Cold hardiness mechanisms and modeling: existing approaches and future avenues

Summary

Cold hardiness models are useful tools to predict cold damage in plants, such as those produced by unseasonal temperature cycles or by increased cold exposure. Although development of these models started about five decades ago, their applications remain limited. We describe the main paradigms driving the different types of cold hardiness models (empirical to process‐based), their similarities and differences. Among the existing paradigms, process‐based models are built to translate physiological mechanisms into mathematical functions over a broad range of climatic conditions, thus making them more accurate for studying the effect of climate change. Different approaches have been developed in predicting cold hardiness: (1) empirical relationships between temperature and cold hardiness; (2) phenological processes controlling acclimation and deacclimation rates; (3) phenological and physiological processes predicting cold hardiness through the osmo‐hydric approach; and (4) molecular regulation driving the metabolic drivers of cold hardiness. For the first three approaches, we describe the context, the experimental and field observations that defined their frameworks as well as their limitations. To increase the realism of cold hardiness models, we describe the potential of a fourth approach, based on the perception of environmental signals, how it translates into cold acclimation/deacclimation and provide recommendations to develop this framework.

Introduction

In temperate, mountain, and boreal areas, cold hardiness is an important driver of a plant's fitness through its survival (essentially maximum cold hardiness in winter) and reproduction (in relation to phenology). In the future, climate models predict a temperature rise of 4–6°C over the next century, contrasting sharply with past temperature changes when a similar rise took c. 20 000 yr (IPCC, ²⁰²³). These rapid changes are jeopardizing the link between climate and plant adaptation. One critical yet counterintuitive consequence of global warming relates to freezing risks, as warming trends may impair the cold acclimation process that enables plants to survive extreme winter temperatures (Charrier et al., ^2015a). How plants will adapt to winter temperatures now and in the future will help to predict vegetation feedback on climate (Rammig et al., ²⁰¹⁰; Lambert et al., ^{2022, 2023}; Wang et al., ²⁰²⁵).

Simultaneous change in climate mean and variability can have detrimental effects on vegetation, especially regarding cold stress, although the threat of freezes depends on the geographical region (Augspurger, 2013; Zohner et al., ²⁰²⁰; Cohen et al., ²⁰²³). On the one hand, global change would delay cold acclimation and hasten cold deacclimation under warmer conditions. On the other hand, an increase in temperature variability would increase the likelihood of extreme weather events at a given date (Reyer et al., ²⁰¹³; Thornton et al., ²⁰¹⁴), that would affect vulnerable plants.

According to Levitt's (1980) stress physiology paradigm, the ability to survive low temperature can be divided into two main processes (Charrier et al., ²⁰¹¹): freeze avoidance (avoidance of exposing sensitive organs to subzero temperature and limiting ice formation), and freeze tolerance (ability to withstand subzero temperatures without suffering damage, cold hardiness hereafter). The ability of plant water to supercool by limiting ice nucleation activity within (intrinsic nucleation) or upon (extrinsic nucleation) their tissues defines freeze avoidance. Although ice nucleation activity does not exhibit a clear link with plant physiological or phenological stage, some critical factors have been identified, such as flavonoids (Kasuga et al., ²⁰⁰⁸), osmotic potential (Gusta et al., ²⁰⁰⁴), structure (Lintunen et al., ²⁰¹³), ice nucleation active bacteria (Lindow et al., ¹⁹⁷⁸), and antifreeze proteins (Griffith & Yaish, 2004). Despite its relevance, especially with respect to late freeze events, ice nucleation has not been modeled in plant tissues, but the molecular dynamics of heterogeneous ice nucleation could provide interesting avenues (Glatz & Sarupria, 2018). In either case (avoidance or tolerance), the maximum cold hardiness of tissues is not constitutive and must be gained through acclimation and lost through deacclimation in a timely manner to survive winters.

Freeze risks can also be categorized according to their period of occurrence: early, mid‐winter, and late freezes. Early freeze damage can result from cold weather setting in after heat waves in late summer and early autumn (Charrier & Améglio, 2011; Ferguson et al., ²⁰¹⁴; Chang et al., ²⁰¹⁵; Charrier et al., ²⁰²¹). In winter, warm spells induce insufficient acclimation or mid‐winter deacclimation (Kalberer et al., ²⁰⁰⁶; Kovaleski, ²⁰²⁴). Attempts to transfer populations from drier and warmer regions to colder environments through assisted migration have raised concerns about cold hardiness, for example massive damage to Pinus pinaster from Portugal introduced in southern France (Benito‐Garzón et al., ²⁰¹³). Freeze risks in mid‐winter are currently the best integrated into species distribution predictions, as for the maximum hardiness zones developed by the USDA. The freeze safety margin, that is the difference between maximum cold hardiness and absolute minimum temperature, decreases sharply at the cold edge of tree species distribution (Baranger et al., ²⁰²⁴). During late winter and early spring, flushing buds are particularly susceptible to damage by low temperatures (Gu et al., ²⁰⁰⁸; Chamberlain et al., ²⁰¹⁹; Kirchhof et al., ²⁰²⁵).

The complexity of the factors involved in cold acclimation has led to the development of various models in many functional types, such as grasses, deciduous and evergreen woody perennials. Cold hardiness models differ in their purpose (e.g. descriptive, understanding, or predictive), defining the input and simulated variables and their underlying paradigms. This viewpoint aims to describe the concepts of these models and identify their potential and limitations in predicting freeze risk in unprecedented conditions, such as those imposed by global change, and propose new approaches to cold hardiness modeling, notably by analogy with phenological modeling.

Main concepts of cold hardiness models

Existing models are distributed according to a gradient of increasing complexity (Fig. 1; Table 1), which is linked to their degree of realism (the extent to which the model describes and explains real‐life causal phenomena) and generality (the ability of the model to maintain accuracy across a broad range of input variables, in different locations and/or over time, particularly outside the calibration range), as defined by Levins (1966). A first, more empirical approach relies directly on statistical relationships and/or historical data to predict cold hardiness (Aniśko et al., ¹⁹⁹⁴). Environmental factors such as air temperature and photoperiod are used as explanatory variables to directly estimate cold hardiness or changes in it. A second approach (called integrated models) incorporates the effect of the phenological cycle on cold hardiness, thus increasing the realism and generality of the model (Hänninen, 2016). A third approach focuses on a lower level of organization, integrating the role of physiological variables on cold hardiness (Charrier et al., ^2013a). A fourth approach has yet to be developed and could be further downscaled by explicitly modeling signal perception and transduction, resulting in metabolic changes. This would decouple the phenological cycle from cold acclimation and facilitate the integration of environmental signals and the accumulation of their impact throughout the plant's life cycle via legacy and memory effects. Finally, all these approaches predict plant vulnerability to low temperature, which still need to be compared with stress exposure to predict a realistic risk of low temperature damage.

Fig. 1.

Approaches developed to model plant cold hardiness across a spectrum of complexity, in relation to the number of intricate processes involved.

Table 1.

Cold hardiness models and their properties.

Model type	Species	Predicted variable	Input variables	Acclim.	Deacc.	Reacc.	Thermal time/phenology	Modulation by phenology	Legacy	Reversibility	Perception	Source
Empirical	Triticum aestivum	Biomass loss	Air temperature									Sharpley & Williams (1990)
	Deciduous woody plants	Survival temperature	Air temperature (sum + mean), day of year (DOY)	x	x	x	No	No	No	Yes	No	Aniśko et al. (1994)
	Pinus sylvestris	Cold hardiness	Air temperature, photoperiod, pH of cell effusate	x	x	x	No	No	No	Yes	No	Taulavuori et al. (1997)
	Brassica napus, Brassica rapa	Winter survival	Multiple, derived from air and soil temperatures and precipitation	x	x	x	Yes	No	No	Yes	No	Waalen et al. (2013)
	Prunus cerasus	Cold hardiness	Air temperature				No	No	No	No	No	Salazar‐Gutiérrez & Chaves‐Cordoba (2020)
	Triticum aestivum, Triticosecale × Wittmack	Winter survival	Multiple, derived from temperature and snow depth	x	x	x	No	No	No	No	No	Rapacz et al. (2022)
	Vitis vinifera	Cold Hardiness	Air temperature	x	x	x	No	No	No	Yes	No	(Jones et al., 2024)
	Vitis spp.	Cold hardiness	Air temperature, DOY				No	No	No	Yes	No	(Wang et al., 2024)
CH‐based	Phleum pratense	Cold hardiness	Air temperature, precipitation	x	x	x	No	No	No	Yes	No	(Thorsen & Höglind, 2010)
	Picea sitchensis	Survival temperature	Crown temperature, photoperiod	x			No	No	No	Yes	No	(Cannell et al., 1985)
	Pinus sylvestris	Cold hardiness	Air temperature	x	x	x	No	No	No	Yes	No	(Repo et al., 1990)
	Pseudotsuga menziesii	Cold hardiness	Air temperature	x	x		Yes	No	No	No	No	(Timmis et al., 1994)
Phenological	Malus domestica	Cold hardiness	Air temperature		x	x	Yes	No	No	Yes	No	(Winter, 1973)
	Cornus sericea	Cold hardiness	Air temperature	x	x	x	Yes	Yes	No	Yes	No	(Fuchigami et al., 1982; Kobayashi et al., 1983; Kobayashi & Fuchigami, 1983a, 1983b)
	Triticum aestivum	Biomass loss	Crown temperature, snow depth, photoperiod	x	x	x	Yes	No	No	No	No	(Ritchie, 1991)
	Pinus sylvestris	Cold hardiness	Air temperature, photoperiod	x	x	x	Yes	Yes	No	Yes	No	(Kellomäki et al., 1992, Kellomaki et al., 1995)
	Pseudotsuga menziesii	Cold hardiness	Air temperature, photoperiod	x	x	x	No	Yes	No	Yes	No	(Leinonen et al., 1995)
	Triticum aestivum, Secale cereale	Cold hardiness	Air temperature, photoperiod	x	x	x	Yes	Yes	No	No	No	(Fowler et al., 1999)
	Medicago sativa	Winter survival and yield loss	Air temperature, photoperiod, snow depth, solar irradiance, soil moisture, precipitation and plant density	x	x	x	Yes	No	No	Yes	No	(Kanneganti et al., 1998a, 1998b)
	Triticum aestivum, Pisum sativum	Cold hardiness	Air temperature	x	x	x	No	No	No	Yes	No	(Lecomte et al., 2003; Castel et al., 2017)
	Triticum aestivum	Cold hardiness	Crown temperature, snow depth	x	x	x	Yes	No	Yes	No	No	(Bergjord et al., 2008)
	Generic (annual and perennial crops)	Biomass loss	Air temperature				Yes	No	Yes	No	No	(Brisson et al., 2009)
	Vitis vinifera, Juglans regia, Camellia sinensis, Pseudotsuga menziesii	Cold hardiness	Air temperature	x	x	x	Yes	Yes	No	Yes	No	(Ferguson et al., 2011, 2014; Charrier et al., 2018b; Kimura et al., 2021; North et al., 2021; Stuke et al., 2024)
	Triticum aestivum, Triticosecale × Wittmack	Biomass loss	Air temperature				Yes	No	No	No	No	(Zheng et al., 2014)
	Vitis spp.	Cold hardiness	Air temperature, DOY				Yes	Yes	No	Yes	No	(Londo & Kovaleski, 2017)
	Pinus sylvestris, Juglans regia	Cold hardiness	Air temperature, photoperiod	x	x	x	Yes	Yes	No	Yes	No	(Leinonen, 1996; Charrier et al., 2018b)
	Vitis riparia	Cold hardiness	Air temperature				Yes	Yes	No	Yes	No	(Londo & Kovaleski, 2019)
	Winter cereals	LT50 (individual)	Crown temperature, photoperiod	x	x	x	Yes	Yes	?	No		(Byrns et al., 2020)
	Prunus cerasus	Cold hardiness	Air temperature, Water content (WC)		x		Yes	No	No	No	No	(Hillmann et al., 2021)
	Vitis spp.	Cold hardiness	Air temperature				Yes	Yes	No	Yes	No	(Kovaleski et al., 2023)
Physiological	Juglans regia	Cold hardiness	Air temperature, starch, soluble carbohydrates and water contents	x	x	x	No	No	Yes	Yes	No	(Poirier et al., 2010)
	Deciduous woody plants	Cold hardiness	Soluble carbohydrates and water contents	x	x	x	No	No	Yes	Yes	No	(Charrier et al., 2013a; Baffoin et al., 2021)
	Juglans regia	Cold hardiness	Air temperature, photoperiod Starch, soluble carbohydrates and water contents (for initialization)	x	x	x	Yes	Yes	Yes	Yes	No	(Charrier et al., 2018a; Charrier & Améglio, 2024)

Cold hardiness as the response variable

Plant cold hardiness is a phenotypic trait that can be modified by environmental stimuli, mainly low temperatures and short days (Welling & Palva, 2006). Cold hardiness thus varies depending on the season, due to environmentally induced changes in gene expression, which result in physiological, biochemical, and anatomical changes primarily through the osmotic control of the intracellular freezing point (Charrier et al., ^2013a), plasma membrane composition (Uemura et al., ²⁰⁰⁶), and cell wall thickness (Takahashi et al., ²⁰²¹).

Depending on the species, cold hardiness per se is not easily measurable and is sometimes based on the consequences of low temperature stress after the winter period. This is particularly true in herbaceous species (e.g. winter cereals and other annual crops), exhibiting cold damage as a disturbance (sensu Grime, ¹⁹⁷³), that is partial or total loss of biomass. In these species, a good cold hardiness index thus relates to the survival (or the lack of recovery) of the whole individual, the destruction of plant tissues, and the resulting yield (generally referred to as ‘winter kill’).

In woody perennial species, cold hardiness can be measured regularly during the winter period through different techniques such as lethal temperature for 50% of the individuals, the temperature of intracellular exotherm by thermal analysis (LTE), or progressive damage by relative electrolyte leakage, or visual scoring. Depending on the measured phenotype, the cold hardiness index can constitute a threshold that, once exceeded, marks the development of damage and jeopardizes survival (survival, biomass loss, LTE). A more quantitative cold hardiness index, as measured by the electrolyte leakage or visual scoring methods, may allow a more gradual assessment of damage intensity, therefore predicting cold damage with a thinner resolution, still missing the link with survival.

One peculiar case is freezing avoidance, that is the ability to resist cold temperatures by maintaining supercooled water. The process of ice nucleation depends on the plant tissue and developmental stage, as well as the nature of the nucleus (e.g. ice‐nucleation‐active bacteria), other biophysical conditions in plants (e.g. solute concentration, wettability, cell wall properties, and physical barriers), and atmospheric conditions (Kirchhof et al., ²⁰²⁵). However, current cold hardiness models consider ice nucleation to be a deterministic process that occurs at a fixed temperature (e.g. 0, −2, or −4°C), whereas experimental evidence has shown that its stochasticity depends on both biotic and abiotic factors (Kirchhof et al., ²⁰²⁵).

Although plant mortality is complex and difficult to measure (see e.g. Leopold, ¹⁹⁷⁸; Filip et al., ²⁰⁰⁷; Anderegg et al., ²⁰¹², for drought stress), the ability of meristematic cells, particularly those in the shoot apical meristem, to divide and form new vegetative organs is crucial for predicting resilience to stressful events (Thomas, 2013). Current methods cannot easily determine the sequence of damage leading to plant mortality; they only allow measurements after the stressful event has occurred. Nondestructive monitoring techniques, such as micro‐dendrometers or acoustic emission analysis (Charrier et al., ^{2017, 2021}; Lamacque et al., ²⁰²²), are therefore needed to assess low‐temperature damage. This would allow a continuous assessment of damages, required to elucidate the dual role of freeze intensity and duration as simulated by the death time model (Faber et al., ²⁰²⁴).

As the mechanisms driving cold hardiness continue to be elucidated, the effects of variables continue to be quantified through experimental evidence or modeling efforts. Depending on the biology of the species and the purpose of the model, cold hardiness can be modeled at a single point in time (static models, e.g. Charrier et al., ^2013a), dynamically over a short period (less than a year; Cannell et al., ¹⁹⁸⁵), or over multiple years (Leinonen, 1996). In perennial species, simulations may require reinitialization at fixed dates (Cannell et al., ¹⁹⁸⁵; Timmis et al., ¹⁹⁹⁴). In winter cereals and crops, the dynamics typically begin at the sowing date and continue until harvest (Bergjord et al., ²⁰⁰⁸; Byrns et al., ²⁰²⁰).

Predicting cold hardiness

In static models, cold hardiness is predicted using variables measured on the same date (Charrier et al., ^2013a; Jones et al., ²⁰²⁴), a few days later (Proebsting, 1963; Andrews & Proebsting, ¹⁹⁸⁶), or up to several months earlier (Aniśko et al., ¹⁹⁹⁴; Poirier et al., ²⁰¹⁰; Rapacz et al., ²⁰²²). In multiple regression analysis, a subset of variables is predefined and measured, and the most informative index is selected through a calibration process that requires it to significantly improve prediction accuracy (stepwise procedure; Aniśko et al., ¹⁹⁹⁴). Over the past decade, partial least squares regressions and machine learning algorithms have significantly increased the number of variables included in analyses, even when there is no prior knowledge of their effect. For example, the NYUS.2 model uses 117 features in its training process, including cultivar features and hourly temperature‐based features such as daily and cumulative temperature descriptors, as well as exponential and reverse exponential weighted moving averages (Wang et al., ²⁰²⁵). While these models generally lead to highly accurate predictions (RMSE lower than 2°C, and even 1°C in some cases), they lack realism, which can restrict their validity to the climate range in which they were developed. Empirical models also usually lack reversibility, that is the ability to allow deacclimation and reacclimation throughout the modeled period; this is only achieved through the fluctuation of driving variables.

Based on a large series of experimental measurements and field observations, Repo et al. (1990) developed a simple, cold hardiness‐based, dynamic model that integrates reversibility through the concept of stationary cold hardiness in order to predict realistic cold hardiness mechanisms (Kalberer et al., ²⁰⁰⁶). In order to enable reversibility in simulations, this model uses stationary cold hardiness and the rate of change in cold hardiness. The prevailing air temperature determines stationary cold hardiness, and the rate of change is proportional to the difference between prevailing and stationary cold hardiness. Based on these assumptions: (1) under a constant air temperature, cold hardiness attains the stationary level determined by that temperature; and (2) under fluctuating temperatures, cold hardiness follows changes in air temperature (which determine the stationary level). To mitigate the impact of rapid air temperature fluctuations on cold hardiness, a time constant (τ) governs the rate of change. The higher the value of τ, the slower the predicted changes in cold hardiness. The first‐order model predicts a rapid change that slows for small differences between the prevailing and stationary cold hardiness levels (Repo et al., ¹⁹⁹⁰; Leinonen, ¹⁹⁹⁶). Leinonen et al. (1995) introduced a second‐order model using two time constants corresponding to two stationary levels of cold hardiness to address the exceptionally complex air temperature response of cold hardiness in Douglas‐fir during dormancy. The second time constant is called ‘asymptotic cold hardiness’. As endodormancy progresses, the effect of the second time constant decreases until it finally vanishes, meaning that at the beginning of ecodormancy, cold hardiness is again modeled with one time constant.

Predictor variables driving cold hardiness

The variables used in current models are described according to the following framework: description of the variable, its perception and molecular regulation, integration into current models, and limitations.

Temperature

Air temperature is the most commonly used input variable for predicting cold hardiness, either directly on the change in cold hardiness or indirectly through its effect on phenological processes (Fig. 2). Directly, the signal of low temperatures enhances cold hardiness, whereas the indirect effect of thermal times (e.g. chill and growing degree‐day accumulation) relates to phenological cycles and other physiological processes, which then affect the dynamics of cold hardiness.

Fig. 2.

Seasonal dynamics of biotic and abiotic factors involved in cold hardiness (CH) during successive phenological stages (Lign, lignification; EnDI, endodormancy induction; EnDR, endodormancy release; EcoD, ecodormancy; and growth). Variation occurs across the year in environmental factors (temperature and photoperiod), sensing of short and long days (SD, LD), and low, freezing, and warm temperatures (LT, FT, and WT), leading to changes in water content (WC), starch to soluble carbohydrates (TSC) interconversion, and conferring cold hardiness to plants.

As there are no known specific temperature sensors in plants, temperature perception occurs at multiple levels (Penfield, 2008; Kerbler & Wigge, ²⁰²³). The main perception pathways involve modulation of plasma membrane fluidity; cytoskeletal stability; activation of channels that allow calcium (Ca²⁺) to enter the cell; conformation of certain proteins; enzymatic activity within different metabolic reactions; and the expression, transcription, and translation of various genes (Ruelland & Zachowski, 2010). Cold temperatures, in particular, affect the rigidity of fatty acid aliphatic chains, thereby exerting physical constraints on membrane proteins such as calcium (Ca²⁺) channels. Calcium can then trigger a cold response via the expression of C‐REPEAT BINDING/DEHYDRATION‐RESPONSIVE ELEMENT BINDING 1 FACTORS (CBF/DREB1), which activate the promoters of COLD‐REGULATED (COR) genes. As acclimation is a process that occurs against the temperature gradient, it provides an interesting avenue for studying the effects of temperature sensing.

Although maximum cold hardiness in the middle of winter does not usually depend on the lowest recorded temperature (Aitken et al., ¹⁹⁹⁶; Larcher & Mair, ¹⁹⁶⁸; Morin et al., ²⁰⁰⁷; Charrier et al., ^2013b; however, see Vitra et al., ²⁰¹⁷), the direct effect of low temperatures on the rate of change in cold hardiness has long been documented (Haberlandt, 1875; Schaffnit, ¹⁹¹⁰; Irmscher, ¹⁹¹²; Chandler, ¹⁹¹³; Gassner & Grimme, ¹⁹¹³). The effective temperature threshold for stem acclimation varies between species but is generally below 5–10°C (Harvey, 1922; Greer et al., ²⁰⁰⁰). The acclimation rate increases with decreasing temperature (Dantuma & Andrews, 1960; Pogosian & Sakai, ¹⁹⁶⁹), and daily temperature fluctuations also play a role thanks to the nonlinear response between temperature and cold hardiness (Wang et al., ²⁰²⁴). In general, the lesser the cold hardiness (i.e. more vulnerable), the greater the effect of low temperatures on the rate of change in cold hardiness (Luoranen et al., ²⁰⁰⁴) via a nonlinear relationship (Greer et al., ²⁰⁰⁰) that depends on the phenological stage (Tumanov, 1969).

Temperature can also be integrated over extended periods, creating new variables that are used to model cold hardiness. Warm temperatures are integrated to simulate a developmental path (Réaumur, 1735). This metric is referred to as growing degree‐days (GDDs) or growing degree‐hours, depending on the timescale of measurement used for integration, and its accumulation is associated with ‘forcing’ by the environmental conditions (i.e. promotion of growth). GDDs are calculated using different functions, but generally at their simplest as a linear response (with or without added thermal limits of base temperature and maximum temperature), or sometimes based on sigmoid or bell‐shaped curves. Similarly, cold temperatures can be integrated over time to produce a metric referred to as chilling accumulation. Many methods exist for integration of low temperatures, from simple sums of time under a threshold (Weinberger, 1950) to more complex, physiology‐informed models that use a combination of multiple functions (e.g. Dynamic model; Fishman et al., ¹⁹⁸⁷). As parts of the process are unknown, the timing of start for integration of warm or cold units remains an arbitrary aspect of the accumulation, which largely affects multilocation analyses (Fernandez et al., ²⁰²⁰). Both forcing and chilling are often parts of models to predict shifts in phenology and physiology, such as endodormancy release and budbreak (see further below), but units of GDD and chilling have also been used to predict cold hardiness directly without considering phenological stages (Timmis et al., ¹⁹⁹⁴; Wang et al., ²⁰²⁴).

Actual organ temperature is of particular importance in order to simulate plant developmental stages and cold hardiness, as well as to predict freeze risks at the intersection of hazard vulnerability and exposure. Season, weather patterns, and geographical aspects, among other factors, define the air temperature in any given location. Generally, standard meteorological stations providing the air temperature records used for modeling purposes are usually 1.5–2 m tall, installed in open fields. These observations can be scaled and transformed into gridded data that may be useful for regional approaches to modeling (e.g. Jones et al., ²⁰²⁴). However, air temperature can differ from that actually experienced by the plants due to different factors (Fig. 3; Charrier et al., ^2015a). Important factors affecting a plant's energy balance to consider are: the growth form and height (e.g. tree vs shrub or grass); the presence of snow cover, which buffer plants from fluctuating temperature; and the openness of the surrounding vegetation, which can create microclimates (Cellier, 1984, 1993; Leuning & Cremer, ¹⁹⁸⁸; Jordan & Smith, ¹⁹⁹⁴; Kirchhof et al., ²⁰²⁵). Additional factors influencing energy balance within a plant are size, shape, and color of individual structures (Vitasse et al., ²⁰²¹; Peaucelle et al., ²⁰²²). Therefore, although air temperature is largely used in modeling efforts, additional factors may be necessary to accurately describe the environment experienced by plants. In crop models, the temperature of the crown or the soil surface is a better proxy for environmental drivers of cold hardiness. The ALFACOLD model thus uses air temperature, photoperiod, snow depth, solar irradiance, soil moisture, precipitation, and plant density to more accurately predict organ temperature (Kanneganti et al., ^{1998a, 1998b}).

Fig. 3.

Temperature experienced by plants depends on growth form and environment. (a) Small trees close to the tree line in mountains (here Abies balsamea at 1600 m altitude on Mt Washington, in New Hampshire, USA) may be (b) completely covered by snow during winter. (c) Temperatures at the soil surface have lower daily variations compared with air temperatures, and snow cover maintains temperatures at warm, subzero temperatures during winter. (d) In managed cranberry (Vaccinium macrocarpon) marshes, (e) temporary flooding is used to form ice layers on the top of plants, preventing their canopies from experiencing extremely low air temperatures.

In addition to the direct effects of low temperatures, the formation of ice in plant tissues induces a number of physiological effects that are not fully incorporated into current models. The only exception is the cold shock effect (Winter, 1973). Freezing can be perceived through a mechanical signal induced by ice volumetric expansion, an osmotic signal induced by ice's extremely negative chemical potential, or oxidative or acoustic signals linked to gas bubble formation within frozen sap (Charrier et al., ^2015b; Charra‐Vaskou et al., ^{2016, 2023}). A biophysical approach can simulate the water flows induced by ice within plant tissues during freeze–thaw cycles, resulting in pronounced contraction of the elastic cells of the bark (Bozonnet et al., ²⁰²⁴). Ultimately, the disorders can be of two kinds, the relative importance of which varies according to the period under consideration: oxidative stress during endodormancy (Beauvieux et al., ²⁰¹⁸), and dehydration stress, delaying growth resumption during ecodormancy.

Photoperiod

Due to the Earth's tilt and its revolution around the Sun, day length varies depending on the time of the year and latitude. Because of the decreased amount of energy arriving on the surface in extratropical regions, temperatures start decreasing as a response to decreasing day lengths in the autumn, with the opposite effect occurring in the spring. However, energy storage on the surface results in lags of about a month between the minimum and maximum temperature when compared with the minimum and maximum day length (Fig. 2). Considering this dynamic, plants can respond to decreasing or increasing day lengths in order to respond to upcoming changes in temperature. Given the stability in day length within a region, using the day of year within models can be viewed as a proxy for photoperiod within a latitude (e.g. within Aniśko et al., ¹⁹⁹⁴; Wang et al., ²⁰²⁴; Londo & Kovaleski, ²⁰¹⁷). However, this would require a parameter for latitude associated with the date.

Photoperiod perception is mediated by various receptors (phytochromes, cryptochromes, and phototropins), which change their conformation and cellular localization when stimulated by light, enabling them to activate or inhibit processes (Chen et al., ²⁰⁰⁴; Zhou et al., ²⁰⁰⁷). As photoreceptors usually revert to their inactive state when not stimulated by light, night length is thus more physiologically relevant. The CONSTANS gene and the CO protein lie at the interface between different activation/inhibition of signal transduction pathways, and their rhythmic expression is a marker of the circadian clock (Jackson, 2009). Consequently, CO expression levels control the expression of genes involved in various seasonal processes such as flowering in Arabidopsis thaliana (Samach et al., ²⁰⁰⁰; Wigge et al., ²⁰⁰⁵; Yamaguchi et al., ²⁰⁰⁵) or growth cessation in Populus trichocarpa (Böhlenius et al., ²⁰⁰⁶).

The initial stage of cold acclimation is affected by a decrease in both photoperiod and temperature (Aronsson, 1975; Christersson, ¹⁹⁷⁸). Short days can induce cold acclimation even at mild temperatures (15°C) in many species (Sakai & Yoshida, 1968; Schwarz, ¹⁹⁷⁰; Buchanan et al., ¹⁹⁷⁴; Charrier & Améglio, ²⁰¹¹). A threshold photoperiod of 11 h initiates the first stage of cold acclimation in Pinus radiata (Greer & Warrington, 1982). However, considering the root system, soil temperature is the only driver (Johnson & Havis, 1977; Kaurin et al., ¹⁹⁸²; Ryyppö et al., ¹⁹⁹⁸).

Although photoperiod and temperature can have cumulative effects, it is difficult to discern their respective effects in natural conditions due to their parallel dynamics. Furthermore, they interact with each other in both cold acclimation and phenological processes (Heide, 1974; Johnsen et al., ^{2005a, 2005b}). For example, two independent pathways govern endodormancy release: one regulated by photoperiod (and enhanced by warm temperatures) and the other regulated by cold temperatures (Tanino et al., ²⁰¹⁰). Therefore, similarly to temperature, direct effect on the change in cold hardiness and indirect effects of day length on phenological cycle can be acknowledged by their inclusion within cold hardiness prediction models, although the concept of additive response of cold hardiness to temperature and photoperiod is questionable (Zhang et al., ²⁰⁰³). This may be particularly relevant for annual, evergreen, and photosensitive species, compared with deciduous trees and shrubs, as demonstrated by the beech tree in the study of phenology (Vitasse et al., ²⁰⁰⁹; Basler & Körner, ²⁰¹⁴).

Phenological stage

Only considering the direct effect of air temperature on cold hardiness is a pitfall, although it accurately predicted the seasonal course of cold hardiness in many examples (Repo et al., ¹⁹⁹⁰; Aniśko et al., ¹⁹⁹⁴; Salazar‐Gutiérrez & Chaves‐Cordoba, ²⁰²⁰). Plants have limited potential for acclimation during active growth, even when exposed to cold temperatures that would generally cause acclimation during other phases of the annual cycle. Similarly, if chilling requirements for endodormancy release have not been fulfilled, plants will deacclimate more slowly. For woody perennials, the influence of temperature depends closely on the maturity of tissues (Leinonen, 1996) and dormancy status (Pisek & Schiessl, 1947; Scheumann & Schönbach, ¹⁹⁶⁸; Howell & Weiser, ¹⁹⁷⁰; Tumanov et al., ¹⁹⁷³; Fuchigami et al., ¹⁹⁸²; Kovaleski et al., ²⁰¹⁸). For herbaceous plants, vernalization (Kanneganti et al., ^{1998a, 1998b}; Fowler et al., ¹⁹⁹⁹; Bergjord et al., ²⁰⁰⁸) or leaf number (Lecomte et al., ²⁰⁰³; Castel et al., ²⁰¹⁷) determines their cold hardiness (Gabbrielli et al., ²⁰²²).

Phenological stage is currently modeled as a black box, although phytohormones, such as abscisic acid (ABA) and gibberellic acid (GA), regulate winter dormancy (seed and bud) and cold hardiness. Their relative abundance regulates phenological processes through antagonist activity; GAs promote growth during ecodormancy, whereas ABA is involved in stress response (endodormancy induction, drought stress), triggering cold acclimation (Gusta et al., ²⁰⁰⁵) and preventing deacclimation (Kovaleski & Londo, 2019). As cold acclimation and dormancy involve ABA (Arora et al., ²⁰⁰³; Welling & Palva, ²⁰⁰⁶) and osmolytes (Beauvieux et al., ²⁰¹⁸; Baffoin et al., ²⁰²¹), their molecular control is partly shared, for example, through the interaction between DORMANCY ASSOCIATED MADS‐BOX (DAM) and genes regulated by CBF/DREB1 (Ding et al., ²⁰²⁴).

Phenological dynamics are influenced by temperature accumulation (forcing and chilling) or photoperiod (explored above), and are thus modeled using those environmental variables. The relation between phenology and competence for hardening calls for the development of integrated models, that is, models which combine phenological and cold hardiness dynamics (Fig. 1; Hänninen, ²⁰¹⁶). Transitions in phenology may be considered an intermediate step between physical environmental variables and the resulting cold hardiness. The first integrated model was introduced by Winter (1973) for Malus domestica, covering only the deacclimation phase, and Fuchigami et al. (1982) for Cornus sericea through the entire annual cycle (Degree Growth Stage; Kobayashi et al., ¹⁹⁸³; Kobayashi & Fuchigami, ^{1983a, 1983b}). In annual crops, developmental stages are also integrated through the prediction of successive phenological stages (coleoptile, tillering, or leaf number; Lecomte et al., ²⁰⁰³) and phenophases (autumn dormancy and vernalization; Kanneganti et al., ^{1998a, 1998b}).

Transitions between phenological stages are usually modeled using a sequential paradigm, in which the phenological variable is related to the completion of the stage (i.e. expressed as a ratio of a dynamic variable to a critical threshold for progression to the next stage), although, when considering molecular regulation, a more realistic formalism could be provided by the overlap concept (i.e. heat requirements being reduced as chilling requirements are fulfilled; Cannell & Smith, ¹⁹⁸³). In crop models, developmental stages can also be simulated (i.e. leaf number in wheat; Lecomte et al., ²⁰⁰³).

Once the phenological model (linear or cyclic) has been simulated, its effect on acclimation and deacclimation can be addressed. The concept of hardening competence (C_h) mediates the constraints caused by the phenological stage to the rate of change in cold hardiness (Leinonen, 1996). C_h starts at 0 (no hardening competence) during growth, and it increases to reach 1 (full hardening competence) during lignification and dormancy induction. During the endodormancy stage, C_h remains at its maximum value and declines progressively during ecodormancy as forcing units accumulate. C_h attains the value of zero again at growth onset or slightly after. Another paradigm, deacclimation kinetics, considers that the rate of deacclimation at similar deacclimating temperatures increases with chilling accumulation over the winter (Kovaleski, 2022; North & Kovaleski, ²⁰²⁴). Although these paradigms were developed in different species, they are both based on experimental measurements and provide accurate predictions of cold hardiness. In other words, models with quite big differences in their ecophysiological assumptions have been provided to be accurate. This calls for further examinations of the biological realism of the models: do all of the differences between the models reflect real ecophysiological differences between different species?

Linking the hardening competence to dormancy induction would set the beginning of the accumulation of chilling units (Charrier, 2023). In walnut trees, cold acclimation mainly occurs during dormancy induction and stabilizes during endodormancy release (Fig. 4). Later in winter, once the chilling requirements have been fulfilled, the deacclimation rate is similar to that during acclimation, with an offset of 15°C. The hysteretic loop that divides cold hardiness into acclimation during dormancy induction and deacclimation during ecodormancy release provides an easy way to correlate dormancy and acclimation processes, albeit indirectly.

Fig. 4.

The cold hardiness dependency on dormancy. Dormancy, represented by the mean time to budbreak (MTB), shows an association with cold hardiness when separated into three different phases.

Physiological variables

Although phenological variables accurately describe the seasonal changes in C_h, phenological models assume that the potential for hardening is always optimal; in other words, that growing conditions have no effect on the ability to cold acclimate (Charrier et al., ²⁰²¹). However, in addition to environmental and phenological variables, several other factors influence the plant's cold hardiness. These include soil fertility (Aronsson, 1980; Luoranen et al., ²⁰⁰⁸), soil temperature (Charrier & Améglio, 2024), drought (Chen & Li, 1978; Colombo et al., ²⁰⁰¹), defoliation and girdling (Poirier et al., ²⁰¹⁰), air pollutants (Taulavuori et al., ²⁰⁰⁵), UV radiation (Taulavuori et al., ²⁰¹²), and microbiome characteristics (Lindow, 2023), among other aspects. Furthermore, it has been demonstrated that: (1) plants can acclimate and deacclimate to cold under conditions that do not release endodormancy (Schwarz, 1970; Charrier & Améglio, ²⁰¹¹); (2) growth cessation is not a strict prerequisite for cold acclimation (Arora et al., ¹⁹⁹²); and (3) the influence of external environmental factors can be bypassed by modulating internal physiology (Poirier et al., ²⁰¹⁰; Charrier & Améglio, ²⁰¹¹).

The artificial modulation of water and carbon levels by girdling and defoliation treatments affected the cold hardiness of different branches within the same tree (Poirier et al., ²⁰¹⁰). Furthermore, controlling the timing of leaf fall while keeping the trees under cold deprivation demonstrated the essential role of water and carbon status (Charrier & Améglio, 2011; Charrier et al., ^2018a). Based on primary metabolism, that is water status and carbon balance, these variables are likely to be affected by the ability to modulate through source–sink relations and/or hydraulic architecture. This physiological mechanism is relevant across all tree organs (root system, trunk, stem, and bud; Charrier et al., ^2013a) and through a wide range of deciduous tree species (Baffoin et al., ²⁰²¹). Another physiological indicator, pH of cell effusate, was used to predict cold hardiness, though with limited physiological insights (Taulavuori et al., ¹⁹⁹⁷).

The control of cold acclimation by temperature‐dependent physiological variables (i.e. osmo‐hydric model) provides a static view of freezing risks. The osmo‐hydric model is now being developed into a dynamic version that simulates variations in physiological variables on a daily timescale (Fig. 5) through: (1) the balance between root water uptake and evaporative losses (Charrier & Améglio, 2024); and (2) the interconversion between starch, soluble sugars, and respiration losses using temperature as an input variable (Charrier et al., ^2018a).

Fig. 5.

Simulations of cold hardiness in the walnut tree in Clermont–Ferrand (France) according to three approaches: a model based on temperature only (in green, thermic; Ferguson et al., ²⁰¹¹); a model based on temperature and photoperiod (in blue, photothermic; Leinonen, ¹⁹⁹⁶); and a model based on temperature‐induced changes in water and soluble carbohydrates (in yellow, osmo‐hydric; Charrier et al., ^2018b). Actual measurements of cold hardiness are shown as black diamonds.

As the physiological processes that drive cold hardiness are common to many perennial angiosperms, this approach, based on carbon metabolism and water status, could also be applicable to other species (Baffoin et al., ²⁰²¹). However, the inclusion of additional processes may be necessary. For example, species capable of cortical photosynthesis, such as beech trees, can assimilate carbon on sunny winter days, providing a consistent carbohydrate supply for acclimation (Berveiller et al., ²⁰⁰⁷). The same may be true for evergreen species at the cost of enhanced dehydration (Chang et al., ²⁰²¹).

Cumulative stress can affect cold hardiness the following winter by impacting carbon balance (Thomas & Blank, 1996; Poirier et al., ²⁰¹⁰). For instance, defoliation (whether induced by drought or by pests) during the growing period (e.g. in the paradormancy stage) affects carbon reserves when growth resumes, as photosynthetic activity may be limited (Thomas & Blank, 1996; Wargo, ¹⁹⁹⁶). Freeze–thaw cycles may also affect hydraulic architecture and water distribution (Charrier et al., ^2013b). Modulating water status and carbon reserves under different conditions (along environmental gradients and at various stress intensities) should provide useful information for increasing the realism of cold hardiness modeling.

New avenues in cold hardiness modeling

Each of the existing approaches has its own advantages and limitations. In the current context, machine learning‐based models are highly accurate for deciphering molecular regulations and for predictive purposes, making them ideal for predicting risk in their intended area and culture. However, their accuracy in other contexts is limited, which can be challenged by unprecedented environmental conditions. To further increase the realism of cold hardiness models, the first task is to improve the description of simplified processes and their interactions (phenological cycles, carbon metabolism, and water status), and the second is to integrate other relevant ones (perception and transduction of environmental signals, determinism of ice nucleation; Wisniewski et al., ²⁰¹⁸). Notably, the lagged effect and resilience to stressful events need an integrative approach, combining ecophysiological and molecular data to characterize both legacy (Anderegg et al., ²⁰¹⁵) and memory effects (Thellier & Lüttge, 2013; Crisp et al., ²⁰¹⁶; Lämke & Bäurle, ²⁰¹⁷). Although the legacy effect has been characterized for ecophysiological functions that explain cold hardiness such as carbon reserves (Poirier et al., ²⁰¹⁰), vulnerability to embolism (Hacke et al., ²⁰⁰¹), and leaf phenology (Fu et al., ²⁰¹⁴), memory incorporates the change in perception of the plant affecting the dynamic response to maintain physiological homeostasis by recalling stored information (Thellier & Lüttge, 2013).

A knowledge gap exists in the translation between transcriptional regulation and functional consequences in response to stress (Wisniewski et al., ²⁰¹⁸). In this regard, the characterization of a marker of cold hardiness that would be central to the predictive models might allow for the validation of a physiology‐based modeling approach. Another important step toward realism would be to provide a mechanistic description of how environmental factors are perceived and transduced into the metabolic and cellular changes that confer cold hardiness. A desired approach should describe the molecular cascade from the perception of environmental factors to cold hardiness, integrating hormone signaling networks, transcription, translation, catalytic activity, and metabolic responses.

Based on molecular time series data, the interactions between photoreceptors, the 24 h circadian clock, and gene expression were used to successfully model the photoperiod sensor of Arabidopsis flowering (Salazar et al., ²⁰⁰⁹). Interestingly, as it is well conserved across species, a focus on the CBF pathway was proposed to integrate environmental and internal signals in a suggested modeling framework from Arabidopsis to crops (Chew & Halliday, 2011). Building from this idea, a molecular‐based model for cold acclimation/deacclimation would include complex interactions between ABA, GA, CBF genes, and dormancy‐associated genes, with functions similar to FLC that maintain dormancy and cold hardiness during winter, while they are inhibited upon prolonged exposure to cold temperatures. Different studies have indeed suggested such interactions in perennial species for MADS‐box genes, including DORMANCY ASSOCIATED MADS‐BOX (DAM) in Rosaceae fruit trees, SHORT VEGETATIVE PHASE‐LIKE (SVL) genes in Populus, and SHORT VEGETATIVE PHASE 2 (SVP2) in kiwifruit (Fig. 6; Falavigna et al., ²⁰¹⁹; Ding et al., ²⁰²⁴).

Fig. 6.

Proposed signaling network for the molecular‐based model for cold hardiness. Ambient low temperatures and freezing temperatures, and photoperiod through the circadian clock, induce the expression of C‐REPEAT BINDING FACTORS (CBF) genes, which in turn activate the expression of COLD REGULATED (COR) genes. Prolonged exposure to cold temperatures inhibits the expression of DORMANCY ASSOCIATED MADS‐BOX (DAM)/SHORT VEGETATIVE PHASE‐like (SVL)/FLOWERING LOCUS C (FLC) genes, which maintain the dormancy stage, partly through the regulation of phytohormones gibberellins (GA) and abscisic acid (ABA). FLOWERING LOCUS T (FT) was shown to maintain growth and prevent dormancy onset under warm temperatures and long‐day photoperiod. Lines that end in an arrowhead or a line represent activation or inhibition, respectively.

Models integrating circadian clock and CBF genes show that the clock also has an important role in temperature signal transduction (Keily et al., ²⁰¹³). Moreover, temperature sensing has been particularly well described and modeled for the process of vernalization and bolting in Arabidopsis species based on the memory of repression of the FLOWERING LOCUS C (FLC) gene upon prolonged cold exposure (Berry & Dean, 2015). Mathematical models built on the extensive knowledge of the thermosensory network were successfully used to predict how temperatures regulate the expression of FLC and the subsequent rate of vernalization in A. thaliana (Angel et al., ²⁰¹⁵; Antoniou‐Kourounioti et al., ^{2018, 2023}). Recently, machine learning‐based models were used to define metabolomic and transcriptomic biomarkers for extreme climate resilience or bud dormancy stages (Vimont et al., ²⁰¹⁹; Dussarrat et al., ²⁰²²).

ABA and GA dynamics are key players regulating fate switching in seed and bud dormancy (Maurya & Bhalerao, 2017) and increased cold tolerance (Gusta et al., ²⁰⁰⁵), although cold acclimation and endodormancy release do not completely overlap (Arora et al., ¹⁹⁹⁷; Charrier et al., ²⁰¹¹). Mathematical models were proposed to simulate seed and bud dormancy based on phytohormone signaling (Topham et al., ²⁰¹⁷; Vimont et al., ²⁰²¹) and could easily be adapted to cold acclimation. The explicit integration of ABA dynamics and the changes in its regulatory network would help to explore the trade‐off between dormancy and stress tolerance (Volaire et al., ²⁰²³), freeze and drought responses (Charrier et al., ²⁰²¹), as well as legacy and memory effects.

Developing such a model would require measurements of phenological, physiological, metabolomic, and transcriptomic variables under natural and controlled conditions over a wide range of environmental conditions (temperature, night length, and their fluctuations) supplemented by dedicated experiments (stress exposure at different stages). Some data and models are already available for fruit and forest trees, and we support the construction of a global experimental and theoretical research network to tackle the acquisition of the necessary knowledge to take it to the next level. This modeling approach will improve our understanding of the molecular mechanisms underlying freezing tolerance, while enabling predictions of plant acclimation to emerging climates and phenotypic plasticity.

This approach not only provides a better understanding of the underlying mechanisms and predictions of consequences but also opens up new perspectives on characterizing genetic variability in regulating cold hardiness at interspecific and intraspecific scales. It is indeed possible to integrate both photoperiod and temperature sensing using a small subset of genes and different signaling pathways to predict flowering, for example (Wilczek et al., ²⁰⁰⁹; Satake et al., ²⁰¹³). Another difference can lie in the synthesis of isoenzymes with slightly different optimal temperatures. For instance, in walnut trees, isoenzymes involved in carbon metabolism may account for differences in the temperature response across genotypes. Further integration of genotypic differences can be approached through two means: in vitro experiments that functionally characterize the response dynamics of enzymes to temperature, and in silico fitting procedures to capture genetic variation. Sensitivity analysis is indeed a relevant procedure to identify adapted genotypes and species and define local ideotypes.

At the interspecific level, metabolic pathways differ, particularly with regard to the identity of the compounds that accumulate. Extending the scope of the model to other species would thus require characterizing the link between cold hardiness and the various compounds that accumulate during acclimation. Therefore, a more generic approach would be to use an integrative, albeit mechanistic, marker of cold hardiness. In this sense, osmotic potential at the cellular level reflects the ability of the cells to tolerate extracellular freezing and is a promising candidate, provided that the measurements reflect intracellular osmotic concentration (Tyler et al., ¹⁹⁸¹; Ravari et al., ²⁰²³). However, while the concept of a generic cold hardiness marker based on osmotic potential may be applicable to most tree species that tolerate frost‐induced dehydration (Arora, 2018), alternative strategies of cold resistance involve different compounds, including organic acids, amino acids, unsaturated lipids, dehydrins, and antifreeze proteins (Chang et al., ²⁰²¹), and thus the entire model architecture would require redesigning (e.g. in other functional types, such as conifers or herbaceous species).

Conclusions

Cold hardiness models play a crucial role in understanding how different species adapt to low temperatures and predicting current and future risks. Although recent weather hazards have brought spring damage and false springs to the forefront of the research agenda, the issue of acclimation during heatwaves in autumn may become more prevalent, indicating that the entire phenological cycle is of interest. The existing models can be categorized into four main approaches along an observational to mechanistic understanding gradient: empirical, cold hardiness‐based, phenological, and physiological models. Each of these approaches encompasses a wide array of species, variables, and intrinsic processes. However, they can be differentiated based on three key properties: reversibility, their connection to the annual cycle, and legacy effects (Table 1).

The most suitable model is contingent upon its intended purpose. For local genotypes (e.g. varieties or provenances) and current climate, empirical models can be effective, provided that phenological cycles remain unperturbed. When integrating phenological processes, cold hardiness models, primarily developed for cold environments, could prove useful in warm winter areas where endodormancy issues arise. Alternatively, physiological models, though potentially less accurate, can offer insights into broader processes thanks to an increased realism.

One of the principal challenges in refining the realism of cold‐hardiness models is the inclusion of numerous intricate processes. This endeavor faces several hurdles, including a lack of convergence during calibration, the significant amount of data required for calibration, and the necessity of conducting experiments under controlled conditions. As researchers attempt to downscale to elemental processes, they often encounter barriers that require interdisciplinary approaches, crossing from biology to (bio‐)chemistry and physics. Essentially, higher‐scale models serve as empirical representations of the processes that occur at lower scales.

The development of regulatory approaches remains in its infancy, likely requiring inductive reasoning initially. The interplay between signal (environmental perception and signal transduction), function (primary and secondary metabolism), and retroaction (legacy and memory effects) should be understood. Such model architecture can be developed thanks to numerous in situ observations, dedicated experiments, and practical measures, such as minimizing the trade‐off between the number of variables measured, parameters calibrated, and the ease of model application and dissemination.

The effective dissemination of cold hardiness models can be facilitated through tools like Excel spreadsheets, R scripts, and online applications to a larger public. These resources enable researchers and practitioners to utilize models with greater ease, increasing their practical applicability while continuing to enhance the understanding of complex cold hardiness systems. As the field progresses, ongoing research will need to balance the intricacies of biological systems with the demand for practical and accessible modeling approaches.

Competing interests

None declared.

Author contributions

GC contributed to conceptualization, visualization, and writing of the original draft, as well as reviewing and editing the manuscript. APK, HH and BW contributed to visualization, reviewing and editing the manuscript.

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