Dr R. C. Dewar Unité de Bioclimatologie, INRA Centre de Bordeaux, BP 81, 33883 Villenave d¢Ornon Cedex, France. Fax: + 33 5 56 84 31 35.
We explore the extent to which a simple mechanistic model of short-term plant carbon (C) dynamics can account for a number of generally observed plant phenomena. For an individual, fully expanded leaf, the model predicts that the fast-turnover labile C, starch and protein pools are driven into an approximate or moving steady state that is proportional to the average leaf absorbed irradiance on a time-scale of days to weeks, even under realistic variable light conditions, in qualitative agreement with general patterns of leaf acclimation to light observed both temporally within the growing season and spatially within plant canopies. When the fast-turnover pools throughout the whole plant (including stems and roots) also follow this moving steady state, the model predicts that the time-averaged whole-plant net primary productivity is proportional to the time-averaged canopy absorbed irradiance and to gross canopy photosynthesis, and thus suggests a mechanistic explanation of the observed approximate constancy of plant light-use efficiency (LUE) and carbon-use efficiency. Under variable light conditions, the fast-turnover pool sizes and the LUE are predicted to depend negatively on the coefficient of variation of irradiance. We also show that the LUE has a maximum with respect to the fraction of leaf labile C allocated to leaf protein synthesis (alp), reflecting a trade-off between leaf photosynthesis and leaf respiration. The optimal value of alp is predicted to decrease at elevated [CO2]a, suggesting an adaptive interpretation of leaf acclimation to CO2. The model therefore brings together a number of empirical observations within a common mechanistic framework.
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Physiologically based plant models can be used, broadly speaking, in two ways. They can be used in an applied sense to address particular problems, such as assessing the impacts of climate change or management practices on crop yield. They can also be used in an explanatory sense, to provide a coherent interpretation of one or more experimental observations in terms of the underlying mechanisms (Thornley & Johnson 1990). Our objective in this paper is to propose, within the framework of a simple model of plant carbon (C) dynamics, a mechanistic interpretation of the following three general observations.
First, it is well documented for many crop and tree species that net primary productivity (NPP) per unit of radiation absorbed by the canopy (i.e. the light-use efficiency, LUE) is approximately constant during vegetative growth when water supply is not limiting (e.g. Monteith 1977; Gallagher & Biscoe 1978; Linder 1985; Landsberg et al. 1997). Secondly, NPP per unit of C assimilated by gross photosynthesis (i.e. the carbon-use efficiency, CUE) has also been observed to remain approximately constant during vegetative growth in a number of diverse species (e.g. Gifford 1991, 1994; Ryan et al. 1994). (Here NPP is defined as the difference between gross canopy photosynthesis and whole-plant respiration.)
The first observation (approximately constant LUE) has been used by modellers as a basis for predicting NPP at scales ranging from a few hectares (e.g. Kirschbaum et al. 1994) to the entire global land surface (e.g. Potter et al. 1993). The second observation (approximately constant CUE) is often used by modellers as a basis for scaling up from canopy photosynthesis to whole-plant NPP. The third observation (leaf acclimation to light) is widely adopted by modellers as a basis for scaling up leaf photosynthesis to the canopy, often using the optimization argument that, for a given amount of canopy photosynthetic N, canopy photosynthesis is maximized when the relative distribution of leaf photosynthetic N is proportional to <I> (e.g. Field 1983; Hirose & Werger 1987; Sellers et al. 1992; Sands 1995a,b; but see de Pury & Farquhar 1997).
Recently, all three observations have been interpreted as the outcome of an optimal balance between photosynthesis and maintenance respiration with respect to variations in the absolute distribution of leaf photosynthetic N (Dewar 1996; Haxeltine & Prentice 1996). However, while optimization theories may have a useful range of realism, and offer an adaptive or evolutionary interpretation, they are ultimately limited because they do not predict the time-scale on which the optimization applies, and often cannot be refined in a systematic way when invalidated by experiments (Thornley 1997). What would be more useful, but is currently lacking, is a mechanistic explanation of these three observations.
We analyse the behaviour of a simple mechanistic model of plant C dynamics. Our analysis was motivated by the hypothesis that, on a time-scale of days to weeks within the growing season, the rapid-turnover plant pools (e.g. labile C, starch and protein, with turnover times of hours to days) are in an approximate or moving steady state which tracks the average environmental conditions on that time-scale. We use the model to evaluate this hypothesis quantitatively, and to examine its consequences for leaf acclimation to light and the behaviour of LUE and CUE. We also examine the consequences of the model for the interpretation of leaf acclimation to elevated atmospheric [CO2]. Although the model is formulated for a horizontally uniform stand of vegetation, the basic results also apply to individual plants.
Single leaf submodel
We first consider the seasonal C dynamics of a single, fully expanded leaf (Fig. 1a). Units and symbol definitions are given in Table 1. The model, adapted from Thornley (1977), McCree (1982) and Thornley & Johnson (1990, p. 285), represents the labile C, starch and protein pools of an individual leaf: Wlc, Wls and Wlp, respectively (all in kg C m–2 leaf); the amount of non-degradable cell wall material in the leaf is assumed to remain constant after leaf expansion, and is not represented explicitly. In this highly aggregated model, Wlc represents both the C in soluble carbohydrates (e.g. sucrose) used as the energy source for protein synthesis and the C in amino acids used as the building blocks for protein synthesis; a more detailed model would separate the roles of these two components. The rates of change of the dynamic leaf pools are given by
Table 1. . Symbol definitions. Abbreviations: C = carbon, c.l.r. = constant light regime, CUE = carbon-use efficiency, LAI = leaf area index, LUE = light-use efficiency, PAR = photosynthetically active radiation, v.l.r. = variable light regime, < X > = average value of X over 24 h (c.l.r.) or 20 d (v.l.r.), * = steady-state value. Subscripts: lc = leaf labile C, ls = leaf starch, lp = leaf protein, rc = root labile C, rs = root starch, rp = root protein, rx = root non-degradable structure
P is the instantaneous rate of gross leaf photosynthesis, fractions als and 1 –als of which are allocated to leaf starch and labile C, respectively; U is the utilization rate of leaf labile C, a fraction alp of which is allocated to leaf protein synthesis, the remaining fraction (1 –alp) being exported out of the leaf; klp and kls are the leaf protein and starch turnover rates, respectively. As depicted in 1Fig. 1a, leaf respiration (Rlp) is treated as a single synthetic process associated with leaf protein synthesis, which has efficiency Ylp.
P is described by a non-rectangular hyperbolic light-response curve,
where α is the quantum yield, I is the photosynthetically active radiation (PAR) absorbed by the leaf, θ is a dimensionless number between 0 and 1 which determines the shape of the light-response curve, and Pm is the light-saturated value of P. We assume that Pm is proportional to the leaf protein content and to leaf intercellular [CO2]:
where ca is the atmospheric [CO2], β is the ratio of leaf intercellular [CO2] to ca which is assumed constant (Wong, Cowan & Farquhar 1979), and kc is a carboxylation constant. Although it would be more realistic to use a non-linear dependence of Pm on leaf intercellular [CO2] (e.g. Farquhar, von Caemmerer & Berry 1980), and to include CO2 effects on the quantum yield (e.g. Thornley 1991; his eqn 2f), the simplified CO2 response incorporated into Eqn
2b captures the essential behaviour we wish to describe.
The utilization rate of leaf labile C is assumed to be proportional to the leaf labile C pool:
where klc is a rate constant. We denote by E the rate at which leaf labile C is exported out of the leaf (Fig. 1a):
According to Eqns
4, E depends only on Wlc and so does not include source–sink interactions between the leaf and other parts of the plant. Source–sink interactions might be incorporated by making labile C export depend on source–sink gradients in labile C concentration (e.g. Thornley 1972a,b; Dewar 1993). As with Eqn
2b, however, Eqns
4 are sufficient to convey the essential ideas we wish to describe.
We now extend the above single-leaf model into a whole-plant model. To do so, we assume that the entire leaf canopy is fully expanded. Thus, the whole-plant model describes vegetative growth during the period following canopy expansion and prior to leaf senescence, when all the C exported out of the leaves enters non-foliar parts of the plant, here collectively termed ‘roots’. The root submodel may also include stem material, although for our purposes the stem need not be represented explicitly. As in the leaf submodel, the root submodel represents labile C, starch and protein pools (Fig. 1b): Wrc, Wrs and Wrp, respectively (all in kg C m–2 ground); to account for structural root growth at certain times during the growing season, the root submodel includes an additional dynamic pool representing non-degradable (cell wall) structure, Wrx (kg C m–2 ground). The rates of change of the root pools are given by
where the notation in Eqns
5a–c is analogous to that in Eqns
1a–c. For trees and other perennials, in which significant starch accumulation may occur in roots and stems at certain times during the growing season for use in leaf expansion the following spring (e.g. Hansen 1967; Hansen & Grauslund 1973; Ericsson & Persson 1980), we will examine the idealized case where root starch turnover is zero (krs = 0). In Eqn
5a, Pr is the total canopy export of labile C to ‘roots’:
where L is the cumulative leaf area index (LAI) from the top of the canopy, Lc is the total canopy LAI and E is given by Eqn
4; Ur is the rate at which root labile C is used for the synthesis of root protein (fraction arp) and root structure (fraction 1 –arp), and is given by
analogous to Eqn
3. In Eqn
5d, Yrx is the efficiency for the synthesis of root structure; the associated respiration rate is Rrx (Fig. 1b).
We have not specified a functional role for root protein, although a consistent treatment of foliage and roots might make nutrient uptake depend on root protein (e.g. Clarkson 1985), just as photosynthesis depends on leaf protein. Nutrient dynamics are outside the scope of the present study, although they might be incorporated into the above scheme by making the utilization rates of leaf and root labile C depend on both labile C and nutrient levels (e.g. Thornley 1972b; Mäkelä & Sievänen 1987; Dewar 1993). Water limitation is also excluded but might be incorporated along the lines of Dewar (1993).
MODEL BEHAVIOUR AT THE LEAF SCALE
Constant ‘square-wave’ light regime
First we analyse the modelled behaviour of a single leaf in the case where I, the leaf absorbed PAR, is a constant (denoted by Id) during the daylight period of length h, and zero otherwise. This analysis helps in understanding the behaviour of the model under more realistic light conditions (see ‘Variable light regime’ below).
Figure 2(a) shows the simulated 24 h averages of the total leaf non-structural carbohydrates (TNC = labile C + starch) and leaf protein, for Id = 50 and 100 W PAR m–2 leaf over a 12 h day and with other parameter values given in Table 2. From arbitrary initial pool sizes of 1 g C m–2 leaf at t = 0, the 24 h average TNC and protein pools reach steady-state values after a time τ≈ 20–40 d. These steady-state values correspond to the solution of the leaf submodel when the 24 h averages of Eqns
where < > denotes 24 h averages and the asterisk denotes steady-state values. The equilibration time-scale τ is determined by the dynamics of the leaf protein pool (which has the longest turnover time, 2·3/klp = 5·75 d), and satisfies τ >> 2·3/klp. In the steady state, the behaviour of the model simplifies because of the underlying balance between the C fluxes into and out of the dynamic leaf pools (Eqns
7a–c). We now analyse the steady-state behaviour in more detail.
Three principal results emerge. First, from Eqns
2a–b, 4 and 7a–c it can be shown that, in the steady state, the 24 h average export rate of leaf labile C, < E >*, is proportional to the 24 h average leaf absorbed PAR:
where s is the dimensionless parameter combination
fd = h/24 h is the daylight fraction and < I > = fdId is the 24 h average leaf absorbed PAR. Note that, in the steady state, < E >* is equal to the net rate of leaf C uptake from the atmosphere. Let us therefore define the leaf LUE on a given time-scale by
(i.e. leaf C export per unit of absorbed PAR), where < E > and < I > are time-averages over the specified time-scale (in this case, 24 h). Equation
8a then implies that, under the constant light regime, the 24 h steady-state leaf LUE, ɛl*, is independent of < I > :
As 2Fig. 2b illustrates, initially ɛl is much less than ɛl* but it approaches ɛl* on the time-scale τ≈ 20–40 d.
Secondly, it can also be shown from Eqns
7a–c that, in the steady state, < E >* is proportional to the 24 h average rate of gross leaf photosynthesis, < P >*:
Defining the leaf CUE on a given time-scale by
(i.e. leaf C export per unit of gross C uptake), it follows from Eqn
10 that under the constant light regime, the 24 h steady-state leaf CUE, ϕl*, is independent of < P >*:
Figure 2(b) illustrates how the leaf CUE approaches the steady-state value ϕl* after a time τ≈ 20–40 d.
Thirdly, from Eqns
7a–c, 8a and 10, it can be shown that, under the constant light regime, the 24 h average leaf labile C, starch and protein pools acclimate to steady-state values which are proportional to < I > :
This last prediction is illustrated by the two simulations in Fig. 2(a). Here, the initial values of the leaf labile C, starch and protein pools are the same for the two simulations, but the steady-state values for the simulation at Id = 100 W m–2 leaf are twice those for the simulation at Id = 50 W m–2 leaf. This behaviour reflects the greater rate of gross leaf photosynthesis, and hence greater production rate of leaf labile C, starch and protein, at the higher irradiance.
The predicted linear response of < E >* to < I > in the steady state (Eqn
8a) contrasts with the underlying instantaneous response of P to I which is non-linear (Eqn
2a). The emergence of linearity from non-linearity occurs because a sustained increase in Id leads to a higher leaf protein content (Fig. 2a), and hence to an increase in light-saturated photosynthesis (Eqn
2b), which tends to desaturate the instantaneous light response of P. This point is illustrated in 2Fig. 2c, which shows that although the instantaneous relationship between P and I is non-linear, the steady-state relationship between < P >* and < I > (to which the leaf acclimates) is linear. This result may be verified analytically by combining Eqns
8a–11b to give
where Pd = < P >*/fd is the constant daylight value of P, and Id = < I >/fd is the constant daylight value of leaf absorbed PAR.
A corollary of Eqns
11b is that both the steady-state leaf CUE and LUE depend on allocation to leaf protein synthesis (alp). As illustrated in Fig. 3, the steady-state leaf CUE declines with increasing alp, reflecting the fact that, in the steady state, the leaf protein recycling loop is a net emitter of C through the respiration rate Rlp (Fig. 1a). In contrast, the steady-state leaf LUE has a maximum at an ‘optimal’ value of alp which can be calculated analytically:
corresponding to a maximum leaf LUE of
where ν is the dimensionless parameter
This maximum reflects the fact that, as leaf protein allocation increases, the gain in leaf photosynthesis associated with increased leaf protein content is eventually offset by increased respiration costs associated with leaf protein synthesis. Figure 3 shows that alp,opt decreases as atmospheric [CO2] increases; this result is discussed in relation to leaf acclimation to CO2 in a later section.
What are the key model features on which the above leaf-scale predictions depend? The analytical expression for ɛl* (Eqn
9b) is specific to our choice of leaf photosynthesis model (P, Eqn
2a), which we chose for simplicity. However, it may be shown more generally that ɛl* is always independent of < I > provided that P has the following homogeneity property as a function of instantaneous leaf absorbed PAR (I) and leaf protein content (Wlp):
for any value of λ. It is a common feature of the present model (Eqn
2a) and more comprehensive models of leaf photosynthesis (e.g. Farquhar et al. 1980; Farquhar 1989) that P is a homogeneous function of I and light-saturated photosynthesis (Pm); Eqn
15 then follows if Pm is proportional to Wlp, as assumed here (Eqn
2b) and as is suggested by the strong linear relationship commonly observed between Pm and leaf N content (Gulmon & Chu 1981; Evans 1989). The analytical expressions for < Wlc >*, < Wls >* and < Wlp >* as functions of ɛl* and < I > (Eqns
12a–c) are general and depend only on the homogeneity property of P. The analytical expression for ϕl* (Eqn
11b) is even more general, being independent of any assumptions for P; this result simply reflects the fact that the total efflux of C from the leaf (< Rlp > + < E >) must equilibrate to a value equal to the influx of C (< P >) which, because < Rlp > ∝ < E > (both fluxes being constant fractions of U by assumption), implies that < E >*∝ < P >*.
Variable light regime
We now consider the more realistic behaviour of a single leaf when the leaf absorbed PAR (I) varies both diurnally and daily through the growing season. We ran numerical simulations of the individual leaf submodel driven by the daily incident radiation (MJ PAR m–2 ground day–1) measured near Canberra, Australia (35°21’ S) from 1 July 1984 to 1 July 1985 (Fig. 4a) during the Biology of Forest Growth experiment (Benson, Landsberg & Borough 1992). We assumed a constant leaf inclination of 60° to convert the incident radiation from a per ground area to a per leaf area basis (giving a conversion factor of 1/2). We also assumed that I varies sinusoidally during each day,
where hj and Id(j) are, respectively, the daylength and the mean daylight PAR absorbed by the leaf, on day j. Although there are more realistic models for the diurnal distribution of leaf absorbed PAR (e.g. Wang & Jarvis 1990), Eqn
16 is sufficient to illustrate the essential features of the model’s behaviour under a variable light regime. The analytical results derived below (Eqns
19) are in fact applicable to any temporal distribution of leaf PAR.
The behaviour of the leaf submodel under the constant light regime (Fig. 2) indicates a characteristic equilibration time of the order of 20 d. For the variable light regime therefore we consider the time-averaged behaviour of the model over 20 d periods, in the expectation that on this time-scale the fast-turnover pools acclimate to the corresponding time-averaged light environment. In the following discussion, < X > denotes the running 20 day average of X.
4Figure 4b shows the seasonal variation of < Wlp >, the 20 d running average leaf protein content, obtained from a dynamic simulation of the model under the light regime in 4Fig. 4a. In 4Fig. 4b, < Wlp > is compared to < Wlp >*, the previously derived steady-state approximation (Eqn
12c) which here corresponds to approximating the simulated light regime within each 20 d period by a constant ‘square wave’ having the same average leaf absorbed PAR, < I >, and daylength, < h >. For this comparison, < I > in Eqn
12c was evaluated with a time lag of 6 d relative to < Wlp >* in order to take approximate account of the leaf protein turnover time. Although the seasonal trends in < Wlp > and < Wlp >* are similar, confirming our expectation that acclimation to light occurs on a time-scale of about 20 d, < Wlp > is significantly lower than <Wlp>*. (Similar results were obtained for the leaf labile C and starch pools; data not shown). This discrepancy occurs because <Wlp>* neglects variations within each 20 d period in the daylight value of I about its 20 d average, < Id >. Because of the saturating shape of the photosynthetic light-response curve, variations in I below < Id > have a greater effect on leaf photosynthesis than variations above < Id > ; thus < P >, and hence < Wlp >, are reduced by light variability.
We now analyse more quantitatively the effect of this subscale variation in I (i.e. on time-scales less than 20 d). If, as a reasonable approximation, we ignore variations in h about < h >, then the 20 d average rate of gross leaf photosynthesis is given by
where < fd > = < h >/24 h is the average daylight fraction, and < Pd > is the average daylight P. If we also ignore variations in Wlp about < Wlp >, then < Pd > can be approximated by expanding P = P(I, Wlp) as a Taylor series in I about < Id >, and then averaging the result. Keeping terms up to second order in the moments of the light distribution, it can be shown that
where P and ∂2P/∂I2 on the right-hand side are evaluated at I = < Id > and Wlp = < Wlp >. The negative effect of subscale variation in I, described by the second term in Eqn
17b (∂2P/∂I2 < 0), is expressed as the sum of diurnal and daily contributions; νdi is the (dimensionless) coefficient of diurnal variation of I about Id(j), its mean daylight value on day j; for the illustrative example of sinusoidal variation (Eqn
16), νdi is a constant given by
where σdi2 is the diurnal variance of I about Id(j); νda is the coefficient of daily variation of Id(j) about < Id >, its average value over each 20 d period:
where σda2 is the daily variance of Id(j) about < Id >.
Supposing now that the 20 d average leaf labile C, starch and protein pools still follow an approximate steady state, but one that is modified by subscale light variability, we set the 20 d averages of Eqns
1a–c to zero (see Eqns
7a–c) and substitute Eqns
17a–b for < P >. With leaf LUE (ɛl) defined as in Eqn
9a (where < E > and < I > now denote 20 d averages rather than 24 h averages), it can then be shown that the 20 d steady-state leaf LUE under a variable light regime (which we denote by ɛl,v*) is given by:
where the constant x is the solution to the equation
The moving steady-state leaf protein content, taking into account subscale light variability (which we denote by < Wlp,v >*), is then found to be
which differs from Eqn
12c only in the replacement of ɛl* by ɛl,v*. 4Figure 4b shows that, while the dynamic leaf protein content, < Wlp >, is poorly approximated by < Wlp >* (which neglects subscale light variability), it is well approximated by < Wlp,v >*. Analogous to the results for leaf protein content shown in 4Fig. 44b, Fig. 4c shows that, while the dynamic leaf LUE, ɛl, is poorly approximated by ɛl*, it is well approximated by ɛl,v*.
What then do these analytical results for ɛl,v* and < Wlp,v >* tell us about the behaviour of the model under variable light conditions? The surprising conclusion from Eqns
18a–b is that ɛl,v* remains independent of < I >; in other words, the time-averaged rate of leaf C export remains linearly proportional to the time-averaged leaf absorbed PAR even under variable light conditions and in spite of the non-linear photosynthetic light response. This conclusion turns out to depend crucially on the homogeneity property of P (in deriving Eqn
18b, we used Eqn
15 with λ = 1/Wlp). The only effect of light variability on ɛl,v* is to make ɛl,v* depend negatively on the normalized variance of leaf PAR (i.e. νdi2 + νda2). When νdi = νda = 0 (constant ‘square wave’ light regime) it may be verified that ɛl,v* equals ɛl*, but in general ɛl,v* is less than ɛl* (Fig. 4c) because of the non-linearity in P (i.e. because ∂2P/∂x2 < 0). The general dependence of ɛl,v* on subscale light variability is illustrated in 4Fig. 4d.
The conclusion from Eqn
19 is that, even under variable light conditions, leaf protein is driven into an approximate or moving steady state that is proportional to the time-averaged leaf absorbed PAR; similar behaviour is exhibited by leaf labile C and starch (data not shown). The only effect of light variability on < Wlp,v >* is to introduce an additional negative dependence of < Wlp,v >* on the normalized variance of leaf PAR (Fig. 4b).
Because light variability reduces the average rates of leaf C export and gross photosynthesis in the same proportion, the steady-state leaf CUE is the same under the variable and constant light regimes (ϕl*, Eqn
11b). 4Figure 4c shows that the dynamic leaf CUE, ϕl, is well approximated by ϕl*. Because allocation to leaf protein synthesis (alp) was set to its optimal value (alp,opt, Eqn
14a) in the simulation, alp decreases slightly with increasing daylength (data not shown); because ϕl* depends negatively on alp (Fig. 3), the seasonal variation in alp leads to a slight mid-summer maximum in ϕl* (Fig. 4c).
In 4Fig. 4c, ɛl,v* has a more pronounced mid-summer maximum. This seasonal trend mainly reflects the positive dependence of leaf LUE on daylength (Fig. 4d), rather than any direct dependence on the leaf absorbed PAR. The positive dependence of leaf LUE on daylength reflects the saturating nature of the P–I curve; leaf photosynthesis per unit leaf absorbed PAR is greatest when the mean daylight leaf PAR is low, which, for a given value of < I >, occurs when daylength is greatest. There is a minor additional seasonal variation in ɛl,v* through its negative dependence on νda (Fig. 4d), with νda tending to be smaller (and hence ɛl,v* larger) during the period of maximum incident PAR (Fig. 4a). Thus, while the leaf LUE does not depend directly on the time-averaged leaf absorbed PAR, these two variables may be correlated in time through associated seasonal changes in daylength and light variability.
Differences in daylength and light variability also occur between different sites, and may contribute to some of the variation in reported values of LUE (e.g. Prince 1991). To illustrate the potential magnitude of this source of spatial variation in LUE, we calculated ɛl,v* at two sites of contrasting latitude – Esperance, Tasmania, Australia (43°16’ S) and Chiang Mai, Thailand (17° N) – at which daily radiation data were collected in 1985 and 1986, respectively (McMurtrie, Landsberg & Linder 1989). The same values were used for the other model parameters at each site (Table 2). As for the Canberra site (Fig. 4c), there was good agreement between ɛl,v* and the dynamic leaf LUE (ɛl) at both Esperance and Chiang Mai (data not shown). 4Figure 4e shows the predicted seasonal course of ɛl,v* at the two sites. Because of the difference between the two sites in the annual average value of νda (0·50 at Esperance, 0·20 at Chiang Mai), the predicted annual average leaf LUE at Esperance is 15% lower than that at Chiang Mai. In addition, the seasonal variation in ɛl,v* at Esperance is more marked than at Chiang Mai, reflecting the difference between the two sites in the seasonal variation of daylength. These predictions do not include site differences in the coefficient of diurnal variation (νdi) or in other parameter values of the model.
Finally in this section, we emphasize that the analytical results derived for the variable light regime (Eqns
19) are not specific to our assumption that light varies sinusoidally during the day (Eqn
16), which we adopted for illustrative purposes only. These analytical results can be applied to any temporal distribution for leaf PAR defined by given values of < I >, νdi and νda, as might be derived, for example, from measurements of leaf PAR at different positions within a plant canopy or from model estimates of leaf PAR based on separate treatment of diffuse and direct radiation (e.g. Norman 1980; Wang & Jarvis 1990; Boote & Loomis 1991; de Pury & Farquhar 1997).
MODEL BEHAVIOUR AT THE PLANT SCALE
We now consider the implications of the above leaf-scale results for LUE and CUE at the plant or stand scale, when the root labile C, starch and protein pools also follow an approximate steady state. To do this, we make the additional assumption that the entire canopy is fully expanded. We also assume that the parameters affecting leaf LUE and CUE are constant through the canopy. We define the plant LUE (ɛp) and CUE by
where < Sc > is the time-averaged PAR absorbed by the canopy, < Pc > is the time-averaged rate of gross canopy photosynthesis:
and < Gp > is the time-averaged whole-plant NPP, defined as the difference between gross canopy photosynthesis and total plant respiration:
First we consider the constant ‘square wave’ light regime. When all the leaves throughout the canopy follow an approximate steady state, the difference between the first two terms on the right-hand side of Eqn
20c equals < Pr >*, the total canopy export rate of C, which is itself proportional to < Sc > and to < Pc >* on account of Eqns
Then, setting the 24 h averages of Eqns
5a–c to zero, it can be shown that, when all the plant labile C, starch and protein pools follow an approximate steady state, the plant LUE and CUE are, respectively, independent of < Sc > and < Pc > and proportional to their leaf-level counterparts (ɛl* and ϕl*):
The approximate steady-state behaviour of the fast-turnover root pools is central to this result; the important feature here is that the respiration rates associated with the synthesis of root protein and structure (< Rrp > and < Rrx >) equilibrate to values that are proportional to labile C import from the canopy (< Pr >), the latter flux being proportional to < Sc > and < Pc > because of the approximate steady-state leaf behaviour through the canopy (Eqn
Figure 5 illustrates the dependence of the steady-state plant LUE on allocation to root protein synthesis (arp); a qualitatively similar dependence on arp is obtained for the steady-state plant CUE (data not shown). As root protein allocation increases, plant LUE decreases, reflecting the fact that, in the steady state, the root protein recycling loop is a net emitter of C through the flux Rrp (Fig. 1b), and so contributes negatively to the plant LUE. This prediction may have implications for the carbon cost of nutrient uptake.
Figure 5 also illustrates the sensitivity of plant LUE to root starch turnover. With root starch turnover switched off (krs = 0), root starch (like root structure) continues to accumulate through the growing season, and only the root labile C and protein pools exhibit approximate steady-state behaviour; in this case, the steady-state plant LUE and CUE are given by Eqn
21a but with:
where ars is the fraction of imported labile C allocated to root starch. The effect of switching off root starch turnover therefore is to increase the plant LUE and CUE, all else being equal. Note that when root starch turnover is switched on (see Eqns
21a–b), the steady-state plant LUE and CUE are independent of the value assumed for krs. While no parameter tuning has been attempted, the range of values of plant LUE in Fig. 5 are physiologically reasonable, and commensurate with typical measured values for crops (e.g. 1·1–1·4 g C MJ–1 PAR, Russell, Jarvis & Monteith 1989; 0·4–2·4 g C MJ–1 PAR, Prince 1991) and trees (e.g. 1·0–1·7 g C MJ–1 PAR, Hunt 1994).
It is straightforward to show that, under a variable light regime, Eqn
21a for ɛp* generalizes to
so that our analysis of leaf LUE under variable light conditions extends straightforwardly to plant LUE. Like leaf CUE, the steady-state plant CUE is the same under the variable and constant light regimes (ϕp*, Eqn
21a). We also note that, although the whole-plant model has been formulated for a horizontally uniform stand of vegetation, the basic results of our analysis concerning plant LUE and CUE also apply to individual plants. In the latter case, the expressions for < Sc > and < Pc > in Eqn
20b may be re-interpreted as the sums of leaf absorbed PAR and gross leaf photosynthesis over an individual plant; likewise, < Gp > and the root pools may be redefined on a per plant basis.
LEAF ACCLIMATION TO ATMOSPHERIC [CO2]
As Fig. 3 illustrates, the steady-state leaf LUE is maximized at an optimal value of leaf protein allocation (alp,opt, Eqn
14a) which decreases as [CO2]a increases. The maximum in leaf LUE reflects a trade-off between leaf photosynthesis and leaf respiration with respect to variations in alp. Because [CO2]a directly affects photosynthesis but has no direct effect on respiration in the model, elevated [CO2]a shifts the point of optimal balance towards lower values of alp. The CO2 response of alp,opt therefore suggests an adaptive interpretation of leaf acclimation to CO2 in terms of the ‘goal’ of maximizing the leaf C export per unit of leaf absorbed PAR. According to this interpretation, the extent of leaf acclimation to CO2 can be quantified in terms of the extent to which alp attains its optimal value at a given [CO2]a.
To illustrate the implications of this interpretation at the leaf scale, let us consider a leaf that is fully acclimated to a [CO2]a of 350 cm3m–3, so that alp(350) = alp,opt(350). Then, under a doubling of [CO2]a to 700 cm3 m–3, we consider the extremes of zero and full acclimation to CO2. With zero acclimation to CO2 we have alp(700) = alp,opt(350) and with full acclimation to CO2 we have alp(700) = alp,opt(700). Table 3 shows the predicted steady-state leaf responses to doubled [CO2]a for both extremes, under the constant light regime. With no acclimation to CO2, leaf protein allocation is unchanged and all pools increase by 22%, reflecting the direct, positive effect of elevated [CO2]a on photosynthesis. In contrast, with full acclimation to CO2, there is a 26% decrease in leaf protein allocation (Fig. 3) which offsets the direct effect of CO2 on photosynthesis, leading to changes in leaf protein, labile C and starch of – 28%, – 2% and + 11%, respectively; full acclimation therefore leads to a marked decrease in the ratio of proteins to total non-structural carbohydrates (TNC) from 1·74 to 1·20 (Table 3), which compares with a reported decrease from c. 2 to 1 in the average protein:TNC ratio across 27 species (Poorter et al. 1997). Thus, while the predicted CO2 response of leaf labile C and starch is almost always positive, the predicted response of leaf protein per unit area to doubled [CO2]a can be positive or negative, depending on the extent to which leaf protein allocation attains its optimal value at 700 cm3 m–3.
Table 3. . Predicted steady-state leaf responses to a doubling of [CO2]a from 350 to 700 cm3 m–3, under zero and full acclimation to CO2. Predictions are given for a constant light regime with Id = 100 W PAR m–2 leaf over a 12 h day, assuming full acclimation to CO2 at 350 cm3 m–3 Other parameter values are given in Table 2. TNC = labile C + starch. The steady-state values at 350 cm3 m–3 correspond to those in Fig. 2 (solid curves)
Figure 6 illustrates the general dependence of steady-state plant LUE and CUE on [CO2]a. With zero (full) acclimation to CO2, and with the default parameter values in Table 2, a doubling of [CO2]a from 350 to 700 cm3 m–3 leads to an increase in plant LUE of 22% (34%) (Fig. 6a), and in plant CUE of 0% (20%) (Fig. 6b), respectively. These responses reflect equivalent percentage increases in the steady-state leaf LUE and CUE (Table 3). In 6Fig. 6b, increases in plant CUE at elevated [CO2]a are associated entirely with decreased leaf protein allocation (Fig. 3), and so there is no response with zero acclimation to CO2. In 6Fig. 6a, increases in plant LUE at elevated [CO2]a reflect, in addition to decreased leaf protein allocation, the direct positive effect of CO2 on photosynthesis, and so there is a significant response (+ 22%) even with zero acclimation to CO2. In 6Fig. 6b, the default value of plant CUE at 350 cm3 m–3 is physiologically reasonable, and commensurate with measured values which indicate that, for a diverse range of species, 40–60% of gross photosynthesis is consumed in respiration at current CO2 levels (e.g. Gifford 1994; Ryan et al. 1994).
The present model is not intended as a definitive representation of plant responses to light and CO2, but rather as a ‘toy’ model whose purpose is to examine the extent to which some simple assumptions about plant C dynamics can account for a number of generally observed plant phenomena. The model’s simplicity enables its behaviour to be studied analytically, providing general insights beyond those obtained from numerical simulations alone.
Light and carbon use efficiencies
Our analysis suggests that, under quite general conditions, the fast-turnover plant pools are driven into an approximate or moving steady state that is proportional to the time-averaged leaf or canopy absorbed PAR, and that this behaviour provides a mechanistic explanation of the observed approximate constancy of plant LUE (Monteith 1977; Gallagher & Biscoe 1978; Linder 1985; Landsberg et al. 1997) and plant CUE (Gifford 1991, 1994; Ryan et al. 1994). Equally, the model also provides a mechanistic basis for interpreting deviations of plant LUE and CUE from strict constancy (Figs 4c–e) and for incorporating the effects of nutrient and water limitation on LUE and CUE.
For example, Lamaud, Brunet & Berbigier (1996) estimated canopy LUE from CO2 flux measurements above and within a coniferous forest in south-west France. They found that the canopy LUE was significantly larger during cloudy conditions, when the ratio of diffuse to direct radiation was larger. Qualitatively, this observation might be explained by the model in terms of the positive effect on leaf LUE of a decrease in light variability (νdi) during periods when the fraction of diffuse radiation is larger.
In contrast, the model suggests that CUE is likely to be a more robust parameter than LUE. Because photosynthesis provides the substrate for respiration, these two fluxes are likely to remain in approximate balance in the long term.
Leaf acclimation to light
Our dynamic interpretation of leaf acclimation to light (Fig. 2a) appears to be consistent with studies showing that, when fully expanded foliage formed in the sun is shaded, light-saturated photosynthesis decreases to a value similar to that for foliage formed in the shade (Björkman 1981; Pons & Pearcy 1994; Brooks, Hinkley & Sprugel 1994; Brooks, Sprugel & Hinkley 1996). Also, given that leaf photosynthetic proteins account for a significant fraction of leaf N content (Field & Mooney 1986), the predicted mid-summer peak in the seasonal pattern of leaf protein content (Fig. 4b) is qualitatively consistent with measured seasonal changes in leaf N content in temperate deciduous trees (Reich, Walters & Ellsworth 1991). Without parameter fine-tuning, the predicted maximum leaf protein content shown in 4Fig. 4b (≈ 15 g C m–2 leaf) is physiologically reasonable, corresponding to a seasonal maximum leaf N content of ≈ 3 g N m–2 leaf (assuming a protein N:C ratio of 0·2). A similar seasonal pattern is predicted for leaf starch (data not shown), in qualitative agreement with measurements of leaf starch seasonal dynamics in Pinus sylvestris (Ericsson 1979).
However, in interpreting observed leaf N dynamics, we stress that the present carbon-based model only quantifies the ‘demand’ for leaf protein N, and does not take into account the effects of limiting N supply or non-photosynthetic leaf N. Several experimental studies of crop and tree species (e.g. Garcia et al. 1988; Dalla-Tea & Jokela 1991) suggest that the effects of limiting N supply on plant growth occur primarily through decreased allocation to leaf area growth, and hence decreased canopy light interception, rather than through decreased plant LUE or leaf photosynthetic capacity. Therefore, the present analysis of LUE and leaf protein on seasonal time-scales may still be applicable under N limitation, but longer term effects of N limitation on growth would require leaf area dynamics also to be modelled.
The prediction that the steady-state leaf protein content is proportional to the time-averaged leaf absorbed PAR (< I >) is also in qualitative agreement with observed spatial profiles of leaf N content and leaf irradiance within canopies (Hirose & Werger 1987; Hirose et al. 1988; Ackerly 1992; Evans 1993; Pons et al. 1993). Observed deviations from strict proportionality between total leaf N content and leaf irradiance may be reconciled with Eqns
19 when account is taken of non-photosynthetic N in leaf structure (Kull & Jarvis 1995). Our model also predicts that the steady-state leaf non-structural carbohydrate contents are linearly related to < I >, in qualitative agreement with numerous experimental studies (e.g. Ammerlaan et al. 1986; Sicher & Kremer 1986; Fjeld 1992; Merlo et al. 1994; Reyes et al. 1996). However, few studies appear to have measured the response of non-structural carbohydrate contents over a continuous range of leaf irradiances, and so the predicted proportionality to < I > remains open to further experimental test. We have not considered other aspects of leaf acclimation to light, such as effects of irradiance on leaf thickness (Björkman 1981).
Relationship to interpretation in terms of optimal leaf N content
Recently it has been proposed that leaf acclimation to light, and the observed approximate constancy of LUE and CUE, can all be interpreted as the outcome of an optimal balance between leaf photosynthesis and maintenance respiration, being, respectively, saturating and linear functions of leaf photosynthetic N content (Dewar 1996; Haxeltine & Prentice 1996). In the present model, leaf respiration (Rlp) is treated as a single synthetic process associated with protein synthesis. From inspection of 1Fig. 1a one sees that in the steady state (when protein synthesis balances protein turnover), < Rlp >* is proportional to leaf protein content and so corresponds to leaf maintenance respiration. Using this correspondence, it can then be shown that the optimal balance between photosynthesis and maintenance respiration analysed by Dewar (1996) and Haxeltine & Prentice (1996) corresponds to the special case of the present analysis where leaf protein allocation (alp) is optimal.
Our analysis extends that of Dewar (1996) and Haxeltine & Prentice (1996), by showing that leaf acclimation to light, and the approximate constancy of plant LUE and CUE, are predicted to occur more generally for any fixed value of alp– and not just when alp is optimal – as a consequence of the approximate steady-state behaviour of the fast-turnover plant pools. Therefore, our model provides a more general mechanistic interpretation of these observations that does not depend on optimization arguments. In particular, unlike optimization theories, the time-scale on which such behaviour occurs is predicted by the model (Figs 2a–b & 4b-c), deviations of LUE and CUE from strict constancy can be interpreted, and the effects of light variability can be analysed in a systematic way. Clearly, the present model highlights the need for further understanding of how alp is regulated, and in particular the extent to which alp remains fixed or optimal under different environmental conditions.
Leaf acclimation to CO2
The special case when leaf protein allocation is optimal leads to an adaptive interpretation of leaf acclimation to CO2, in terms of the goal of maximizing the leaf C exported per unit of leaf absorbed PAR. In the model, elevated [CO2]a has two competing effects on leaf protein per unit area: a positive effect through the direct CO2 response of photosynthesis, and a negative effect through the CO2 response of optimal leaf protein allocation. As a result, leaf protein per unit area can increase or decrease under elevated [CO2]a, depending on the degree to which alp attains its optimal value (Table 3). Observed plant responses to elevated [CO2]a on time-scales of weeks to months show a similar variety of behaviours. For example, leaf N content per unit area has been observed to increase in some experiments and decrease in others (Luo, Field & Mooney 1994). In experiments on a range of species grown under 700 cm3 m–3, Gifford (1991) observed increases in plant CUE ranging from 0·5% to 15·6%; this compares with predicted increases in plant CUE of 0–20% over the range from zero to full acclimation to CO2. Thus, the present model is a potentially useful tool for interpreting some of the variety of observed short-term leaf and plant CO2 responses.
In accordance with our remarks in the ‘Introduction’, however, our adaptive interpretation of leaf acclimation to CO2 is limited in scope, like all optimization theories, and highlights the need for a more mechanistic understanding of how leaf protein allocation is regulated. The present model may nevertheless provide a useful framework for incorporating the mechanisms underlying plant CO2 responses, including longer term CO2 responses that involve changes in allocation between leaf area and root biomass, and changes in soil mineral nutrient availability (Eamus 1996; Medlyn & Dewar 1996).
We have shown that leaf acclimation to light, and the observed approximate constancy of plant LUE and CUE, may be interpreted mechanistically in terms of the approximate steady-state behaviour of the fast-turnover plant pools that is predicted to occur on time-scales of days to weeks, even under variable light conditions. Within the framework of the present model, this interpretation leads to testable predictions of LUE and CUE, and of plant carbohydrate and protein pools as functions of several physiological and environmental factors. The model also suggests an adaptive interpretation of leaf acclimation to elevated [CO2]a in terms of the CO2 dependence of optimal leaf protein allocation. The model may be extended to describe nutrient and water limitations, and to incorporate additional mechanisms, such as source–sink interactions, in a systematic way.
We thank Andrew Friend, Bart Kruijt, Olevi Kull and John Thornley for helpful comments and criticisms. RCD and REM received financial support from the dedicated grants scheme of the National Greenhouse Advisory Committee, and the Australian Research Council. BEM received financial support from the European Union through the ECOCRAFT project (Contract no. ENV4-CT95–0077).
Present address: Dr R. C. Dewar, Unité de Bioclimatologie, INRA Centre de Bordeaux, BP 81, 33883 Villenave d′Ornon Cedex, France. Fax: + 33 5 56 84 31 35.