New understanding on phloem physiology and possible consequences for modelling long-distance carbon transport

Authors

  • P. E. H. Minchin,

    Corresponding author
    1. ICG-III Phytosphaere, Forschungszentrum Juelich, D 52425 Juelich, Germany;
      Author for correspondence: P. E. H. Minchin Tel: +49 2461 61 8687 Fax: +49 2461 61 2492 Email: p.minchin@fz-juelich.de
    Search for more papers by this author
  • A. Lacointe

    1. INRA, UMR PIAF, Site de Crouelle, 234 av. du Brézet, F-63039 Clermont-Ferrand cedex 2, France
    Search for more papers by this author

Author for correspondence: P. E. H. Minchin Tel: +49 2461 61 8687 Fax: +49 2461 61 2492 Email: p.minchin@fz-juelich.de

Summary

Most current models of assimilate carbohydrate partitioning are based on growth patterns observed under a range of experimental conditions, from which a set of empirical rules are derived to simulate partitioning. As a result, they are not good at extrapolating to other conditions; this requires a mechanistic approach, which only transport-resistance (TR) models currently provide. We examine an approach to incorporating recent progress in phloem physiology into the TR approach, which leads to a ‘minimalist’ Münch model of a branched system with competing sinks. In vivo whole-plant measurements have demonstrated that C-flow rates are dependent not only on the properties of the sink, but also on the properties of the whole transport system, and the detailed dynamics of this behaviour is mimicked by the proposed model. This model provides a sound theoretical framework for an unambiguous definition of sink and source strengths, with sink priority being an emergent property of the model. Further developments are proposed, some of which have already had limited application, to cope with the complexity of plants; the emphasis is on a modular approach, together with the importance of choosing the appropriate scale level for both structure and function. Whole-plant experiments with in vivo measurement of the phloem dynamics will be needed to help with this choice.

Introduction

All higher plants possess a very highly developed vascular network linking the plant organs. This network is essential for transfer of both organic and inorganic moieties, as well as being a signalling network coordinating organ function. Xylem is responsible for transfer of water from roots to the shoot, as well as mineral ions absorbed from the soil and various organic compounds synthesized within the roots. Phloem is the major pathway for transfer of carbohydrates from their site of synthesis in mature leaves to sites of utilization (growth and respiration) and storage, as well as other leaf-derived moieties, and for mineral ion recycling. There is often an interplay between these two flows, for example at times of low fruit transpiration phloem flow into fruit brings more water than is used in growth, so the excess is dealt with by a xylem backflow (Lang, 1990).

This review addresses only the phloem system, as carbohydrate is the major substrate for all plant growth and consequently a common starting point for resource-based modelling of plant growth. Over the past 10 years there has been a huge increase in interest in phloem function (reviewed by van Bel, 2003). The purpose of the present review is to suggest how some of this understanding may be relevant to modelling plant growth, and especially to understanding of carbon partitioning between competing sinks. We concentrate on transport and partitioning, which are closely interlinked, the weak point in functional–structural models of plant growth (Lacointe, 2000).

Mechanistic modelling of any system needs to be done with a thorough awareness of the processes involved. But it also needs to be done with carefully applied knowledge of what to include and what to leave out. If too much detail is incorporated, a model can be made to perform in almost any way – especially when there are many parameters estimated by fitting to observations. Data-based modelling, developed by Young (1999), provides a method of identifying the dominant processes to include in a minimalist (parsimonious) description of a complex system, and is central to the approach suggested in this review.

Mechanistic modelling of carbon transport and partitioning

A short overview of current growth models

The currently accepted theory of phloem transport was put forward by Ernst Münch in 1928. This states that phloem transport involves a conduit consisting of a file of sieve tubes lined with a semipermeable membrane. Phloem loading within the source region generates a high solute concentration, resulting in osmotic flow of water into the sieve tube, creating a high hydrostatic pressure. At the sink, solute unloading results in a much lower local hydrostatic pressure. The pressure gradient between source and sink drives a bulk flow of sieve-tube sap from source to sink. Münch did not expand on the detail of phloem loading or unloading.

Although the Münch theory is widely accepted, very few of the many growth models published to date are based on it. The transport-resistance (TR) models (proposed by Thornley, 1972; Thornley & Johnson, 2000) are the most mechanistic. The other three main classes of C-allocation models are empirical models; growth rule-based models; and source–sink relationship-based models, all three incorporating partitioning algorithms derived from observational data (Marcelis et al., 1998; Lacointe, 2000; Le Roux et al., 2001). The algorithmic approach is effective in simulating C partitioning within the range of conditions appropriate to the calibration data, but is much less successful for extrapolating into other conditions, and gives no insight as to the processes involved. Several authors have questioned the need to refer to transport processes to simulate carbohydrate partitioning adequately (Heuvelink, 1996; Bancal & Soltani, 2002). But DeJong (1999) pointed out that ‘… dry matter partitioning does not direct the growth of the tree but is the result of the growth and development of the organs that make up the tree’. We believe that to advance beyond the calibration data sets to different locations and environments, and to model alternative management practices, a mechanistic understanding of partitioning is essential.

Although an extended discussion is outside the scope of this paper, an approach commonly found in C-based models needs to be mentioned here, involving partial delocalization of sink function by first removing respired C from the pool available for distribution between the sinks (Lacointe, 2000). For example, Grossman & DeJong's (1994) PEACH model first removes all the plant's requirement for maintenance respiration from the pool of available C, and what remains is then partitioned between the plant organs. This idea comes from the work of Crapo & Ketellapper (1981), who found that under starvation conditions, root-maintenance respiration continued while growth respiration reduced and even ceased. This demonstrates that the limited C within the root cells was utilized preferentially in maintenance respiration, but shows nothing about its transport to the root. The carbohydrate utilized in maintenance respiration had to be transported to the root, which would have involved phloem transport. Partitioning within the cells between maintenance and growth respiration, possibly by a similar mechanism, such as the one proposed below for phloem transport, occurs after the carbohydrate has been delivered to the root, not before. There is no reason to suggest that an organ's C requirement for maintenance is imported independently of other C needs. It is the total carbohydrate requirement of a sink, supplied by a single phloem pathway, that drives phloem flow and partitioning. Sink competition involves a sink's ability to attract carbohydrate, and this includes that which is respired.

Transport-resistance models

The Münch theory requires the sieve tubes to be lined with a continuous semipermeable membrane. The laminar flow between source and sink, described by Poiseuille's equation, is proportional to the hydrostatic pressure difference between source and sink (ΔP), and inversely proportional to the pathway length (L) (Minchin et al., 1993; Lacointe, 2000). With Van't Hoff's osmotic equation describing the relationship between solute concentration C and the osmotically generated hydrostatic pressure P, the solute flow rate can be expressed as:

image((Eqn 1) )

where the resistance R is proportional to L. This is the basis of the TR models formulated by Thornley (1972). Assuming a similar TR description for nitrogen flow between root and shoot, combined with bisubstrate (C, N) growth kinetics, the well known shoot : root balance follows (Lacointe, 2000 and references therein). Dewar (1993) showed that this qualitative balance still holds when more complex N cycling involving both xylem and phloem is included. Distance effects, such as a sink being supplied by the nearest available source, are qualitatively explained by the resistance term (Lacointe, 2000). Thornley (1998) has since put forward a strong argument that such a minimal TR model needs to be the starting point for all more complex models, as this incorporates the ‘only two significant processes: transport and biochemical conversion’.

However, the TR model has not been widely used so far. It has been used to simulate tobacco growth (Wann et al., 1978; Wann & Raper, 1984), and to simulate the dynamics of shoot : root ratio and of reserve storage/mobilization in pruned trees (Berninger et al., 2000). Deleuze & Houllier (1997) developed a single-substrate (C) version to simulate stem radial growth (reaction–diffusion model). More recently, Nikinmaa et al. (2004) extended the reaction–diffusion approach to branched trees. At the plantation or ecosystem level it has been used for forest or grassland growth simulation (Rastetter et al., 1991; Thornley, 1991, 1999; Luan et al., 1996).

One reason why TR models have remained essentially theoretical is the difficulty in estimating the model parameters. However, a more fundamental limitation may be involved: the organs are considered as ‘bulk’ compartments, and phloem-loading and -unloading processes are ignored, making concentrations in equation 1‘average’ substrate concentrations of that organ, which do not reflect the concentrations within sieve tubes. Hence the sieve-tube dynamics cannot be appropriately expressed. As a result, equation 1 describes a diffusive rather than a mass-flow process; while qualitatively preserving many features, it significantly alters the orders of magnitude of speeds and flow rates, as mass flow is much more efficient than diffusion at moving material. A consequence is that the resistance R in equation 1 is effectively an inverse diffusive coefficient, so cannot be derived from anatomical investigation of the vascular tissues, which would be a way to estimate sieve-tube resistance parameters, as done by Sheehy et al. (1995). Furthermore, as within-organ compartmentalization is ignored, selective cross-membrane processes, or the resulting local gradients between sieve tubes and surrounding tissues, which are believed to be of major relevance to actual C fluxes (see below), cannot be incorporated. However, these limitations originate in the description of the organs as bulk compartments, not in the TR concept itself. These limitations can be overcome using a slightly more detailed description, as described below.

Recent advances

Recent progress in phloem physiology

Recently there have been significant advances in understanding of the processes involved in phloem transport, from the route of loading, realization that the pathway is not just a passive conduit, through plasmodesma involvement, to identification and localization of the protein transporters involved (Lalonde et al., 2003; van Bel, 2003). Techniques of molecular biology have led to an explosion in detail about the membrane transporters involved (sugar, water and ions). Here we briefly discuss only those advances that we see as relevant to modelling plant growth.

Several refinements to the Münch theory have been proposed, but the basic mechanism is still thought to be bulk solution flow driven by an osmotically generated pressure gradient. Phloem loading is now understood to involve either an apoplastic or a symplastic step, or a combination of these, between the site of carbohydrate synthesis and the sieve tubes involved in long-distance transport. Apoplastic loading involves flow of carbohydrate, usually sucrose, from the leaf parenchyma into the apoplastic space in the vicinity of the vascular tissue, from where it is actively taken up across the cell membrane into the sieve-tube companion-cell complex. With symplastic loading, carbohydrate flow from parenchyma tissue to the sieve-tube companion-cell complex is entirely through the plasmodesmata, not involving crossing a membrane, probably driven by diffusion. Many temperate-climate woody plants, including most of the tree species, are symplastic loaders (Turgeon & Medville, 1998). This is a very active research area with much interest in the function of plasmodesmata and what drives carbohydrate flow from parenchyma into the sieve tubes within symplastic phloem-loading species. While the specific details of generation of the sieve-tube pressure gradient is unlikely to be important in modelling plant growth, there is evidence for a correlation between the loading route and subsequent net lateral loss along the transport phloem (van Bel, 1996).

Phloem unloading into most tissues is currently thought to be symplastic, through plasmodesmata linking the cells in the sink region. So the end of the symplastic flow is not the terminal sieve elements, but within the receiver cells, with the sink osmotic pressure being kept low by metabolic utilization of the carbohydrate or conversion into less osmotically active polymorphic forms (starch, fructans). There is a growing body of evidence that the region of highest flow resistance is not within the transport phloem linking sources and sinks, but is within the symplastic pathway of the receiver cells (Gould et al., 2004a). Even within developing seeds, where the daughter tissue is symplastically isolated from the parent, making apoplastic transfers essential, phloem unloading from the terminal sieve tubes is symplastic into the seed coat tissues. But, whatever the route of unloading, the kinetics will be saturable.

Early in the study of phloem physiology, high metabolic rates associated with the vascular tissue were thought to be evidence for the driving force for phloem transport being generated along the entire length of the sieve tubes. This is now thought to be associated with the continuous simultaneous leakage and reloading along the transport pathway (Minchin & Thorpe, 1987; Thorpe & Minchin, 1996; van Bel, 2003). The sieve-tube solute concentration can exceed 1m and, as biological membranes are not perfect, it is not surprising that there is leakage into the surrounding apoplast. Reloading is essential if the conduit is not to lose its entire contents before delivery to the terminal sink, although this apoplastic carbohydrate is essential in maintaining stem tissues, and is also probably an important route for the storage and remobilization of carbohydrate within stems, trunks and roots of all plants. Plant species using symplastic phloem loading within the source leaves have been found to have a relatively higher potential for utilization of leaked photosynthate by the stem than found in species utilizing apoplastic loading. If this interpretation is correct, the former will have a tendency for greater lateral sinkiness than the latter, which will tend to favour terminal sinks more. Such a difference could give rise to substantial differences in relative growth rates and affect architectural traits (van Bel, 1996). A direct consequence of leakage and remobilization along the entire length of the transport phloem is to buffer changes in sieve-tube content brought about by changes in supply or demand. For example, a sudden reduction in supply is counteracted by reduced unloading flux with continuation of reloading from the apoplastic pool. This tends to decouple the source and sink for as long as there is an apoplastic pool able to be reloaded into the sieve tubes. (Minchin et al., 1983). The apoplastic pool may be replenished by remobilization of carbohydrate stored within the nearby ray or parenchyma cells.

The sieve elements are intimately associated with the companion cells, both being derived from a common mother cell. During maturation the sieve elements lose almost all their cytoplasm and organelles to become highly dependent on their adjacent companion cell, forming the sieve element–companion cell complex. Phloem loading involves carbohydrate transport into the sieve element–companion cell complex and through specialized plasmodesma between sieve elements and companion cells. The plasmodesmata allow the passage of a wide range of molecular species between these cells. Phloem sap has been shown to contain at least 150 different proteins: mRNA and RNA fragments, as well as carbohydrates and small molecules (van Bel, 2003). The roles, if any, played by this wide range of molecules in the coordination of plant growth and development are still quite uncertain, and these molecules may simply be the products of unselective loss from companion cells (Oparka & Santa Cruz, 2000), although there is growing evidence that short strands of RNA transported within the phloem sap are involved in gene silencing within the sink tissue (Yoo et al., 2004). Ayre et al. (2003) have shown that, while smaller sized metabolites from the companion cells do enter the translocation stream indiscriminately, subsequent membrane leakage and selective reloading has the effect of cleaning out such material from the long-distance pathway. Phloem-transported macromolecules have the potential to alter sink function by modulating the unloading kinetics, but this is still to be demonstrated.

Numerical modelling of Münch's basic hypothesis has demonstrated that, while this mechanism is able to account for both the observed specific mass flow and the transport speed within herbaceous plants, files of sieve tubes longer than several metres are not able to support observed flow rates (Thompson & Holbrook, 2003). Lang's (1979) suggestion of short files of contiguous sieve tubes acting as a relay, with unloading and reloading necessary to move photoassimilate from one short conduit to the next, would overcome this problem. This has not yet been confirmed, and will be difficult to distinguish from continuous unloading and reloading along the pathway.

New research tools

Molecular biology has added a large number of new methods for looking at the detailed processes involved in phloem transport, as well as providing transgenic plants which enable the whole-plant response to the complete knockout, or the downregulation or upregulation of specific genes. These approaches have confirmed the importance of apoplastic phloem loading; of continuous reloading along the pathway; and of the sucrose transporters within sink tissues. A large body of new mechanistic detail supporting Münch's original hypothesis has resulted, which has recently been reviewed by van Bel (2003).

For modelling purposes we need measurements of the dynamic behaviour of the intact phloem system, which can be obtained only by noninvasive methods, as the phloem system is very sensitive to mechanical probing. Hence noninvasive measurement techniques are playing an increasingly important role in phloem physiology. Confocal microscopy allows in vivo observation of phloem function (Wright & Oparka, 1997), while nuclear magnetic resonance has been used to image functional phloem (Peuke et al., 2001). Short-lived isotope techniques, specifically C-11, allow in vivo measurement of phloem transport with a very fine time resolution (seconds), but is limited by its short decay time (t½ 20.4 min; Minchin & Thorpe, 2003). In vivo measurement has been fundamental to the development of the minimalist Münch-based model discussed in the following section.

Aphids feed on phloem sap, so their ability to find and connect their stylets into individual functioning sieve tubes has long been exploited as a means of sampling phloem sap. Wright & Fisher (1980) showed how a manometer can be attached to the stylet of an aphid; more recently Gould et al. (2004b) used the pressure probe continuously to monitor the hydrostatic pressure within functional sieve tubes while applying experimental treatments to the phloem system. This work has produced results consistent with Münch's hypothesis of osmotically driven bulk flow.

Another potentially useful tool in phloem studies is metabolic control theory, developed by Kacser & Burns (1973) and Fell (1997), to provide a sound theoretical framework for studies of controlling flow through complex metabolic systems. This resulted in radically new ways of understanding flux control, and has demonstrated that the concepts of limiting factors and bottlenecks are not appropriate. A top-down form of this theory (Quant, 1993), involving lumping the parts of the overall system, has been developed. Using this approach on a simple plant system consisting of a single source and a single sink, it was found that the source leaf had 80% of the control of photosynthate flux between leaf and sink (Sweetlove et al., 1998; Farrar & Jones, 2000), and this high control by the source leaf was maintained for plants deprived of N, cooled, or at low light levels (Sweetlove & Hill, 2000). This approach has not been used to look at flows with multiple sinks; such work is badly needed, as there is a general belief that it is the sinks that determine C flows into the individual sinks.

A possible mechanistic model of carbon transport and partitioning

The model

In this section we outline a very simple TR model of phloem transport from one source to two sinks (Minchin et al., 1993), which is easily extended to more complex source/sink configurations, incorporating bulk flow within a perfect conduit connecting to multiple saturable sinks; we then propose further possible developments of this approach.

The model is based on:

  • the pathway of C flow is the phloem;
  • the osmotic driving force is generated by active loading of solutes, and all the carbohydrate utilized by a sink for growth, storage or respiration must first be transported to that organ via the phloem;
  • sink function is nonlinear in the substrate.

The model consists of a source with a specified sieve-tube solute concentration (phloem loading was not part of the model initially proposed, but this has been rectified by Bancal & Soltani, 2002), linked to two sinks via a common pathway resistance representing the leaf petiole and stem, which then splits into two separate pathways supplying each sink. Each pathway has its own value of flow resistance, and each sink is described by a saturable Michaelis–Menten utilization func-tion (Fig. 1a; Minchin et al., 1993) and associated resistance. The pathway is assumed to be nonleaking, and osmotic water flow occurs only at the source of sink regions. Assuming that solution flow through the sieve tubes is laminar, and therefore described by Poiseuille's equation which states that flow is proportional to the pressure difference and Van't Hoff's osmotic equation stating that the osmotic pressure is proportional to the solute concentration, then for steady-state flow we obtain a pair of equations involving the unknown solution concentrations at the two sinks. As these are nonlinear simultaneous equations numerical methods are used to solve for the sink concentrations, then the flow into each of the sinks can be calculated, and flow out of the source is the sum of these. For full details see Minchin et al. (1993). With more sources and sinks, linked by any pathway geometry, a similar set of nonlinear simultaneous equations involving the steady-state sink concentrations is similarly derived.

Figure 1.

One-source, two-sink phloem model. (a) Schematic diagram of single source and two sinks connected via a common resistance and a resistance associated with each sink. (b) Photosynthate fluxes F1 and F2 into the two competing sinks supplied by a common source, depicted in (a), as a function of source photosynthate concentration. The partitioning fractions P1 and P2 were calculated from these fluxes as the ratio of a specific to the total flux. The calculation was based on the model of Minchin et al. (1993) with a common pathway resistance of 1.0 × 1013 mol m−6 s and two sinks defined by (R, Km, Vm) taking values of (1 × 1013, 100, 1 × 10−9) and (1 × 1013, 100, 2 × 10−9) (units: Km, mol m−3; Vm, mol s−1).

With the one-source, two-sink geometry shown in Fig. 1a, we find that this approach predicts a change in solute distribution between the two sinks when the source supply is varied (Fig. 1b), with the sink described by the larger Vm (sink 2, Fig. 1a) receiving a decreased fraction of the total flow with decreasing supply. This is exactly what is seen on shading a plant shoot, with shading causing a change in partitioning between root and shoot (Minchin & Thorpe, 1996). The experimental result is for shade to decrease partitioning to the root, so we identify the root as the sink with the higher Vm.

The experimental plant was a barley seedling. Photosynthate produced in the mature leaves would first have been transported into the stem, where the phloem flow bifurcates between the roots and the growing parts of the shoot (meristem and young leaves), giving a geometry similar to that shown in Fig. 1a, with the shoot being one sink and the root the other. In this plant example, it is unlikely that the flow resistance associated with roots and shoot will be equal, but simulation with the proposed model shows that equality of the sink resistances is not a necessary prerequisite to obtaining a distribution change.

Grusak & Lucas (1985) reported changes in partitioning to two developing leaves of sugar beet when slow cooling was applied to the petiole of the labelled source leaf. They suggested that the change in partitioning was induced by a signal perceived by the crown region at the site of bifurcation of the translocation stream between the two sinks. Pickard et al. (1993) reported a similar redistribution between alternative sinks, pea pods in this case, when phloem transport from a labelled petiole of a pea plant was temporally interrupted by cold or electric shock. But Minchin et al.'s (1993) model predicts that changes in the common pathway resistance will induce changes in the distribution of flows between the two sinks. It is accepted that cold causes partial blockage of the phloem pathway, so this appears to be a simple explanation of this observed change in partitioning between alternative sinks.

The experiments mentioned above demonstrate that flow into a sink is not simply a function of the sink, but depends on the source, pathway and sink properties. That is, flow into a sink is a property not just of the sink, but of the entire pathway (cf. metabolic control theory). Also, the model mimics the changes in distribution of flow between shoot and root that were seen in barley seedlings. This immediately raises the issue: what do we mean by source and sink strength?

Sink strength, sink priority and source strength

The first quantitative definition of sink strength, suggested by Warren-Wilson (1972), was the flow per unit time into the sink. But, as we have seen above, this is a function not just of the sink but also of the source and pathway. If flow rate into a sink is not a property of the sink alone, then the sink is not usefully described by the observed flow into it. Wareing & Patrick (1975) suggested a definition of sink strength as the potential import into a sink when the source and pathway are not limiting import. In the above formalism, this is given by Vm. DeJong & Grossman (1995) used fruit thinning to eliminate source limitations, in order to measure the genetically determined potential maximum growth rate of peach fruit, which in our model would be represented by Vm. In practice measurement of Vm is not straightforward, as it involves estimation of an asymptotic flow which is probably never actually reached. This is well illustrated in Fig. 1, where even with a high solute supply neither sink has reached its potential Vm level, demonstrating that it can be difficult, if not impossible, to induce maximal flows experimentally.

Sink priority describes the preferential supply of available photosynthate between competing sinks. When supply is decreased, the sink that suffers the greater reduction in supply is the sink of lower priority. Shading the shoot of our experimental barley seedling (see above) would have reduced the supply of available carbohydrate, so the observed decrease in fractional supply to the root was interpreted as the root having a lower priority than the shoot. But in our model the sink with the lower priority is the sink with the higher Vm (Fig. 1b). Hence, for our model to have the same behaviour as our experimental plant, we were forced to identify the root as the sink with the higher Vm. This hypothesis immediately offered us the opportunity to test it – if we experimentally reduce Vm of the root to a value below that of the shoot, the model predicts that we would reverse the shoot : root priorities. Vm is a measure of the amount of enzyme present, while Km is a property of the enzyme itself and is independent of the amount present. Also, Vm can be expected to be sensitive to temperature. So we can experimentally reduce Vm simply by pruning off part of the root, or by cooling the entire root. Both these predictions have been verified (Minchin & Thorpe, 1996), giving this model further credibility.

This simple model offers a sound basis for the concept of sink strength and sink priority, both being concepts that have been used within the source–sink literature but, because they lacked a theoretically rigorous definition, have often been confused and used in contradictory ways.

The existence of a sink hierarchy has been shown by experiments involving manipulating the supply or demand of photosynthate (Wright, 1989; Wardlaw, 1990), and this is summarized by the priority rank ordering. The currently accepted ordering is:

seeds > fleshy fruit parts = shoot apices and leaves > cambium > roots > storage

That is, seeds have the highest priority and carbohydrate storage the lowest. There has been confusion in the horticultural literature between sink strength and sink priority, with suggestions that a high-priority sink is associated with a large sink strength. Seeds are high-priority sinks, but seeds are small, and at no stage will they be a large sink. Fruit flesh growth rates are much larger than that of the seeds, so the sink strength (Vm) of the flesh must be greater than that of the seeds, but the seeds have the higher priority for available carbohydrate.

Bancal & Soltani (2002) correctly stated that representing the source by a constant solute concentration is not realistic. This concentration can be expected to be affected by the rate of flow into the sinks. They suggested using a source activity, defined as the supply rate which, for conservation reasons, must equal the total flow through the common pathway. In this approach there is no need explicitly to describe phloem loading. The solute concentration within the sieve tubes at the source affects the flow, and this must take a value to ensure that the photosynthetic supply rate matches the rate of flow out of the sink region. This source supply rate can then be defined as the source strength. It is independent of the sinks or the pathway, and uniquely defines the source's ability to supply photosynthate. And, as pointed out by Bancal & Soltani (2002), the total flow into all the sinks must always equate with this definition of source strength.

Further developments

Bancal & Soltani (2002) also extended Minchin et al.'s (1993) model to include changes in sap viscosity with solute concentration, which is not a small effect, with viscosity changing by a factor of two over a solute concentration range of 0.5–1.0 m. They went on to demonstrate that realistic values of pathway resistance are important only for physiologically unrealistically low sieve-tube concentrations (< 0.1 m). This is consistent with the large body of data supporting the conclusion that transport resistance does not normally exert any control on sink growth (reviewed by Gifford & Evans, 1981); Minchin et al. (1993) showed that flow resistance usually has little effect on phloem flow into sinks functioning at or near saturation, the exception being for large changes in resistance, as seen experimentally with cold inhibition. There is experimental evidence to support the idea that sinks normally operate at or near saturation (Farrar & Williams, 1990; Farrar & Minchin, 1991), in that they are not able immediately to utilize extra carbohydrate made available by removing other sinks. It takes time for sinks to alter their ability to utilize more photosynthate (Minchin et al., 1997), which has been attributed to expression of more enzyme associated with carbohydrate metabolism.

Bidel et al. (2000) have developed a model of root growth based on similar concepts, with root growth represented by extension and lateral growth driven by nonlinear unloading. With their model, by altering the unloading parameters they were able to mimic growth of a tap root verses a fibrous root, and to produce determinate and indeterminate root growth. While no attempt was made to fit their model to specific root-growth data, they showed that such a simple model was able qualitatively to display a range of quite different growth behaviours that are actually encountered within a range of plants. No similar model for shoot growth has been attempted, but our expectation is that this would be equally successful.

The level of plant detail needed for the work of Bidel et al. (2000) cannot possibly be usefully carried through to whole-plant modelling, so a modular approach, as found in Daudet et al. (2002), is needed. In this work a leaf, segment of stems and fruit are treated as individual modules that are then assembled to produce a realistic architecture and the module equations, incorporating water relations and xylem interactions, linked via their boundary conditions. This approach would link very neatly with the current advances with L-system models of plant growth and development, which also work with plant modules (or metamers) (Prusinkiewicz & Lindenmayer, 1990; Allen et al., 2004). But technical issues involving the huge amount of calculation involved quickly arise. Daudet et al. (2002) took their approach as far as current computing power permitted; while developments in alternative software may temporally ease the computational issues, they will continue to limit this approach.

The modular approach reduces computation significantly, so investigations are needed into working with larger modules such as clusters of leaves, longer lengths of stem, entire root systems, etc. Simplification of the mechanistic processes may be also be appropriate: issues of how much mechanistic detail needs to be known in order to describe growth at the whole-plant level are important, as the temptation to build in too much detail is always there. Whole-plant experiments must be the deciding point as to the dominant processes needed to describe the macro-behaviour of metamers and their mutual interactions. In vivo measurement of the interaction dynamics within whole plants and concepts of data-based modelling leading to parsimonious models (Young, 1999) become essential, requiring a change in modelling philosophy from a reductionist to a holistic approach.

Ancillary

Advertisement