Exploring mechanisms of xylem cell wall patterning with dynamic models

Vascular plants employ an extensive vascular system, known as the xylem, to transport water from their roots to their leaves. Xylem vessels have thick secondary cell wall reinforcements that help them withstand the pressures generated during water transport. These cell wall reinforcements are deposited in a variety of intricate patterns, depending on the type of xylem. Protoxylem has a ringed or spiral cell wall pattern, to allow further elongation, while metaxylem has a pattern of regularly spaced gaps for radial transport in an otherwise solid wall. These patterns are determined by underlying patterns of cortical microtubules, which guide the deposition of cell wall material. Establishing the microtubule pattern, in turn, requires localised activity so-called Rho of Plant (ROP) proteins. ROPs are a type of small GTPase, molecular switches with an active and an inactive form. Microtubules and small GTPases also interact in other important processes, including the development of complex cell shapes. In this thesis, I aim to improve our understanding of the way in which ROPs and microtubules generate the xylem patterns using dynamic models. Chapters 2 and 3 focus on ROP patterning, whereas chapters 4 and 5 deal with microtubule dynamics.Xylem patterning requires the formation of spotted (metaxylem) or banded (protoxylem) ROP patterns. The molecular properties of ROPs and other small GTPases make them particularly suited to spontaneously form patterns through so-called reaction-diffusion mechanisms. Many models for GTPase patterning exist, but most of them only yield a pattern with just one single stable cluster. Initially, these models may give rise to multiple clusters, but in the long run these clusters compete with each other until only one remains. Beyond the standard reaction-diffusion ingredients of self-activation and an inactive form that diffuses faster than the active form, these polarisation models have one more ingredient in common: conservation of the total amount of GTPase. In chapter 2, we show that a pattern of stable, coexisting clusters can be obtained when this so-called mass conservation is broken by adding GTPase turnover. We also show that adding negative feedback through GAPs (GTPase activating proteins) --proteins that inactivate GTPases by activating their GTP hydrolysis activity-- allows for coexistence, even in mass conserved models. To study the mechanism driving coexistence, we construct an ordinary differential equation model, as a  simplified version of the full partial differential equation model. With this simplified model, we show that turnover stabilises coexistence by providing a constant local source of GTPase to smaller clusters, effectively providing a GTPase redistribution, while GAP feedback makes larger clusters lose relatively more GTPase.While reaction-diffusion systems that allow coexistence reproduce the spotted metaxylem pattern well, they do not reproduce ringed or spiral protoxylem patterns, because they miss an intrinsic orienting force. In chapter 3, we investigate the possibility that this orienting force might be provided by a microtubule-based diffusion restriction of active ROP. Experimental studies have found that in metaxylem microtubules form a diffusion barrier to ROP proteins. Since protoxylem patterning starts with a transversely oriented microtubule array, we model the barrier effect as a longitudinal ROP diffusion restriction. That way, we show that this diffusion restriction is able to orient the ROP pattern into protoxylem-like bands, as long as inactive ROPs can bypass the barriers more easily than active ROP. Similarly, spiral ROP patterns can be obtained by making the direction of the diffusion restriction oblique. Furthermore, by analysing the diffusion term, we show that the orienting mechanism only affects patterns with curved features. This implies that existing banded or spiral patterns cannot be reoriented by a change in the direction of diffusion restriction.With these new insights into ROP patterning, we change our focus to questions related to microtubule patterning. Many structures formed by the plant cortical microtubule array require a uniform distribution of microtubules. This homogeneity requirement appears to be at odds with the experimental finding that most microtubules nucleate from existing microtubules, which introduces a positive feedback loop on local microtubule density, leading to inhomogeneous arrays. In chapter 4, we use a combination of experiments and a simplified one-dimensional microtubule model to study how this inhomogeneity problem could be avoided. To that end, we investigate two hypotheses: (1) a local limitation of the amount of tubulin available for microtubule growth, and (2) a locally limited availability of nucleation complexes for making new microtubules. Our simulations reveal that the first option requires an unrealistically low tubulin diffusion coefficient. Our experiments show that nucleation complexes are preferentially inserted near microtubules, and that these insertions saturate locally with microtubule density. Using detailed simulations of nucleation complex behaviour at the cell membrane we show that the local saturation of nucleation complex insertion with microtubule density allows for the formation of homogeneous arrays as well as structured protoxylem arrays.In the simplified one-dimensional model, all microtubules are perfectly aligned, while in reality, microtubule orientations are more variable. In chapter 5, we therefore use two-dimensional microtubule simulations to study protoxylem band formation. With these simulations, we show that protoxylem band formation in a biologically realistic time frame requires three important ingredients: (1) sufficient co-alignment of the initial microtubule array with the underlying ROP pattern, (2) sufficiently flexible microtubules with random fluctuations in their growth direction, and (3) a sufficiently realistic nucleation implementation in line with our findings from chapter 4. The first requirement may be biologically satisfied by the orienting effect of the microtubule array on the ROP pattern (as discussed in chapter 3). However, the co-alignment that could reasonably be expected from this orienting effect is only sufficient in our simulations if both microtubule flexibility and realistic nucleation are taken into account. Microtubule flexibility can correct small mismatches in the initial co-alignment. The realistic nucleation contains (limited) positive feedback on microtubule density, which helps maintaining the band regions and speeds up the separation process.Our findings pave the way for future research integrating ROP and microtubule models to elucidate the interplay between the two in xylem patterning and other processes.