^{1}, Gastone C. Castellani

^{1,2,a)}, Enrico Giampieri

^{1}, Daniel Remondini

^{1}and Leon N Cooper

^{2}

### Abstract

Dual phospho/dephosphorylation cycles, as well as covalent enzymatic-catalyzed modifications of substrates are widely diffused within cellular systems and are crucial for the control of complex responses such as learning, memory, and cellular fate determination. Despite the large body of deterministic studies and the increasing work aimed at elucidating the effect of noise in such systems, some aspects remain unclear. Here we study the stationary distribution provided by the two-dimensional chemical master equation for a well-known model of a two step phospho/dephosphorylation cycle using the quasi-steady state approximation of enzymatic kinetics. Our aim is to analyze the role of fluctuations and the molecules distribution properties in the transition to a bistable regime. When detailed balance conditions are satisfied it is possible to compute equilibrium distributions in a closed and explicit form. When detailed balance is not satisfied, the stationary non-equilibrium state is strongly influenced by the chemical fluxes. In the last case, we show how the external field derived from the generation and recombination transition rates, can be decomposed by the Helmholtz theorem, into a conservative and a rotational (irreversible) part. Moreover, this decomposition allows to compute the stationary distribution via a perturbative approach. For a finite number of molecules there exists diffusion dynamics in a macroscopic region of the state space where a relevant transition rate between the two critical points is observed. Further, the stationary distribution function can be approximated by the solution of a Fokker-Planck equation. We illustrate the theoretical results using several numerical simulations.

This work was partially supported by the Italian Ministry of Research and University and National Institute of Nuclear Physics. G.C.C. wants to acknowledge the Fondazione CarisBO, and both G.C.C. and A.B. want to acknowledge the support of the Institute for Brain and Neural Systems at Brown University.

I. INTRODUCTION

II. DUAL PHOSPHORYLATION/DEPHOSPHORYLATION ENZYMATIC CYCLES

III. THE STATIONARY DISTRIBUTION

IV. NUMERICAL SIMULATIONS

V. CONCLUSIONS

### Key Topics

- Fokker Planck equation
- 13.0
- Biochemical reactions
- 9.0
- Chemical reactions
- 9.0
- Chemical kinetics
- 8.0
- Critical point phenomena
- 7.0

##### B01J

## Figures

Scheme of the double enzymatic cycle of addition/removal reactions of chemical groups via Michelis-Menten kinetic equations as shown in Eq. (1) in the case of phosphoric groups.

Scheme of the double enzymatic cycle of addition/removal reactions of chemical groups via Michelis-Menten kinetic equations as shown in Eq. (1) in the case of phosphoric groups.

Stationary distributions for the *A* and states in the double phosphorylation cycle when detailed balance (13) holds with and . In the top figure we set the reaction velocities and (symmetric case), whereas in the bottom figure we increase the and value to 1.15. The number of molecules is *N* _{ T } = 40. The transition from a unimodal distribution to a bimodal distribution is clearly visible.

Stationary distributions for the *A* and states in the double phosphorylation cycle when detailed balance (13) holds with and . In the top figure we set the reaction velocities and (symmetric case), whereas in the bottom figure we increase the and value to 1.15. The number of molecules is *N* _{ T } = 40. The transition from a unimodal distribution to a bimodal distribution is clearly visible.

In grey we show the region where components of the vector field (11) are ≃1 using the parameter values of Fig. 2 (bottom). The blue lines enclose the region where the first component is nearby 1, whereas the red ones enclose the corresponding region for the second component.

In grey we show the region where components of the vector field (11) are ≃1 using the parameter values of Fig. 2 (bottom). The blue lines enclose the region where the first component is nearby 1, whereas the red ones enclose the corresponding region for the second component.

Plot of the rotor field for the potential *H* using the following parameter values: *v* _{ M1} = *v* _{ M2} = 1, , *N* _{ T } = 40 and (left picture) or (right picture).

Plot of the rotor field for the potential *H* using the following parameter values: *v* _{ M1} = *v* _{ M2} = 1, , *N* _{ T } = 40 and (left picture) or (right picture).

(Left picture): Plot of the zero-order approximation for the probability distribution using the decomposition (17) for the vector field associated to the CME. (Right picture): Plot of the stationary distribution computed by directly solving the CME (2). We use the parameter values of the case I in Table I.

(Left picture): Plot of the zero-order approximation for the probability distribution using the decomposition (17) for the vector field associated to the CME. (Right picture): Plot of the stationary distribution computed by directly solving the CME (2). We use the parameter values of the case I in Table I.

The same as in Fig. 5 using parameter values of case II in Table I.

The same as in Fig. 5 using parameter values of case II in Table I.

The same as in Fig. 5 using the parameter values of the case III in Table I.

The same as in Fig. 5 using the parameter values of the case III in Table I.

Current vector field computed by using definiton (5) and the stationary solution of the CME with case IV parameters. The distribution is bimodal (cf. Figure 9) and the current lines tend to be orthogonal to the distribution gradient near the maximal value.

Current vector field computed by using definiton (5) and the stationary solution of the CME with case IV parameters. The distribution is bimodal (cf. Figure 9) and the current lines tend to be orthogonal to the distribution gradient near the maximal value.

The same as in Fig. 5 using parameter values of case IV in Table I.

The same as in Fig. 5 using parameter values of case IV in Table I.

## Tables

Main parameters used in the simulations.

Main parameters used in the simulations.

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