Understanding the discrete genetic toggle switch phenomena using a discrete ‘nullcline’ construct inspired by the Markov chain tree theorem

Abstract

Nullclines provide a convenient way of characterising and understanding the behaviour of low dimensional nonlinear deterministic systems, but are, perhaps not unsurprisingly, a poor predictor of the behaviour of discrete state stochastic systems in the low numbers regime. Such models are appropriate in many biological systems. In this paper we propose a graphical discrete 'nullcline-like' construction, inspired by the Markov chain tree theorem, and investigate its application to the original genetic toggle switch, which is a feedback interconnection of two mutually repressing genes. When the feedback gain (the 'cooperativity') is sufficiently large, the deterministic system exhibits bistability, which shows itself as a bimodal stationary distribution in the discrete stochastic system for sufficiently large numbers. However, at small numbers a third mode appears corresponding to roughly equal numbers of each molecule. Without cooperativity, on the other hand (i.e. low feedback gain), the deterministic system has just one stable equilibrium. Nevertheless, the stochastic system can still exhibit bimodality. In this paper, we illustrate that the discrete 'nullclines' proposed can, without the need to calculate the steady state distribution, provide an efficient graphical way of predicting the shape of the stationary probability distribution in different parameter regimes, thus allowing for greater insights in the observed behaviours.

Autoregulation of switching behavior by cellular compartment size

“…In light of these results, we hypothesize that this phenomenon holds generally for inhibitory systems, regardless of dimensionality. Establishing this result without recourse to expensive simulation will require new analytical tools in nonlinear stochastic systems that thus far remain out of reach despite decades of research (15,16,39,(49)(50)(51). Nonetheless, the consistency of the results documented here suggests that sizedependent regulation mechanisms can readily exist in general biochemical systems.…”

Autoregulation of switching behavior by cellular compartment size

“…Moreover, we conjecture that the results extend to systems with different analytic forms for the inhibitory reaction rates beyond Hill functions, provided the geometry of the nullclines in the mean-field description is approximately preserved. Establishing this result without recourse to expensive simulation will require new analytical tools in nonlinear stochastic systems that so far remain out of reach despite decades of research ( 15 , 16 , 18 , 45 , 54 – 56 ). Nonetheless, the consistency of the results documented here suggests that size-dependent regulation mechanisms can readily exist in general biochemical systems.…”

Analysis and Control of Genetic Toggle Switches Subject to Periodic Multi-Input Stimulation

“…System (8) with the static relations ( 11)-( 12) and diffusion dynamics across the cell membrane ( 5)-( 6) can be represented in block form as in Figure 4. The cell membrane acts as a linear (non-symmetrical) first order low-pass filter for the signals u aTc (t) and u IPTG (t) with a cut-off frequency that In our analysis we make the following simplifying assumption.…”

“…Different approaches have been presented to control the response of genetic toggles switches. Examples include methods based on piecewise affine approximations [6], pulse shaping of the external inputs based on monotone systems theory [7], and the analysis of the stationary probability distributions of the outputs in different working conditions [8].…”