Temperature dependence of two parameters in a photosynthesis model


  • R. Leuning

    Corresponding author
    1. CSIRO Land and Water, FC Pye Laboratory, PO Box 1666, Canberra, ACT, 2601, Australia
      R. Leuning. E-mail: ray.leuning@csiro.au
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R. Leuning. E-mail: ray.leuning@csiro.au


The temperature dependence of the photosynthetic parameters Vcmax, the maximum catalytic rate of the enzyme Rubisco, and Jmax, the maximum electron transport rate, were examined using published datasets. An Arrehenius equation, modified to account for decreases in each parameter at high temperatures, satisfactorily described the temperature response for both parameters. There was remarkable conformity in Vcmax and Jmax between all plants at Tleaf < 25 °C, when each parameter was normalized by their respective values at 25 °C (Vcmax0 and Jmax0), but showed a high degree of variability between and within species at Tleaf > 30 °C. For both normalized Vcmax and Jmax, the maximum fractional error introduced by assuming a common temperature response function is < ± 0·1 for most plants and < ± 0·22 for all plants when Tleaf < 25 °C. Fractional errors are typically < ± 0·45 in the temperature range 25–30 °C, but very large errors occur when a common function is used to estimate the photosynthetic parameters at temperatures > 30 °C. The ratio Jmax/Vcmax varies with temperature, but analysis of the ratio at Tleaf = 25 °C using the fitted mean temperature response functions results in Jmax0/Vcmax0 = 2·00 ± 0·60 (SD, n = 43).


leaf temperature (°C)


leaf temperature (K)


reference temperature (298·2 K)


Vcmax0, the maximum catalytic rate of the enzyme Rubisco at Tl and T0


Jmax0, the maximum electron transport rate at Tl and T0


activation energy


deactivation energy


entropy term


ideal gas constant.


The model of C3 photosynthesis first described by Farquhar, von Caemmerer & Berry (1980; referred to as FCB model hereafter) has been adopted almost universally in models describing CO2 exchange at multiple scales ranging from individual leaves (e.g. Harley et al. 1992), plant canopies (e.g. Leuning et al. 1995; Harley & Baldocchi 1995; De Pury & Farquhar 1997; Wang & Leuning 1998), through to landscapes in Global Climate Models (Sellers et al. 1996).

Many authors have shown that the FCB accurately describes leaf-level photosynthesis, provided values of the eight parameters of the model are known. There is considerable variation between and within species in two key parameters, Vcmax and Jmax, although some of this variation may be ascribed to differences in leaf nitrogen concentration (Field 1983; Leuning, Cromer & Rance 1991; Harley et al.; 1992; Kellomäki & Wang 1997). Another problem in utilizing the FCB model at the leaf scale is that Vcmax and Jmax are temperature dependent, and the dependence varies significantly between and within species.

Variation in the temperature dependence of the parameters in the FCB model also affects our ability to model photosynthesis at the canopy scale. Wang & Leuning (1998) incorporated the FCB model into a model of canopy photosynthesis and energy partitioning, and Leuning, Dunin & Wang (1998) found that the model accurately simulated the exchanges of heat, water vapour and CO2 for wheat fields. In a subsequent sensitivity analysis, Wang et al. (2001) showed that a maximum of only four parameters in the canopy model (including Vcmax) could be estimated independently from micrometeorological measurements of heat, water vapour and CO2 fluxes over crops and pastures. Their analysis used previously published temperature functions and parameter values to describe the temperature dependence of Vcmax and the other parameters in the FCB model. Without these assumptions, the problem of estimating parameters for the non-linear canopy model would have been extremely difficult. Similar problems will confront others wishing to utilize the highly parameterized FCB model at all spatial scales.

We may thus ask whether it is possible to utilize the biochemical FCB model in models of canopy photosynthesis when there are significant differences in the temperature dependence of Vcmax and Jmax between and within plant species? To answer this question, the present note examines published data for the temperature dependence of Vcmax and Jmax to determine whether a single temperature response function, with fixed parameter values for each of Vcmax and Jmax, may be used for modelling purposes, and if so, what degree of error does this introduce into these parameters?

Temperature dependence of vcmax and jmax

Following Farquhar et al. (1980) and Leuning (1997), the temperature dependence of Vcmax is described using a modified Arrhenius equation which allows for the decrease in Vcmax above an optimum temperature, namely:.


where Vcmax0 is the value of Vcmax at the reference temperature (T0 = 298·2 K), C = 1 + exp[(SvT0 − Hd)/(RT0)], Ha is the energy of activation, Hd is the energy of deactivation and Sv is an entropy term. A similar equation is used to describe the temperature dependence of Jmax. The notation Vcmax* and Jmax* is used for Vcmax and Jmax normalized to their respective values at T0.

Data sources

Date sources and parameter values for Eqn 1 that are used in this analysis are presented in Table 1. In most cases, the parameters reported by the authors were used directly, but for Dreyer et al. 2001) and Bunce (2000), data were extracted from their graphs of Vcmax and/or Jmax versus temperature, and parameters in Eqn 1 were estimated using a non-linear parameter estimation algorithm. This procedure ensured a common analysis framework.

Table 1.  Parameter values and data sources used in analysis. Hav, HdvSvv are the activation and deactivation energies and an entropy term for Vcmax, whereas Haj, HdjSvj are the corresponding terms for Jmax
(J mol−1)
(J mol−1)
(J mol−1
(J mol−1)
(J mol−1)
(J mol−1 
  • 1

    Numbers refer to 33 and 100% nitrogen treatments.

  • 2

    Refer to Wohlfahrt et al. (1999) for species identification.

 608004010001285298·2 48500193000 622Acer pseudoplatanusDreyer et al. (2001)
 697004040001285298·2 50400154000 494Betula pendula 
 75400175000 559298·2 65300129000 420Fagus sylvatica 
 88000 90000 293298·2 86900139000 461Fraxinus excelsior 
 89000265000 851298·2 521007140002280Juglans regia 
 67600144000 451298·2 46100280000 888Quercus petraea 
 61100247000 778298·2 55900 88300 284Quercus robur 
 52672206083 650298·2 46337198924 650Pinus radiata (33)Walcroft et al. (1997) 1
 45027203594 650298·2 45977199345 650Pinus radiata (100) 
 52750202600 669298·2 61750185600 621Pinus sylvestrisWang et al. (1996)
116300202900 650298·2 79500201000 650CottonHarley et al. (1992)
 58520   0  0298·2 37000220000 710SpinachFarquhar et al. (1980)
 55389204839 650298·2 48270203549 650Pinus sylvestrisLeuning (unpublished)
 65330   0  0298·2 55000   0  0TobaccoBernacchi et al. (2001)
 88434199390 656293·2 65649195860 643daglWohlfahrt et al. (1999) 2
149061198000 656293·2 78754200000 643kopy(Monte Bodone)
 61304202538 656293·2 44386196168 643nast 
 64490200000 656293·2 51014197551 643plat 
 79294200415 656293·2 56292198050 643plme 
 60940199571 656293·2 61521192521 643povi 
 57098204000 656293·2 57101199247 643poau 
 60600202000 656293·2 37407201000 643rhal 
118599197500 656293·2115191191300 643trmo 
 53017202000 656293·2 68707199000 643trpr 
 54051201186 656293·2 67161194000 643trfl 
 68000201000 656293·2 55465199521 643treu 
102568201194 656293·2 57329198922 643vamy 
109740199500 656293·2 41052199400 643acmiWohlfahrt et al. (1999)
105945197250 656293·2 56324198582 643alvu(Passeier valley)
 54726201500 656293·2 55973197000 643armo 
102570200000 656293·2 57330197100 643cavu 
130500194800 656293·2112049190676 643feru 
 61304202583 656293·2 44386196168 643nast 
 98400196900 656293·2 66000194000 643raac 
 83300200000 656293·2 89742196268 643rual 
103100198800 656293·2 54452195872 643tral 
 56796202000 656293·2 44281196000 643trre 
 59316202000 656293·2 55034197263 643aluvWohlfahrt et al. (1999)
 56700199500 656293·2 47790195900 643brme(Stubai valley)
 56935199500 656293·2 49995195337 643gesy 
 51300200000 656293·2 43594196200 643plme 
106500199666 656293·2 52739195575 643seva 
 57154200645 656293·2 52576196000 643vaul 
 571533279751064·6298·2   Hordeum vulgareBunce (2000)
 65472188903 613·7298·2   Brassica rapa(Tgrowth= 15 °C)
 83516205910 674·4298·2   Vicia faba 
 97029122165 405·7298·2   Chenopodium album 
 60264394103 1273298·2   Helianthus annuus 
 80652172070 560·1298·2   Lycopersicon aesculentum 
224470225431 772·6298·2   Glycine max 
 58114288788 931·7298·2   Abutilon theophrasi 
 62615276990 890298·2   Hordeum vulgareBunce (2000)
 635904155511341298·2   Brassica rapa(Tgrowth= 25 °C)
 64325228180 737298·2   Vicia faba 
 70945213601 683·8298·2   Chenopodium album 
 78830148290 483·7298·2   Helianthus annuus 
 87103126990 415·7298·2   Lycopersicon aesculentum 
 64959123386 400298·2   Glycine max 
 85361122247 400298·2   Abutilon theophrasi 

Figure 1 shows the temperature dependence of Vcmax* and Jmax*. The mean curve was constructed by calculating Eqn 1 at 5 °C intervals using the parameter values for each entry in Table 1, and then fitting Eqn 1 to the resultant means at each Tleaf using a non-linear parameter estimation procedure. Also shown are the extreme curves from the dataset: cotton with a temperature optimum near 40 °C for Vcmax* and 35 °C for Jmax* (Harley et al. 1992) and Scots pine, with respective temperature optima of 25 °C for Vcmax* and 23 °C for Jmax* (Wang, Kellomäki & Laitinen 1996). At temperatures less than 30 °C, the standard deviation of the means are well within the two extreme curves, and from these results it appears acceptable to use a single temperature response curve for all species up to 30 °C. This is consistent with the view that Rubisco kinetics are conserved among higher plants, and consistent with the curve of Vcmax* versus temperature obtained through careful measurements on transgenic tobacco plans by Bernacchi et al. (2001; see Fig. 1a). This curve is identical to the mean of the published datasets for Tleaf < 25 °C. Parameter values for the mean curves for Vcmax* and Jmax* in Fig. 1 are presented in Table 2.

Figure 1.

Temperature response curves for Vcmax* and Jmax* derived from data sources given in Table 1. (a) Vcmax*, mean, curve for 59 entries, the upper extreme represented by cotton (Harley et al. 1992), the lower extreme represented by Scots pine (Wang et al. 1996); and the response for transgenic tobacco observed by Bernacchi et al. (2001). (b) Jmax*, mean curve from 43 entries, the upper extreme represented by cotton, the lower extreme by Scots pine.

Table 2.  Parameter values for Eqn 1 for the mean curves Vcmax *and Jmax *shown in Fig. 1
Ha 73637 50300J mol−1
Hd149252152044J mol−1
Sv  486  495J mol−1 K−1

Figure 1 shows there is significant variation in the temperature response of both Vcmax* and Jmax* between plants at leaf temperatures above 30 °C, but that the variation is small at lower temperatures. To minimize the number of parameters to be estimated from data, it is desirable to use a single temperature response curve for each of Vcmax* and Jmax* when modelling the behaviour of leaves, whole canopies or at larger scales. What is the likely error in adopting this approach? There is no simple answer to this question, as it may be possible to obtain a reasonably good fit to data using a fixed temperature response function, but allowing other parameters to vary in a non-linear model. For example, it would be possible to use the mean curve for Vcmax* in Fig. 1a to describe data from cotton obtained at temperatures over 30 °C, provided a very high value of Vcmax0 were used. However, this would lead to unrealistically high values of Vcmax at temperatures below 25 °C.

The degree of mismatch in Vcmax, at any given temperature, between the mean model and the actual curve describing the data depends on the point at which the data and the model are matched. In the case of the temperature response function Eqn 1, the value of Vcmax0 obtained by using the mean model, Vcmax02, is related to the correct value, Vcmax01, and the matching temperature Tm according to


where the subscripts 1 and 2 refer to the correct and mean model parameters, respectively, and Tm, T0 are in K.

Figure 2 shows the maximum errors for Vcmax* and Jmax* at the leaf scale arising from the data shown in Fig. 1. The errors are calculated as

Figure 2.

Fractional errors in (a) Vcmax*and (b) Jmax*, obtained when the envelope curves for cotton and Scots pine are approximated by the mean curves in Fig. 1, with Tm = 25 °C in Eqn 2.


where m(T) and o(T) are the values obtained from the model (mean curve) and the observations at temperature T, and o(Tm) is the observation (and model) value at the matching temperature. When Tm = 25 °C, ?ɛ? < 0·22 for both Vcmax* and Jmax* for the outer envelope of the data as represented by cotton and Scots pine (Table 3). The errors increase rapidly at T > 30 °C as Vcmax and Jmax continue to increase for some leaves, whereas they decrease for others. These values are the extremes, and for most plants in the datasets examined, the absolute fractional errors are < 0·10 at leaf temperatures below 30 °C when Tm = 25 °C. This level of error may be acceptable for modelling purposes when leaf temperatures are below 30 °C, but a single temperature response curve will not be suitable for leaf temperatures over 30 °C.

Table 3.  Estimated maximum errors in Vcmax *and Jmax *for temperatures < 25 °C and < 30 °C, when the matching temperature Tm = 25 °C
Scots pineCottonScots pineCotton
< 25 °C> −0·20< 0·21> −0·20< 0·22
< 30 °C< 0·43> −0·72< 0·41> −0·36

Wullschleger (1993) obtained a strong relationship between Vcmax and Jmax from results for 109 species reported in the literature. As noted by Leuning (1997), this correlation is convenient because it reduces by one the number of parameters that need to be specified when modelling photosynthesis. Leuning (1997) rescaled Wullschleger's data to a common temperature using the response functions for Jmax and Vcmax for cotton obtained by Harley et al. (1992), and obtained a mean Jmax/Vcmax = 2·68 at 20 °C. This corresponds to Jmax/Vcmax = 2·04 at the reference temperature of 25 °C used in the present paper. To check this relationship, values of Jmax0 and Vcmax0 were calculated at Tleaf = 25 °C using Eqn 1 with the parameter values from Table 1 when information for both Jmax and Vcmax was available. Figure 3a shows that the slope of the linear regression of Jmax0 versus Vcmax0 is 1·92 and R2 = 0·55. The mean ratio Jmax/Vcmax and the standard deviation of the mean for the 43 datasets are plotted as a function of temperature in Fig. 3b. There is considerable scatter in the results, as indicated by the standard deviations, but the mean Jmax/Vcmax = 2·00 ± 0·60 (SD) at 25 °C. Both this, and the previous estimate, are close to the value of 2·04 obtained from the re-analysis of Wullschleger's (1993) data by Leuning (1997).

Figure 3.

(a) Relationship between Jmax0 and Vcmax0 at leaf temperature of 25 °C derived from 43 data sets (y = 1·92x, R2 = 0·55); (b) mean and standard deviations of the ratio Jmax/Vcmax as a function of leaf temperature from the 43 entries analysed (y = 4·384 − 0·1378x + 0·0017x2, R2 = 0·9982).


Functions describing the temperature response of the photosynthetic parameters Vcmax and Jmax, normalized to their values at 25 °C (Vcmax* and Jmax*), shows remarkably little variation between different species at Tleaf < 30 °C. Above this temperature, variation is large and depends on species and on temperatures experienced by the plants during growth.

A modified Arrhenius function (Eqn 1), with a single set of parameters for each of Vcmax* and Jmax* (Table 2), may be used to describe the temperature dependence of Vcmax* and Jmax* with absolute fractional errors of < 0·22 at Tleaf < 25 °C for all plants examined. These errors are the extremes, and for most plants in the dataset examined, the absolute fractional errors are < 0·10 at leaf temperatures below 30 °C, when a matching temperature of 25 °C is chosen.

Analysis of the ratio Jmax/Vcmax using the mean temperature response functions for Jmax andVcmax results in a mean value for Jmax0/Vcmax0 = 2·00 ± 0·60 at Tleaf = 25 °C.


I thank Dr Adrian Walcroft, Manaaki Whenua – Landcare Research, New Zealand, for suggesting a detailed analysis of the temperature response of photosynthetic parameters. Dr Ying Ping Wang, CSIRO Atmospheric Research, provided valuable comments on an earlier draft of the manuscript.