Deterministic and Stochastic Models of Arabidopsis thaliana Flowering
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Abstract
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). Steadystate 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 (\(\textit{AP}1\)). Twodimensional 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.
Keywords
Arabidopsis flowering Gene regulatory network Deterministic–stochastic linear stability Ordinary delay differential equations1 Introduction
Arabidopsis thaliana is a small, annual flowering plant in the Brassicaceae (mustard) family which is a favourite model organism for plant biology research due mainly to its small size, simple genome and rapid life cycle. The transition from vegetative to reproductive development, which is an initiation of flower growth, is crucial for the life cycle of any angiosperm plant like Arabidopsis thaliana (Krizek and Fletcher 2005; Ó’Maoiléidigh et al. 2014; Wang et al. 2014) as flowering on time is a key factor to achieve reproductivity of these plants. Physiological and environmental conditions of the plant regulate the timing of transition for the optimal reproductive achievement, and their reactions are integrated into a complex GRN which monitors and regulates this transition (Kardailsky et al. 1999; Levy and Dean 1998; Wellmer and Riechmann 2010). Genes and their regulatory interactions are significant factors in biological systems at the molecular level since the understanding of their impact on each other’s regulation is crucial to comprehend the response of gene disturbances on flowering time (Valentim et al. 2015). Recently, the dynamics of Arabidopsis flowering time regulation has been studied using a systems approach along with experimental data to understand the effect of the genes on flowering of Arabidopsis thaliana (Daly et al. 2009; Jaeger et al. 2013; Pullen et al. 2013; Valentim et al. 2015; Wang et al. 2014).
Numerous genes appear to be acting as flowering time regulators of Arabidopsis thaliana (Ryan et al. 2015), and different pathways have been constructed to reveal the flowering of this plant (Amasino 2010; Greenup et al. 2009; Kardailsky et al. 1999; Yant et al. 2009). This complex network of many interacting genes can be dynamically modelled using systems with many equations (Jaeger et al. 2013; Valentim et al. 2015; van Mourik et al. 2010; Wang et al. 2014). In this study, we consider the deterministic dynamic model of delay differential equations (DDEs) describing the flowering of the Arabidopsis species proposed by Valentim et al. (2015). This model involves core set of gene–regulator interactions, while protein–protein interactions are not explicitly included. The model is based on a feedback loop, constructed with eight genes, where six of them are internal: Apetala1 (AP1), Leafy (LFY), Suppressor of Overexpression of Constants 1 (SOC1), AgamousLike 24 (AGL24), Flowering Locus T (FT) and FD. The other two genes are considered as external inputs: Short Vegetative Phase (SVP) and Flowering Locus C (FLC).
System behaviour of the GRNs usually cannot be understood heuristically due to the complexity of interactions in organisms. We propose a different approach by simplifying the network and studying its behaviour. Stability analysis is used to study the properties of the GRN and threshold in flowering. Moreover, such analysis provides a reliability test and more insights into the behaviour of GRN’s elements.
Our stability analysis produces conditions which include the biological parameters. Such parameters are difficult to determine from the experiment, and one of our aims was to provide specific ranges for individual coefficients that secure stable solutions. To overcome this issue of complexity, we reduce the differential equation system by decoupling some concentrations before simplifying the new system using network motifs that capture essential characteristics of the floral transition. Examples of reduced Arabidopsis thaliana GRNs can be seen in the study of Pullen et al. (2013), where a complex flowering time pathway included in the model of Jaeger et al. (2013) was simplified by focusing on essential flowering genes. Following these papers, we produce a subsystem of our network with three different motifs.
Indeed, it is known that the floral meristem identity genes have an important role to control the floral meristem specification while the flower development process is starting (Irish 2010; Levy and Dean 1998; Simon et al. 1996). Thus, this minimal regulatory network consists of the main floral meristem identity genes of Arabidopsis thaliana: \(\textit{AP}1\), \(\textit{LFY}\), \(\textit{FT}\) and \(\textit{FD}\) where \(\textit{AP}1\) is the dominant regulatory concentration of floral initiation with \(\textit{LFY}\) in Arabidopsis thaliana (Irish 2010; Wellmer and Riechmann 2010) and has a key role between floral induction to flower formation, being a junction of flowering in the GRN (Kaufmann et al. 2010). On the other hand, \(\textit{FT}\) induces flowering of Arabidopsis as an inhibitor and acts similarly with \(\textit{LFY}\). Additionally, activation tagging isolates it (Kardailsky et al. 1999). Moreover, \(\textit{FT}\) and transcription factor \(\textit{FD}\) affect each other in the meristem as a combined activator (Wang et al. 2014). The aim of this subsystem is to construct parameterdependent stability conditions that reflect essential behaviour of the complex network.
Another aim of this study is to investigate the properties of the simplified Arabidopsis flowering model modified with stochastic perturbations. The motifs are reflecting the essential behaviour of the complex network and can capture the significant behaviour of the full Arabidopsis flowering model and can investigate necessary conditions (threshold values of the concentrations) for the flowering initiation. The advantage of this approach is based on the realistic description of gene effects and their interactions on flowering of Arabidopsis. New sufficient conditions of mean square stability are obtained analytically for this simplified model using Lyapunov function. Analytical and numerical investigations of the stability are performed with respect to concentrations and noise terms.
This paper is organised as follows: in Sect. 2, the main features of the deterministic dynamic model of Arabidopsis flowering introduced in Valentim et al. (2015) are recalled, and analytical and numerical investigations of its steady state are both conducted. Section 3 provides a simplified deterministic model by decoupling some concentrations in the full model. A comparative numerical investigation of both models is also given. Deterministic motifs of the simplified model are presented in Sect. 4 along with an analytical investigation of their steady state and their stability. Stochastic perturbations of the motifs are investigated in Sect. 5 using Lyapunov functions to obtain sufficient conditions for their mean square stability. Finally, our concluding remarks are given in Sect. 6, while further technical information can be seen in “Appendix”.
2 Deterministic Model
Description and range for the parameters in the dynamic model
Parameters  Description  Range 

\( \beta _i \), \((i=1,2,\ldots , 12)\)  Maximum transcription rate  \([0.001, 200]\,\hbox {nM}*\hbox {min}^{1}\) 
\(K_i\), \((i=1,2,\ldots , 16)\)  Abundance at half maximum transcription rate  [0.001, 2000] nM 
\(d_i\), \((i=1,2,\ldots , 6)\)  Gene products degradation rate  \([0.001, 1]\,\hbox {min}^{1}\) 
\(\Delta \)  Delay  [0, 1] days 
Model parameters, estimated from experimental gene expression data using a polynomial fit (Valentim et al. 2015)
Parameters  Estimated values  Parameters  Estimated values (nM)  Parameters  Estimated values 

\(\beta _{1}\)  \(99.8\,\hbox {nM}\times \hbox {min}^{1}\)  \(K_{1}\)  9.82  \(K_{13}\)  7.9 nM 
\(\beta _{2}\)  \(5\,\hbox {nM}\times \hbox {min}^{1}\)  \(K_{2}\)  700  \(K_{14}\)  125 nM 
\(\beta _{3}\)  \(10\,\hbox {nM}\times \hbox {min}^{1}\)  \(K_{3}\)  10.1  \(K_{15}\)  0.63 nM 
\(\beta _{4}\)  \(22\,\hbox {nM}\times \hbox {min}^{1}\)  \(K_{4}\)  346  \(K_{16}\)  985 nM 
\(\beta _{5}\)  \(2.4\,\hbox {nM}\times \hbox {min}^{1}\)  \(K_{5}\)  842  \(d_{1}\)  \(0.86\,\hbox {min}^{1}\) 
\(\beta _{6}\)  \(0.79\,\hbox {nM}\times \hbox {min}^{1}\)  \(K_{6}\)  1011  \(d_{2}\)  \(0.017\,\hbox {min}^{1}\) 
\(\beta _{7}\)  \(64\,\hbox {nM}\times \hbox {min}^{1}\)  \(K_{7}\)  695  \(d_{3}\)  \(0.11\,\hbox {min}^{1}\) 
\(\beta _{8}\)  \(0.52\,\hbox {nM}\times \hbox {min}^{1}\)  \(K_{8}\)  1182  \(d_{4}\)  \(0.0075\,\hbox {min}^{1}\) 
\(\beta _{9}\)  \(189\,\hbox {nM}\times \hbox {min}^{1}\)  \(K_{9}\)  2.4  \(d_{5}\)  \(0.001\,\hbox {min}^{1}\) 
\(\beta _{10}\)  \(8.5\,\hbox {nM}\times \hbox {min}^{1}\)  \(K_{10}\)  4.8  \(d_{6}\)  \(0.1\,\hbox {min}^{1}\) 
\(\beta _{11}\)  \(100\,\hbox {nM}\times \hbox {min}^{1}\)  \(K_{11}\)  909  \(\Delta \)  0.5 day 
\(\beta _{12}\)  \(51\,\hbox {nM}\times \hbox {min}^{1}\)  \(K_{12}\)  501 
The behaviour of system (1) is simulated in Fig. 2 using the parameters in Table 2 and the experimental data used in Valentim et al. (2015). The initial conditions are taken from the experimental data. The sharp rise in \(\textit{AP}1\) from 13 to 17 days after germination can be interpreted as a predictor of flowering.
As is known from laboratory experiments (see Krämer 2015), Arabidopsis thaliana is an annual plant and its flowering is limited to approximately two to four weeks after germination. In this context, a nontrivial stable steady state can be seen as an attracting point for the flowering process. Hence, in the next section, we turn to the analysis of the steady state of the flowering model to determine its behaviour, give conditions on its initiation and investigate the terminal stages of the flowering process.
2.1 Steady State and Stability Analysis of the Deterministic Model
Steady states of the system represent equilibrium points about which the dynamics can be studied using linear stability analysis. It helps to describe the behaviour of a delayed system solution by considering the trajectories in a phase space of all dependent variables. As mentioned previously, we interpret a stable steady state as an attractor for the flowering process. Therefore, if the Arabidopsis flowering is successful, then there exists at least one strictly positive stable steady state.
\({\bar{x}}_1\)  \({\bar{x}}_2\)  \({\bar{x}}_3\)  \({\bar{x}}_4\)  \({\bar{x}}_5\)  \({\bar{x}}_6\) 

121.567  452.395  827.835  1113.882  86,881.258  2.037 
Linearisation of the nonlinear system (1) is required to analyse the local stability of this dynamic model at its steady states \(({\bar{x}}_{1},{\bar{x}}_{2},{\bar{x}}_{3},{\bar{x}}_{4},{\bar{x}}_{5},{\bar{x}}_{6})\). Stability analysis is used to establish threshold conditions on the model parameters for the flowering of the plant. Therefore, we analyse the linear stability of the model in detail, and explicit conditions for local stability are formulated using the Routh–Hurwitz criterion. This gives the following theorem, for which further details can be seen in “Appendix A.2”.
Theorem 1
In summary, the conditions in Theorem 1 show that the local stability of system (1) at the steady state depends on values of parameters and concentrations. Given the high dimensionality of the parameter space, it is a difficult task to fully describe regions where stability holds. Nonetheless, it is worth noting that the delay \(\tau \) does not influence stability in this particular system. No bifurcation has been numerically detected in the parameter ranges considered in this work.
To reduce the number of parameters, we now introduce a simpler system which reproduces the essential behaviour of system (1). Therefore, we performed local parameter sensitivity analysis to figure out the most important parameters in GRN (see “Appendix C”), which are \(\beta _{1}\), \(\beta _{4}\), \(\beta _{5}\), \(K_{1}\), \(K_{4}\), \(K_{5}\) and \(d_1\), and all belong to the first two equations. For this purpose, we consider subsystems and analyse their stability to understand the behaviour of system (1).
3 Deterministic Model of the Simplified Network
The complex large regulatory network represented in (1) can be simplified while still saving its core structure. By decoupling some concentrations, it is possible to reduce the number of differential equations of the large system. One can see from the analysis in the previous section that the main contribution to the dynamics is from protein concentrations related to \(\textit{AP}1\), \(\textit{LFY}\) and \(\textit{SOC}1\). Indeed, from the structure of system (1), it is seen that \(x_4\), \(x_5\) and \(x_6\) can be computed explicitly from the knowledge of \(x_2\), \(x_3\) and the external outputs. Hence, we focus the analysis on these genes to investigate how they contribute to the regulation of \(\textit{AP}1\).
The numerical solution of the nondecoupled variables \(\textit{SOC}1\), \(\textit{LFY}\) and \(\textit{AP}1\) in system (5) is compared with the numerical solution of system (1) in Fig. 5. The convergence of \(x_1\) is affected by the constant values used for \(x_4, x_5, x_6\). Using the steadystate values, which represent the highest concentrations for these variables, leads to a slightly faster converging graph for \(\textit{AP}1\) that can be seen in Fig. 5 on the right. This result shows that decoupling some concentrations on the system can still capture the essential behaviour of the complex network for these nonconstant variables.
Linear stability of the simplified model and explicit conditions for local stability at its steady states (\({\bar{x}}_1\), \({\bar{x}}_2\), \({\bar{x}}_3\)) are formulated using the Routh–Hurwitz criterion in Theorem 2, and further details can be seen in “Appendix A.3”.
Theorem 2
4 Deterministic Models of Motifs
To further reduce the complexity of system (1), we use the approach in Pullen et al. (2013) and reduce system (5) from three to two equations to understand the essential characteristics of the floral transition by considering the two components, \(\textit{LFY}\) and \(\textit{AP}1\), which constitute the minimal set for enabling the transition to floral meristem (Mandel et al. 1992). Here, we model minimal regulatory networks of core components consisting of the protein concentrations for \(\textit{LFY}\), \(\textit{AP}1\), \(\textit{FT}\) and \(\textit{FD}\). We consider the simplified subsystem proposed in Figure 1(b) in Pullen et al. (2013) to establish the essential characteristics of the floral transition. From system (5), one can integrate the third equation to obtain \(x_3\) in terms of \(x_1\) and \(x_2\) and the various constant inputs, which also depend on \(\textit{FT}\) and \(\textit{FD}\). The simplified system is represented in Fig. 4 and results from considering constant \(\textit{SOC}1\) concentrations (\(\dot{x}_3 = 0\)). The reason we use these four genes is: \(\textit{AP}1\) and \(\textit{LFY}\) are key floral meristem identity genes in the network of Arabidopsis flowering (Irish 2010; Wellmer and Riechmann 2010) and \(\textit{FT}\) induces flowering through the activation of these two genes in a feedforward circuit (Kardailsky et al. 1999) where \(\textit{FD}\) has a significant role for \(\textit{FT}\) signalling.
The aim of the first and second subsystems is to analyse the effect of input variables on \(\textit{AP}1\) and \(\textit{LFY}\), individually. The third subsystem is aimed to obtain the effect of input variables when they have an equal action on both main concentrations. The parameters in Table 2 are used to investigate the behaviour of the input variables whether they play an inhibitor or an activator role.
4.1 Steady States of Motifs
4.2 Deterministic Stability of Motifs
4.3 Numerical Results for Deterministic Steady States and Stability of the Motifs
Subsystem 1 (\(F_1=1\)). Figure 8 shows the presence of a double root at \(F_2 = 0.0431\) from which two distinct strictly positive equilibria emanate for \(0.04317 < F_2 \le max\{F_2\}\). Hence, when no action of \(\textit{FT}\)\(\textit{FD}\) on \(\textit{LFY}\) is present, the inhibition of \(\textit{FT}\) on \(\textit{AP}1\) starts at the value of \(F_2=0.04317\) and activation can be seen for \(F_2 >1\). Moreover, the behaviour of subsystem 1 is similar to system (1) for \(F_2 \ge 0.04317\). The best match with the numerical solution of system (1) occurs for \(F_2\) just above 1, with a best match of the steadystate value at \(F_2 = 1.0476\). This in turn indicates an activation action of \(\textit{FT}\) on \(\textit{AP}1\).
Subsystem 2 (\(F_2=1\)). A similar situation is seen in this case (Fig. 9). The numerical result for this subsystem indicates that in the absence of action of \(\textit{FT}\) on \(\textit{AP}1\), the inhibition of \(\textit{FT}\)\(\textit{FD}\) on \(\textit{LFY}\) starts at the double root \(F_1 = 0.05185\), from which it originates one stable and one unstable positive steady states. The behaviour of subsystem 2 is similar to system (1) for \(F_1 \ge 0.05185\), while the best match with the numerical solution of system (1) can be seen in the activation of \(\textit{FT}\)\(\textit{FD}\) on \(\textit{AP}1\) for \(F_1\) just above 1, with a best match of the steadystate value at \(F_1 = 1.3445\). In view of such information, we use \(F_1\) and \(F_2\) external input variables as an activator of the \(\textit{LFY}\) and \(\textit{AP}1\) in subsystem (7) to be able to obtain a compatible behaviour with system (1).
The unstable steady state can be regarded as the threshold values of the concentrations for the flowering of Arabidopsis thaliana. As a consequence, if flowering is processing for some time which means the concentrations have already reached their threshold values for the flowering, then the values of concentrations can move away from an unstable steady state and converge to a nontrivial stable one, which shows the same flowering behaviour as in the full model.
As can be seen in Fig. 14, in which the \(\textit{AP}1\) value is chosen just over its threshold (0.24nM), if the initial value of \(\textit{LFY}\) is lower than 1.25nM, then no flowering is observed. This is in agreement with the findings of van Dijk and Molenaar (2017), according to which mutants with knockeddown \(\textit{LFY}\) may not flower. A lower threshold of \(\textit{LFY}\)\(\approx 1nM\) was estimated for the flowering of these mutants (see van Dijk and Molenaar 2017, SI Figure 9).
5 Stochastic Models of Motifs
To obtain more realistic representations of the behaviour of biological systems, it is appropriate to work with stochastic differential equations (SDEs), which can be obtained by incorporating noise terms into deterministic models. The aim of this section is to introduce and study for the first time SDEs for the behaviour of Arabidopsis flowering.
5.1 Stochastic Motifs with Additive White Noise
On the other hand, the behaviour of the stochastic model (18) is more complex and depends on the initial conditions and the amount of noise in each of the concentrations. So, it is not certain whether it reaches nontrivial (passing the subthreshold for the flowering) or trivial (nonflowering) stable equilibria, a phenomenon known as ”stochastic switching” (Ullah and Wolkenhauer 2011). We show the behaviour of stochastic model (18) with a timevarying histogram to see the change of the behaviour. The initial values are fixed as (0.2, 1.2), which lie between unstable and trivial stable steady states for the parameter values from Table 2. The implementation has been performed 100 times with a fixed constant noise of \(5\%\) (\(\sigma _i=0.05\)).
As can be seen from Fig. 15, stochasticity can change the behaviour of the system. The solutions are initially concentrated around the initial values, and then, they are separated into two different realisations. At the end, they converged around either trivial or nontrivial stable solutions with a considerable proportion. This shows that successful solutions for the Arabidopsis flowering can be obtained by using stochastic equations system even if the initial values are under the threshold value.
5.2 Stochastic Motifs with Multiplicative White Noise
In contrast to the previous subsection, where the possibility of successful flowering was depending only on the amount of noise terms and initial values of the concentrations, here we assume that the amplitude of noise depends on the state of the system. More precisely, stochastic perturbations of the variables around their equilibrium values are assumed to be of white noise type and proportional to the distances of \(AP1 (x_1)\) and \(LFY (x_2)\) from the steadystate values \({\bar{x}}_1\) and \({\bar{x}}_2\). The question whether the dynamical behaviour of model (7) is influenced by stochastic effects is investigated by looking at the asymptotic stochastic stability of equilibrium points.
The aim is to determine the flowering domain of the stochastic motifs with multiplicative white noise. These can be obtained by using a Lyapunov function approach, centred at the origin or at a nontrivial steady state of the system. This allows to obtain necessary stability conditions which depend on the noise parameters \(\sigma _i\).
Let us show that the trivial solution \(x=0\) of system (20) is locally asymptotically stable in probability. Using a stochastic stability approach from Khasminskii (2011), we derive that there exists a noisedependent domain around \({\bar{x}}=0\) for which asymptotic stability holds. This domain thus corresponds to nonflowering conditions for the Arabidopsis thaliana GRN modelled by system (20).
Theorem 3

\(0 \le \sigma _i < \sqrt{2d_i}\), \(i=1,2\),
Proof
6 Conclusion
In this paper, we considered a dynamic model of Arabidopsis flowering introduced by Valentim et al. (2015). This model is reconstructed with Hill functions to emphasise the importance of these functions and their effects on the concentrations. An analytical study of the deterministic model and its steady state for the full system was performed. The stability analysis was used to establish the conditions for initiating the transition to flowering. The steady states are calculated numerically with the estimated parameters taken from Valentim et al. (2015). The analysis results have shown that the system has only one positive stable steady state and that the time for which \(\textit{AP}1\) reaches the steady state is in agreement with the observed flowering time between 20 and 30 days. The Routh–Hurwitz criterion has been used to provide local stability conditions which characterise the existence of this stable steady state; details are given in “Appendix”.
Given the complexity of the system, more precise conditions have been formulated by considering subsystems which focus on the dynamics of essential elements. According to our analysis for the full system, three genes, \(\textit{SOC}1\), \(\textit{LFY}\) and \(\textit{AP}1\), have a strong effect on the flowering of Arabidopsis. The network has been simplified by decoupling. Analytical solution of the simplified system is still difficult; however, it illustrates specific pathways of inhibition and activation. By using these pathways, we reconstruct three different subsystems suggested in Jaeger et al. (2013) and Pullen et al. (2013). This allowed us to derive necessary and sufficient conditions for the existence of the positive steady states of these subsystems that represent the dynamics and cooperativity of the Arabidopsis flowering time regulation system. The most important floral identity genes, \(\textit{AP}1\) and \(\textit{LFY}\), are used to investigate the flowering where they are regulating each other, and the results are confirmed by experiments (Liljegren et al. 1999). The necessary and sufficient conditions for the local stability of the deterministic model have then been determined analytically, and the stability ranges are established with the estimated parameters and compared with the numerical solutions. The numerical results have confirmed that these subsystems can capture the essential behaviour of the full model by estimating the \(\textit{FT}\)\(\textit{FD}\) inhibition/activation effects on \(\textit{LFY}\) and \(\textit{AP}1\), and also they help to investigate the conditions (threshold values) for the initiation of flowering, which cannot be obtained from the full model. Moreover, stochastic motifs, which are extended from the deterministic ones by adding additive and multiplicative white noise terms, have been developed to obtain more realistic description of gene effects and their interactions on the behaviour of Arabidopsis flowering. The effects of stochasticity on the steadystate regimes have been observed. The numerical solutions show that the flowering behaviour of the system does not only depend on the initial values, state variables and parameters of the stochastic system but also the amount of noise terms, where the noise can change behaviour of the stability region from nonflowering to flowering through a stability switch even if the initial values are lower than the threshold values.
Our analyses, being in a good agreement with the experimental findings, bring further insights into the roles of \(\textit{LFY}\) and \(\textit{AP}1\) and provide the opportunity to explore different pathways for flowering.
Notes
Acknowledgements
EH would like to thank the Republic of Turkey Ministry of National Education for the PhD scholarship.
Supplementary material
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