Temperature robustness in Arabidopsis circadian clock models is facilitated by repressive interactions, autoregulation, and three-node feedbacks

Highlights

• Temperature dependence is incorporated in a range of circadian clock models.

• Key network features allowing temperature compensation are identified.

• These features are tested via a set of simple randomly parameterised networks.

Abstract

The biological interactions underpinning the Arabidopsis circadian clock have been systematically uncovered and explored by biological experiments and mathematical models. This is captured by a series of published ordinary differential equation (ODE) models, which describe plant clock dynamics in response to light/dark conditions. However, understanding the role of temperature in resetting the clock (entrainment) and the mechanisms by which circadian rhythms maintain a near-24 h period over a range of temperatures (temperature compensation) is still unclear. Understanding entrainment and temperature compensation may elucidate the principles governing the structure of the circadian clock network. Here we explore the design principles of the Arabidopsis clock and its responses to changes in temperature. We analyse published clock models of Arabidopsis, spanning a range of complexity, and incorporate temperature-dependent dynamics into the parameters of translation rates in these models, to discern which regulatory patterns may best explain clock function and temperature compensation. We additionally construct three minimal clock models and explore what key features govern their rhythmicity and temperature robustness via a series of random parameterisations. Results show that the highly repressive interactions between the components of the plant clock, together with autoregulation patterns and three-node feedback loops, are associated with circadian function of the clock in general, and enhance its robustness to temperature variation in particular. However, because the networks governing clock function vary with time due to light and temperature conditions, we emphasise the importance of studying plant clock functionality in its entirety rather than as a set of discrete regulation patterns.

Introduction

Circadian rhythms are the result of positive and negative feedbacks in a network of transcription factors, which regulate the mRNA and protein levels. These complex systems have been modelled in diverse organisms (Schmelling and Axmann, 2018, Podkolodnaya et al., 2017, Fathallah-Shaykh et al., 2009) in order to understand the crucial factors driving the circadian oscillations, and in some cases to determine whether clock network structure can determine functionality (Rand et al., 2004, Rand et al., 2006).

Empirically-driven mathematical modelling of the plant Arabidopsis had its genesis in the ODE model presented by Locke et al. (2005a) in 2005, which characterised the interaction between two clock genes and their associated proteins. Since then, several ODE-based models have been built which reflect the increasing complexity of the clock as new biological discoveries arise (Locke et al., 2005a, Locke et al., 2005b, Locke et al., 2006, Pokhilko et al., 2010, Pokhilko et al., 2012, Pokhilko et al., 2013, Fogelmark and Troein, 2014, De Caluwé et al., 2016). Throughout this process, mathematical models have been useful tools to help generate and test hypotheses about the underlying structure of the plant system and to understand its dynamics (Bujdoso and Davis, 2013).

The mathematical models proposed to date typically seek to characterise the time evolution of both mRNA and protein levels within plant cells (Locke et al., 2005a, Locke et al., 2005b, Locke et al., 2006, Pokhilko et al., 2010, Pokhilko et al., 2012, Pokhilko et al., 2013, Fogelmark and Troein, 2014, De Caluwé et al., 2016) by describing the production and degradation rates of both gene products. A combination of first-order interactions and Hill functions are commonly used to characterise the interactions within the system (Locke et al., 2005a, Locke et al., 2005b, Locke et al., 2006, Pokhilko et al., 2010, Pokhilko et al., 2012, Pokhilko et al., 2013, Fogelmark and Troein, 2014, De Caluwé et al., 2016). The models are usually parametrized by fitting to experimental observations in a qualitative fashion, principally by matching circadian period, amplitude and phase of gene expression. However, details of the clock mechanism differ between the models, as do the details of the statistical parameter fitting, leading some models to appear to offer a better quantitative fit than others.

Mathematical models for Arabidopsis have focused mainly on characterising the plant clock in response to light. Temperature, however, has received less attention and yet its characterization in the plant model system is of increasing importance in the context of global climate change (Avello et al., 2019). Experimental evidence has shown that under constant light conditions, and at a constant temperature, circadian rhythms display a relatively invariant period over a large range of temperatures (Salomé et al., 2010, Gould et al., 2006). This is known as temperature compensation. Oscillations with a $Q_{10}$ of period in the range 0.8–1.2 are considered to manifest this feature of compensation (Akman et al., 2008). An explanation of this distinctive characteristic of circadian rhythms is based on the hypothesis that there is a resulting balance of network reactions having opposite behaviours (Hastings and Sweeney, 1957, Ruoff et al., 2005). Alternatively, temperature compensation has been explained by the hypothesis that independent molecular mechanisms have evolved in order to establish compensation (Akman et al., 2008).

In Arabidopsis, several clock components have been identified as playing a key role in temperature compensation (Salomé et al., 2010, Gould et al., 2006). However, it is not clear which interactions within the circadian plant network might explain the dynamics of temperature compensation. In biological networks, attempts have been made to explain oscillatory dynamics by analysing the feedback and feedforward loops that form these networks (Tsai et al., 2008, Alon, 2007). A feedback loop is defined as positive if the number of negative interactions within the loop is even, or negative if this number is odd, and their basic functionalities can facilitate bistability and promote sustained oscillations, respectively (Alon, 2007). Feedback loops play different roles within a network, and have been classified broadly with regards to their shape (Fig. 1) and functionality (Alon, 2007). The feedforward loop is a three-component circuit of interactions with edges having inhibitory or activator roles. This circuit is formed by a component that regulates directly and indirectly (through the other component) a target component. Thus, a total of eight possible classes of feedforward loops can be defined (Fig. 2). For a description of their dynamical functions, see Mangan and Alon (2003).

Feedback and feedforward substructures within biological networks are examples of regulation patterns in real networks which occur more frequently than would be expected in random networks with the same number of nodes and edges (Wong et al., 2011). In systems biology, these “regulation patterns” would be described as “network motifs” (Alon, 2006). We use the former wording in this study, to avoid ambiguity.

The underlying principle is that, if the regulation pattern has been retained over evolutionary time, then it is likely to confer a fitness advantage to the organism (Alon, 2006). These classes of structures were first detected for the E. coli transcription network (Shen-Orr et al., 2002), and from then the interest turned on the detection of their dynamical functions in different organisms and different biological networks (Milo et al., 2002). In Arabidopsis, regulation pattern analysis carried out under simulated constant light conditions has recently helped to explain the dynamics observed in certain clock mutants (Joanito et al., 2018).

In this work, we analysed the structures of a range of plant clock models proposed over the past 15 years, and added temperature dependence in order to gain insights into thermal robustness of circadian oscillations. We centred our analysis on the transcription regulatory interactions. We asked whether regulation patterns commonly found in other transcription networks are involved in temperature-dependent mechanisms in the plant circadian system. To gain a better understanding of the key patterns driving clock function and its robustness to temperature variation, we also performed a large-scale simulation study based on random parameterisations of three simplified models, following Joanito et al. (2018). Taken together, our results support the idea that the highly repressive role of transcription factors in the plant clock contributes to its robustness to temperature changes. Our results also show that a simple pattern-based analysis is insufficient to describe dynamics induced by light and temperature; rather, such networks should be analysed as a whole, and as dynamic rather than static networks.