjun Qiu, Chao Tang, Liming Wang, Zhi-Xin Wang, and Jianhua Xing for discussions and Melissa Vellela for reading the manuscript. Q.H. is supported by a scholarship from the China Scholarship Council.

I. INTRODUCTION

II. THE KINETIC MODEL

A. Steady state analysis and bistability

B. The kinetic equations and fixed points

1. Approximate root near

2. Approximate root near

III. THE MATHEMATICAL FORMULATION IN TERMS OF SINGULAR EQUATIONS FOR ULTRASENSITIVITY

A. A nonsingular counterpart

1. Fixed point (0,1)

2. Fixed point (1,0)

3. The interior fixed point

B. Further analysis near the singular fixed points

C. Canonical competition behavior in dual PdPC

IV. HOMOGENEOUS VERSUS HETEROGENEOUS DUAL PdPC

A. Zeroth-order limit

B. Further simplified mathematical model

V. STOCHASTIC APPROACH TO DUAL PdPC

VI. DISCUSSION

### Key Topics

- Biochemical reactions
- 12.0
- Enzyme kinetics
- 9.0
- Enzymes
- 8.0
- Stochastic processes
- 6.0
- Differential equations
- 5.0

## Figures

Functions (filled circles) and (open squares) from Eq. (10) have three intersections. The two interactions, one at 0.99 and one near 0.4, are clearly visible. To see the third one near 0.0004, we note that and the slopes of and at the origin are and , respectively. Thus there is a third intersection near the origin.

Functions (filled circles) and (open squares) from Eq. (10) have three intersections. The two interactions, one at 0.99 and one near 0.4, are clearly visible. To see the third one near 0.0004, we note that and the slopes of and at the origin are and , respectively. Thus there is a third intersection near the origin.

Two examples of as the solution to Eqs. (22a) and (22b). The lower line: , , , and ; therefore, . The top curve: , , , and ; therefore .

Two examples of as the solution to Eqs. (22a) and (22b). The lower line: , , , and ; therefore, . The top curve: , , , and ; therefore .

Molecular competition in homogeneous dual PdPC system according to Eqs. (19a) and (19b) or equivalently Eqs. (17a) and (17b) with and . Each panel has two nullclines whose intersections are the steady state of the system: open circles are unstable and filled circles are stable.

Molecular competition in homogeneous dual PdPC system according to Eqs. (19a) and (19b) or equivalently Eqs. (17a) and (17b) with and . Each panel has two nullclines whose intersections are the steady state of the system: open circles are unstable and filled circles are stable.

For and and with small and , the and nullclines are very near the , as they are left of the two vertical asymptotes shown in (a). The intersection between two nullclines, however, is near the if and near the if . Thus, as the function of defined in the text, there is an ultrasensitive transition between and shown in (b).

For and and with small and , the and nullclines are very near the , as they are left of the two vertical asymptotes shown in (a). The intersection between two nullclines, however, is near the if and near the if . Thus, as the function of defined in the text, there is an ultrasensitive transition between and shown in (b).

The dynamics of dual PdPC is confined on the triangle shown in (a): and . For heterogeneous dual PdPC operating under zeroth-order kinetics, the vector field in the triangle is a constant given in Eq. (33). The dynamics moves from the domain to its boundaries. When it is on the edge, the dynamics follows the direction shown in (b): if , the steady state is (0,0,1) and if , the steady state is (0,1,0). The same applies to the edge with . The dynamics confined on (near) the edge of is more delicate. See the text for more discussions.

The dynamics of dual PdPC is confined on the triangle shown in (a): and . For heterogeneous dual PdPC operating under zeroth-order kinetics, the vector field in the triangle is a constant given in Eq. (33). The dynamics moves from the domain to its boundaries. When it is on the edge, the dynamics follows the direction shown in (b): if , the steady state is (0,0,1) and if , the steady state is (0,1,0). The same applies to the edge with . The dynamics confined on (near) the edge of is more delicate. See the text for more discussions.

A stochastic kinetic model for dual PdPC with kinase and phosphatase catalyzed reactions. One has the transition rates going right, left, up, and downward being , , , and , respectively. For heterogeneous dual PdPC with zeroth-order kinases and phosphatases, all are constant as in Eq. (33). However, for shared kinase and phosphatase, there are competitions for kinase the phosphatase when they are saturated. Hence the are no longer independent of . In fact, , , , and .

A stochastic kinetic model for dual PdPC with kinase and phosphatase catalyzed reactions. One has the transition rates going right, left, up, and downward being , , , and , respectively. For heterogeneous dual PdPC with zeroth-order kinases and phosphatases, all are constant as in Eq. (33). However, for shared kinase and phosphatase, there are competitions for kinase the phosphatase when they are saturated. Hence the are no longer independent of . In fact, , , , and .

## Tables

Steady state probability distribution of the stochastic model given in Fig. 6 for homogeneous dual PdPC with , , , and . The distribution is bimodal with peaks at upper-right and lower-left corners representing the and .

Steady state probability distribution of the stochastic model given in Fig. 6 for homogeneous dual PdPC with , , , and . The distribution is bimodal with peaks at upper-right and lower-left corners representing the and .

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