Modeling pollen tube growth

Abstract

The frequency and amplitude of oscillatory pollen tube growth can be altered by changing the osmotic value of the surrounding medium. This has motivated the proposition that the periodic change in growth velocity is caused by changes in turgor pressure. Using mathematical modeling we recently demonstrated that the oscillatory pollen tube growth does not require turgor to change but that this behavior can be explained with a mechanism that relies on changes in the mechanical properties of the cell wall which in turn are caused by temporal variations in the secretion of cell wall precursors. The model also explains why turgor and growth rate are correlated for oscillatory growth with long growth cycles while they seem uncorrelated for oscillatory growth with short growth cycles. The predictions made by the model are testifiable by experimental data and therefore represent an important step towards understanding the dynamics of the growth behavior in walled cells.

Plant cell growth and expansion is driven by the turgor pressure which provides the physical force necessary to expand the rigid cell wall. While turgor is an essential prerequisite for growth, no direct correlation was found between the magnitude of the turgor pressure and the growth rate of the cell in the pollen tube, a unidirectionally expanding cell.¹ This observation seems counterintuitive given that a simple physical relation, formulated by Lockhart's equation, predicts a linear relationship between turgor and growth rate.² The question of whether and how turgor and water movement regulate the growth process in pollen tubes has therefore garnered a lot of attention and is at the center of considerable controversy.^3-8 The relevance of this discussion is not limited to the case of the pollen tube, since the mechanical principles that govern this process apply to all walled cells including those of plant, fungal and bacterial origin.^9-14

In a recent modeling study,¹⁵ we showed that if the physical properties of the cell wall are allowed to vary, experimental results are recovered for the correlation between the turgor and the growth rate of oscillatory pollen tubes. Specifically, turgor and growth rate are correlated for oscillatory growth with long growth cycles while they are uncorrelated for oscillatory growth with short growth cycles. Since the Lockhart equation was used to obtain these numerical results, this shows that experimental data produced by us and others are consistent with the role of the turgor as a driving force despite the apparent lack of correlation between growth rate and turgor pressure. Additional simulations showed that a direct relation between instantaneous growth rate and pressure can be observed when the turgor pressure varies faster than the cell wall's physical properties.¹⁶

Our theoretical results stem from the particular mechanism chosen to model the dynamics of the cell wall, but should also hold for other threshold mechanisms controling cell growth. In the pollen tube, new cell wall assembly and expansion of the existing wall material occurs at the apical end of the cell. Much of the newly added cell wall material is delivered to the growing surface region through secretion. In our model, the assembly of new cell wall volume through vesicle deposition is triggered at a set growth rate, independent of the turgor value, and is accompanied by a sudden reduction in growth rate. The time necessary to return to this threshold growth rate after each episode of renewal depends on the turgor. The higher the turgor, the higher the average growth rate, and, accordingly the shorter the growth cycle. However, the higher the turgor, the closer the average growth rate will be to the threshold growth rate. Eventually, the average growth rate will asymptotically approach the threshold growth rate inducing renewal, without ever surpassing it (Fig. 1).

Figure 1.

Influence of turgor on oscillation period (blue) and average growth rate (green) as predicted by the model described in Kroeger et al. 2011. At a low turgor regime, the average growth rate is influenced significantly by a change in turgor (A,B), whereas at a high turgor regime the average growth rate approaches a maximum value in asymptotic manner (C).

The predictions made by our model reflect experimental observations made by ourselves and others thus supporting the claim that turgor pressure plays the role of the driving force for expansion but not necessarily that of the regulator of its dynamics. By assuming that the turgor pressure is constant during an oscillatory growth cycle, our model leaves the role of instantaneous growth rate regulator to the cell wall and its relatively faster varying physical properties. Indeed, the cell wall at the tip of the growing tube is ideally suited for the regulation of the growth rate as it can, and does, regulate its physical properties locally.^17-19 The alternative hypothesis, on the other hand, which states that turgor pressure at the apex oscillates in time⁵ is based on the visual monitoring of the changing diameter of the subapical region of the tube and hitherto it has not been informed by other methods. Since changes in shape such as the observed swelling can equally be achieved by altered distributions of the mechanical properties in the wall²⁰ complementary methods to measure turgor need to be used. Micro-indentation and the use of a turgor pressure probe have successfully demonstrated that in the distal region of the pollen tube turgor temporal variations do not occur or are below the noise level.^1,21 However, since the insertion of the pressure probe into the tip of a growing pollen tube is extremely challenging, more indirect methods may be required to measure pressure in the growing apex or to determine putative pressure gradients over spatial distances. Extracting velocity fields by tracking intracellular fluorescent markers²² in oscillating pollen tubes, and coupling them to hydrodynamic equations may have the potential to yield values for pressure and its possible fluctuations inside growing pollen tubes, but this concept awaits experimental testing.

Given the wealth of experimental as well as theoretical information published on pollen tube growth in recent years, many of the key components necessary to induce oscillatory behavior in elongating pollen tubes are emerging.^23-26 Furthermore, many different mathematical models now simulate oscillations.^27-29 While certain models produce oscillations without involving the cytoskeleton,²⁷ others do so without explicitly changing cell wall properties.^28,29 While all components have been observed to oscillate in concert, i.e., in a phase-locked manner,²³ not all of these components are essential for generating the oscillatory behavior, particularly if the growth environment is changed. For example, temporal changes in vesicle density in the apical cytoplasm have been observed without growth.³⁰ Understanding the complex interactions between all the contributors to pollen tube growth thus remains a challenge. However, with the emergence of new data and the improvement of methods producing distinctive results, pollen tube growth may soon be described with the same accuracy, on a biophysical level, as for example the behavior of individual neurons.³¹ Given the knowledge of the ion channels present in individual excitable cells such as neurons, it is possible to predict, at least in a statistical manner, their responses to various electrophysiological but also optic or mechanical stimuli.³² In the case of the pollen tube, such a precise description may enable us to predict its growth and signaling responses to different environmental perturbations. Ongoing efforts to identify ion channels in pollen, therefore, contribute important pieces to the puzzle.³³ A working description of the mechanical and chemical interaction involved in tip growing cells could foster the understanding of cell growth under very different circumstances, even of those involved in seemingly unrelated processes such as bone remodeling. In more immediate terms, it has the potential to provide a biological function for the pollen tube growth rate oscillations, a concept that is has been elusive ever since the oscillations were first observed.

Kroeger JH, Zerzour R, Geitmann A. Regulator or driving force? The role of turgor pressure in oscillatory plant cell growth. PLoS ONE. 2011;6:e18549. doi: 10.1371/journal.pone.0018549.