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