Multi-objective Optimisation of Gene Regulatory Networks: Insights from a Boolean Circadian Clock Model

The gene regulatory networks that comprise circadian clocks modulate biological function across a range of scales, from gene expression to performance and adaptive behaviour. These timekeepers function by generating endogenous rhythms that can be entrained to the external 24-hour day-night cycle, enabling organisms to optimally time biochemical processes relative to dawn and dusk. In recent years, computational models based on differential equations, and more recently on Boolean logic, have become useful tools for dissecting and quantifying the complex regulatory relationships underlying the clock’s oscillatory dynamics. Optimising the parameters of these models to experimental data is, however, non-trivial. The search space is continuous and increases exponentially with system size, prohibiting exhaustive search procedures, which are often emulated instead via grid-searching or random explorations of parameter space. Furthermore, to simplify the search procedure, objective functions representing fits to individual experimental datasets are often aggregated, meaning the information contained within them is not fully utilised. Here, we examine casting this problem as a multi-objective one, and illustrate how the use of an evolutionary optimisation algorithm — the multi-objective evolution strategy (MOES) — can significantly accelerate the parameter search procedure. As a test case, we consider an exemplar circadian clock model based on Boolean delay equations — dynamic models that are discrete in state but continuous in time. The discrete nature of the model enables us to directly compare the performance of our optimiser to grid searches based on enumeration of the parameter space at a fixed resolution. We find that the MOES generates near-optimal parameterisations in computation times which are several orders of magnitude faster than the grid search. As part of this investigation, we also show that there is a distinct trade-off between the performance of the clock circuit in free-running and entrained photic environments. Importantly, runtime results indicate that the use of multi-objective evolutionary optimisation algorithms will make the investigation of larger and more complex models computationally tractable.