In order to successfully reproduce, plants must sense changes in their environment and flower at the correct time. Many plants utilize day length and vernalization, a mechanism for verifying that winter has occurred, to determine when to flower. Our study used available temperature and day length data from different climates to provide a general understanding how this information processing of environmental signals could have evolved in plants. For climates where temperature fluctuation correlations decayed exponentially, a simple stochastic model characterizing vernalization was able to reconstruct the switch-like behavior of the core flowering regulatory genes. For these and other climates, artificial neural networks were used to predict flowering gene expression patterns. For temperate plants, long-term cold temperature and short-term day length measurements were sufficient to produce robust flowering time decisions from the neural networks. Additionally, evolutionary simulations on neural networks confirmed that the combined signal of temperature and day length achieved the highest fitness relative to neural networks with access to only one of those inputs. We suggest that winter temperature memory is a well-adapted strategy for plants’ detection of seasonal changes, and absolute day length is useful for the subsequent triggering of flowering.
Citation: Zhao L, Richards S, Turck F, Kollmann M (2020) Information integration and decision making in flowering time control. PLoS ONE 15(9): e0239417. https://doi.org/10.1371/journal.pone.0239417
Editor: Hector Candela, Universidad Miguel Hernández de Elche, SPAIN
Received: May 2, 2019; Accepted: September 7, 2020; Published: September 23, 2020
Copyright: © 2020 Zhao et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper and its Supporting Information files.
Funding: M.K and F. T received the DFG (Deutsche Forschungsgemeinschaft) grant KO3442-9. https://www.dfg.de/en/ The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Plants must make correct flowering time decisions in a noisy environment in order to successfully reproduce. As key environmental signals, day length and temperature are processed by plants’ genetic networks for detecting seasonal changes. The core genes and their interplays have been well understood in the model plant, Arabidopsis thaliana [1, 2], as shown in Fig 1. The gene FLOWERING LOCUS T (FT) merges signals from both day length and temperature, and its encoded protein eventually induces the flowering . The expression of FT genes is promoted by the expression of the gene CONSTANS (CO), whose gene products are produced about 12 hours after dawn and quickly degrade in the dark [1, 3]. Thus, the condition that day length is long enough to produce stable CO proteins is necessary for initiating the flowering of the so-called long-day plants [1, 3, 4]. In particular, for winter annuals of Arabidopsis thaliana [5, 6], vernalization is required to cease expression of FlOWERING LOCUS C (FLC), the inhibitor of FT. Vernalization involves the exposure of plants to a prolonged period of cold temperature, which induces histone modifications on the epigenetic level for silencing FLC [7–9]. The repressed FLC allows FT to be expressed under long days. Similarly, in perennial Arabis alpina, the orthologs of FLC, PERPETUAL FLOWERING 1 (PEP1), downregulates the orthologs of FT, AaFT1 and AaFT3, and needs to be silenced by vernalization in order for the perennials to flower in the right time . Moreover, for perennial Arabidopsis halleri, Nagano et.al  demonstrated the role of cooperation between the oscillations of temperature and day length in adaptation to seasonal changes. Thus, the integration of signals from temperature and day length is crucial for both annual and perennial plants to make flowering decisions. Despite the qualitative understanding of the genetic regulation of flowering time, it is unclear from an information point of view why plants have evolved vernalization from fluctuating winter temperatures and how it relates to day length in flowering decisions.
In Arabidopsis Thaliana, long days promote the expression of FLOWERING LOCUS T (FT). The vernalization process also promotes its expression by turning-off its repressor FLOWERING LOCUS C (FLC).
Many theoretical studies have contributed to the understanding of the vernalization mechanism. It was shown to be an inheritable and stable epigenetic switch for the expression of FLC [11–15]. Dodd et.al  developed a stochastic model for Schizosaccharomyces pombe showing that gene expressions bistability can be established by accumulating histone modifications, which acted as an epigenetic memory. The work of Angel et al.  extended their approach by incorporating histone modifications of FLC in Arabidopsis thaliana and investigating how the different epigenetic states could be controlled. Several studies have reported that FLC repression was cell-autonomous and that cold temperature memory was encoded by the fraction of cells with repressed FLC [8, 9, 16, 17]. Due to the positive feedbacks that lead to adding more of the same type of histone modifications, a particular cell would mostly have one type of modifications at FLC. To understand how plants utilize fluctuating temperature in vernalization, Antoniou-Kourounioti et al. developed a model by incorporating thermosensing on multiple timescales and suggested that the sensing was broadly distributed in plants .
Investigations of flowering time regulation in Arabidopsis exploited mathematical modeling and experiments. Wilczeck et al. developed a model to incorporate the impacts of temperature, day length, and vernalization on flowering initialization in different accessions of Arabidopsis thaliana [5, 19]. Their analyses yielded a photothermal measure of plant states which was able to accurately predict flowering time of Arabidopsis Thaliana field plants. Another dynamic model described interplays between FT and FLC in Arabidopsis halleri to study impacts of temperature on flowering decisions . It was able to reproduce the observed seasonal expression changes and estimate the climate change-induced reduction of flowering season. Hepworth et al. reported that the spikes of temperatures above 15°C may have deleterious consequences for vernalization . These studies have significantly refined our understanding of the effects of temperature and day length on Arabidopsis flowering, but the mechanisms employed by a variety of plants in various environments cannot be described by the same processes . A method leading to a more general understanding of climate information processing would promote understanding of flowering-time decision making for a greater variety of plants.
Our study focuses on the extraction of available information from climate data (temperature and day length) and its usefulness in making precise flowering decisions. We first established a simple stochastic model for vernalization in perennials, which showed that the idealized expression patterns of FT or FLC can be reconstructed for temperate climates due to exponentially decaying correlation in temperature fluctuations. The stochastic model does not apply to other climates where temperature fluctuations correlate differently. To relax this restriction on climate properties, we employed artificial neural networks to learn idealized gene expression patterns from several climate datasets. We showed that, in temperate city Cologne, the neural network models trained solely on temperature memory roughly reconstructed the idealized expression patterns of FLC. However day length data was required to resolve the danger of incorrect flowering time decisions based on a local optimum in September rather than April. Further, to simulate the evolutionary adaptation to environmental conditions, individual neural networks were used in a simulation of evolution for plants with access to temperature, day length, or both. Simulations with different mutation rates and population sizes showed a persistent selective advantage for the neural networks with access to the combined temperature and day length data.
Materials and methods
The temperature and day length data of several climate regions (Table 1) were retrieved from NOAA  and PTAFF . 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.
Master equation and Hermite polynomial
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 , for the reaction , the probability p of having n molecules of B can be described as Eq (1) for time 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.
Neural network models and data features
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 , which is 15 days earlier than the peak of FT . 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 idealized expression levels of FLC and FT for Arabidopsis perennials in the northern hemisphere.
The Neural Networks Toolbox in Matlab  was used to build the classification and regression models.
Evolution of individual neural networks
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  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.
Expression patterns reproduced from modeling key reactions
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 with 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 , while vernalization requires typically around 6 weeks in Cologne  and even three months in North Sweden . 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).
The temperature dynamics consist of the seasonal changes and the daily temperature fluctuations, which were fitted by a second order Fourier series. The dynamical data were obtained by averaging 93 years of temperatures in temperate city Cologne. The first day in the plot was January 1st.
(a): The vernalization was simplified as birth-death process for actively modified cells. n stands for the number of active cells, the production rate βT(t) depends on the temperature T(t) and λ is the degradation rate. Solving the model led to the distribution p(n, t) which is parameterized by β and λ; (b): For the data from Cologne, the autocorrelation in daily temperature decays exponentially, the bootstrapping was performed by using block length of 50 days; (c): The probability p(n, t) distribution over number of active cells (in total 60 cells in the simulation) on 1st January and 15th April; (d): The switching FLC expression behavior was constructed from the probabilities of having less than 30 active cells over two years, with green areas for potential flowering seasons.
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): (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 (3) with and . Here, we denoted by the expected memorized temperature. The parameters σ2 and τ denote the averaged variance and autocorrelation time of temperature fluctuations respectively. Hm(⋅) 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 Nmax = 60 cells were available and Nc = 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 (4) Maximizing this objective is equivalent to maximizing the probability of flowering (i.e. having at least Nc active cells) in flowering season and non-flowering (i.e. having at most Nc 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 Nc 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.
Long-term cold temperature and short-term day lengths together as a robust signal
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 . 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.
Three years of temperatures were used to test the regression models. In the second year, one can observe a long temperature spike starts from late October to early November. This is in correspondence with a predicted local minimum in Fig 6.
(a): Cologne, fitting result of the idealized FLC expressions using 42 days of temperature as the input features gave a local minimum in September due to similarity between spring and autumn; (b): Cologne, fitting result from 42 days of temperature and 2 days of day lengths with eliminated local minima; (c): Kahului, fitting result from 42 days of temperatures; (d): Kahului, fitting result from 42 days of temperature and 2 days of day lengths.
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.
Evolutionary simulation favors integration of temperature and day length
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.
Distributions of group offspring proportions for mutation rates of 0, 0.05, and 0.2, for group sizes of 10, 50, and 200 individuals. “B” stands for group with access to both temperature and day length, “T” for temperature and “D” for day length.
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.
Our study verified that the temperature and day length in temperate regions like Cologne have the information premise for plants such as Arabidopsis to establish the epigenetic switch vernalization. For different climates, the vernalization requires different memory spans due to different temperature seasonal properties. The memory spans depended on the autocorrelations in temperature fluctuations, which decayed differently from climate to climate. For regions with flat temperature dynamics like San Fransisco and Hawaii, the autocorrelations decay slower than regions like Cologne, and have autocorrelation length above 100 days, which might be the reason to fail the establishment of vernalization. Our stochastic model, describing the dynamics of the number of cell with repressed FLC, was able to integrate the temperature dynamics as well as the temperature fluctuations. For regions with exponential decays in temperature fluctuation autocorrelations, the switch behavior of FLC in Arabidopsis perennials can be reconstructed. Further, without requirements on climate properties and system details, our machine learning approach showed that the idealized expression patterns of FLC can be robustly reconstructed by the combination of prolonged cold and short-term day lengths (e.g. the absolute day lengths from two consecutive days). The strategy of combining long-term cold and short-term day length is proven to be also favored by an evolution simulation where neural networks were regarded as the agents of plants for processing climate information.
Although in natural environments, temperate plants need to cope with other signals such as ambient temperature using additional genes like FLM , it might indicate the backbone of flowering mechanism is that the plants utilized long-term temperature to detect seasonal changes and used absolute day lengths to decide the eventual flowering days. The reason is that, for temperate climates, a cold winter is guaranteed and it is easier to track the absolute day length than the variations in day lengths which is 4min in maximum from day to day. It could also be a good strategy for cold regions like Norway since the autocorrelation decays similarly to Cologne, but due to lower temperature and shorter summer, the fast life cycles such as summer annuals of Arabidopsis was adapted . In this case, ambient temperature and light intensity might play a more important role in plants’ vegetative or reproductive timing. In tropical regions where locate the most diverse and abundant plant species on earth, more factors have to be taken into consideration and merely temperature and day length are not sufficient for flowering decision making. For instance, flowering is mostly rain-season dependent, which might play a more critical role than the temperature and day length as they contain less seasonal information than that of temperate regions.
By investigating flowering decision making from an information point of view, our study suggested that, for temperate regions, cold winter memory and short term of day length can serve as a robust strategy for plants to determine flowering season.
S1 File. Details of data preprocessing, proofs of analytical solution to stochastic model, regression of the idealized FLC expressions based on different climates.
S1 Code. It includes the code autocorrelation.py for analyzing temperature data, optimization.py for optimizing the flowering objective, evolution.py for the evolutionary simulations and neuralnets.m for cleansing temperature data and training neural networks.
- 1. Andrés F, Coupland G. The genetic basis of flowering responses to seasonal cues. Nature reviews Genetics. 2012;13(9):627–39. pmid:22898651
- 2. Song YH, Ito S, Imaizumi T. Flowering time regulation: photoperiod- and temperature-sensing in leaves. Trends in plant science. 2013; p. 1–9. pmid:23790253
- 3. Bratzel F, Turck F. Molecular memories in the regulation of seasonal flowering: From competence to cessation; 2015. pmid:26374394
- 4. Turck F, Fornara F, Coupland G. Regulation and Identity of Florigen: FLOWERING LOCUS T Moves Center Stage. Annual Review of Plant Biology. 2008;59(1):573–594. pmid:18444908
- 5. Wilczek AM, Roe JL, Knapp MC, Cooper MD, Lopez-Gallego C, Martin LJ, et al. Effects of genetic perturbation on seasonal life history plasticity. Science (New York, NY). 2009;323(5916):930–4. pmid:19150810
- 6. Hyun Y, Vincent C, Tilmes V, Bergonzi S, Kiefer C, Richter R, et al. A regulatory circuit conferring varied flowering response to cold in annual and perennial plants. Science. 2019;363(6425):409–412. pmid:30679374
- 7. Csorba T, Questa JI, Sun Q, Dean C. Antisense COOLAIR mediates the coordinated switching of chromatin states at FLC during vernalization. Proceedings of the National Academy of Sciences. 2014;111(45):16160–16165. pmid:25349421
- 8. Angel A, Song J, Yang H, Questa JI, Dean C, Howard M. Vernalizing cold is registered digitally at FLC. Proceedings of the National Academy of Sciences. 2015;112(13):4146–4151. pmid:25775579
- 9. Angel A, Song J, Dean C, Howard M. A Polycomb-based switch underlying quantitative epigenetic memory. Nature. 2011;476(7358):105–8. pmid:21785438
- 10. Nagano AJ, Kawagoe T, Sugisaka J, Honjo MN, Iwayama K, Kudoh H. Annual transcriptome dynamics in natural environments reveals plant seasonal adaptation. Nature Plants. 2019;5(1):74–83. pmid:30617252
- 11. Mukhopadhyay S, Nagaraj VH, Sengupta AM. Locus dependence in epigenetic chromatin silencing. BioSystems. 2010;102(1):49–54. pmid:20655355
- 12. Kelemen JZ, Ratna P, Scherrer S, Becskei A. Spatial epigenetic control of mono- and bistable gene expression. PLoS Biology. 2010;8(3). pmid:20305717
- 13. David-Rus D, Mukhopadhyay S, Lebowitz JL, Sengupta AM. Inheritance of epigenetic chromatin silencing. Journal of Theoretical Biology. 2009;258(1):112–120. pmid:19174167
- 14. Sedighi M, Sengupta AM. Epigenetic chromatin silencing: Bistability and front propagation. Physical Biology. 2007;4(4):246–255. pmid:17991991
- 15. Dodd IB, Micheelsen Ma, Sneppen K, Thon G. Theoretical analysis of epigenetic cell memory by nucleosome modification. Cell. 2007;129(4):813–22. pmid:17512413
- 16. Song J, Angel A, Howard M, Dean C. Vernalization—a cold-induced epigenetic switch. Journal of cell science. 2012;125(Pt 16):3723–31. pmid:22935652
- 17. Satake A, Iwasa Y. A stochastic model of chromatin modification: cell population coding of winter memory in plants. Journal of theoretical biology. 2012;302:6–17. pmid:22381539
- 18. Antoniou-Kourounioti RL, Hepworth J, Heckmann A, Duncan S, Qüesta J, Rosa S, et al. Temperature Sensing Is Distributed throughout the Regulatory Network that Controls FLC Epigenetic Silencing in Vernalization. Cell Systems. 2018;7(6):643–655.e9. pmid:30503646
- 19. Chew YH, Wilczek AM, Williams M, Welch SM, Schmitt J, Halliday KJ. An augmented Arabidopsis phenology model reveals seasonal temperature control of flowering time. The New phytologist. 2012;194(3):654–65. pmid:22352314
- 20. Satake A, Kawagoe T, Saburi Y, Chiba Y, Sakurai G, Kudoh H. Forecasting flowering phenology under climate warming by modelling the regulatory dynamics of flowering-time genes. Nature communications. 2013;4:2303. pmid:23941973
- 21. Hepworth J, Antoniou-Kourounioti RL, Bloomer RH, Selga C, Berggren K, Cox D, et al. Absence of warmth permits epigenetic memory of winter in Arabidopsis. Nature communications. 2018;9(1):639. pmid:29434233
- 22. Topham AT, Taylor RE, Yan D, Nambara E, Johnston IG, Bassel GW. Temperature variability is integrated by a spatially embedded decision-making center to break dormancy in Arabidopsis seeds. Proceedings of the National Academy of Sciences. 2017;114(25):6629–6634. pmid:28584126
- 23. Weigel D. Natural Variation in Arabidopsis: From Molecular Genetics to Ecological Genomics. Plant Physiology. 2012;158(1):2–22. pmid:22147517
- 24. NOAA. Climate data online;. Available from: https://www.ncdc.noaa.gov/cdo-web/.
- 25. ptaff. Sunrise, sunset, daylight in a graph;. Available from: https://ptaff.ca/soleil/?lang=en_CA.
- 26. Whiting D, Roll M, Vickerman L. Plant Growth Factors: Light; 2016.
- 27. WILSON D, COOPER JP. Effect of light intensity during growth on leaf anatomy and subsequent light-saturated photosynthesis among contrasting Lolium genotypes. New Phytologist. 1969;68(4):1115–1123.
- 28. Powles SB. Photoinhibition of Photosynthesis Induced by Visible Light. Annual Review of Plant Physiology. 1984;35(1):15–44.
- 29. Gillespie DT. A rigorous derivation of the chemical master equation. Physica A: Statistical Mechanics and its Applications. 1992;188(1):404–425. https://doi.org/10.1016/0378-4371(92)90283-V.
- 30. Risken H. Fokker-Planck Equation. Berlin, Heidelberg: Springer Berlin Heidelberg; 1996. Available from: https://doi.org/10.1007/978-3-642-61544-3_4.
- 31. Gardiner CW. Handbook of stochastic methods: for physics, chemistry and the natural sciences. Springer; 2004.
- 32. Walczak AM, Mugler A, Wiggins CH. Analytic methods for modeling stochastic regulatory networks. In: Computational Modeling of Signaling Networks. Springer; 2012. p. 273–322.
- 33. Weisstein EW. Hermite Polynomial. From MathWorld—A Wolfram Web Resource; Available from: http://mathworld.wolfram.com/HermitePolynomial.html.
- 34. doubllle (https://mathoverflow.net/users/57344/doubllle). An interesting calculation of derivative;. MathOverflow. Available from: https://mathoverflow.net/q/179031.
- 35. Aikawa S, Kobayashi MJ, Satake A, Shimizu KK, Kudoh H. Robust control of the seasonal expression of the Arabidopsis FLC gene in a fluctuating environment. Proceedings of the National Academy of Sciences of the United States of America. 2010;107(25):11632–7. pmid:20534541
- 36. MATLAB. R2014a. Natick, Massachusetts: The MathWorks Inc.; 2014.
- 37. Bergstra J, Breuleux O, Bastien F, Lamblin P, Pascanu R, Desjardins G, et al. Theano: a CPU and GPU Math Expression Compiler; 2010.
- 38. Adhikari KN, Buirchell BJ, Sweetingham MW. Length of vernalization period affects flowering time in three lupin species. Plant breeding. 2012;131(5):631–636.
- 39. Duncan S, Questa J, Irwin J, Dean C, Holm S, Grant A. Seasonal shift in timing of vernalization as an adaptation to extreme winter. eLife. 2015;4(JULY2015):1–11. pmid:26203563
- 40. Posé D, Verhage L, Ott F, Yant L, Mathieu J, Angenent GC, et al. Temperature-dependent regulation of flowering by antagonistic FLM variants. Nature. 2013. pmid:24067612