^{1}and Kingshuk Ghosh

^{1,a)}

### Abstract

We study stochastic dynamics of two competing complexation reactions (i) A + B↔AB and (ii) A + C↔AC. Such reactions are common in biology where different reactants compete for common resources – examples range from binding enzyme kinetics to gene expression. On the other hand, stochasticity is inherent in biological systems due to small copy numbers. We investigate the complex interplay between competition and stochasticity, using coupled complexation reactions as the model system. Within the master equation formalism, we compute the exact distribution of the number of complexes to analyze equilibrium fluctuations of several observables. Our study reveals that the presence of competition offered by one reaction (say A + C↔AC) can significantly enhance the fluctuation in the other (A + B↔AB). We provide detailed quantitative estimates of this enhanced fluctuation for different combinations of rate constants and numbers of reactant molecules that are typical in biology. We notice that fluctuations can be significant even when two of the reactant molecules (say B and C) are infinite in number, maintaining a fixed stoichiometry, while the other reactant (A) is finite. This is purely due to the coupling mediated via resource sharing and is in stark contrast to the single reaction scenario, where large numbers of one of the components ensure zero fluctuation. Our detailed analysis further highlights regions where numerical estimates of mass action solutions can differ from the actual averages. These observations indicate that averages can be a poor representation of the system, hence analysis that is purely based on averages such as mass action laws can be potentially misleading in such noisy biological systems. We believe that the exhaustive study presented here will provide qualitative and quantitative insights into the role of noise and its enhancement in the presence of competition that will be relevant in many biological settings.

We thank Ken Dill, Steve Presse, Gerhard Stock, and Rob Phillips for long standing collaboration, and Hong Qian for many inspiring and helpful discussions. K.G. acknowledges support from the Research Corporation for Science Advancement as Cottrell scholar and PROF grant from the University of Denver.

I. INTRODUCTION

II. STOCHASTIC MODEL FOR TWO COMPETING REACTIONS

A. Exact equilibrium distribution

B. Equations of motion for different moments

III. RESULTS AND DISCUSSION

A. Quantifying equilibrium fluctuations and correlations

1. Coefficient of variation can be high

2. Simpler approximate analytical result for fluctuation

3. Competition enhances fluctuation

4. Competition promotes correlation

5. Equilibrium constants can be highly heterogeneous

6. Mass action prediction deviates from the exact average

B. Determination of dynamical quantities

1. Mean-field time evolution equations fail to predict correct averages

2. Modified time evolution equations predict correct averages and fluctuations

IV. CONCLUSION

### Key Topics

- Biochemical reactions
- 32.0
- Enzyme kinetics
- 18.0
- Proteins
- 12.0
- Molecular fluctuations
- 10.0
- Equations of motion
- 7.0

##### C12

## Figures

Phase diagrams of f 1 for (a) M = 50, x 1 = 1000, x 2 = 20 000, and for (b) N 1 = 50, x 1 = 1000, x 2 = 20 000. Different colors denote different f 1 percentages (blue: 50% < f 1 < 75%, green: 75% < f 1 < 100%, red: 100% < f 1 < 150%, black: f 1 > 150%).

Phase diagrams of f 1 for (a) M = 50, x 1 = 1000, x 2 = 20 000, and for (b) N 1 = 50, x 1 = 1000, x 2 = 20 000. Different colors denote different f 1 percentages (blue: 50% < f 1 < 75%, green: 75% < f 1 < 100%, red: 100% < f 1 < 150%, black: f 1 > 150%).

Probability distribution comparison under competitive (case (i)) conditions (solid line) and non-competitive (case (ii)) conditions (dashed line) with M = 50, N 1 = N 2 = 500, and x 1 = x 2 = 1000 (case (i): ⟨m 1⟩ = 25.0, σ1 = 3.45; case (ii): , ).

Probability distribution comparison under competitive (case (i)) conditions (solid line) and non-competitive (case (ii)) conditions (dashed line) with M = 50, N 1 = N 2 = 500, and x 1 = x 2 = 1000 (case (i): ⟨m 1⟩ = 25.0, σ1 = 3.45; case (ii): , ).

Phase diagrams of for (a) M = 50, x 1 = 1000, x 2 = 1000, and for (b) N 1 = 50, x 1 = 1000, x 2 = 1000. Different colors denote different η values (blue: 2.0 < η < 4.0, green: 4.0 < η < 6.0, red: 6.0 < η < 8.0, black: η > 8.0).

Phase diagrams of for (a) M = 50, x 1 = 1000, x 2 = 1000, and for (b) N 1 = 50, x 1 = 1000, x 2 = 1000. Different colors denote different η values (blue: 2.0 < η < 4.0, green: 4.0 < η < 6.0, red: 6.0 < η < 8.0, black: η > 8.0).

Plots of f 1 and as a function of β = N 2/N 1 with x 1 = x 2 = 1000 and varying values of M (blue lines: M = 5; green lines: M = 25; red lines: M = 50). The maximum value between N 1 and N 2 is 10 000 and the other is derived based on the value of β (e.g., for β = 0.5, N 2 = 5000, and N 1 = 10 000; for β = 2, N 2 = 10 000, and N 1 = 5000).

Plots of f 1 and as a function of β = N 2/N 1 with x 1 = x 2 = 1000 and varying values of M (blue lines: M = 5; green lines: M = 25; red lines: M = 50). The maximum value between N 1 and N 2 is 10 000 and the other is derived based on the value of β (e.g., for β = 0.5, N 2 = 5000, and N 1 = 10 000; for β = 2, N 2 = 10 000, and N 1 = 5000).

Plots of f 1 and as a function of α = M/N 2 with x 1 = x 2 = 1000 and varying values of N 1 (blue lines: N 1 = 5; green lines: N 1 = 25; red lines: N 1 = 50). The maximum value between M and N 2 is 10 000 and the other is derived based on the value of α (e.g., for α = 0.5, M = 5000, and N 2 = 10 000; for α = 2, M = 10 000, and N 2 = 5000).

Plots of f 1 and as a function of α = M/N 2 with x 1 = x 2 = 1000 and varying values of N 1 (blue lines: N 1 = 5; green lines: N 1 = 25; red lines: N 1 = 50). The maximum value between M and N 2 is 10 000 and the other is derived based on the value of α (e.g., for α = 0.5, M = 5000, and N 2 = 10 000; for α = 2, M = 10 000, and N 2 = 5000).

Plot of R 2 as a function of N 2 for a fixed value of M = 50 with (a) x 1 = x 2 = 1 and (b) x 1 = x 2 = 1000. The blue line represents N 1 = 2, the green line N 1 = 10, the red line N 1 = 20, and the black line N 1 = 50.

Plot of R 2 as a function of N 2 for a fixed value of M = 50 with (a) x 1 = x 2 = 1 and (b) x 1 = x 2 = 1000. The blue line represents N 1 = 2, the green line N 1 = 10, the red line N 1 = 20, and the black line N 1 = 50.

Joint distribution of K 1 and K 2 for M = 10 000, N 1 = N 2 = 5000, and x 1 = x 2 = 100. Different color points denote different probabilities (blue: P(K 1, K 2) < 0.25%; red: 0.25% < P(K 1, K 2) < 3%; black: P(K 1, K 2) > 3%) and the green square denotes the coordinates of ⟨K 1⟩ = 145.9 and ⟨K 2⟩ = 145.9; K 1 = K 2 = 100 would be the single value predicted by mass action laws.

Joint distribution of K 1 and K 2 for M = 10 000, N 1 = N 2 = 5000, and x 1 = x 2 = 100. Different color points denote different probabilities (blue: P(K 1, K 2) < 0.25%; red: 0.25% < P(K 1, K 2) < 3%; black: P(K 1, K 2) > 3%) and the green square denotes the coordinates of ⟨K 1⟩ = 145.9 and ⟨K 2⟩ = 145.9; K 1 = K 2 = 100 would be the single value predicted by mass action laws.

Time evolution of the average (Gillespie simulation = black circles; analytical solution with fluctuation = black line; analytical mass action solution = green line) and standard deviation (Gillespie simulation = red circles; analytical solution with fluctuation = red line) of the number of complexes m 1 (left panel) and m 2 (right panel) with M = 5, N 1 = 10, N 2 = 5, k 1f = 0.01 s−1, k 1b = 1 s−1, k 2f = 1 s−1, k 2b = 0.5 s−1, and initial conditions of m 1(t = 0) = 0, and m 2(t = 0) = 0. Equilibrium values are: ⟨m 1⟩ exact = 0.11, ⟨m 1⟩ massaction = 0.13, σ1, exact = 0.32, ⟨m 2⟩ exact = 3.8, ⟨m 2⟩ massaction = 3.6, σ2, exact = 0.8.

Time evolution of the average (Gillespie simulation = black circles; analytical solution with fluctuation = black line; analytical mass action solution = green line) and standard deviation (Gillespie simulation = red circles; analytical solution with fluctuation = red line) of the number of complexes m 1 (left panel) and m 2 (right panel) with M = 5, N 1 = 10, N 2 = 5, k 1f = 0.01 s−1, k 1b = 1 s−1, k 2f = 1 s−1, k 2b = 0.5 s−1, and initial conditions of m 1(t = 0) = 0, and m 2(t = 0) = 0. Equilibrium values are: ⟨m 1⟩ exact = 0.11, ⟨m 1⟩ massaction = 0.13, σ1, exact = 0.32, ⟨m 2⟩ exact = 3.8, ⟨m 2⟩ massaction = 3.6, σ2, exact = 0.8.

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