Digital clocks: simple Boolean models can quantitatively describe circadian systems

Abstract

The gene networks that comprise the circadian clock modulate biological function across a range of scales, from gene expression to performance and adaptive behaviour. The clock functions by generating endogenous rhythms that can be entrained to the external 24-h day–night cycle, enabling organisms to optimally time biochemical processes relative to dawn and dusk. In recent years, computational models based on differential equations have become useful tools for dissecting and quantifying the complex regulatory relationships underlying the clock's oscillatory dynamics. However, optimizing the large parameter sets characteristic of these models places intense demands on both computational and experimental resources, limiting the scope of in silico studies. Here, we develop an approach based on Boolean logic that dramatically reduces the parametrization, making the state and parameter spaces finite and tractable. We introduce efficient methods for fitting Boolean models to molecular data, successfully demonstrating their application to synthetic time courses generated by a number of established clock models, as well as experimental expression levels measured using luciferase imaging. Our results indicate that despite their relative simplicity, logic models can (i) simulate circadian oscillations with the correct, experimentally observed phase relationships among genes and (ii) flexibly entrain to light stimuli, reproducing the complex responses to variations in daylength generated by more detailed differential equation formulations. Our work also demonstrates that logic models have sufficient predictive power to identify optimal regulatory structures from experimental data. By presenting the first Boolean models of circadian circuits together with general techniques for their optimization, we hope to establish a new framework for the systematic modelling of more complex clocks, as well as other circuits with different qualitative dynamics. In particular, we anticipate that the ability of logic models to provide a computationally efficient representation of system behaviour could greatly facilitate the reverse-engineering of large-scale biochemical networks.

Multi-level Modeling of Light-Induced Stomatal Opening Offers New Insights into Its Regulation by Drought

“…[87]). The recent successful Boolean model of circadian clocks [88] makes the construction of such an integrated gene regulatory and signal transduction network model feasible.…”

The Role of Genotype in the Predictability of Dynamical Behavior in Complex Model Gene Regulatory Networks

“…Negative feedback loops (analogous to gene expression suppression) have been associated with oscillating behavior in many biological systems [47], including E. coli [48], the Neurospora circadian oscillator [49], the cell cycle [50], the lac operon system [51] the p53-mdm2 system [52] and NF-kappa B translocation [53]. Boolean approaches have also been used to model circadian oscillations [54], which are the most studied example of biological periodic oscillations.…”

Mathematical modeling of an oscillating gene circuit to unravel the circadian clock network of Arabidopsis thaliana

“…To date no less than seven distinct models of the plant clock have been published ( Figure 1 ), and the most recent kinetic model (Pokhilko et al, 2012) and linear time invariant (LTI) model (Herrero et al, 2012) have highlighted the complexity in this system. These models, alongside other approaches, such as Boolean modeling (Akman et al, 2012), have various strengths in predicting the system. The first model that placed the plant clock in molecular terms was in 2005 (Locke et al, 2005a), where the authors presented an approach to inferring models when available molecular-genetic data are sparse and noisy.…”