The temperature and day length data of several climate regions (Table 1) were retrieved from NOAA [24] and PTAFF [25]. Temperatures were recorded as the daily maximum and minimum, and data from different stations are considered distinct. The mean of daily maximum and minimum is regarded as the daily average temperature. To account for the effect of noise in daily light quantity [26–28] on the day length, Gaussian noise was used to corrupt the day length data and simulate real variations due to weather conditions.

Chemical master equations are used to model the probabilistic states of chemical reactions over time [29–31]. Following the previous work of modeling birth-death process [32], for the reaction ϕ ⇌ λ β B, the probability p of having n molecules of B can be described as Eq (1) for time t. ∂ t p ( n , t ) = β ( p ( n - 1 , t ) - p ( n , t ) ) - λ n p ( n , t ) + λ ( n + 1 ) p ( n + 1 , t ) (1)

In equilibrium, p(n) is Poisson distributed (see Supplement Section 3.1 in S1 File). In our study, a modified form of this model was used with B representing the cellular state of having active modifications at the FLC locus. Different from the basic model, the production rate is adapted to be temperature dependent. Since the temperature recordings are time series, the temperature dependence of the model make it able to integrate temperature properties. This modification necessitates the use of Hermite polynomials [33, 34] to solve the master equation (see Supplement Section 3.1 in S1 File).

The original daily average temperature from different regions have been used for the analytical deductions based on the master equation.

Artificial neural networks was used as a complementary method to extract information from temperature and day length. A feedforward neural network is comprised of a number of neurons to transmit information in only one direction, from the input data through hidden neurons to output neurons. Each neuron can be regarded as either a computing node or a decision maker which outputs a decision by weighing and transforming the information it receives from upstream neurons. A detailed formal description of neural networks is in Supplementary section 4 in S1 File.

Fully connected feed-forward neural networks with one hidden layer were used to classify different time windows in each year and regress the idealized expression patterns of FT and FLC in Arabidopsis perennials.

For the classification, each year was shrank to 360 days for simplicity. For example, it will be divided into 12 windows corresponding to 12 months if the window size is set to 30 days. The neural networks then need to determine which month the input window belongs to, given 30 days of temperatures within the window. The daily temperatures were summarized by daily maximum and minimum.

For the regression of gene expressions, the target can be idealized patterns of FT or FLC. Specifically, the expression pattern of FT is characterized by a normal distribution p(t), peaking in April every year as shown in Fig 2, where t denotes certain day of a year. And the FLC is featured as an upside-down normal distribution centered at middle of March [35], which is 15 days earlier than the peak of FT [20]. To learn the expression patterns from the climate data, the input features consist of daily temperature maxima and minima and day lengths of the past days. That is, the expression level on a specific day is determined by the plant’s memory of temperature and day length of the past days.

The Neural Networks Toolbox in Matlab [36] was used to build the classification and regression models.

Plant selection pressure and evolution were simulated using neural networks to represent individual plants, with weights of individual networks analogous to plant genotypes (see details in Supplement Section 5 in S1 File). A group (population) of networks trained on distinct subsets of the same climate data were used to evaluate the flowering decision-making strategies of plants based on climate information. There were three populations in each simulation, each with access to only temperature, only day length, or both. Each generation of individuals was trained on randomly selected subsets of climate data from Cologne. At the end of each reproductive cycle, fitness was measured by the Kullbach-Leibler distance between target and neural network-predicted gene expression level, and mutations (in the form of neural network weight perturbations) were introduced at specific rates which varied across simulations for the next generation. The simulation procedure is summarized in S8 Fig in S1 File. It was implemented in Python 2.7, and the Theano package [37] was used to optimize individual networks. The results of each simulation were reported as the proportion of individuals from each of the three populations throughout the entire simulation, which would be greater for neural networks with higher fitness leading to better reproductive odds.

Plants need to avoid the disastrous effect of flowering at the wrong time as a reaction to sudden and sustained temperature fluctuations. We hypothesize that long-cold based vernalization is not only for capturing the winter cold temperature but also for canceling this effect and sensing the winter robustly. The vernalization machinery can be interpreted that plants have evolved biochemical processes to capture and accumulate the information in cold temperature that is a reliable signal in temperate regions. Inspired by the machinery, we modeled the process of having n cells with active histone modifications, which is driven by real temperature to reconstruct the idealized expression patterns of FLC/FT of Arabidopsis perennials.

In the following, we denoted daily average temperature as T(t) for day t ∈ [1, ⋯, 365]. To investigate the seasonal changes and fluctuations in real temperature, it is decomposed into three parts as T ( t ) = T ¯ + 〈 T ( t ) 〉 + δ T ( t ) with T ¯ denoting the average yearly temperature, 〈T(t)〉 the seasonal temperature changes, and δT(t) the remaining temperature fluctuations. The temperature dynamics comprising of 〈T(t)〉 and δT(t) are shown in Fig 3 for Cologne. The Fourier fitting of 〈T(t) was detailed Supplementary section 2 in S1 File. The temperature fluctuation on a specific day typically correlates with its neighboring days. And the longer periods of unseasonally cold or warm temperatures may confuse the plants more than the shorter periods. Therefore, we analyzed the autocorrelation times, which quantify the correlation length in a time series, in the fluctuations of temperature from five different regions (Table 1). The autocorrelation times in Cologne, Auckland and Oslo decay exponentially, and decay faster in Auckland than in Cologne and Oslo (Supplementary Section 2 in S1 File, S3a, S3b Fig in S1 File). This is consistent with experimental observations that some plants in Auckland required only two weeks of vernalization [38], while vernalization requires typically around 6 weeks in Cologne [5] and even three months in North Sweden [39]. Having verified the exponential decay of temperature fluctuations, without loss of generality, we used the climate data of Cologne for the successive modeling, where the autocorrelation in averaged yearly-cycle temperature fluctuations decays exponentially with an approximate half life of 4 days (Fig 4b). To further evaluate the exponential fitting of the fluctuation decay, the bootstraping of temperature data with block length of 50 showed that the fitted coefficients of the exponential function indeed located in the bootstrapped confidence interval(further details in Supplementary Section 2 in S1 File).

To investigate the effect of temperature dynamics on vernalization, we modeled the number of cells which have their histones all with active modifications as a death-birth process. The assumption of having all active modifications in a cell relied on the fact that the fraction of histones in one state affects modifications in their own vicinity, which makes histone modification a relative fast process. As shown in Fig 4A, the probability of having n active cells at time t is denoted as p(n, t): ∂ t p ( n , t ) = β T ( t ) ( p ( n - 1 , t ) - p ( n , t ) ) - λ n p ( n , t ) + λ ( n + 1 ) p ( n + 1 , t ) , (2) with βT(t) the temperature dependent production rate and λ the temperature-independent degradation rate. Due to the fact that the autocorrelation time of temperature fluctuations decays exponentially, the stochastic differential equation has a concise form of solution given by p ( n , t ) = e a - b α n n ! H n ( b + 2 α 2 2 α ) . (3) with a ≔ β 2 σ 2 2 λ ( λ + τ - 1 ) and b = T ¯ β λ + β D ( λ , t ). Here, we denoted by D ( λ , t ) = ∫ - ∞ t 〈 T ( t ) 〉 e - λ ( t - t ′ ) d t ′ the expected memorized temperature. The parameters σ² and τ denote the averaged variance and autocorrelation time of temperature fluctuations respectively. H_m(⋅) denotes the mth Hermite polynomial. The parameterization for a, b, and α is detailed in the S1 File.

To reconstruct the expression patterns, we need to define an objective for getting the optimal reaction rates based on the probability p(n, t). We reduced the number of cells for computational feasibility and assumed that N_max = 60 cells were available and N_c = 30 cells were sufficient to turn off the expression of FLC on the plant level. The objective function of flowering probability in time window between March and June and non-flowering probability between July and next February can be defined as F ( β , λ ) = ∫ M a r J u n ∑ n = N c N m a x p ( n , t ) d t + ∫ J u l F e b ∑ n = 0 N c - 1 p ( n , t ) d t (4) Maximizing this objective is equivalent to maximizing the probability of flowering (i.e. having at least N_c active cells) in flowering season and non-flowering (i.e. having at most N_c active cells) during non-flowering season. The optimal reaction rates which maximized the objective led to time dependent optimal probability sequences which were capable of reproducing the idealized expression pattern of FLC that is switched off during flowering season. In Fig 4C, two different time points are chosen to show the typical probability distributions in flowering and non-flowering seasons. On 1st January, the most density (99.2%) of the probability located below the critical value of 30 cells, whereas on 15th April most density (92.2%) was distributed above the critical value. As shown in Fig 4D that the probability of having at least N_c active cells at different time of a yearly cycle preserved the idealized expression pattern of FLC, which was active during the non-flowering season and then gradually switched off in flowering season.

Under the condition that the autocorrelation length of temperature fluctuations in time decays exponentially, it was shown that the idealized expression patterns of FLC could be rebuilt from the stochastic model. By relaxing the autocorrelation condition, we would also like to investigate the effect of climate information on flowering time decision using machine learning, which is typically not requiring great details of the system thus can be more broadly applied to different climates.

To make flowering time decisions upon environmental cues such as temperatures and day lengths, plants are essentially information processing units for extracting critical environmental signals in order to survive by making the correct transit to the reproductive state. We employed artificial neural networks to approximate the information processing in plants by predicting the flowering season. The networks were trained to learn the idealized expression pattern of FLC from temperatures and day length from different climates, and the results relied on climate data in Cologne. The approach was broken into two tasks: to determine the effective memory length of determining season and to reconstruct the idealized expression pattern of FLC of Arabidopsis perennials.

The first step was to determine the number of days in the past that the plants need to remember in order to recognize the current season. It was cast into a classification problem as for given consecutive L days of temperature, the neural networks learned to classify which time window the temperature belonged to. The results showed that for Cologne, the prediction achieved MCC score of 0.964 for temperature memory of over 40 days. And increasing the memory length did not increase the score accordingly. The result was consistent with experimental result [5]. This reflected an expected tradeoff between sufficiently long memory to reduce the fluctuations of temperature signal (variance) and the loss of season specific averaged temperature if the memory stretches beyond the length of the season (bias). The detailed classification result can be found in S1 File.

Having determined the effective memory length, we were able to construct the input features of the neural networks for fitting the idealized expression pattern of FLC of Arabidopsis perennials. With only the temperature memory fed into the neural networks, the trained neural network was tested on three years of temperature as shown in Fig 5. Local minima arose in autumn as shown in Fig 6A, which resulted from the temperature similarity between spring and autumn. Especially for the second test year, a clear local minimum was observed in October, which has a correspondence to the high temperature spike in October as indicated in Fig 5. The local minima could potentially give a high chance of making wrong decision to flower in autumn. In order to remove the local minima and have a robust detection of spring, two days of day-length signal were added to the input features. It can be seen from Fig 6B that the local minima are eliminated, leading to more precise regression of the FLC expression pattern. The precise and robust reconstruction of the FLC signal was critical for precise flowering decisions in the right season. As a comparison, the climate data from Kahului, Hawaii was used as well to learn the FLC pattern. Fig 6C showed that the temperature memory only features led to a high-error fitting, which probably due to both the flat seasonal changes and the long autocorrelation in temperature fluctuations as shown in S4e Fig in S1 File. The integration of two days of day lengths increased the regression but still being noisy. The result may provide an explanation that vernalization in natural tropical climates was not established.

The neural networks based method was broadly applicable to different climates (the results for other climates are shown in the S1 File), unlike the stochastic model which required more system details and climate properties. Its fitting results from the climates data reflected that for temperate regions, long-term temperature and short-term day length together were deployed as a robust signal for determining the flowering transition, while for other regions, merely temperature and day length signals were not sufficient to have a mechanism such as the interplay between FLC and FT.

To investigate the possible effects of evolution on the role of temperature and day length in achieving the idealized expression pattern of FT, we conducted simulations using neural networks as plants’ agents. In each simulation, three groups of virtual plants were given access to either temperature, day length, or both. The Kullbach-Leibler Divergence (KLD) between the fitted and idealized expression patterns was used as the fitness measure. Fitnesses were calculated for individuals from each group and normalized to the population of each generation to have a fitness probability for each individual. Individuals then reproduced according to their fitness probability, that is, individuals with higher fitnesses had more offspring accordingly. Multiple offspring were possible for each individual, and offspring had access to the same input type as their parents (temperature, day length, or both). For each reproduced generation, a fixed number of mutations, represented by randomly selected neural network weights, were applied to each individual. Reproduction continued until 500 cycles were simulated or until all surviving individuals were the offspring of only one of the three groups. Each simulation was repeated 50 times, and the proportion of offspring per group throughout the entire simulation were tallied to estimate fixation probability.

To simulate the effects of strongly or weakly deleterious mutations, simulations were run with various numbers of mutations per generation. To investigate the effects of population size and genetic drift, the number of individuals per group was varied. The results of the simulations are shown in Fig 7, as violin plots of group offspring proportions under different mutation rates and population sizes.

The group with access to both temperature and day length generally had the most offspring, and the group with access to day length alone had the least. The temperature-only group performed somewhere between the two others. Consistent with the basic principles of genetic drift, larger population sizes resulted in less variance in the fixation probability. Possibly due to the small number of input variables in the neural network, the group with access to day length alone performed notably worse as the mutation rate increased.