^{1}, Qing Nie

^{2}, Miao He

^{3,a)}and Tianshou Zhou

^{1,4,a)}

### Abstract

We present a simple yet effective method, which is based on power series expansion, for computing exact binomial moments that can be in turn used to compute steady-state probability distributions as well as the noise in linear or nonlinear biochemical reaction networks. When the method is applied to representative reaction networks such as the ON-OFF models of gene expression, gene models of promoter progression, gene auto-regulatory models, and common signaling motifs, the exact formulae for computing the intensities of noise in the species of interest or steady-state distributions are analytically given. Interestingly, we find that positive (negative) feedback does not enlarge (reduce) noise as claimed in previous works but has a counter-intuitive effect and that the multi-OFF (or ON) mechanism always attenuates the noise in contrast to the common ON-OFF mechanism and can modulate the noise to the lowest level independently of the mRNA mean. Except for its power in deriving analytical expressions for distributions and noise, our method is programmable and has apparent advantages in reducing computational cost.

This work was partially supported by Grant Nos. 91230204 (T.Z.), 30973980 (T.Z.), 11005162 (J.Z.), 2010CB945400 (T.Z.), 10451027501005652 (J.Z.), 20100171120039 (J.Z.), 2012J2200017 (J.Z.), P50GM76516 (Q.N.), R01GM67247 (Q.N.), and DMS-1161621 (Q.N.).

I. INTRODUCTION

II. GENERAL THEORY

III. APPLICATIONS

A. A two-species reactionnetwork

B. Common gene models

C. Gene models of promoter progression

D. Gene auto-regulatory models

E. A signaling system with autocatalytic kinase

IV. CONCLUSION AND DISCUSSION

### Key Topics

- Biochemical reactions
- 32.0
- Probability theory
- 16.0
- Signal generators
- 16.0
- Proteins
- 11.0
- Genetic networks
- 7.0

##### C12

##### G06F19/00

## Figures

Shown is that LNA gives approximate noise whereas our method gives exact noise. (a) The noise strength in two-species system (17) with is taken as a function of the parameter λ_{1}. The other parameters are λ_{2} = 10, τ_{1} = 2, τ_{2} = 1. (b) The noise strength in the two-species system (17) with is taken as a function of the parameter λ_{2}. The other parameters are λ_{1} = 1, τ_{1} = 2, τ_{2} = 1.

Shown is that LNA gives approximate noise whereas our method gives exact noise. (a) The noise strength in two-species system (17) with is taken as a function of the parameter λ_{1}. The other parameters are λ_{2} = 10, τ_{1} = 2, τ_{2} = 1. (b) The noise strength in the two-species system (17) with is taken as a function of the parameter λ_{2}. The other parameters are λ_{1} = 1, τ_{1} = 2, τ_{2} = 1.

Schematic of gene expression models considering promoter activity (active or inactive): (a) two-stage gene model and (b) three-stage gene model.

Schematic of gene expression models considering promoter activity (active or inactive): (a) two-stage gene model and (b) three-stage gene model.

Multi-state gene expression models and chromatin template-controlled noise in mRNA: (a) Schematic of a multi-inactive-state model of gene expression; (b) schematic of a multi-active-state model of gene expression; (c) the mean noise strength ratio as a function of two experimentally-measureable indices τ_{ on } and τ_{ off }, where both and are obtained from one of 1000 sets of randomly sampling τ_{1}, …, τ_{ K } with the fixed τ_{ on } and τ_{ off } for the fixed *K* ≡ *L* = *R* = 10 (where *L* and *R* represent the number of OFF states and ON states, respectively). The other parameters are μ = 20, δ = 1, *K* = 10; (d), (e), and (f) shown is the histogram of (red) and (green), where (d), (e) and (f) correspond to points D, E and F labeled in (c), respectively.

Multi-state gene expression models and chromatin template-controlled noise in mRNA: (a) Schematic of a multi-inactive-state model of gene expression; (b) schematic of a multi-active-state model of gene expression; (c) the mean noise strength ratio as a function of two experimentally-measureable indices τ_{ on } and τ_{ off }, where both and are obtained from one of 1000 sets of randomly sampling τ_{1}, …, τ_{ K } with the fixed τ_{ on } and τ_{ off } for the fixed *K* ≡ *L* = *R* = 10 (where *L* and *R* represent the number of OFF states and ON states, respectively). The other parameters are μ = 20, δ = 1, *K* = 10; (d), (e), and (f) shown is the histogram of (red) and (green), where (d), (e) and (f) correspond to points D, E and F labeled in (c), respectively.

The effect of multi-state mechanism on the noise in gene expression. (a) Schematic of multi-inactive-active-state model of gene expression. (b)–(d) The noise strength as a function of the parameter *k* ≡ λ_{4} = λ^{′} _{4} = λ_{5} = λ^{′} _{5} = λ_{6} = λ^{′} _{6}. The other parameter values used in computation are λ_{1} = 2, λ_{2} = 1, λ_{3} = 1, γ_{2} = 1, γ_{3} = 1, μ = 5 and (b) γ_{1} = 1, (c) γ_{1} = 2, (d) γ_{1} = 1.7.

The effect of multi-state mechanism on the noise in gene expression. (a) Schematic of multi-inactive-active-state model of gene expression. (b)–(d) The noise strength as a function of the parameter *k* ≡ λ_{4} = λ^{′} _{4} = λ_{5} = λ^{′} _{5} = λ_{6} = λ^{′} _{6}. The other parameter values used in computation are λ_{1} = 2, λ_{2} = 1, λ_{3} = 1, γ_{2} = 1, γ_{3} = 1, μ = 5 and (b) γ_{1} = 1, (c) γ_{1} = 2, (d) γ_{1} = 1.7.

The effect of feedback mechanisms on the noise in gene expression. (a) Schematic of auto-activation model of gene expression. (b) Schematic of auto-repression model of gene expression. (c) The noise strength as a function of the positive feedback strength (*a*) for system (62) , where the parameters are λ = γ = 1, μ = 5, δ = 0.1. (d) The noise strength as a function of the negative feedback strength (*r*) for system (71) , where the parameters are λ = γ = 1, μ = 5, δ = 0.1.

The effect of feedback mechanisms on the noise in gene expression. (a) Schematic of auto-activation model of gene expression. (b) Schematic of auto-repression model of gene expression. (c) The noise strength as a function of the positive feedback strength (*a*) for system (62) , where the parameters are λ = γ = 1, μ = 5, δ = 0.1. (d) The noise strength as a function of the negative feedback strength (*r*) for system (71) , where the parameters are λ = γ = 1, μ = 5, δ = 0.1.

(a) Schematic of the enzymatic futile cycle reaction mechanism. (b) The noise strength as a function of the total number *N* _{ T }, showing that there is an extreme, where the parameters are *k* _{1} = 1, *k* _{−1} = 1, *k* _{2} = 10, *k* _{−2} = 0.01.

(a) Schematic of the enzymatic futile cycle reaction mechanism. (b) The noise strength as a function of the total number *N* _{ T }, showing that there is an extreme, where the parameters are *k* _{1} = 1, *k* _{−1} = 1, *k* _{2} = 10, *k* _{−2} = 0.01.

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