Purely stochastic binary decisions in cell signaling models without underlying deterministic bistabilities

Abstract

Detection of different extracellular stimuli leading to functionally distinct outcomes is ubiquitous in cell biology, and is often mediated by differential regulation of positive and negative feedback loops that are a part of the signaling network. In some instances, these cellular responses are stimulated by small numbers of molecules, and so stochastic effects could be important. Therefore, we studied the influence of stochastic fluctuations on a simple signaling model with dueling positive and negative feedback loops. The class of models we have studied is characterized by single deterministic steady states for all parameter values, but the stochastic response is bimodal; a behavior that is distinctly different from models studied in the context of gene regulation. For example, when positive and negative regulation is roughly balanced, a unique deterministic steady state with an intermediate value for the amount of a downstream signaling product is found. However, for small numbers of signaling molecules, stochastic effects result in a bimodal distribution for this quantity, with neither mode corresponding to the deterministic solution; i.e., cells are in "on" or "off" states, not in some intermediate state. For a large number of molecules, the stochastic solution converges to the mean-field result. When fluctuations are important, we find that signal output scales with control parameters "anomalously" compared with mean-field predictions. The necessary and sufficient conditions for the phenomenon we report are quite common. So, our findings are expected to be of broad relevance, and suggest that stochastic effects can enable binary cellular decisions.

Fig. 1.

A schematic representation of dueling positive and negative feedback loops stimulated upon receptor binding to stimulatory or inhibitory ligands. The negative regulator can shut off signaling by inactivating the receptor-associated signaling complex (negative feedback), whereas the positive regulator could prevent this inhibitory interaction and increase or continue production of downstream signaling products (positive feedback).

Fig. 2.

Bimodal stochastic solutions distinct from the unique deterministic solution. Histogram showing the bimodal distribution of the protected agonists at steady state (red) for a situation when there are 10 agonist (A₁) and 10 antagonists (A₂). The corresponding single steady state solution of the deterministic ODEs (blue) is also shown. The other parameter values are: k₁ = 1, k₂ = 1, k₃ = 100, k₄ = 1, k₅ = 100, k_D = 1 (all s⁻¹), and statistics were collected over 5,000 trajectories obtained by using the Gillespie algorithm. The result is robust to variations in the parameter values as long as there is strong feedback.

Fig. 3.

A purely stochastically driven transition. Histograms showing the distribution of the protected agonists at steady state (red) and corresponding steady state solution of deterministic ODEs (blue) for different amounts of agonist and antagonist. All other parameters are identical to that in Fig. 2.

Fig. 4.

Results from the minimal model. Non-mean-field scaling in the limit of large positive feedback (k₂ → ∞): The average values of X species, 〈x〉, at steady state obtained from Gillespie simulations scale with (1/log(1 + Mk₁/k₃)) instead of the mean field scaling variable Nk₁/k₃. The values of the parameter k₂ is 100 s⁻¹ (i.e., a large value). k₁ = 0.0012 s⁻¹ and M = 20 are held fixed as k₃ and N are varied. The solid line is a plot of the scaling function shown in Eq. 19.

Fig. 5.

Stochastic fluctuations can enable cellular decisions. Schematic representation showing that irreversibility, branching, and dueling feedback loops associated with intrinsic time scales, when combined with stochastic effects, can result in distinct functional decisions for each cell. A deterministic treatment would mask this ability of cells to make decisions.