Journal of Experimental Botany, Vol. 52, No. 365, pp. 2313-2322,
December 1, 2001
© 2001 Oxford University Press
Original Papers |
Sink feedback regulation of photosynthesis in vines: measurements and a model
1 Unité de Bioclimatologie, INRA, Centre de Bordeaux, BP81, 33883 Villenave d'Ornon CEDEX, France
2 Unité d'Agronomie, INRA, Centre de Bordeaux, BP81, 33883 Villenave d'Ornon CEDEX, France
Received 30 January 2001; Accepted 6 July 2001
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Abstract |
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An experimental and modelling study of source–sink interactions in Vitis vinifera L., cv. Cabernet Sauvignon, rooted cuttings under non-limiting environmental conditions with a 12 h photoperiod is presented here. After 4 h, measured photosynthesis, stomatal conductance and leaf carbohydrate content reached maximum values. Over the remainder of the photoperiod, photosynthesis and stomatal conductance decreased continuously, whereas leaf carbohydrate content remained relatively constant. Because the experiment took place in a non-limiting environment, the results suggest that stomatal regulation of photosynthesis was mediated by an internal factor, possibly related to sink activity. A simple 1-source, 2-sink model was developed to examine the extent to which the data could be explained by a hypothetical sink-to-source feedback mechanism mediated by carbohydrate levels in either the mesophyll, the source phloem or the phloem of one of the two sinks. Model simulations reproduced the data well under the hypothesis of a phloem-based feedback signal, although the data were insufficient to elucidate the detailed nature of such a signal. In a sensitivity analysis, the steady-state response of photosynthesis to sink activity was explored and predictions made for the partitioning of photosynthate between the two sinks. The analysis highlights the effectiveness of a phloem-based feedback signal in regulating the balance between source and sink activities. However, other mechanisms for the observed decline in photosynthesis, such as photoinhibition, endogenous circadian rhythms or hydraulic signals in the leaf cannot be excluded. Nevertheless, it is concluded that the phloem-based feedback model developed here may provide a useful working hypothesis for incorporation into plant growth models and for further development and testing.
Key words: Vitis vinifera L., photosynthesis, sink, feedback, model.
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Introduction |
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For many plant species, the activities of carbon source and sink organs appear to be closely co-ordinated such that a balance is maintained between the source supply and the sink demand (Wardlaw, 1990
; Ho, 1992; Foyer et al., 1995). In vines, for example, defoliation (source limitation) leads to lower grape growth and lower yield (Candolfi-Vasconcelos and Koblet, 1990; Foyer et al., 1995), while removal or thinning of grapes (sink limitation) leads to lower rates of photosynthesis (Iacono et al., 1995; Foyer et al., 1995). Understanding the mechanistic basis of source–sink interactions and their integrated outcome at the whole-plant level is of particular relevance to vines, where practices such as pruning, leaf and shoot removal, and crop load adjustment are commonly performed in order to enhance fruit quality.
Currently, however, the mechanisms underlying sink–source interactions observed at the whole-plant level are not well understood. On the one hand, in vitro experiments have demonstrated that mesophyll carbohydrate concentration, which depends on the local balance between assimilation and export, can modify the expression of photosynthetic gene promoters (Koch et al., 1992, 1996; Sheen, 1994; Jang and Sheen, 1994). On the other hand, a continuous, diurnal decline in photosynthesis from mid-morning is widely-observed in vines (Downton et al., 1987; Chaves et al., 1987; Chaumont et al., 1994), even under non-limiting environmental conditions (Correia et al., 1990, 1995), suggestive of sink-limitation, and yet leaf sucrose levels do not increase concomitantly but instead remain relatively stable throughout the day (Correia et al., 1990; Chaumont et al., 1994). In the latter case, one possible interpretation is that photosynthetic rates decline in direct response to a build-up of carbohydrate in the plant other than in the mesophyll, including in the sinks themselves, via a currently unidentified biochemical signal.
The objective here is to develop and evaluate a simple mathematical representation of source–sink interactions for eventual use in vine growth models suitable for management applications. Previous authors have developed mechanistic models incorporating the effect of source activity on carbohydrate transport to, and unloading by, competing sinks (Thornley, 1972; Dewar, 1993; Minchin et al., 1993). Here the model of Minchin et al., consisting of one source and two sinks, is extended to incorporate a feedback effect of sink activity on source activity (Minchin et al., 1993). Following Minchin et al., the purpose here is not to build a mechanistic model incorporating all the perceived biochemical detail, but to seek the minimal amount of mechanistic detail required to account for the phenomenon of sink–source interactions observed in whole-plant experiments. This modelling approach is also appropriate from a pragmatic viewpoint, in terms of developing a practical model for addressing management problems.
In view of the above-mentioned uncertainty regarding the nature of the feedback mechanism, three alternative hypotheses are examined in the model, according to which photosynthesis is negatively related to carbohydrate concentration in either the mesophyll, the source phloem or in the phloem of one of the two sinks. In order to provide data to evaluate the model, the diurnal courses of leaf photosynthesis, stomatal conductance and leaf carbohydrate content of a single illuminated source leaf in Cabernet-Sauvignon cuttings were measured. Recalling the modelling objective noted above, the purpose here was simply to examine the extent to which the data could be reproduced by a semi-empirical representation of sink–source feedback interactions, rather than to determine experimentally the precise nature of these interactions in all their detail.
After describing the experimental methods and results, the model is presented and, for each feedback hypothesis, the simulated dynamic behaviour of leaf photosynthesis and carbohydrate content is compared with the experimental data. Finally, the predicted steady-state response of photosynthesis to sink activity is examined, in order to explore the general behaviour of the model, and to generate further model predictions for future testing.
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Materials and methods |
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Data were collected in a controlled environment on eight rooted cuttings of Vitis vinifera L. cv. Cabernet-Sauvignon with similar leaf areas, during a 12 h period within the grape maturation phase. Two types of plant were considered, those having fruits and those where the fruits had aborted at an earlier stage of development, in order to assess whether the diurnal course of photosynthesis depends on the nature of the sinks (respectively fruits+roots, and roots only). 150 cm3 of nutritive solution (Pouget and Delas, 1984
) were supplied each day during the entire development stage prior to and including the measurement period. Plants were kept in darkness for 48 h before the measurement period in order to condition the eight plants as uniformly as possible. Comparisons with plants not previously subjected to 48 h of darkness confirmed that the initialization protocol did not affect the subsequent gas exchange results (data not shown). A single leaf was exposed to constant, saturating light (1200 µmol m-2 s-1) during a 12 h photoperiod. The rest of the plant was kept in darkness using a cover which permitted gas exchange. Each plant thus consisted of a single source (illuminated leaf) and multiple sinks (fruits+roots, or roots only). This protocol simplified the analysis of carbon assimilation by, and export from, the source. The closed environment of the plants was kept as constant as possible. Temperature, humidity and CO2 concentration around the sampled leaf were respectively 23.9±1.6 °C, 1.8±0.24 kPa, and 373±10 ppm.
Leaf photosynthesis and stomatal conductance were measured using a portable infrared gas analyser and a Parkinson leaf cuvette (Ciras-1, Broad Automatic Leaf Version, PP systems, Hitchin, Herts, UK). The rate of photosynthetic CO2 uptake (A) and the intercellular CO2 concentration (ci) were calculated (Caemmerer and Farquhar, 1981). Leaf carbohydrate contents were analysed by the enzymatic method on 6 mm diameter foliar discs (Kunst et al., 1984). Carbohydrate export rates were estimated by carbon balance as the rate of net photosynthesis minus the net change in leaf carbohydrate content. CO2 response curves of photosynthesis were measured after 2, 6 and 10 h of exposure to saturating light. At the end of the photoperiod, pressure chamber measurements of leaf water potentials were made on leaves that were illuminated, shaded and exposed to natural (greenhouse) light (respectively -0.25±0.14, -0.168±0.04 and -0.25±0.07 MPa), and showed no significant differences between illuminated and shaded leaves.
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Results |
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As shown in Fig. 1
, the daily time-courses of photosynthesis (A) and stomatal conductance (gs) were similar for both types of plant sinks (fruits+roots, and roots only). During the growth phase prior to the gas exchange measurements, plants without fruits developed a larger root system (4 g dry weight) than plants with fruits (2 g root dry weight, 4 g fruit dry weight). Thus it appeared over time that the extra root growth in plants without fruits acted as a compensatory sink, to the extent that the diurnal decline in A was independent of the nature of the overall sink. Figure 2A shows the diurnal patterns of A and gs averaged over all plants, which were used for the model evaluation.
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) and fruits+roots (•).
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). (C) Leaf carbon export rate (
) and CO2 assimilation (Maximum values of A and gs were reached after 4 h of continuous, saturating light. Thereafter, A decreased continuously from 9 to 5 µmol m-2 s-1 during the remainder of the photoperiod, while gs decreased by two-thirds of its maximum value (Fig. 2A
).
Leaf total carbohydrates reached a maximum value coinciding with the maximum rate of photosynthesis, decreased over the next 2 h, but then remained relatively stable during the remainder of the photoperiod (Fig. 2B).
Leaf carbon export rate (E) was calculated by carbon balance as the difference between A and the net change in leaf carbohydrate carbon content. E reached a maximum value after 5 h of light (Fig. 2C), then adjusted to a value close to the rate of carbon assimilation (reflecting the stable value of leaf carbohydrate content).
The initial slope of the A/ci curve, corresponding to the carboxylation efficiency, did not change significantly during the day, while the CO2-saturated rate of photosynthesis showed a progressive decline (Fig. 3). However, during the day the operating point was confined to the CO2-limited portion, so that the observed decline in A could be attributed almost exclusively to the observed decline in gs.
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The lack of correlation between leaf photosynthesis and total leaf carbohydrate content is similar to that found by other authors (Rodrigues et al., 1993
; Chaumont et al., 1994; Correia et al., 1990). The gas exchange results suggest that stomatal regulation of photosynthesis occurs, preventing carbohydrate accumulation in leaves. As the experiment was conducted under controlled, non-limiting environmental conditions, this regulation would appear to be mediated by some internal factor in the plant, one possibility for which might be a build-up of carbohydrate in the plant other than in the source leaf, due to sink limitation. However, as Correia et al. noted, while total leaf carbohydrate content remained stable, a feedback role for some component of leaf carbohydrates cannot be discounted (Correia et al., 1990). In view of the experimental results, and of the overall objective, a simple model was developed to examine further the possibility that the decline in A and gs is regulated by sink demand. The specific aim was to determine the extent to which the data could be reproduced by a hypothetical carbohydrate-mediated feedback mechanism, either in the mesophyll or elsewhere, and to explore the behaviour of the model more generally through sensitivity analysis in order to generate further testable predictions.
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The model |
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Symbol definitions and units are given in Table 1
. Following Minchin et al. (Minchin et al., 1993), a model consisting of one source compartment (illuminated leaf) and two sink compartments (e.g. grapes and roots), which are linked by phloem vessels of volume V0 for the source and V1, V2 for the sinks (Fig. 4) is considered. The source is divided further into mesophyll and phloem compartments, in order to examine the possibility that photosynthesis is regulated by one of these components of total leaf carbohydrate content (Correia et al., 1990). There are thus four dynamic carbon pools (mol C): Cm (source mesophyll), C0 (source phloem), C1 (sink 1 phloem), and C2 (sink 2 phloem). The corresponding phloem carbohydrate concentrations (mol C m-3 phloem) are given by si=Ci/Vi (for i=0, 1, 2). In the model of Minchin et al., s0 is a fixed parameter (Minchin et al., 1993); in the present model s0 is a dynamic variable.
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Carbohydrates are assimilated by the source mesophyll during photosynthesis at rate A (as described below in equation 5
), and actively loaded into the source phloem at a rate L which depends positively on the mesophyll carbohydrate content. Assuming Michaelis–Menten loading kinetics (Komor, 2000), then
 | (1) |
), the transport rates of photosynthate in the branches leading to each sink are given by
 | (2) |
 | (2b) |
0, 1 and 2 are the phloem transport resistances of, respectively, the common pathway and the branch to sinks 1 and 2, and where sij=si-sj denotes concentration differences. Equations 2a and 2b are based on the Münch hypothesis for flow driven by an osmotically-generated pressure gradient. Substituting the relationships si=Ci/Vi (for i=0, 1, 2) into equation 2 introduces an explicit dependence on the phloem volumes Vi, which can be partially absorbed into the definition of the transport resistances i, as described in the Appendix.
The transport rate of photosynthate in the common pathway is then given by
 | (3) |
Ui, the rate of phloem unloading in sink i (i=1, 2), is described by Michaelis–Menten kinetics with a maximum rate vmi and Michaelis constant kmi:
 | (4) |
). Carbon assimilation is then given by:
 | (5) |
is adopted here as a simple, empirical inhibitory response (Thornley and Johnson, 1990) for which no mechanistic interpretation is assumed.
The rate of change of each pool is then calculated as the difference between the relevant input and output fluxes:
 | (6a) |
 | (6b) |
 | (6c) |
 | (6d) |
Numerical simulations were performed using Euler's integration method with a time step of 10 min.
Evaluation of the three feedback hypotheses
For each feedback hypothesis measured and predicted, values of photosynthesis (A) were compared. Measured total leaf carbohydrate content was compared with the value of Cm+C0 predicted by the model. Model pools and parameter values were therefore expressed on an equivalent per unit leaf area basis. Table 2 (columns 2–4) gives the parameter values and initial pool sizes, corresponding to the best fit for A in each case. An attempt was made to minimize the number of fitted parameters by fixing the phloem unloading parameters and initial values of the carbon pools. In view of the scope of this study, no attempt was made to model the transitory period of stomatal opening and establishment of photosynthesis observed during the first hour of the photoperiod (Fig. 2).
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It was impossible to reproduce the data using hypothesis 1, according to which photosynthesis is regulated locally by leaf mesophyll carbohydrate Cm. Under this hypothesis, the predicted diurnal decline in A was always accompanied by a strong increase in total leaf carbohydrate content (Cm+C0), whereas the measured value of leaf carbohydrate content remained relatively stable from t=6 h until the end of the photoperiod, as illustrated in Fig. 5A
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) carbon assimilation rates, and simulated (—) and observed (•) leaf carbohydrate content for the data shown in Fig. 2In contrast, under hypotheses 2 (Fig. 5B
) and 3 (Fig. 5C), it was possible to find a reasonable fit of the model to the data for both A and Cm+C0, consistent with the conjecture that photosynthesis may respond to a more distant signal coming either from the source phloem supplying the sinks, or from the sinks themselves. The data are not sufficient to decide between hypotheses 2 and 3.
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Steady-state behaviour of the model
A sensitivity analysis of the model was performed, with the aim of exploring more widely the behaviour of the model under hypotheses 2 and 3, and of generating testable predictions that might be used to design future experiments. The simulations presented so far illustrate the transient, dynamic behaviour of the model over a 12 h period, under constant environmental conditions, starting from a given initial condition (Fig. 5
). Over a longer period (approximately 100 h), the model eventually reaches a steady state, independent of the initial conditions. Although the steady state is unlikely to be reached under natural conditions, it is useful to explore the sensitivity of the model in the steady state because this eliminates any transient dependence on the initial conditions, thus revealing more clearly the intrinsic properties of the model. (The same approach was adopted by Minchin et al., 1993).
Dependence of source activity on sink demand
Figure 6A shows the steady-state response of photosynthesis (A) to variation in the demand of either sink 1 (vm1) or sink 2 (vm2) under hypothesis 3 (feedback signal from sink 1). When the demand of the signalling sink (vm1) is varied, while that of the non-signalling sink (vm2) is held fixed, the model predicts that carbon assimilation is stimulated when demand is increased, and suppressed when demand is reduced. Significantly, the same qualitative response occurs when the demand of the non-signalling sink (vm2) is varied, while that of the signalling sink (vm1) is held fixed. For example, according to the model, an increase in the sink demand of grapes would lead to a stimulation of photosynthesis, even if it is the roots that generate the feedback signal. Alternatively, according to Fig. 6A, fruit removal from the plant, simulated by decreasing vm2 and 2 to zero, would result in a decrease in steady-state carbon assimilation to 5.2 µmol m-2 s-1. These results illustrate the interdependence of source and sink carbon pools in the model, mediated by the sink–source feedback mechanism and by linkage through the phloem transport network.
In quantitative terms, the feedback response of assimilation is predicted to be stronger when the demand of the signalling sink is increased, rather than the demand of the non-signalling sink. This difference in feedback response arises from the existence of the phloem resistance in the branch to the non-signalling sink (2). When this resistance is set to zero, the effect of varying sink demand on source activity is predicted to be the same for both sinks (data not shown).
Under hypothesis 2 (feedback signal from source phloem), the same qualitative behaviour is found for the feedback effect of sink demand on carbon assimilation (Fig. 6B). Quantitatively, the feedback response of A to varying sink demand is the same regardless of which of the two sinks is modified.
Carbon partitioning to sinks
Following Minchin et al., an investigation of the steady-state dependence of carbon partitioning between the two sinks (F1 versus F2) on various model parameters was made (Minchin et al., 1993). For the most part, it was found that the addition of a dynamic feedback mechanism on assimilation rate did not alter in any qualitative way the partitioning behaviour found before (Minchin et al., 1993). For example, under hypothesis 3, when the demand of sink 1 (vm1) is increased with vm2 held fixed, F1 increases and s1 decreases, while s2 varies to maintain an almost constant flux F2 into sink 2 (cf. Minchin et al., 1993, their Fig. 5). Also, when the source activity in this model (Amax) is decreased in a system with two inequivalent sinks of fixed demand, say, with vm1<vm2, the decrease in F1 is smaller than that of F2, reflecting the greater degree of saturation of sink 1, so that F1/F2 increases as source activity declines (Fig. 7A, cf. Minchin et al., 1993, their Fig. 9, in which s0 was varied).
With respect to partitioning behaviour, the only qualitative difference from the earlier model (Minchin et al., 1993) lies in the predicted response of F1 and F2 to variation in the common pathway resistance 0. Minchin et al. predicted a reduction in both F1 and F2 as 0 is increased (Minchin et al., 1993, their Fig. 7), because the source concentration s0 in their model is fixed. By contrast, in this model s0 is a dynamic variable. As a result there is little or no change in F1 and F2 as ~0 varies (Fig. 7B). An increase in 0 leads to a build-up of carbohydrate in the source phloem (s0) which compensates the increase in 0 such that the fluxes F1 and F2 and their total (F0) are virtually unchanged in the steady state. Artificially fixing s0 in the present model would lead to the behaviour predicted previously (Minchin et al., 1993).
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Discussion |
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The authors' main motivation for developing the present model has been the need for a simple representation of source–sink interactions for use in whole-plant models generally, and models of vine functioning in particular. In order to be of practical use in viticultural applications, and in view of current uncertainty regarding the underlying mechanisms, such a model should capture the essential features of source–sink interactions without attempting to represent the underlying biochemistry in great detail. The simple substrate-based source–sink transport models (Thornley, 1972
; Dewar, 1993; Minchin et al., 1993) therefore provide an appropriate starting point. In extending the model of Minchin et al. (Minchin et al., 1993), nothing was assumed about the nature of the feedback mechanism other than that it is mediated by plant carbohydrate levels.
The experimental results support earlier findings (Correia et al., 1990; Chaumont et al., 1994), that leaf photosynthesis undergoes a diurnal decline during which total leaf carbohydrate content remains relatively constant. Correia et al. attributed the decline in photosynthesis to photoinhibition (Correia et al., 1990), finding that both electron transport and carboxylation efficiency declined by 35% and 33%, respectively, between morning and afternoon under excessive light (1450 µmol m-2 s-1). However, photoinhibition is unlikely to play a major role under normal environmental conditions (Chaumont et al., 1994). Chaumont et al. found no evidence of photoinhibition during the mid-morning decline of photosynthesis, either in field-grown vines exposed to high PAR from morning under temperate or semi-arid climates, or in greenhouse vines under moderate temperatures (Chaumont et al., 1994, 1997).
A modest diurnal decrease in the CO2-saturated rate of photosynthesis was observed (11% between hours 2 and 10 of the photoperiod). This may or may not be associated with photoinhibition, although there was no significant change in the initial slope of the A/ci curve. Because the operating point of the source leaf was confined to the CO2-limited portion of the A/ci curve, the observed decline in photosynthesis under ambient conditions occurred largely through stomatal limitation, with photoinhibition unlikely to have played a significant role. The fact that these results were obtained under controlled, non-limiting environmental conditions suggests that stomatal conductance is regulated by some internal factor other than plant water stress. A similar decline in photosynthesis in well-watered grapevines has been observed before (Correia et al., 1990).
The sink–source feedback model is able to reproduce our experimental data, consistent with the view that the internal signal is linked to plant carbohydrate levels and thus to sink demand. These data and model simulations suggest that the feedback signal is unlikely to originate in the mesophyll (hypothesis 1), but may originate in the phloem of the source (hypothesis 2) or sinks (hypothesis 3). However, this conclusion is dependent on the mathematical form chosen to represent the negative feedback of carbohydrate levels on carbon assimilation (equation 5). It is possible that a different form, such as a threshold-delay response to mesophyll sucrose content, could reproduce the data. Hypothesis 1, therefore, cannot be excluded entirely, although the underlying mechanism for stomatal regulation by mesophyll sucrose content remains obscure.
Because only one leaf was illuminated in this experiment, one might have expected the plants to be source-limited rather than sink-limited. However, the following order-of-magnitude estimate of the sink–source activity ratio suggests otherwise. Daily assimilation by the source leaf was estimated at 4.3 mmol C d-1, based on an average photosynthetic rate of 10 µmol C m-2 s-1 over a 12 h photoperiod and a leaf area of 100 cm2. This level of source activity compares with an estimate of total sink demand of 4.2 mmol C d-1, based on a total sink dry weight of 5 g with a 50% C content, a daily relative growth rate during the maturation phase of 1% for grapes (Ollat and Gaudillère, 1998) with a similar relative growth rate assumed for roots, and assuming that sink growth and maintenance respiration account for 50% of total sink demand. Therefore, source supply and sink demand are of the same order of magnitude, consistent with a feedback mechanism acting to maintain a balance between source and sink activities.
One might still question the reality of a sink feedback mechanism, on the basis that if such a mechanism were operating in plants with a single illuminated leaf, then in plants with many illuminated leaves the surplus of assimilates would lead to an unreasonably large down-regulation of photosynthesis. To address this concern, the daily cumulative PAR intercepted by whole plants under natural (greenhouse) light conditions was estimated. The plants used in the experiments had 7–10 leaves with a mean total leaf area of 720 cm2. Using external PAR data collected at INRA Bordeaux during the 20 d prior to the experiment (January 2000, mean daylength=9 h), and applying a measured greenhouse transmission fraction of 0.62, it was estimated that the whole plants intercepted an average of 0.38 mol PAR d-1 during this period. This is less than the 0.52 mol PAR d-1 intercepted by the single illuminated leaf (incident PAR=1200 µmol m-2 s-1, photoperiod=12 h, leaf area=100 cm2). This comparison suggests that during the period concerned, the whole-plant source activity under natural light conditions was unlikely to have been greater than that for the single illuminated leaf, and therefore that the degree of down-regulation in whole plants would not have been unreasonably large.
Although other mechanisms for the observed diurnal decline in photosynthesis, such as photoinhibition, endogenous circadian rhythm, or local hydraulic signals in the leaf cannot be excluded entirely, carbon build-up in the sinks coupled with sink-to-source signalling appears to be the simplest mechanistic model able to describe these data.
It is suggested that, regardless of the underlying mechanism, a phloem-based feedback mechanism in the form of equation 5 provides a useful working hypothesis for incorporating source–sink interactions into plant models, which can be developed and tested further. The effectiveness of such a phloem-based feedback mechanism was illustrated by the predicted steady-state response of photosynthesis to sink demand (Fig. 6A, B). Under hypothesis 3, for example, the model predicts a positive feedback between sink demand and photosynthesis, regardless of the identity of the sink emitting the hypothetical signal, and of the sink whose demand varies. The model effectively describes a complete interdependence between all pools, i.e. between different sinks, as well as between sinks and source, mediated by sink-to-source signalling and by linkage through the phloem transport network.
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Conclusion |
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Incorporation of a dynamic sink–source feedback mechanism into the model of Minchin et al. (Minchin et al., 1993
) introduces a positive dependence of carbon assimilation on sink demand, but does not alter the manner in which this carbon is partitioned between competing sinks. On the basis of the data from this study, other mechanisms for the observed diurnal decline in photosynthesis, such as photoinhibition, endogenous circadian rhythm or local hydraulic signals in the leaf cannot be excluded. Nevertheless, the phloem-based feedback model presented here is in reasonable agreement with observations, is open to further development and testing, and provides a useful working hypothesis for incorporation into plant growth models. |
Appendix 1 – Volume dependence |
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In equations (2a)
and (2b) for the transport fluxes F1 and F2, the units of s0, s1 and s2 are mol C m-3 phloem and the units of i are mol C m-6 s. Substituting si=Ci/Vi (for i=0, 1, 2) where Ci and Vi are, respectively, the relevant phloem carbohydrate pool (mol C) and volume (m3), equations (2a) and (2b) can be written as:
 | (A1) |
 | (A2) |
~i=ix
(i=0, 1, 2) are rescaled transport resistances with units of mol C s. The rescaled parameters ~i, rather than the i, were fitted to the experimental data. In the absence of direct estimates of the volume ratios V0/V1 and V0/V2, different values of V0/V1 and V0/V2 were considered over the range 0.1 to 1.0. Satisfactory model fits as in Fig. 5B, 5C were obtained in all cases (data not shown). The results shown in Figs 5–7 are for V0/V1=V0/V2=1.
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Acknowledgments |
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We thank Nathalie Ollat and Philippe Pieri for useful discussions and comments.
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Notes |
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3 To whom correspondence should be addressed. Fax: +33 5 57 12 24 20. E-mail:dewar{at}bordeaux.inra.fr
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References |
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