Deterministic and Stochastic Models of Arabidopsis thaliana Flowering

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Experimental studies of the flowering of Arabidopsis thaliana have shown that a large complex gene regulatory network (GRN) is responsible for its regulation. This process has been mathematically modelled with deterministic differential equations by considering the interactions between gene activators and inhibitors (Valentim et al. in PLoS ONE 10(2):e0116973, 2015; van Mourik et al. in BMC Syst Biol 4(1):1, 2010). However, due to complexity of the model, the properties of the network and the roles of the individual genes cannot be deducted from the numerical solution the published work offers. Here, we propose simplifications of the model, based on decoupling of the original GRN to motifs, described with three and two differential equations. A stable solution of the original model is sought by linearisation of the original model which contributes to further investigation of the role of the individual genes to the flowering. Furthermore, we study the role of noise by introducing and investigating two types of stochastic elements into the model. The deterministic and stochastic nonlinear dynamic models of Arabidopsis flowering time are considered by following the deterministic delayed model introduced in Valentim et al. (2015). Steady-state regimes and stability of the deterministic original model are investigated analytically and numerically. By decoupling some concentrations, the system was reduced to emphasise the role played by the transcription factor Suppressor of Overexpression of Constants1 (textit{SOC}1) and the important floral meristem identity genes, Leafy (textit{LFY}) and Apetala1 (textit{AP}1). Two-dimensional motifs, based on the dynamics of textit{LFY} and textit{AP}1, are obtained from the reduced network and parameter ranges ensuring flowering are determined. Their stability analysis shows that textit{LFY} and textit{AP}1 are regulating each other for flowering, matching experimental findings. New sufficient conditions of mean square stability in the stochastic model are obtained using a stochastic Lyapunov approach. Our numerical simulations demonstrate that the reduced models of Arabidopsis flowering time, describing specific motifs of the GRN, can capture the essential behaviour of the full system and also introduce the conditions of flowering initiation. Additionally, they show that stochastic effects can change the behaviour of the stability region through a stability switch. This study thus contributes to a better understanding of the role of textit{LFY} and textit{AP}1 in Arabidopsis flowering.