Abstract
In the mathematical modeling of biochemical reactions, a convenient standard approach is to use ordinary differential equations (ODEs) that follow the law of mass action. However, this deterministic ansatz is based on simplifications; in particular, it neglects noise, which is inherent to biological processes. In contrast, the stochasticity of reactions is captured in detail by the discrete chemical master equation (CME). Therefore, the CME is frequently applied to mesoscopic systems, where copy numbers of involved components are small and random fluctuations are thus significant. Here, we compare those two common modeling approaches, aiming at identifying parallels and discrepancies between deterministic variables and possible stochastic counterparts like the mean or modes of the state space probability distribution. To that end, a mathematically flexible reaction scheme of autoregulatory gene expression is translated into the corresponding ODE and CME formulations. We show that in the thermodynamic limit, deterministic stable fixed points usually correspond well to the modes in the stationary probability distribution. However, this connection might be disrupted in small systems. The discrepancies are characterized and systematically traced back to the magnitude of the stoichiometric coefficients and to the presence of nonlinear reactions. These factors are found to synergistically promote large and highly asymmetric fluctuations. As a consequence, bistable but unimodal, and monostable but bimodal systems can emerge. This clearly challenges the role of ODE modeling in the description of cellular signaling and regulation, where some of the involved components usually occur in low copy numbers. Nevertheless, systems whose bimodality originates from deterministic bistability are found to sustain a more robust separation of the two states compared to bimodal, but monostable systems. In regulatory circuits that require precise coordination, ODE modeling is thus still expected to provide relevant indications on the underlying dynamics.
Highlights
In the last decades, the potential of mathematical modeling for the analysis of biological systems has widely been recognized
One of the most frequently used deterministic approaches consists in ordinary differential equations (ODEs), which are based on the phenomenological law of mass action
We consider ODEs based on the law of mass action, which has originally been formulated by Guldberg and Waage
Summary
The potential of mathematical modeling for the analysis of biological systems has widely been recognized. One of the most frequently used deterministic approaches consists in ordinary differential equations (ODEs), which are based on the phenomenological law of mass action They provide a dynamic and quantitative description of spatially homogenous systems. Randomness plays a major role in signaling and regulation, where the copy number of the involved components is small and noise in gene expression is significant They are major application fields for stochastic models in systems biology (Tsimring, 2014). Unlike a couple of other studies on this topic, we will regard mesoscopic systems which are not close to the thermodynamic limit We will discuss these aspects in the context of a simple gene regulatory system, using it as a platform for identifying general factors which influence the comparability between these kinds of deterministic and stochastic models
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