Interplay between stochasticity and negative feedback leads to pulsed dynamics and distinct gene activity patterns

Samuel Zambrano, Marco E. Bianchi, Alessandra Agresti, and Nacho Molina

Phys. Rev. E 92, 022711 – Published 14 August 2015

Abstract

Gene expression is an inherently stochastic process that depends on the structure of the biochemical regulatory network in which the gene is embedded. Here we study the dynamical consequences of the interplay between stochastic gene switching and the widespread negative feedback regulatory loop in a simple model of a biochemical regulatory network. Using a simplified hybrid simulation approach, in which only the gene activation is modeled stochastically, we find that stochasticity in gene switching by itself can induce pulses in the system, providing also analytical insights into their origin. Furthermore, we find that this simple network is able to reproduce both exponential and peaked distributions of gene active and inactive times similar to those that have been observed experimentally. This simplified hybrid simulation approach also allows us to link these patterns to the dynamics of the system for each gene state.

Received 14 November 2014
Revised 13 March 2015

DOI:https://doi.org/10.1103/PhysRevE.92.022711

Authors & Affiliations

Samuel Zambrano^*

San Raffaele University, Via Olgettina 58, 20132 Milan, Italy and Division of Genetics and Cell Biology, San Raffaele Scientific Institute, Via Olgettina 60, 20132 Milan, Italy

Marco E. Bianchi

San Raffaele University, Via Olgettina 58, 20132 Milan, Italy

Alessandra Agresti

Division of Genetics and Cell Biology, San Raffaele Scientific Institute, Via Olgettina 60, 20132 Milan, Italy

Nacho Molina^†

SynthSys Centre, University of Edinburgh, Mayfield Road, EH9 3JD Edinburgh, United Kingdom

^*zambrano.samuel@hsr.it
^†nacho.molina@ed.ac.uk

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Vol. 92, Iss. 2 — August 2015

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Images

Figure 1
Dynamics of a network with stochastic gene switching and a negative feedback loop. (a) Model diagram showing the activator $A$ , the inhibitor $I$ , and the gene that can be active $G$ or inactive $\bar{G}$ . (b) Stochastic simulation of a trajectory of the system showing the free activator fraction $A / A_{tot}$ [blue (gray)], the gene state $G$ (step gray line), and the relative amount of inhibitor $I / A_{tot}$ [orange (light gray)] displaying pulses. For the gene state, an active time $t_{on}$ and an inactive time $t_{off}$ are displayed. Simulations were performed with the parameters $d_{C} = 1.5 \times 10^{- 4} s^{- 1}$ , $k_{a} = 5.1 \times 10^{- 6} s^{- 1}$ , $k_{d} = 1.4 \times 10^{- 3} s^{- 1}$ , $d_{I} = 2.8 \times 10^{- 4} s^{- 1}$ , $k_{I} = 11.2 s^{- 1}$ , $k_{off} = 1.4 \times 10^{- 7} s^{- 1}$ , and $k_{on} = 5.8 \times 10^{- 7} s^{- 1}$ .
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Figure 2
Stochastic gene switching is responsible for most of the dynamic variability in the network. (a) Evolution in time of the free activator $A$ using fully stochastic [orange (light gray)] and simplified hybrid [blue (gray)] simulations. Both display similar pulses while the fully deterministic simulation (dashed line) remains at a stable fixed point. Distribution of the values of (b) $A$ and (c) $I$ for both the simplified hybrid [blue (gray)] and fully stochastic [orange (light gray)] simulations of the biochemical network. Distribution of (d) the amplitudes and (e) the periods of the pulses for the simplified hybrid [blue (gray)] and fully stochastic [orange (light gray)] models. The distribution of the activator and the pulse parameters are very similar for the simplified hybrid and the fully stochastic systems. The chosen parameter values are shown in the caption of Fig. 1. (f) Box plot showing the Kullback-Leibler divergence $D_{KL}$ between the distributions of the magnitudes considered obtained from simplified hybrid simulations and fully stochastic simulations for 1000 combinations of random parameters; the box represents the 25th, 50th, and 75th percentiles while the lines above and below the box represent the 1st and 99th. Dots represent the $D_{KL}$ values obtained for the parameter combination used in (a)–(e) and the 1000 additional parameter combinations were obtained by varying them randomly up to one order of magnitude above and below those values. The maximum $D_{KL}$ value is displayed as a horizontal dashed line. Overall, for most parameter combinations $D_{KL}$ values are low.
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Figure 3
The three nullclines that can be associated with the simplified hybrid model: the nullcline for $\frac{d A}{d t} = 0$ (dashed line) and the nullclines for $\frac{d I}{d t} = 0$ with $G = 1$ (black line above) and $G = 0$ (black line below). A trajectory of the simplified hybrid model is also depicted [orange (light gray)]. There are two intersections between these three curves (black diamonds); a simple graphical analysis shows that these points are stable. Trajectories of the simplified hybrid model jump between these two fixed points. The chosen parameter values are shown in the caption of Fig. 1.
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Figure 4
Distinct gene activity patterns are reproduced with different model parameters. Distributions of (a) the gene active times $t_{on}$ and (b) the inactive times $t_{off}$ with the highest and the lowest CVs [average in blue (gray) and orange (light gray), respectively] obtained by varying randomly the parameters given in the caption of Fig. 1 one order of magnitude above or below their value. (c) Box plot showing the distributions of the ratio between the parameter values and the reference values shown in the caption of Fig. 1, producing exponential distributions [blue (gray)] and peaked distributions [orange (light gray)]; the box represents the 25th, 50th, and 75th percentiles while the lines above and below the box represent the 1st and 99th. Distributions of the eigenvalues with larger absolute value $λ_{fast}$ of (d) the fixed point $(A_{0}^{*}, I_{0}^{*})$ (obtained when the gene is inactive) and (e) the fixed point $(A_{1}^{*}, I_{1}^{*})$ (obtained when the gene is active), producing exponential distributions [blue (gray)] and peaked distributions [orange (light gray)].
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