•This paper presents two models of carrier-dependent long-distance auxin transport in stems that represent the process at different scales.
•A simple compartment model using a single constant auxin transfer rate produced similar data to those observed in biological experiments. The effects of different underlying biological assumptions were tested in a more detailed model representing cellular and intracellular processes that enabled discussion of different patterns of carrier-dependent auxin transport and signalling.
•The output that best fits the biological data is produced by a model where polar auxin transport is not limited by the number of transporters/carriers and hence supports biological data showing that stems have considerable excess capacity to transport auxin.
•All results support the conclusion that auxin depletion following apical decapitation in pea (Pisum sativum) occurs too slowly to be the initial cause of bud outgrowth. Consequently, changes in auxin content in the main stem and changes in polar auxin transport/carrier abundance in the main stem are not correlated with axillary bud outgrowth.
When a plant of an apically dominant species such as garden pea (Pisum sativum) is decapitated (its main shoot apex is removed) several previously inhibited buds are released to form new axillary shoots (e.g., Morris et al., 2005; Dun et al., 2006; Waldie et al., 2010). In pea, as well as many other plant species, this outgrowth of the axillary buds after decapitation can be substantially inhibited by the presence of auxin (Sachs & Thimann, 1967). However, this inhibition of bud outgrowth by exogenous auxin is rarely complete, as revealed by investigations of whether auxin treatment of decapitated plants can prevent the earliest developmental stage of bud release (Morris et al., 2005; Beveridge et al., 2009; Ferguson & Beveridge, 2009). Moreover, as auxin cannot move acropetally in buds, it must act indirectly. There is considerable evidence that auxin signalling affects cytokinin and strigolactone concentrations and that these can act directly in buds to affect outgrowth (Brewer et al., 2009; Ferguson & Beveridge, 2009; Dun et al., 2012). In addition, based on auxin canalization theory, the relative flux of auxin down the stem, compared with that from axillary buds, has been suggested to trigger bud outgrowth directly or indirectly (Prusinkiewicz et al., 2009; Balla et al., 2011; Domagalska & Leyser, 2011). Importantly, in all of these potential mechanisms of auxin regulation of bud outgrowth after decapitation, auxin transport and/or concentration in the stem adjacent to axillary buds is considered of critical importance (Prusinkiewicz et al., 2009).
Given the suggested importance of auxin depletion in the stem in stimulating the outgrowth of an adjacent bud after decapitation, it is essential that we test whether changes in auxin flux or concentration in the stem adjacent to axillary buds are actually correlated with the timing of initial bud outgrowth. Our previous study on polar auxin transport and bud outgrowth (Morris et al., 2005) is perhaps the only study that has incorporated the temporal and/or spatial resolution to test this correlation. Bud outgrowth and radiolabelled auxin transport (3H-indole-3-acetic acid) were measured at a fine temporal and spatial scale in tall garden pea plants where the growing buds and the shoot tip were separated by a relatively large distance (Morris et al., 2005). The experiments provided surprising evidence that the initial growth of axillary buds after decapitation is not correlated with a local depletion in stem auxin concentration (Morris et al., 2005). Experimental data indicated that the radiolabelled auxin moves slowly down the plant in a wave; the peak of this auxin wave moves at c. 0.8–1.0 cm h−1. Axillary bud growth commenced almost immediately after decapitation at the node just below the decapitation site (node 7) and within c. 5 h at node 2, which was c. 20 cm below the decapitation site (Morris et al., 2005). This indicates that the trigger for initial bud outgrowth moves down through the stem at c. 4 cm h−1 or faster. The conclusion was drawn that auxin moves too slowly to be the cause of the initial bud outgrowth, and that some other mechanism must be responsible (Morris et al., 2005; Ferguson & Beveridge, 2009). These conclusions are supported by observations that while decapitation promotes bud outgrowth in pea, stem girdling or pharmacological methods that cause equivalent changes in stem auxin content to those caused by decapitation do not always induce bud outgrowth even in wildtype plants (Morris et al., 2005; Ferguson & Beveridge, 2009).
Given that these results do not support the long-standing dogma that changes in auxin concentrations or transport near buds is the initial, albeit indirect, trigger for outgrowth, we decided to use a modelling approach to investigate the process of long-distance auxin transport in more detail. For example, although the peak of the auxin wave moved at c. 0.8–1.0 cm h−1, the front appeared to move faster and the tail slower, and we wanted to test whether this kind of phenomenon might be able to provide an explanation for how changes in auxin concentration may affect bud outgrowth. Long-distance auxin transport studies in pea and Arabidopsis are also not consistent with an auxin transport limitation/canalization hypothesis because stems of these species appear to have a robust ability to transport both normal and relatively large amounts of auxin (Brewer et al., 2009). Again, we need to understand whether the long-distance auxin transport in stems is consistent with a carrier-limited auxin transport system such as is required for canalization. Various models of auxin transport within the apical meristem (e.g., Heisler & Jönsson, 2007; Kramer, 2008), and within and between metamers (Prusinkiewicz et al., 2009) have been presented, but we are not aware of any existing models that represent long-distance auxin transport along a whole stem and which are tested against empirical auxin transport data.
In this paper, we present and contrast two models that represent auxin transport along a stem at different degrees of abstraction. The more detailed intracell model allowed us to explore the effects of different underlying biological assumptions on predicted patterns of auxin transport and signalling. In particular, we identify three contrasting scenarios about underlying processes of auxin transport. These scenarios vary in the extent to which passive transport, active transport and limiting transporter numbers impact on the transport profile observed. We show that each scenario results in different predicted patterns of auxin transport, with varying implications for hormonal signalling, and discuss which scenario is most likely to be relevant to auxin transport in stems, based on the match with experimental data. The scenario that best matches biological data from pea and Arabidopsis is also consistent with our simple segment-based compartment model.
Model I: segment-based compartment model
This simple model represents the stem as a series of connected compartments, with each compartment si representing the amount of auxin in one 2-mm segment of the stem. As we assume the cross-sectional stem area to be constant, this can also represent the concentration per segment volume. The segment length of 2 mm was chosen to correspond with original data (Morris et al., 2005). Auxin is transported basipetally in shoots by an active transport mechanism. Thus, in the model, auxin can move down from one segment to the next, but not up, as shown in Fig. 1. This means that
for i = 1, 2,…, n, where n is the number of compartments included in the model, fi is the auxin flux from compartment i to compartment i + 1 (i.e., the amount of auxin moving from one compartment to the next in one time step) and si is the amount of auxin in segment i. We assumed that, for all segments, the amount of auxin moving out of a given segment into the next is a linear function of the amount of auxin in the segment, so
for i = 1, 2,…, n, where α (the transfer rate, proportion of auxin moved to the next segment per min) is a constant parameter. See Table 1 for a summary of model parameters and variables.
Table 1. Summary of parameters and variables of segment-based compartment model and particle-based mechanistic intracell model
Auxin transfer rate (proportion per min)
Proportion of cell with transporters
Relative diffusion speed of auxin particles when not being transported
Number of transporters in a cell
Filament density (proportion of cell occupied)
The number of particles per transporter
Time a transporter spends inactive while transporting auxin (s)
Amount or concentration of auxin in segment i
Auxin flux from segment i to segment i + 1
Time for auxin particle to move from one cell to the next
The compartment model is thus based on the following assumptions or hypotheses:
1 The transfer rate of both endogenous and radiolabelled auxin is constant within the experimental system considered, and is thus independent of concentration.
2 Over a given time period, auxin flux from one segment to the next is a constant proportion of the total auxin in the tissue of the upper segment.
3 Auxin in any segment is subject to a small amount of metabolism at a constant proportional rate.
4 Metabolized auxin is no longer available for transport; during the experimental period, radiolabelled metabolites remain in the segment in which they are produced.
5 Radiolabelled auxin is loaded into the top segment according to a loading function that depends on time.
6 Endogenous auxin is loaded into the top segment at a constant rate unless decapitation occurs, in which case no endogenous auxin is loaded.
The model was parameterized by simulating the experimental situation where a pulse of 3H-indole-3-acetic acid, described here as marked auxin, travelled down through the stem, and then adjusting the α parameter (the transfer rate) to fit the experimental data that showed the marked auxin peak moving at c. 0.9 cm h−1 in the first 2 h. That is, we used the early time points to derive α and then validated the model by running it out to the later time points (Fig. 2a, additional data from Morris et al., 2005; Supporting Information, Movie S1). This gave a value of α = 0.075 (per min). Based on this simple calibration of one parameter, which was first done by eye, and later confirmed by least-squares fitting, our model predictions closely matched the experimental data across all time points, capturing the timing and the spread of the peak as it moved down the stem (Fig. 2).
Based on experimental evidence, we assume that decapitation does not affect the process of auxin transport through the stem (Morris et al., 2005). Using the transfer rate parameter value, α = 0.075, we simulated decapitation and investigated the relationship between the threshold concentration of auxin needed to trigger a decapitation response (such as bud outgrowth) and the rate at which the resulting auxin depletion moves down the stem following decapitation, by assuming the response occurs whenever the unmetabolized auxin concentration falls to some fixed percentage of its original value. Note that metabolized radiolabelled auxin is included in the simulated auxin concentration shown in Fig. 2(a) to enable comparison with the empirical data. This amount of metabolized auxin has only a minor effect on the model as it is kept at low amounts comparable with experimental data (Beveridge et al., 2000; Morris et al., 2005).
Model II: particle-based mechanistic intracell model
The cellular mechanisms involved in auxin transport and action are complex (Petrášek & Friml, 2009; Vanneste & Friml, 2009) and hence the salient features relevant to the model are listed here. Basipetal auxin transport is predominately governed by the asymmetric distribution of efflux carriers known as PIN proteins (Friml et al., 2002, 2003; Benkováet al., 2003; Reinhardt et al., 2003; Blilou et al., 2005). In stems, PIN proteins cycle rapidly between the plasma membrane at the bottom of the cell and endosomes located within the cytoplasm (Geldner et al., 2001; Dhonukshe et al., 2007), and may thus, at a conceptual level, ‘pick up’ auxin particles within the cell and actively transport them towards and across the cell wall in the direction of auxin transport.
The overall idea represented in the intracell model is that both the auxin molecule and an active transporter have to be in the right place at the right time in order to connect and transport auxin. The model represents the stem as a series of connected 0.02-mm-high cells, and auxin as discrete (individual) particles. The model runs with a discrete time step of 0.1 s. Within the cell, auxin particles are assumed to move by diffusion, which is represented in the model by a random walk, where at each time step a particle can move either up or down. Particles cannot move beyond the boundaries of the cell without the assistance of a transporter. When an auxin particle enters the part of the cell where transporters are cycling, it may be ‘picked up’ by an active transporter and taken to the bottom of the cell where it is exported. The transporter is inactive during this transport period and hence cannot pick up new auxin particles. When a transporter releases the auxin particle it has been transporting, it then becomes active (available for auxin) and starts cycling again. Figs 3 and 4 show a series of frames from an animated visualization of the filament cell simulation where all of the auxin particles start at the top of the cell (also see Movie S2). As not every possible process can be represented in a single model, choices must be made about which to represent and which to ignore. The metabolism of auxin was not included in this model. If we assume that metabolism occurs at a constant rate (constant proportion of total) at every point in the stem, and this rate itself is not affected by decapitation, then it will just scale the total amount of auxin rather than significantly affect the general shape of the resulting unmetabolized auxin waves or the relative drop in total auxin content with time after decapitation. Moreover, indole-3-acetic acid is quite stable in the polar auxin transport stream of pea (Beveridge et al., 2000). We did not explicitly and separately include aspects such as active auxin import and auxin diffusion in the apoplast and into a cell (Kramer, 2008). However, conceptually these can be seen as part of the diffusion and active transport processes that are included, as they would be likely to have similar effects on overall transport.
The model has a number of parameters, summarized in Table 1. The parameter sp is the relative diffusion speed of the auxin particles when they are not being transported by transporters; more precisely, the size, relative to the length of the cell, of the step up or down in the random walk. The number of transporters per cell is represented by nt. If the number of particles per cell is fixed, as it was for some of our analyses, the number of transporters can also be defined by the relative parameter npr (the number of particles per transporter), but otherwise, npr will vary as auxin concentrations vary. The distance to which transporters circulate within the cell is termed filament height, and the ratio between filament height and the height of the cell is termed filament proportion, fp. The parameter fd represents the density of the filaments (along which auxin transporter and endosomes are assumed to cycle). The length of time (the ‘delay’) that a transporter remains inactive while transporting the auxin to the next cell is represented by d.
All of the parameters affect the rate of movement of the auxin particles through the cell and the time taken to enter the next cell. For example, the time taken for an auxin particle to move by diffusion into the filament zone depends on the ‘filament proportion’fp, and on the particle diffusion speed sp. The probability that an auxin particle within the filament zone is picked up by a transporter depends on the filament density fd and the proportion of total transporters that are currently active, which in turn depends on d, the length of time a transporter spends inactive while transporting the auxin to the next cell.
1 A cell consists of a lower zone (fp of the cell) where transporters cycle and an upper zone (1 − fp of the cell) where they do not cycle.
2 A cell contains a fixed number of transporters, nt.
3 Auxin particles within the upper zone only move through the passive process of diffusion.
4 Auxin particles within the lower zone move through diffusion but may also be picked up by cycling transporters. The probability of getting picked up is equal to fd multiplied by the proportion of transporters that are currently cycling.
5 If an auxin particle is picked up then it is transported to the bottom of the cell, and then across the cell wall. During this period d, the transporter that picked it up is inactive and unable to pick up other auxin particles.
6 At the end of this period the auxin particle is released (across the cell wall) and the transporter becomes active, starts cycling and is able to pick up auxin particles again.
7 Released auxin particles (not associated with transporters) move across the cell wall, then into the top of the next cell, and then diffuse passively within that cell until they are picked up by another transporter.
Analysis of intracell model
The parameters of the intracell model were varied through a wide range of possible combinations in order to see how different parameter value combinations (representing different assumptions about underlying biological processes) would result in different predicted patterns of auxin transport at both the scale of a single cell and the scale of the stem. As a base for our analysis, we defined the set of parameter values, sp = 0.06, npr = 1.0, fp = 0.5, fd = 1.0, and d = 1, to be the standard parameter value set, since it results in the simulated peak travelling at 9.2 mm h−1, which corresponds to the auxin speed observed in our biological experiment.
For analysis at the scale of the cell, the statistic considered to represent auxin movement is tt or the time taken in seconds for a particle to travel from one cell to the next. For this analysis, a single cell was simulated, and particles leaving the cell were ‘recycled’ and placed back at the top wall of the cell. This recycling does not affect the model in any way other than to simplify computation and analysis, by ensuring that the number of particles in the cell is constant. A record was made of the time taken to travel through the cell each time a particle was recycled. Since it is stochastic, for each particular set of parameter values the model was run for long enough to collect a large sample (n = 2000) of tt values in order to get a reliable estimate of the distribution of tt corresponding to these parameter values. We then looked at the way that the distribution of these tt‘travel times’ or ‘time per cell’ values depended on model parameters. We considered all model parameters, but focused on npr, the ratio between the number of auxin particles and the number of transporters in a cell, as this is the only parameter that varies between cells and across time during a single model run. For analysis at the scale of the stem, the simulation was scaled up to include many consecutive cells, with particles being passed to the following cell after being released by a transporter, instead of being recycled.
Initial and boundary conditions
For the pulse situation, marked auxin concentrations were assumed initially to be zero in all segments or cells. Auxin was then added to the very top segment or cell, according to a sinusoidal ‘input function’ that starts at zero, increases to a maximum at t = 50, and reaches zero again at t = 100. Since we had limited knowledge of whether this input function was a good representation of reality, we also conducted a robustness or sensitivity analysis by determining to what extent input functions of a different shape made a difference to the behaviour of the auxin signal. For the decapitation situation, auxin concentrations were assumed initially to be at a constant maximum value in all cells or segments and the input function was constant at zero, that is, no auxin was supplied to the system for t > 0. Since all results are shown as percentages of the maximum, and the number of transporters per cell, nt, is defined via the relative parameter npr, the actual values used make no difference to the results.
Model I: compartment model
In the segment model, the wave of auxin flattens and spreads out as it moves down the stem (Fig. 2a). The input function and transfer rate α can be set such that the output matches the widespread experimental data at each time point (Fig. 2a, Morris et al., 2005; Movie S1). Decapitation causes the trailing edge of the wave to become less steep over time (Fig. 2b). This provides the variable relationship between signal-response activation speed and auxin threshold concentration observed in Fig. 5. For example, if the threshold is 0.1% (the auxin concentration has to fall by 99.9% to trigger a response), the signal takes 100 min to start, then progresses relatively slowly, speeding up slightly as it moves down the stem. If the threshold is 99.9% (the auxin concentration only has to fall by 0.1% to trigger a response) then the signal starts immediately, progresses relatively quickly and slows as it moves down the stem. Even with this very high threshold, the rate of long-distance signalling is relatively slow compared with the timing of bud outgrowth.
Model II: cell scale
The distribution of time taken in a cell, tt, for the standard set of parameter values used was approximately Poisson-distributed (Fig. S1) with a mean tt value of 7.8 s (corresponding to 462 cells h−1, or 9.2 mm h−1). As the relative number of transporters decreases (which means npr increases; fewer available transporters), the mean time taken per cell increases, as does the variability (Fig. 6a). The underlying cause of the increasing mean is a large spread in the tail of the distribution (Fig. 6b). Note that increasing the relative number of transporters (decreasing npr below 1.0) here has no affect on the mean, standard deviation or distribution, as the transporters are not limiting at these values.
Varying filament proportion, fp, and delay, d, also affect tt (Fig. S2). As fp increases, both the mean and variance of tt decrease, because auxin particles need to diffuse over smaller distances before they reach a carrier. As d increases, the mean of tt increases but the variance remains constant because the time taken and variation involved in finding a carrier remain the same. Similar plots for the other parameters show predictable results; tt decreases and becomes less variable as the filament density, fd, increases and tt also decreases and becomes less variable as the speed, sp, increases (data not shown). The effect of varying transporter availability (npr) after changing other parameters, such as lower filament density or lower filament proportion, or a combination thereof, produced different results to varying it at standard values (Fig. S3).
Varying parameters of the intracell model also had interesting effects on particles moving through a whole stem, particularly in relation to variability and transporter limitation. The shape of the waves does not change much over time for the standard values which represent low intrinsic variability and high transporter availability (Fig. 7a). Values representing more intrinsic variability cause the wave to spread out over time as for the segment model (Fig. 7b). Note that wave speed is much slower here too. By contrast, values representing limiting transporters cause the front of the wave to spread out significantly over time and the back of the peak to incline sharply (Fig. 7c,d).
Different relationships between signal–response–activation distance over time and response thresholds were found for different scenarios (Fig. 8 and data not shown). For the standard values where the shape of the wave does not change much over distance (Fig. 7a), the signal speed is relatively independent of the threshold (Fig. 8a). By enhancing variability, the signal speed depends even more on the threshold (Fig. 8b), because the back of the wave shape changes more over time (Fig. 7b). When transporters are highly limiting, the speed of a signal triggered by falling auxin concentrations does not depend on the threshold at all (Fig. 8c), as the shape of the back edge does not spread at all over time, maintaining its steep shape and moving at a constant rate (Fig. 7c,d). However, if we were to assume that an activation response is somehow being triggered by the front edge of the wave, then the signal–response–activation speed is strongly dependent on the threshold (Fig. 8d), as the shape of the front edge spreads so much (Fig. 7c,d). While this is not biologically relevant for auxin depletion and bud outgrowth following decapitation, it has other possible implications, as discussed later.
Effects of varying input function
We found that, for the compartment model, the shape of the input function had no effect on the speed at which the peak of the auxin pulse moved or on the shape of the auxin wave. For the intracell model, the shape of the input function also had no effect on the speed at which the peak of the auxin pulse moved. By contrast, it did affect the shape of the auxin wave for standard values (where the shape is maintained) but considerably less so for cases where intrinsic variability was high or the number of transporters was limiting (data not shown).
Complementary approaches at different levels of abstraction were used to produce models that matched closely the observed transport profile and dynamics of radiolabelled polar auxin transport in stems. We identified three contrasting scenarios that predict different patterns of long-distance transport in shoots. We conclude that the profile of radiolabelled auxin transport in stems is best explained by a system with considerable intrinsic variability and where the number of transporters is in excess in comparison with the endogenous auxin concentrations in the stem. As such, any difference in auxin transport in stems is likely to be a consequence of differences in auxin supply or in the intrinsic properties of auxin transport, independent of auxin concentration or the quantity of transporters such as PIN proteins. The other main finding from this modelling study is that, while the speed of a signal based on falling auxin concentrations following apical decapitation does depend on the threshold concentration required to trigger a local response, it is still unlikely that auxin depletion following apical decapitation could occur fast enough to be the initial cause of bud outgrowth at basal nodes in pea.
The simple compartment model with a constant transfer rate matches biological data
In addition to an input function, the compartment model has just one parameter and this can be adjusted to provide a good fit to the experimental data. This makes the compartment model a good descriptive and predictive model of behaviour at the scale of the stem. As we discovered, based on the intracell model discussed later, this compartment-based segment model is a good fit to the data because of two main properties. First, it does not represent a scenario where the number of transporters is limiting. Second, transporting a proportion of auxin from one whole segment to the next segment at each time step effectively represents the average sum effect of stochastic processes occurring at the cellular level. The assumption in this model that the auxin transfer rate is constant (the amount transported is proportional to concentration) matches our main finding that the auxin transport system in stems is essentially independent of auxin concentration. As discussed later, this must be true for the stem even though auxin content may regulate components of the auxin transport system, such as via transcriptional regulation of PIN by auxin (Vieten et al., 2005).
Comparison of different auxin transport models
Exploring auxin transport for a range of parameter values in the more detailed intracell model revealed three contrasting scenarios. (This was possible because of the greater biological detail in this model, as discussed in Notes S1) These three scenarios are best discussed by considering that within a pulse wave of auxin there are three kinds of particles which vary in behaviour in different situations: ‘straggler’ particles found in the back part of the wave; ‘leader’ particles found in the front or advance part of the wave; and ‘conformist’ particles found in the middle or main body of the wave.
We initially chose to explore the extent to which the number of auxin transporters limits the auxin transport. In the first scenario, active transport is the main factor in auxin transport (particles spend more time being actively transported than in passive diffusion or other processes) and the number of transporters is limiting. In this case, the shape of the front edge of the wave changes significantly over time but the shape of the back edge does not. This is a direct result of the limiting effect of the transporters. Straggler particles at the back of the wave will find themselves in cells with a relatively low number of particles, and thus a high number of transporters relative to particles. These stragglers will thus be transported more quickly and tend to rejoin the main body of auxin particles. On the other hand, conformist particles within the main body will find themselves in cells with a relatively high number of particles, and thus a low number of transporters relative to particles. These conformist particles will thus on average be transported more slowly and tend to be rejoined by the stragglers. Finally, leader particles that find themselves out in front of the main body of auxin particles will find themselves in cells with a relatively low number of particles, and thus a high number of transporters relative to particles. These leaders will thus be transported more quickly and continue to move even further out in front of the main body. This is the case illustrated in Fig. 7(c) and (d) where the front edge of the pulse wave spreads out dramatically but the back edge is held steep and constant.
A second scenario is that active transport with little variability is the main factor in auxin transport (particles spend most of their time in active transport) and the number of transporters is not limiting. In this case, there is no constraint slowing particles that are in cells with a large number of other particles, and so stragglers, conformists and leaders will all tend to behave the same. There is also little variability among particles in the same cell in the time taken to move to the next cell. Both the back and front edge of the wave of auxin will thus maintain their shape over time as the wave moves down the stem (Fig. 7a).
A third scenario is that the number of carriers are not limiting and the process at the cellular level most affecting auxin transport is inherently stochastic or variable, such as diffusion. As in the second scenario, which also involves nonlimiting carriers, there is no constraint that slows particles that are in cells with a large number of other particles, and so stragglers, conformists and leaders will all tend to behave the same. However, because there is a large amount of variability among particles in the same cell, in the time taken to move to the next cell, some stragglers will tend to fall even further behind and some leaders will draw even further ahead. Under this scenario, both the back and front edges of the wave of auxin will thus spread further and further over time as the wave moves down the stem, and the wave will flatten (Fig. 7b). Although there is considerable stochastic variation at the level of the behaviour of individual components such as auxin molecules in this scenario, the emergent property of the pulse of auxin transport through a cell file is a relatively smooth wave.
The model with excess transporters and intrinsic variation is the best comparison with biological data
The shape of the auxin wave observed in the experimental data (Morris et al., 2005; Fig. 2) seems to best fit the predictions of the third scenario (Fig. 7b) and the simple segment-based compartment model because the wave flattens and spreads out over time and because the back of the peak is not sharp and the leading edge does not race forwards. This indicates that variable processes such as passive diffusion, transporter cycling and/or the time taken for an auxin particle to be bound by a transporter could be among the most important factors in determining the rate of auxin transport. Although we did not model it explicitly, trafficking of auxin through the cell or any other intrinsically variable time-consuming process could also lead to similar outcomes.
The emergent property of the third scenario of intrinsic variation and unlimited carriers also explains why the simple segment-based compartment model is such a good fit with experimental data. The segment model has a constant transfer rate, as would be the case where transporters are not limiting. As only a proportion of auxin is moved at any time step, the concept of stragglers, conformists and leaders is retained in this model as an indirect consequence of transferring a fixed proportion of auxin from each cell at each time step. The fact that the segment-based compartment model is able to match the experimental data adds support to the hypothesis that auxin transport is not carrier-limited and involves considerable intrinsic variation.
Based on auxin addition experiments, we suggested previously that long-distance auxin transport in intact stems of pea and Arabidopsis (e.g., Brewer et al., 2009) is unlikely to be limited by the relative abundance of auxin transporters. This is supported here where modelling transporter limitation showed a sharp back edge and a long leading edge that did not match data and where modelling unlimited transporters produced a close fit with biological data. This is not to suggest that that PIN proteins or auxin transporters do not limit the speed of auxin transport. What this modelling study shows is that it is very unlikely that the natural abundance or quantity of any carriers limits auxin transport (at concentrations at or slightly above concentrations typically found in plants) and so changes in auxin concentration will not affect the speed of transport or the amount of auxin that may be transported (Brewer et al., 2009). It is possible that this situation in the stem may be in contrast to the shoot apical meristems where PIN abundance may be critical to control phyllotaxis. It is important to emphasize that the direction of auxin transport is of critical importance in phyllotaxis.
Although our experimental and modelling data suggest that long-distance auxin transport in stems is not limited by the abundance of carriers, and does depend on some intrinsically variable process within the cell, it is possible that active transport is the inherently variable component that determines the emergent properties of auxin transport. Indeed the actual mechanism by which PIN transports auxin is unknown and it is possible that the time taken to transport an auxin particle through the cell wall is more variable than we hypothesized. Another inherent source of variation may relate to the extent to which PIN transporters within the cell cause auxin to be retained for variable periods (Wabnik et al., 2011). Active transport occurs laterally as well as vertically, and different particles will undergo different amounts of lateral transport; this will also cause variability among particles in vertical transport times (i.e., in polar auxin transport). However, in these cases, transport time would still not be dependent on auxin or PIN concentrations at or near the values observed in intact plants.
Auxin transport is too slow to account for bud outgrowth at considerable distance, even if the change in auxin content required is small
Despite the differences in predicted patterns of auxin transport, the segment model and the intracell model under each of the three scenarios all indicate that auxin depletion following apical decapitation occurs too slowly to be the initial cause of bud outgrowth at basal nodes in pea. In scenario three (intrinsic variability) and the segment model, which are the best representations of the actual biological process, the signal speed is quite strongly dependent on threshold (Figs 5, 8b). This is indeed a property that might be expected for a hormonal signal. As shown in Fig. 5, even if the signal threshold was so sensitive that a 0.1% drop in auxin triggered a response, the signal speed would only be about twice as fast as the main peak and still less than half the speed required to be the cause of initial bud outgrowth. In the other scenarios we modelled, the signal speed would move at the same speed as the main wave and would be less dependent on the threshold required for a response (Fig. 8a,c). Consequently, in all three cases, a reduction in auxin concentrations in the stem adjacent to a bud following decapitation could not be the cause of initial bud outgrowth, thus supporting original interpretations of auxin transport data (Morris et al., 2005; Ferguson & Beveridge, 2009).
Carrier-mediated systems could lead to rapid responses to increased hormone concentrations at a distance
The modelling results presented here show that a positive hormonal signal could move many times faster than its measured chemical peak, if it is triggered by a relatively small increase in hormone concentrations, if its transport is dominated by active transport and if the number of associated transporters is limiting (Fig. 8d). In the case of phyllotaxis, where auxin transport is suggested to be limited and the distances are comparatively short, changes in auxin concentration could be propagated and perceived very efficiently.
Does auxin enhance its own transport in the stem?
Experimental data from Arabidopsis and pea show that the profile of the leading edge of the auxin transport wave is similar at different concentrations of applied auxin, including those that would cause enhanced overall stem auxin content (Brewer et al., 2009). However, it is known that auxin promotes PIN expression and hence can promote its own transport (Vieten et al., 2005). According to our modelling, if auxin enhances its own transport in stems, it must do so at a rapid timescale so as not to measurably affect the bulk auxin flux profile and hence not cause a backlog or alteration in auxin transport even at the leading edge of the wave of enhanced auxin content. If it took any significant amount of time for the increased auxin concentration to cause an increase in carrier numbers, there would still be some delay and thus a build-up or ‘backlog’ of auxin particles within cells at the front edge of the wave. As this is not observed in biological experiments (Brewer et al., 2009), we propose that auxin enhancement of PIN protein abundance in stems is in accordance with a relationship that ensures PIN proteins are always in excess even as auxin concentrations vary.
Further implications for auxin biology in the stem and shoots
Our results indicate a near unlimited potential of the main stem to transport widely varying amounts of auxin. This information, together with the fact that axillary buds can be promoted after decapitation well before auxin concentrations are depleted locally, suggests that a difference in auxin content or transport between bud and stem is not a likely trigger for bud outgrowth in these plants. Although decapitation may change PIN gene expression within a short time period at nodes close to the site of decapitation (Balla et al., 2011), no changes in auxin transport are expected in the stem adjacent to distal buds within the same timeframe, even though buds at these nodes may commence growth (Beveridge et al., 2000; Morris et al., 2005). Instead, if auxin canalization is the stimulus for bud outgrowth (Prusinkiewicz et al., 2009; Balla et al., 2011), it may be driven by enhanced auxin content in the bud itself. Enhanced auxin content in buds as the main driver for bud outgrowth is able to account for the phenotypes described by Prusinkiewicz et al. (2009). Future studies on shoot branching should explore the role of bud-specific genes known to affect bud outgrowth, such as TB1/FC1/BRC1 (Braun et al., 2012; Muller & Leyser, 2011) and their role in regulating the cell cycle in buds and/or enhancing local auxin flux. Moreover, this should be considered in a framework that considers additional plant hormones, strigolactone and cytokinin (Dun et al., 2012), and hormone-independent effects of decapitation (Morris et al., 2005) such as changes in sinks (Morris & Thomas, 1968).
Strigolactone mutants have been reported with enhanced auxin transport (Bennett et al., 2006). Based on modelling evidence from this study and on empirical data (Brewer et al., 2009) we suggest that this effect of strigolactones is most likely a result of an altered supply of auxin to the polar transport stream from the shoot tip, rather than of a change in transport pathway along the length of the stem. The predicted outcome of enhanced loading at the shoot tip would be a general increase in auxin concentrations along the stem (Fig. S4), as is observed empirically with radiolabelled auxin (Bennett et al., 2006; Brewer et al., 2009). Enhanced loading of auxin into this transport stream in these mutants should be tested directly but could be caused by reduced auxin metabolism or enhanced lateral transport of auxin from the treated apical leaves. Feedback up-regulation of auxin content and signalling sometimes reported for strigolactone mutants is consistent with this hypothesis (Brewer et al., 2009).
Interestingly, the constant overall transport rate in the main stem, combined with its intrinsic variation at the cellular level, would cause a dampening effect of differences in the auxin signalling over distance because of the mixing effect of individual molecules during the inherently variable polar auxin transport process. A pulse or depletion in auxin from a closer branch would potentially have a greater affect than one from a distal branch, even though both branches may produce the same quantity of signal transported in the same long-distance pathway.
Our findings here appear to be relevant to the full length of our experimental system, including partially and fully expanded internodes of different ages and up to 6 cm from the shoot tip (Morris et al., 2005; Brewer et al., 2009). Similar rates of IAA transport of c. 1 cm h−1 at warm temperatures are observed across a range of species and a range of transport distances and hence internodes of different developmental ages and cell sizes (Goldsmith, 1977). If auxin transport across the membrane of individual cells is not the rate-limiting step in auxin transport, it could explain why internodes with cells of different sizes and maturity may transport auxin at a similar rate, even though there may be a different number of membranes that must be traversed over a given distance (Eliezer & Morris, 1980). This would provide the plant with a reliable communication system based on distance rather than cell number. It would have obvious advantages for control of growth of axillary buds at physical distances from apical auxin sources, rather than being dependent on effects of cell numbers.