Since the ion channels have a much smaller diameter than the pollen tube lumen, the ion movement in the channel will be convective, whereas it is diffusive in the pollen tube cytosol. The ratio of convective to diffusive motion is given by the Peclet number . In the absence of detailed knowledge on the calcium sink in the apex, we assume that the number of available sites is constant in time, such that the binding reaction depends only on the cytosolic calcium concentration.
The total electrical conductance of the apex was evaluated at pS  , . The factor of 10 between the total conductance of the apex and the conductance of individual calcium channels suggests that the number of channels on the tube apex is very limited. It appears that the stress in the membrane must be significantly smaller than the stress in the cell wall. We thus assume that the stress on the membrane is set by the joint movement of the plasma membrane and the cell wall.
Due to a no-slip condition at the membrane-cell wall interface, the lipids forming the membrane move at the same velocity as the pectin molecules forming the cell wall. The stress in the membrane is set by the velocity of the lipids and the strain rate of the membrane. The tension in the membrane must obey The use of the surface viscosity renders eq. For a strain rate of 0. When soft cell wall material is secreted, it is incorporated into the existing cell wall.
We therefore assume that the cell wall extensibility, i. The rate of change of the extensibility is modeled as a mixing process. This modeling strategy is a simplification of the dynamics in the cell wall that include binding to existing polymers and transport driven by turgor pressure . After a series of secretion events, the cell wall viscosity is assumed to be the average of the original highly viscous portion and the added softer less viscous portion. However, this process will continuously reduce the viscosity, without accounting for enzyme mediated hardening that occurs during cell wall maturation.
A crucial maturation process in the pollen tube cell wall is the de-esterification of pectins by the enzyme pectin methyl esterase that is initiated after the deposition of the polysaccharide at the cellular surface  ,  , . The removal of methyl-groups leaves negatively charged carboxyl groups, that in the presence of calcium ions, leads to the gelation of the polymers . All simulations were carried out using the simple Euler algorithm. We model the growth of pollen tubes by coupling the Lockhart equation for the growth rate to the dynamical equations for the cytosolic calcium concentration and the secretion rate which feeds back onto the cell wall rheology, thus controlling the growth rate theory section; Fig.
The identification of material delivery secretion and hydrodynamics Lockhart equation as the two primary processes controlling tube growth was proposed by Geitmann . We find that the growth rate oscillates with a peak that precedes that of the cytosolic calcium concentration by 2 s Fig.
This is consistent with the magnitude and the order of events measured in oscillating Lilium longiflorum pollen tubes . C Cytosolic calcium concentration in the apex. D Cell wall viscosity. E Phase shifts are visualized by superimposing normalized growth rate black , cytosolic calcium concentration red and cell wall thickness green. To study the effect of global turgor changes on the average growth rate, simulations were carried out with different values of the turgor pressure.
Above 0. For turgor values greater than 0.
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No matter how high the turgor pressure, the average growth rate cannot increase significantly beyond v c and is thus only weakly dependent on the turgor. This difference in behavior can be explained by a mathematical analysis of our threshold model. It shows how the upper bound on the instantaneous growth rate set by our threshold mechanism prevents the average growth rate from increasing despite an increase in the turgor.
The theoretical relationship between the turgor and the average growth rate is obtained by a slow-fast analysis  of our model see Text S1 and yields C Maximal growth rate as a function of the critical growth rate at which the calcium channels open. It appears that the dependence of the average pollen tube growth rate on the turgor pressure P depends on the ratio. If this ratio is much smaller than 1, the average growth rate will be close to v c and essentially independent of P. However, if the ratio is much greater than 1, the average growth rate will be directly proportional to P, i.
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Reducing the secretion rate R results in a decrease of the amplitude of the oscillations. This behavior is due to the threshold dynamics of our model, i. The average growth rate cannot increase beyond some critical value set by the exocytosis mechanism despite a drastic increase in the turgor pressure. For small growth oscillations, the growth rate will always be close to its maximal and threshold value, despite increases in the turgor: the average growth rate will thus be insensitive to the turgor value.
These dynamics can be illustrated by a ball bouncing on the ground. In this case, the threshold for the vertical position of the ball is the ground. The bigger the bounces, the longer they last, and the further the average vertical position is from the ground. Conversely, a ball with small bounces will have an average vertical position very close to the ground. A robust prediction of threshold dynamics is that when the oscillation is amplified, the oscillation period increases, and the average value is moved further from the threshold value corresponding to a decrease in the case of the average growth rate.
In our simulations, a decrease in the period does accompany the increase in average growth rate as the turgor pressure is increased Fig.
This exact behavior was observed upon buffering the pH in the cell wall in Lilium longiflorum pollen tubes, which caused the oscillations to increase both in duration and amplitude . It is inherent to this threshold model that the maximum growth rate of the pollen tube cannot exceed by much the growth rate that induces the opening of the calcium channels Fig. Once the growth rate reaches the value that induces the opening of the calcium channels, a sequence of events is triggered that reduces the growth rate to its minimum value.
As will be seen in greater detail in the following section, the minimal value of the growth rate depends on the amount of calcium ions entering the cytosol when the calcium channels open. In order to test the predictions of our model, we used in vitro growing lily, tobacco and petunia pollen tubes to measure the average growth rate and the period of the growth rate oscillation as a function of the osmotic value of the growth medium.
Previous studies had shown that increasing the external osmotic pressure induces a decrease in the cytosolic turgor pressure in pollen tubes .
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We thus altered the growth medium by modifying either sucrose or mannitol concentrations and recorded the growth rate of germinated pollen tubes Fig. We observed that an increase in the osmotic value of the medium modestly reduced the average growth rate and modestly increased the period of oscillations of individual tubes Fig. These results are in agreement with previous studies of growth under conditions of changing osmotic values . An increase in growth rate upon a turgor pressure increase is predicted by our model as embodied by equation 14 see Text S1 for the derivation.
Accordingly, the size of the increase in growth rate and decrease in period depend, among others, on the turgor pressure and the amplitude of the growth rate oscillations. When the turgor is only slightly higher than the yield pressure, the model predicts a linear relation between turgor pressure and average growth rate. On the other hand, for high turgor pressures, the model predicts that the average growth rate will become asymptotically independent of turgor as the turgor increases and the growth rate oscillation amplitude decreases.
Such an absence of direct correlation between the average growth rate and turgor was reported in Lilium longiflorum . Experimental measure of the instantaneous pollen tube growth rate in Nicotiana tabacum in a medium with 10 mM sucrose A , For mannitol, the data are plotted against the corresponding sucrose concentrations of equimolar solutions. The graphs compile new data with re-analysed data of experiments performed for .
In order to gain a better understanding of the pollen tube oscillator  ,  , we simulated transient perturbations of the turgor pressure. We showed above that the average and maximal values of the growth rate are not very sensitive to global, long lasting changes in the turgor pressure. Is this also true for transient short changes of the turgor pressure? One can predict that if the pressure transient is of sufficient amplitude and with a timescale much shorter than that of the other quantities, the growth rate should correlate with the pressure during the transient.
We simulate a transient increase in the turgor pressure by rendering the turgor time dependent Fig. The turgor is maintained at 0.
This is modeled by the following expression for the turgor pressure given in MPa B Tube growth rate. C Apical cytoplasmic calcium concentration.
The cell wall expansion rate upon a transient stress perturbation cannot be modeled using eqs. For changing values of the pressure, and thus the stress exerted on the cell wall, the strain rate obeys the augmented growth equation that had been established based on the Lockhart equation  Under conditions of slowly changing cell wall stress such as those modeled in the previous section, the augmented growth equation reduces to the Lockhart equation.
Using eq. The perturbation of the turgor pressure is accompanied by a sudden influx of calcium ions Fig. We conclude that contrary to the behavior of the average growth rate upon permanent changes in turgor, for very short and transient turgor variations the instantaneous value of the growth is directly proportional to the turgor.
Crucially, this confirms that the cell wall expansion in pollen tubes obeys the Lockhart i. These simulations help us to understand the lack of sensitivity of the average tube growth rate to global changes in the turgor. Our model suggests that the maximal value of the tube growth rate depends on the membrane strain rate at which the calcium channels open Fig. Specifically, the cell wall expansion rate cannot increase much beyond the strain rate that will open the channels.
If turgor is very high and the oscillations in the growth rate have a small amplitude, the average growth rate will be close to the threshold growth rate and relatively independent of the turgor value. This is despite the fact that the instantaneous growth rate correlates with the turgor during transient turgor changes and despite the fact that the main driving force for the growth is the turgor pressure, as embodied by the Lockhart equation.
Thus our model predicts that the average growth rate will be insensitive to the turgor value if the period of the oscillations is short, but that it is sensitive when the period of oscillations is long. Evidence for both situations, independence and dependence of the average growth rate on turgor, is available Figs. Our explanation for the absence of correlation between the turgor and the average growth rate is based on an upper bound on the growth rate set by the exocytosis mechanism. In this section, we investigate how calcium concentration affects exocytosis and how calcium uptake can increase the period of oscillation.
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To study the effect of an increased uptake of calcium ions into the cytoplasm, we simulate a raise in the cytosolic calcium concentration at a precise moment. The uptake transient, modeled by the equation This can be explained by the response of the growth rate to the calcium uptake Figs. The calcium ions induce an increased exocytosis activity and thus induce a thickening of the cell wall.
This increase in exocytotic activity produces an immediate drop in the growth rate, since the growth rate is proportional to the cell wall stress which is inversely proportional to the cell wall thickness eq. This is consistent with experimental data demonstrating that photoactivation of caged calcium in the cytoplasm of growing pollen tubes causes a transient reduction in the growth rate  , . While the calcium triggered exocytosis can also be expected to cause an overall softening of the cell wall through the addition of new, highly methyl-esterified pectic polymers, our simulations suggest that the increase in thickness has the more immediate effect on the growth rate.
The drop and the subsequent minimum in the growth rate depend on the amount of cell wall material added and are thus directly proportional to the amount of calcium taken up Fig. The time necessary to reach the subsequent maximum depends directly on the minimal value, and thus on the calcium uptake. Most of our simulations were carried out using a constant turgor pressure, implying our assumption that turgor pressure is mostly constant throughout tube growth and not responsible for the oscillations in the tube growth rate.