267740 Splitting Hill-Type Kinetics: Reduced, Modular, Stochastic Simulations
In recent years it has become evident that while stochastic simulations are necessary for many biological models, these simulations are often difficult to implement. Stochastic simulations fully account for the “randomness” or “noise” associated with a system. Such probability-based methods are necessary when the system exists far from the thermodynamic limit, considered a common condition in biological simulations. Unfortunately, many biological models include a large number of components and reactions over wide-ranging timescales. As the system grows, many of the unknown kinetic constants begin to become difficult or impossible to determine.
A common solution to reduce the number of unknown parameters is to bundle many unknowns into a non-linear reaction rate using an approximating method like the Quasi-Steady State Approximation. Such approximations are common in deterministic systems, but implementation in stochastic systems may only be applicable across a narrow set of conditions. Outside of these conditions the approximations fail to accurately describe higher-order statistics (e.g. the variance). This problem makes approximating methods difficult to implement in stochastic systems.
Modification of the non-linear reaction rates allows for higher-order statistics to be accounted for without significantly increasing the complexity of the system. In our particular work, transcription-based repression in a small gene network can be described by Hill-type kinetics. The Hill-type reaction rate has two parameters that can fully account for the mean protein output in the presence of repressor protein molecules. However, while the mean output is accurately described, the 2nd-order statistics do not match well. The Hill reaction rate law is then modified by “splitting” the rate law, effectively adding an additional parameter to take into account higher-order statistics.
Two systems were produced using the Hill-reduction method and the Split-Hill method to show how accuracy is improved and modularity preserved by accounting for higher-order statistics. A negative-feedback loop illustrates the ability of the Split-Hill model to fully and accurately describe results. A bistable switch constructed from two repressor systems demonstrates a tendency for modularity in the Split-Hill systems.
Higher-order statistics are a vital part of stochastic simulations and any reduction method or approximation must take these statistics into account to ensure accuracy and modularity. The Split-Hill model as presented is a modification to the Hill-type reaction rate designed for transcription-based control in gene networks. The results show that the Split-Hill network is accurate and modular in simple gene networks, without significantly increasing the complexity of the model.
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