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Ecophysiologists have long been fascinated by the photosynthetic behaviour of alpine plants, which often have to withstand extreme environmental pressures (Gale, 1972; Friend & Woodward, 1990; Körner, 2003, 2007; Shi et al., 2006). About 8% of the world's land surface is above 1500 m altitude (Körner, 2007). High altitudes can be climatically unusual, often with (for example) low temperatures, strong winds, and now high rates of warming (Körner, 2003; Pepin & Lundquist, 2008; Rangwala & Miller, 2012). Moreover, the low atmospheric pressure provides a set of environmental conditions unique on Earth (Table 1). There has been extensive speculation about altitudinal effects on photosynthesis and, in particular, how to account for the puzzling – but consistently observed – tendencies towards higher carbon dioxide (CO2) drawdown (low ratio of leaf-internal to ambient CO2 partial pressures (ci : ca; hereafter, χ), resulting in low carbon isotope discrimination) and higher carboxylation capacity (Vcmax) with increasing altitude (Gale, 1972; Körner & Diemer, 1987; Friend et al., 1989; Terashima et al., 1995; Bresson et al., 2009; Zhu et al., 2010). At first glance, it might be expected that CO2 assimilation rates would be reduced at high altitudes due to the low partial pressure of CO2 (Friend & Woodward, 1990). However, actual measured photosynthetic rates are usually as high as, or even higher than, those at low altitudes (Mächler & Nösberger, 1977; Körner & Diemer, 1987; Cordell et al., 1999; Shi et al., 2006). One group of hypotheses that attempt to explain the effects of altitude on photosynthetic physiology focuses on the effects of low temperature. It has been argued that alpine plants possess thick leaves as an adaptation to low temperatures, and thus higher leaf nitrogen (N) on an area basis (Narea). Higher Narea is taken to imply higher Vcmax, in turn leading to higher CO2 drawdown due to higher photosynthetic rates (Woodward, 1979; Körner & Diemer, 1987; Friend et al., 1989; Sparks & Ehleringer, 1997). This reasoning assumes that higher Narea in thicker leaves would be associated with higher Vcmax, but this is not necessarily so, as a substantial fraction of leaf N in thick leaves (with low specific leaf area) is located in cell walls rather than in chloroplasts (Onoda et al., 2004). An alternative argument, from the perspective of carbon isotope discrimination, suggests that increased leaf thickness could lengthen the diffusional pathway for CO2 from the atmosphere to the site of carboxylation, and therefore potentially decrease χ (Vitousek et al., 1990). However, low air pressure would be expected to counteract this effect, by allowing CO2 to diffuse more readily through the stomata (Table 1). In any case, no hypothesis based on temperature effects can account for the difference in plant responses to altitudinal and latitudinal gradients, i.e. why the same adaptations in photosynthetic capacity observed on high mountains are not observed in polar regions where growing-season temperatures are also low (Billings et al., 1961; Mooney & Billings, 1961; Billings & Mooney, 1968; Chabot et al., 1972; Zhu et al., 2010). It is moreover worth noting that although low temperatures can depress photosynthesis, measured growing-season leaf temperatures and optimal temperatures for photosynthesis in both alpine and arctic plants are typically only reduced by a few degrees, in contrast with a much larger decline in air temperature with altitude or latitude (Körner & Diemer, 1987; Körner, 2007). The dense canopy structure and crowded leaf arrangement on stems of cushion and prostrate alpine plants create a low boundary-layer conductance and thus allow the maintenance of large differences between the temperatures of leaves and air (Gauslaa, 1984; Körner, 2003; Michaletz et al., 2015). The effect of such morphological adaptations is superimposed on the universal tendency, rooted in the fundamentals of leaf energy balance, for leaf temperatures to be maintained in a narrower range than air temperatures (Campbell & Norman, 1998; Michaletz et al., 2015). A further group of hypotheses suggests that low atmospheric pressure might influence photosynthesis through more direct physiological influences, independently of temperature (Decker, 1959; Billings et al., 1961; Mooney & Billings, 1961). However, despite much previous speculation, and the fact that many biophysical quantities relevant to gas exchange are known to change with air pressure and leaf temperature in a predictable manner (Table 1), effects of those biophysical quantities on plant physiology have not been fully explored. Misconceptions abound in the literature. For example, alpine plants were predicted to be more sensitive to the decreased CO2 concentration (molar mixing ratio) in the Quaternary glacial periods simply because the CO2 partial pressure at high altitudes is low (Street-Perrott et al., 1997). This is incorrect, however, because the partial pressure of O2 is also reduced at high altitudes – implying a reduced photorespiratory burden which counteracts the effect of CO2 concentration on photosynthesis, as previously noted, for example, by Körner et al. (1991) and Terashima et al. (1995). Natural selection implies that plants optimize ecophysiological traits by regulating the allocation of resources to different functions. This principle leads to the least-cost hypothesis and the coordination hypothesis for the optimal photosynthetic behaviour of C3 plants. These hypotheses can be incorporated into the standard (Farquhar et al., 1980) model, thereby potentially generating a unifying explanation and prediction of photosynthetic trait responses to environmental factors (Wang et al., 2016). The composite parameter ξ represents the sensitivity of χ to D and is influenced by both the cost terms. The mathematical form of Eqn 1 is the same as that proposed by Medlyn et al. (2011), which is based on a widely-cited stomatal optimality hypothesis stating that plants minimize E − λA (Cowan & Farquhar, 1977). However, the marginal cost of transpiration (λ) in that expression is not clearly defined. In the least-cost hypothesis, by contrast, the parameter ξ can be expressed explicitly as a function of the cost factors and the effective Michaelis–Menten coefficient of Rubisco (K), which is related to the partial pressure of O2 (O) and the Michaelis–Menten coefficients of Rubisco for CO2 and O2 (KC and KO). The temperature dependencies of KC and KO follow an Arrhenius relationship as described by Bernacchi et al. (2001) and this also gives rise to a temperature dependency of ξ (Prentice et al., 2014). The leaf-to-air VPD (D) is the difference between the vapour pressure in the intercellular spaces and the vapour pressure in the free air beyond the leaf boundary layer. The intercellular vapour pressure is usually assumed to be saturated and is determined by the leaf temperature. The free-air vapour pressure is the actual vapour pressure, which depends on the molar mixing ratio of water vapour in the air and on the atmospheric pressure. The conductance for gas exchange between leaves and air that corresponds to this vapour pressure gradient is composed of stomatal conductance and boundary-layer conductance in series. The boundary-layer conductance is generally many times larger than the stomatal conductance, so differences among leaves in boundary-layer conductance can be compensated by changes in stomatal conductance, allowing the maintenance of optimal χ. By introducing the known altitudinal responses of various key biophysical quantities, we show here how the partial responses of χ, Vcmax and A to atmospheric pressure and leaf temperature along the altitude gradient can be predicted from the equations mentioned earlier. Moreover, these predictions appear to be consistent with the field observed altitudinal trends in χ, Vcmax and A (Table 2; Fig. 1). To separate the effects of pressure and temperature, and also to cover a realistic leaf temperature variation along altitude gradients, we start by listing 10 potential influences of atmospheric pressure on plant physiological processes under constant leaf temperature, which may be a reasonable approximation for herbaceous plants as discussed earlier. Then we impose the additional effects of temperature, assuming that leaf temperature declines with altitude but follows a lapse rate shallower than air temperature due to the general homoeostatic tendency of leaf temperatures (Campbell & Norman, 1998; Michaletz et al., 2015). This approach may be realistic for tree species, whose leaves are situated well above the ground and subject to a potentially high wind speed, and therefore cannot be expected to maintain leaf temperatures near-constant with altitude (Table 1). Among the listed biophysical quantities, VPD and K are key variables predicting an altitudinal response of χ (Eqns 1, 2), whereas Γ*, ca, and PPFD impose further effects on Vcmax and A (Eqns 4, 5). With a constant leaf temperature, the pressure-induced decrease in K and enhancement of D both lead to a lower χ for alpine plants (Eqns 1, 2). As indicated in Table 1, K declines with altitude, due to the reduced partial pressure of O2, thereby increasing the affinity of Rubisco for CO2 and reducing the carboxylation capacity required per mole of carbon fixed (Bresson et al., 2009). However, for a given molar mixing ratio of water vapour to air, lowered atmospheric pressure leads to reduced actual vapour pressure. As the saturated leaf-internal vapour pressure is invariant with atmospheric pressure, this reduction tends to increase leaf-to-air VPD, thereby increasing the water transport required per mole of carbon fixed. According to the least-cost hypothesis, both effects support a shift in the investment of resources towards increased Rubisco capacity and against water transport capacity (Wang et al., 2016). The predicted outcome of a lowered χ with atmospheric pressure can be shown mathematically by differentiation of the expression for optimal χ, which shows that the partial response of χ to decreasing atmospheric pressure is always negative (Supporting Information Notes S1). Our predicted response of χ to pressure is consistent with observations by Körner & Diemer (1987) where leaf temperature was shown to vary only a few degrees (Table 2). After superimposing temperature effects, declining leaf temperature reduces the saturated vapour pressure, and thus decreases the leaf-to-air VPD – leading to a lower cost of water transport, opposite to the effect of air pressure. However, the declining leaf temperature still reduces K and this has the stronger influence, favouring a decline in χ (Table 1). By separating altitudinal and latitudinal trends, Körner et al. (1991) showed that aside from the effect of pressure, lower temperature reduces χ, potentially reinforcing the decline of χ with altitude. The leaf-internal partial pressure of CO2, ci, is the product of χ and ca. Although ci declines with altitude due to declines in both ca and χ, this does not automatically imply an increased limitation of CO2 on photosynthesis. This is because CO2 limitation is also determined by the CO2 compensation point (Γ*), as shown by Eqn 4. If a constant leaf temperature is assumed, Γ* is proportional to the O2 partial pressure and thus changes in proportion to ca (Farquhar et al., 1980) (Table 1). Consequently, a stronger CO2 limitation due to the reduction in χ (not due to ca or Γ*) is expected for alpine plants. After imposing a temperature effect, the decline in Γ*, following an Arrhenius relationship (Bernacchi et al., 2001), is much faster than that of χ (Table 1) and this leads to a weaker CO2 limitation on photosynthesis. It has been suggested that photosynthesis might be influenced by the more rapid diffusion of gases in air at lower pressure (Table 1) (Gale, 1972; Smith & Donahue, 1991; Terashima et al., 1995). We might therefore predict that the consequence of more rapid gaseous diffusion at high altitudes would be a reduction in stomatal density and/or diameter. In reality, both positive (Wagner, 1892; Bonnier, 1895; Paridari et al., 2013) and negative (Körner et al., 1983) responses of stomatal density to altitude increase have been reported, suggesting that some other environmental factors or morphological adaptations might also be involved in determining stomatal density (Körner et al., 1986; Friend & Woodward, 1990). According to the least-cost hypothesis, a relatively lower cost of maintaining carboxylation due to increased affinity to CO2 (lower K) in turn implies an increased Vcmax, as required (by the coordination hypothesis) to achieve an optimal assimilation rate that is set by PPFD. Mathematically, the sensitivity of Vcmax to air pressure based on Eqn 2 (Notes S1) shows that the response is always positive provided K ≫ Γ*. In this response, either enhanced PPFD on clear days or reduced ci is a secondary contributor to the positive response of Vcmax to altitude, whereas the decline in K is the main contributor – being about three times larger than the other contributions. Reduced leaf temperature superimposes a negative effect on Vcmax, which is opposite to the positive effect of pressure decline. This can also be theoretically predicted by the ‘kinetic’ response of biochemical rate parameters (Kc, Ko and Γ*) to temperature and is supported by field observations (Dong et al., 2016). Our predictions are supported by previous observations (Table 2; Fig. 1). Quantitative comparison with Körner & Diemer (1987) is possible because this study reported all of the relevant environmental variables (in addition to altitude) that would be expected theoretically to influence χ and Vcmax (Table 2). Our literature search revealed a number of other studies of altitude effects (Fig. 1) but it was not generally possible to exclude other effects, for example, of changes in leaf temperature or cloudiness (it is worth noting that Körner & Diemer (1987) reported negligible changes in leaf temperature). Therefore, the observed changes in Vcmax are variable (Shi et al., 2006; Fan et al., 2011), but nonetheless consistent with our predicted range (Fig. 1). The coordination hypothesis also allows prediction of the sensitivity of the assimilation rate A to air pressure through Eqn 4 (Notes S1). Table 1 shows how much each pressure-dependent quantity contributes to changes in A under defined reference conditions. The increased diffusion coefficients for water vapour and CO2 may physically affect how stomatal regulation achieves the optimal χ, but should not influence its value, nor the value of A. As discussed earlier, the opposite effects of the declining O2 and CO2 partial pressures approximately cancel each other. Therefore, the sensitivity of A to altitude depends on the competition between the negative effect of reduced χ and the positive effects of enhanced PPFD, due to a shorter path length (enhanced clear-sky transmittivity), and reduced Γ* if leaf temperature declines. Therefore, either a negative or a positive response of A can be expected, depending on the conditions. Referring again to the study by Körner & Diemer (1987), as altitude increases from 600 m to 2600 m, PPFD is predicted to increase by 5.4%, as observed (Table 2). Our predicted change in A is only 0.4%, and Körner & Diemer (1987) reported no significant change (Table 2). Bresson et al. (2009) also found no significant change in A with altitude, while measurements made at constant (low-elevation) CO2 partial pressure showed a consistent increase; this is in line with our prediction of increasing Vcmax with altitude. Bresson et al. (2009) also found increasing Narea with altitude, which is to be expected, given increasing Vcmax. In principle, photosynthesis could be enhanced at high altitudes, if the benefit from increased radiation and reduced photorespiration were to overcome the effect of the reduction of ci. However, reduced photorespiration relies on a reduction in leaf temperature, whereas radiation is also influenced by cloud cover, which in reality can decrease or increase with altitude, depending on latitude and continentality (Barry, 1992). Thus a diversity of trends might be found in a wider sampling of altitudinal gradients in different plant types and climatic regions. Nevertheless, the theoretical analysis presented here provides a first-order explanation for some commonly observed trends in photosynthetic traits along altitudinal gradients. The explanation is derived from a proposed general model to predict photosynthetic rates via eco-evolutionary optimization of photosynthetic traits (Wang et al., 2016). By disentangling the effects of pressure and temperature on a number of variables influencing leaf-level gas exchange, we show that both declining χ and increasing Vcmax can be predicted by air pressure change alone, while superimposed temperature effects typically modify the magnitude of the responses – accounting for why these trends in χ and Vcmax have been so widely observed. The authors thank Vincent Maire for discussions. This research was supported by the National Natural Science Foundation of China (Grant no. 31600388) to H.W. and by an Australian Research Council Discovery grant (DP120103600) to I.C.P. and I.J.W. It represents a contribution to the AXA Chair Programme in Biosphere and Climate Impacts and the Imperial College initiative on Grand Challenges in Ecosystems and the Environment. T.F.K. was supported in part by the Laboratory Directed Research and Development Programme of Lawrence Berkeley National Laboratory under US Department of Energy (Contract no. DE-AC02-05CH11231), and by a Macquarie University research fellowship. H.W. and I.C.P. derived the predictions. H.W. carried out all the analyses, constructed the figures and tables, and wrote the first draft. H.W., I.C.P., T.F.K., I.J.W., T.W.D. and C.P contributed to subsequent drafts. H.W., T.W.D. and I.C.P. summarized altitudinal dependences of gas exchange and various relevant biophysical quantities. I.C.P. and T.F.K contributed to the data analysis. I.J.W. first proposed the least-cost theory, and I.C.P further developed the theory. 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