Survey and synthesis of intra- and interspecific variation in stomatal sensitivity to vapour pressure deficit

Responses of stomatal conductance (g_s) to increasing vapour pressure deficit (D) generally follow an exponential decrease described equally well by several empirical functions. However, the magnitude of the decrease – the stomatal sensitivity – varies considerably both within and between species. Here we analysed data from a variety of sources employing both porometric and sap flux estimates of g_s to evaluate the hypothesis that stomatal sensitivity is proportional to the magnitude of g_s at low D ( ≤ 1 kPa). To test this relationship we used the function g_s = g_sref–m· lnD where m is the stomatal sensitivity and g_sref = g_s at D = 1 kPa. Regardless of species or methodology, m was highly correlated with g_sref (average r² = 0·75) with a slope of approximately 0·6. We demonstrate that this empirical slope is consistent with the theoretical slope derived from a simple hydraulic model that assumes stomatal regulation of leaf water potential. The theoretical slope is robust to deviations from underlying assumptions and variation in model parameters. The relationships within and among species are close to theoretical predictions, regardless of whether the analysis is based on porometric measurements of g_s in relation to leaf-surface D (D_s), or on sap flux-based stomatal conductance of whole trees (G_Si), or stand-level stomatal conductance (G_S) in relation to D. Thus, individuals, species, and stands with high stomatal conductance at low D show a greater sensitivity to D, as required by the role of stomata in regulating leaf water potential.     

Modelling the soil-plant-atmosphere continuum in a Quercus–Acer stand at Harvard Forest: the regulation of stomatal conductance by light, nitrogen and soil/plant hydraulic properties

Our objective is to describe a multi-layer model of C₃-canopy processes that effectively simulates hourly CO₂ and latent energy (LE) fluxes in a mixed deciduous Quercus-Acer (oak–maple) stand in central Massachusetts, USA. The key hypothesis governing the biological component of the model is that stomatal conductance (g_s) is varied so that daily carbon uptake per unit of foliar nitrogen is maximized within the limitations of canopy water availability. The hydraulic system is modelled as an analogue to simple electrical circuits in parallel, including a separate soil hydraulic resistance, plant resistance and plant capacitance for each canopy layer. Stomatal opening is initially controlled to conserve plant water stores and delay the onset of water stress. Stomatal closure at a threshold minimum leaf water potential prevents xylem cavitation and controls the maximum rate of water flux through the hydraulic system. We show a strong correlation between predicted hourly CO₂ exchange rate (r²= 0.86) and LE (r²= 0.87) with independent whole-forest measurements made by the eddy correlation method during the summer of 1992. Our theoretical derivation shows that observed relationships between CO₂ assimilation and LE flux can be explained on the basis of stomatal behaviour optimizing carbon gain, and provides an explicit link between canopy structure, soil properties, atmospheric conditions and stomatal conductance.     

A reinterpretation of stomatal responses to humidity

The stomatal conductance (g) for single leaves and the equivalent canopy conductance for stands of vegetation are often represented in models as empirical functions of saturation vapour pressure deficit or relative humidity. The mechanistic basis of this dependence is very weak. A reanalysis of 52 sets of measurements on 16 species supports the conclusion of Mott & Parkhurst (1991, Plant, Cell and Environment 14, 509–515) that stomata respond to the rate of transpiration (E) rather than to humidity per se. In general, ?g/?E is negative and constant so that the relation between g and E can be defined by two parameters: a maximum conductance g_m obtained by extrapolation to zero transpiration, and a maximum rate of transpiration E_m obtained by extrapolation to zero conductance. Both parameters are shown to be functions of temperature, CO₂ concentration, and soil water content. Exceptionally, transpiration rate and conductance may decrease together in very dry air, possibly because of patchy closure of stomata.     

Stomatal dynamics are limited by leaf hydraulics in ferns and conifers: results from simultaneous measurements of liquid and vapour fluxes in leaves

Stomatal responsiveness to vapour pressure deficit (VPD) results in continuous regulation of daytime gas‐exchange directly influencing leaf water status and carbon gain. Current models can reasonably predict steady‐state stomatal conductance (g_s) to changes in VPD but the g_s dynamics between steady‐states are poorly known. Here, we used a diverse sample of conifers and ferns to show that leaf hydraulic architecture, in particular leaf capacitance, has a major role in determining the g_s response time to perturbations in VPD. By using simultaneous measurements of liquid and vapour fluxes into and out of leaves, the in situ fluctuations in leaf water balance were calculated and appeared to be closely tracked by changes in g_s thus supporting a passive model of stomatal control. Indeed, good agreement was found between observed and predicted g_s when using a hydropassive model based on hydraulic traits. We contend that a simple passive hydraulic control of stomata in response to changes in leaf water status provides for efficient stomatal responses to VPD in ferns and conifers, leading to closure rates as fast or faster than those seen in most angiosperms.     

Combining quantitative trait loci analysis with physiological models to predict genotype‐specific transpiration rates

Transpiration is controlled by evaporative demand and stomatal conductance (g_s), and there can be substantial genetic variation in g_s. A key parameter in empirical models of transpiration is minimum stomatal conductance (g₀), a trait that can be measured and has a large effect on g_s and transpiration. In Arabidopsis thaliana, g₀ exhibits both environmental and genetic variation, and quantitative trait loci (QTL) have been mapped. We used this information to create a genetically parameterized empirical model to predict transpiration of genotypes. For the parental lines, this worked well. However, in a recombinant inbred population, the predictions proved less accurate. When based only upon their genotype at a single g₀ QTL, genotypes were less distinct than our model predicted. Follow‐up experiments indicated that both genotype by environment interaction and a polygenic inheritance complicate the application of genetic effects into physiological models. The use of ecophysiological or ‘crop’ models for predicting transpiration of novel genetic lines will benefit from incorporating further knowledge of the genetic control and degree of independence of core traits/parameters underlying g_s variation.     

Stomatal function,density and pattern,and CO2 assimilation in Arabidopsis thaliana tmm1 and sdd1‐1 mutants

• Stomata modulate the exchange of water and CO₂ between plant and atmosphere. Although stomatal density is known to affect CO₂ diffusion into the leaf and thus photosynthetic rate, the effect of stomatal density and patterning on CO₂ assimilation is not fully understood.
• We used wild types Col‐0 and C24 and stomatal mutants sdd1‐1 and tmm1 of Arabidopsis thaliana, differing in stomatal density and pattern, to study the effects of these variations on both stomatal and mesophyll conductance and CO₂ assimilation rate. Anatomical parameters of stomata, leaf temperature and carbon isotope discrimination were also assessed.
• Our results indicate that increased stomatal density enhanced stomatal conductance in sdd1‐1 plants, with no effect on photosynthesis, due to both unchanged photosynthetic capacity and decreased mesophyll conductance. Clustering (abnormal patterning formed by clusters of two or more stomata) and a highly unequal distribution of stomata between the adaxial and abaxial leaf sides in tmm1 mutants also had no effect on photosynthesis.
• Except at very high stomatal densities, stomatal conductance and water loss were proportional to stomatal density. Stomatal formation in clusters reduced stomatal dynamics and their operational range as well as the efficiency of CO₂ transport.

     

Estimating the sensitivity of stomatal conductance to photosynthesis: a review

A common approach for estimating fluxes of CO₂ and water in canopy models is to couple a model of photosynthesis (A_n) to a semi‐empirical model of stomatal conductance (g_s) such as the widely validated and utilized Ball–Berry (BB) model. This coupling provides an effective way of predicting transpiration at multiple scales. However, the designated value of the slope parameter (m) in the BB model impacts transpiration estimates. There is a lack of consensus regarding how m varies among species or plant functional types (PFTs) or in response to growth conditions. Literature values are highly variable, with inter‐species and intra‐species variations of >100%, and comparisons are made more difficult because of differences in collection techniques. This paper reviews the various methods used to estimate m and highlights how variations in measurement techniques or the data utilized can influence the resultant m. Additionally, this review summarizes the reported responses of m to [CO₂] and water stress, collates literature values by PFT and compiles nearly three decades of values into a useful compendium.     

最优气孔行为理论和气孔导度模拟

气孔调节功能是陆地生态系统碳-水耦合过程中最重要的环节。与即时的气孔导度测量相比, 气孔导度斜率能有效地反映气孔导度对CO₂浓度、饱和水汽压亏缺和光合作用的敏感性, 包含了环境因子对光合作用和临界水分利用效率等的综合影响, 为研究全球变化下陆地生态系统碳-水耦合关系提供了一个简明且综合的框架。气孔导度模型从经验模型、半经验模型发展到机理模型, 经过很多学者的改进, 但是模型参数的生物学意义和变化规律还不明确。鉴于气孔导度斜率方面研究的重要性和研究的不足, 为了加强对气孔导度调节规律的认识, 并减少气孔导度模拟的不确定性, 该文主要综述了长期以来国内外关于最优气孔行为理论和气孔导度模拟方面的研究成果, 其中包括广泛使用的气孔导度模型及参数意义, 讨论影响气孔导度斜率的主要因素以及气孔导度机理模型的应用, 并对最优气孔行为理论和气孔导度模拟方面的研究做了简单展望。     

Optimal stomatal behavior theory for simulating stomatal conductance

Among the most critical processes in simulating terrestrial ecosystem performance is the regulatory role of stomata in carbon and water cycles. Compared with field measurements, the changes in stomatal slope caused by the biophysical environment provide a simple but effective synthetic framework for studying climate-related carbon and water cycling, due to its sensitivity to CO₂, vapor pressure deficit, and photosynthesis. It is also crucial in understanding the effects of climate change on photosynthesis and water use efficiency. Endeavored by numerous scholastic efforts, stomatal conductance models have been improved based on experimental, semi-experimental, and mechanical processes. However, the underlying biological mechanisms and the dynamics of key parameters in these models remain unexplored, especially regarding the changes in stomatal slope. By improving the understanding of the stomata’s regulatory role, we reduced the uncertainty of stomatal conductance simulation. We then synthesized the recent developments and lessons in optimal stomatal behavior theory to simulate stomatal conductance and included an introduction to widely used stomatal conductance models and parameters, the main factors influencing stomatal slopes, and applications of the mechanical stomatal conductance models in different ecosystems. Based on our literature review, we proposed that future research is needed on the optimal stomatal behavior theory and its applications in simulating stomatal conductance.     

Interpretation of an empirical model for stomatal conductance in terms of guard cell function

An empirical model for stomatal conductance (g), proposed by Leuning (1995, this issue) as a modification of Ball, Woodrow & Berry's (1987) model, is interpreted in terms of a simple, steady-state model of guard cell function. In this model, stomatal aperture is a function of the relative turgor between guard cells and epidermal cells. The correlation between g and leaf surface vapour pressure deficit in Leuning's model is interpreted in terms of stomatal sensing of the transpiration rate, via changes in the gradient of total water potential between guard cells and epidermal cells. The correlation between g, CO₂ assimilation rate and leaf surface CO₂ concentration in Leuning's model is interpreted as a relationship between the corresponding osmotic gradient, irradiance, temperature, intercellular CO₂ concentration and stomatal aperture itself. The explicit relationship between osmotic gradient and stomatal aperture (possibly describing the effect of changes in guard cell volume on the membrane permeability for ion transport) results in a decrease in the transpiration rate in sufficiently dry air. Possible extension of the guard cell model to include stomatal responses to soil water status is discussed.     

植物气孔导度的机理模型

Ball-Berry气孔导度模型及其修正模型是评价植物叶片气孔调节的重要工具。该文从CO₂分子在叶片气孔中扩散这个最基本的物理过程出发, 应用物理学中的分子扩散和碰撞理论、流体力学与植物生理学等知识, 严格推导出叶片气孔导度的机理模型。利用美国Li-Cor公司生产的Li-6400光合作用测定仪控制CO₂浓度、湿度和温度, 测量了华北平原冬小麦(Triticum aestivum)的光响应数据和气孔导度数据。拟合结果表明: 推导的气孔导度机理模型较之Ball-Berry气孔导度模型和Tuzet等气孔导度模型, 能更好地描述冬小麦的气孔导度与净光合速率之间的关系。如果用气孔导度的机理模型耦合光合作用对光响应的修正模型, 则耦合模型可以很好地描述华北平原冬小麦叶片气孔导度对光强的响应曲线, 并可直接估算冬小麦的最大气孔导度和对应的饱和光强, 同时可以研究最大气孔导度是否与最大净光合速率同步的问题。拟合结果还表明: 冬小麦在30 ℃、560 μmol·mol^-1CO₂, 或在32 ℃、370 μmol·mol^-1CO₂条件下, 最大气孔导度与最大净光合速率并不同步。     

A stomatal optimization theory to describe the effects of atmospheric CO2 on leaf photosynthesis and transpiration

Background and Aims

Global climate models predict decreases in leaf stomatal conductance and transpiration due to increases in atmospheric CO₂. The consequences of these reductions are increases in soil moisture availability and continental scale run-off at decadal time-scales. Thus, a theory explaining the differential sensitivity of stomata to changing atmospheric CO₂ and other environmental conditions must be identified. Here, these responses are investigated using optimality theory applied to stomatal conductance.

Methods

An analytical model for stomatal conductance is proposed based on: (a) Fickian mass transfer of CO₂ and H₂O through stomata; (b) a biochemical photosynthesis model that relates intercellular CO₂ to net photosynthesis; and (c) a stomatal model based on optimization for maximizing carbon gains when water losses represent a cost. Comparisons between the optimization-based model and empirical relationships widely used in climate models were made using an extensive gas exchange dataset collected in a maturing pine (Pinus taeda) forest under ambient and enriched atmospheric CO₂.

Key Results and Conclusion

In this interpretation, it is proposed that an individual leaf optimally and autonomously regulates stomatal opening on short-term (approx. 10-min time-scale) rather than on daily or longer time-scales. The derived equations are analytical with explicit expressions for conductance, photosynthesis and intercellular CO₂, thereby making the approach useful for climate models. Using a gas exchange dataset collected in a pine forest, it is shown that (a) the cost of unit water loss λ (a measure of marginal water-use efficiency) increases with atmospheric CO₂; (b) the new formulation correctly predicts the condition under which CO₂-enriched atmosphere will cause increasing assimilation and decreasing stomatal conductance.     

The response of stomatal conductance to seasonal drought in tropical forests

Stomata regulate CO₂ uptake for photosynthesis and water loss through transpiration. The approaches used to represent stomatal conductance (g_s) in models vary. In particular, current understanding of drivers of the variation in a key parameter in those models, the slope parameter (i.e. a measure of intrinsic plant water‐use‐efficiency), is still limited, particularly in the tropics. Here we collected diurnal measurements of leaf gas exchange and leaf water potential (Ψ_leaf), and a suite of plant traits from the upper canopy of 15 tropical trees in two contrasting Panamanian forests throughout the dry season of the 2016 El Niño. The plant traits included wood density, leaf‐mass‐per‐area (LMA), leaf carboxylation capacity (V_c,max), leaf water content, the degree of isohydry, and predawn Ψ_leaf. We first investigated how the choice of four commonly used leaf‐level g_s models with and without the inclusion of Ψ_leaf as an additional predictor variable influence the ability to predict g_s, and then explored the abiotic (i.e. month, site‐month interaction) and biotic (i.e. tree‐species‐specific characteristics) drivers of slope parameter variation. Our results show that the inclusion of Ψ_leaf did not improve model performance and that the models that represent the response of g_s to vapor pressure deficit performed better than corresponding models that respond to relative humidity. Within each g_s model, we found large variation in the slope parameter, and this variation was attributable to the biotic driver, rather than abiotic drivers. We further investigated potential relationships between the slope parameter and the six available plant traits mentioned above, and found that only one trait, LMA, had a significant correlation with the slope parameter (R² = 0.66, n = 15), highlighting a potential path towards improved model parameterization. This study advances understanding of g_s dynamics over seasonal drought, and identifies a practical, trait‐based approach to improve modeling of carbon and water exchange in tropical forests.     

Ozone‐induced stomatal sluggishness changes stomatal parameters of Jarvis‐type model in white birch and deciduous oak

• Stomatal ozone flux is closely related to ozone injury to plants. Jarvis‐type multiplicative model has been recommended for estimating stomatal ozone flux in forest trees. Ozone can change stomatal conductance by both stomatal closure and less efficient stomatal control (stomatal sluggishness). However, current Jarvis‐type models do not account for these ozone effects on stomatal conductance in forest trees.
• We examined seasonal course of stomatal conductance in two common deciduous tree species native to northern Japan (white birch: Betula platyphylla var. japonica ; deciduous oak: Quercus mongolica var. crispula ) grown under free‐air ozone exposure. We innovatively considered stomatal sluggishness in the Jarvis‐type model using a simple parameter, s , relating to cumulative ozone uptake (defined as POD : phytotoxic ozone dose).
• We found that ozone decreased stomatal conductance of white birch leaves after full expansion (?28%). However, such a reduction of stomatal conductance by ozone fell in late summer (?10%). At the same time, ozone reduced stomatal sensitivity of white birch to VPD and increased stomatal conductance under low light conditions. In contrast, in deciduous oak, ozone did not clearly change the model parameters.
• The consideration of both ozone‐induced stomatal closure and stomatal sluggishness improved the model performance to estimate stomatal conductance and to explain the dose–response relationship on ozone‐induced decline of photosynthesis of white birch. Our results indicate that ozone effects on stomatal conductance (i.e . stomatal closure and stomatal sluggishness) are crucial for modelling studies to determine stomatal response in deciduous trees, especially in species sensitive to ozone.

     

Pre‐dawn stomatal opening does not substantially enhance early‐morning photosynthesis in Helianthus annuus

Most C₃ plant species have partially open stomata during the night especially in the 3–5 h before dawn. This pre‐dawn stomatal opening has been hypothesized to enhance early‐morning photosynthesis (A) by reducing diffusion limitations to CO₂ at dawn. We tested this hypothesis in cultivated Helianthus annuus using whole‐shoot gas exchange, leaf level gas exchange and modelling approaches. One hour pre‐dawn low‐humidity treatments were used to reduce pre‐dawn stomatal conductance (g). At the whole‐shoot level, a difference of pre‐dawn g (0.40 versus 0.17 mol m^?2 s^?1) did not significantly affect A during the first hour after dawn. Shorter term effects were investigated with leaf level gas exchange measurements and a difference of pre‐dawn g (0.10 versus 0.04 mol m^?2 s^?1) affected g and A for only 5 min after dawn. The potential effects of a wider range of stomatal apertures were explored with an empirical model of the relationship between A and intercellular CO₂ concentration during the half‐hour after dawn. Modelling results demonstrated that even extremely low pre‐dawn stomatal conductance values have only a minimal effect on early‐morning A for a few minutes after dawn. Thus, we found no evidence that pre‐dawn stomatal opening enhances A.     

A coupled model of stomatal conductance, photosynthesis and transpiration

A model that couples stomatal conductance, photosynthesis, leaf energy balance and transport of water through the soil–plant–atmosphere continuum is presented. Stomatal conductance in the model depends on light, temperature and intercellular CO₂ concentration via photosynthesis and on leaf water potential, which in turn is a function of soil water potential, the rate of water flow through the soil and plant, and on xylem hydraulic resistance. Water transport from soil to roots is simulated through solution of Richards’ equation. The model captures the observed hysteresis in diurnal variations in stomatal conductance, assimilation rate and transpiration for plant canopies. Hysteresis arises because atmospheric demand for water from the leaves typically peaks in mid‐afternoon and because of uneven distribution of soil matric potentials with distance from the roots. Potentials at the root surfaces are lower than in the bulk soil, and once soil water supply starts to limit transpiration, root potentials are substantially less negative in the morning than in the afternoon. This leads to higher stomatal conductances, CO₂ assimilation and transpiration in the morning compared to later in the day. Stomatal conductance is sensitive to soil and plant hydraulic properties and to root length density only after approximately 10 d of soil drying, when supply of water by the soil to the roots becomes limiting. High atmospheric demand causes transpiration rates, LE, to decline at a slightly higher soil water content, θ_s, than at low atmospheric demand, but all curves of LE versus θ_s fall on the same line when soil water supply limits transpiration. Stomatal conductance cannot be modelled in isolation, but must be fully coupled with models of photosynthesis/respiration and the transport of water from soil, through roots, stems and leaves to the atmosphere.     

The effect of tree height on crown level stomatal conductance

Variation in stomatal conductance is typically explained in relation to environmental conditions. However, tree height may also contribute to the variability in mean stomatal conductance. Mean canopy stomatal conductance of individual tree crowns (G_Si) was estimated using sap flux measurements in Fagus sylvatica L., and the hypothesis that G_Si decreases with tree height was tested. Over 13 d of the growing season during which soil moisture was not limiting, G_Si decreased linearly with the natural logarithm of vapour pressure deficit (D), and increased exponentially to saturation with photosynthetic photon flux density (Q_o). Under conditions of D = 1 kPa and saturating Q_o, G_Si decreased by approximately 60% with 30 m increase in tree height. Over the same range in height, sapwood‐to‐leaf area ratio (A_S:A_L) doubled. A simple hydraulic model explained the variation in G_Si based on an inverse relationship with height, and a linear relationship with A_S:A_L. Thus, in F. sylvatica, adjustments in A_S:A_L partially compensate for the negative effect of increased flow‐path length on leaf conductance. Furthermore, because stomata with low conductance are less sensitive to D, gas exchange of tall trees is reduced less by high D. Despite these compensations, decreasing hydraulic conductance with tree height in F. sylvatica reduces carbon uptake through a corresponding decrease in stomatal conductance.     

Optimal stomatal conductance in relation to photosynthesis in climatically contrasting Eucalyptus species under drought

Models of stomatal conductance (g_s) are based on coupling between g_s and CO₂ assimilation (A_net), and it is often assumed that the slope of this relationship (‘g₁’) is constant across species. However, if different plant species have adapted to different access costs of water, then there will be differences in g₁ among species. We hypothesized that g₁ should vary among species adapted to different climates, and tested the theory and its linkage to plant hydraulics using four Eucalyptus species from different climatic origins in a common garden. Optimal stomatal theory predicts that species from sub‐humid zones have a lower marginal water cost of C gain, hence lower g₁ than humid‐zone species. In agreement with the theory that g₁ is related to tissue carbon costs for water supply, we found a relationship between wood density and g₁ across Eucalyptus species of contrasting climatic origins. There were significant reductions in the parameter g₁ during drought in humid but not sub‐humid species, with the latter group maintaining g₁ in drought. There are strong differences in stomatal behaviour among related tree species in agreement with optimal stomatal theory, and these differences are consistent with the economics involved in water uptake and transport for carbon gain.     

气孔导度对CO_2浓度变化的模拟及其生理机制

基于气孔运动的生理生化机制重点进行了气孔导度(gs)对CO2浓度变化的响应机制分析,并推导得到气孔导度(gs)对CO2浓度变化响应模型,并以9种植物进行了模型验证。结果表明:随着CO2浓度的升高,气孔导度会逐渐降低,且下降的幅度会随着CO2浓度的升高而逐渐减弱。气孔导度对CO2浓度(Cs)变化的响应模型可以表达为gs=gmax/(1+Cs/Cs0),其中式中gmax是最大气孔导度和Cs0是实验常数。该模型较好地模拟了气孔导度随CO2浓度变化的规律,模型参数具有明确的生理意义,与Jarvis模型和Ball-Berry模型相比,该模型如何实现多种环境因子的耦合有待进一步突破。另外,模型是在短期改变叶片CO2浓度的条件下得出的,在CO2浓度长期胁迫下的适用性也有待进一步确认。     

Responses of stomatal conductance to light, humidity and temperature in winter wheat and barley grown at three concentrations of carbon dioxide in the field

Responses of leaf stomatal conductance to light, humidity and temperature were characterized for winter wheat and barely grown at ambient (about 350 μmol mol^?1 in the daytime), ambient + 175 and ambient + 350 μmol mol^?1 concentrations of carbon dioxide in open‐topped chambers in field plots over a three year period. Stomatal responses to environment were determined by direct manipulation of single environmental factors, and those results were compared with responses derived from natural day to day variation in mid‐day stomatal conductance. The purpose of these experiments was to determine the magnitude of reduction in stomatal conductance at elevated [CO₂], and to assess whether the relative response of conductance to elevated [CO₂] was constant across light, humidity and temperature conditions. The results indicated that light, humidity and temperature all significantly affected the relative decrease in stomatal conductance at elevated [CO₂]. The relative decrease in conductance with elevated [CO₂] was greater at low light, low water vapour pressure difference, and high temperature in both species. For measurements made at saturating light near mid‐day, the ratio of mid‐day stomatal conductances at doubled [CO₂] to that at ambient [CO₂] ranged from 0.42 to 0.86, with a mean of 0.66 in barley, and from 0.33 to 0.80, with a mean of 0.56 in wheat. Day‐to‐day variation in the relative effect of elevated [CO₂] on conductance was correlated with the relative stimulation of [CO₂] assimilation rate and with temperature. Some limitations of multiple linear regression, multiplicative, and ‘Ball–Berry' models as summaries of the data are discussed. In barley, a better fit to the models occurred in individual years than for the combined data, and in wheat a better fit to the models occurred when data from near the end of the season were removed.