Numerical bifurcation analysis of the pattern formation in a cell based auxin transport model

Two-stage patterning dynamics in conifer cotyledon whorl morphogenesis

Background and Aims

Conifer embryos, unlike those of monocots or dicots, have variable numbers of cotyledons, even within the same species. Cotyledons form in a single whorl on a dome-shaped embryo. The closely spaced cotyledons are not found outside this ring, indicating a radial control on where they can form. Polar transport of the hormone auxin affects outgrowth of distinct cotyledons, but not the radial aspect of the whorl or the within-whorl spacing between cotyledons. A quantitative model of plant growth regulator patterning is needed to understand the dynamics of this complex morphogenetic process.

Methods

A two-stage reaction-diffusion model is developed for the spatial patterning of growth regulators on the embryo surface, with a radial pattern (P1) constraining the shorter-wavelength cotyledon pattern (P2) to a whorl. These patterns drive three-dimensional (3-D) morphogenesis by catalysing local surface growth.

Key Results

Growth driven by P2 generates single whorls across the experimentally observed range of two to 11 cotyledons, as well as the circularly symmetric response to auxin transport interference. These computations are the first corroboration of earlier theoretical proposals for hierarchical control of whorl formation. The model generates the linear relationship between cotyledon number and embryo diameter observed experimentally. This accounts for normal integer cotyledon number selection, as well as the less common cotyledon fusings and splittings observed experimentally. Flattening of the embryo during development may affect the upward outgrowth angle of the cotyledons.

Conclusions

Cotyledon morphogenesis is more complex geometrically in conifers than in angiosperms, involving 2-D patterning which deforms a surface in three dimensions. This work develops a quantitative framework for understanding the growth and patterning dynamics involved in conifer cotyledon development, and applies more generally to the morphogenesis of whorls with many primordia.

Localized auxin peaks in concentration-based transport models of the shoot apical meristem

We study the formation of auxin peaks in a generic class of concentration-based auxin transport models, posed on static plant tissues. Using standard asymptotic analysis, we prove that, on bounded domains, auxin peaks are not formed via a Turing instability in the active transport parameter, but via simple corrections to the homogeneous steady state. When the active transport is small, the geometry of the tissue encodes the peaks' amplitude and location: peaks arise where cells have fewer neighbours, that is, at the boundary of the domain. We test our theory and perform numerical bifurcation analysis on two models that are known to generate auxin patterns for biologically plausible parameter values. In the same parameter regimes, we find that realistic tissues are capable of generating a multitude of stationary patterns, with a variable number of auxin peaks, that can be selected by different initial conditions or by quasi-static changes in the active transport parameter. The competition between active transport and production rate determines whether peaks remain localized or cover the entire domain. In particular, changes in the auxin production that are fast with respect to the cellular life cycle affect the auxin peak distribution, switching from localized spots to fully patterned states. We relate the occurrence of localized patterns to a snaking bifurcation structure, which is known to arise in a wide variety of nonlinear media, but has not yet been reported in plant models.

Modeling plant development: from signals to gene networks

Mathematical modeling has become a common tool in plant developmental biology. Indeed, it allows for the prediction of complex and often unintuitive dynamics of the molecular networks driving plant development. This has enabled the test of their possible involvement in robust and specific developmental processes. Modeling has also been fruitful in predicting new interactions within gene networks, such as the Arabidopsis circadian clock. A new challenge is to integrate patterning issues with tissue growth and biomechanics. The development of new tools to gain resolution in data collection as well as new frameworks to confront models and data might provide even more robust predictions.

Auxin-driven patterning with unidirectional fluxes

The plant hormone auxin plays an essential role in the patterning of plant structures. Biological hypotheses supported by computational models suggest that auxin may fulfil this role by regulating its own transport, but the plausibility of previously proposed models has been questioned. We applied the notion of unidirectional fluxes and the formalism of Petri nets to show that the key modes of auxin-driven patterning-the formation of convergence points and the formation of canals-can be implemented by biochemically plausible networks, with the fluxes measured by dedicated tally molecules or by efflux and influx carriers themselves. Common elements of these networks include a positive feedback of auxin efflux on the allocation of membrane-bound auxin efflux carriers (PIN proteins), and a modulation of this allocation by auxin in the extracellular space. Auxin concentration in the extracellular space is the only information exchanged by the cells. Canalization patterns are produced when auxin efflux and influx act antagonistically: an increase in auxin influx or concentration in the extracellular space decreases the abundance of efflux carriers in the adjacent segment of the membrane. In contrast, convergence points emerge in networks in which auxin efflux and influx act synergistically. A change in a single reaction rate may result in a dynamic switch between these modes, suggesting plausible molecular implementations of coordinated patterning of organ initials and vascular strands predicted by the dual polarization theory.

Self-organization of plant vascular systems: claims and counter-claims about the flux-based auxin transport model

The plant hormone auxin plays a central role in growth and morphogenesis. In shoot apical meristems, auxin flux is polarized through its interplay with PIN proteins. Concentration-based mathematical models of the flux can explain some aspects of phyllotaxis for the L1 surface layer, where auxin accumulation points act as sinks and develop into primordia. The picture differs in the interior of the meristem, where the primordia act as auxin sources, leading to the initiation of the vascular system. Self-organization of the auxin flux involves large numbers of molecules and is difficult to treat by intuitive reasoning alone; mathematical models are therefore vital to understand these phenomena. We consider a leading computational model based on the so-called flux hypothesis. This model has been criticized and extended in various ways. One of the basic counter-arguments is that simulations yield auxin concentrations inside canals that are lower than those seen experimentally. Contrary to what is claimed in the literature, we show that the model can lead to higher concentrations within canals for significant parameter regimes. We then study the model in the usual case where the response function Φ defining the model is quadratic and unbounded, and show that the steady state vascular patterns are formed of loopless directed trees. Moreover, we show that PIN concentrations can diverge in finite time, thus explaining why previous simulation studies introduced cut-offs which force the system to have bounded PIN concentrations. Hence, contrary to previous claims, extreme PIN concentrations are not due to numerical problems but are intrinsic to the model. On the other hand, we show that PIN concentrations remain bounded for bounded Φ, and simulations show that in this case, loops can emerge at steady state.

Systems biology approaches to understand the role of auxin in root growth and development

The past decade has seen major advances in our understanding of auxin regulated root growth and developmental processes. Key genes have been identified that regulate and/or mediate auxin homeostasis, transport, perception and response. The molecular and biochemical reactions that underpin auxin signalling are non-linear, with feed-forward and feedback loops contributing to the robustness of the system. As our knowledge of auxin biology becomes increasingly complex and their outputs less intuitive, modelling is set to become much more important. For the last several decades modelling efforts have focused on auxin transport and, latterly, on auxin response. Recently researchers have employed multi-scale modelling approaches to predict emergent properties at the tissue and organ scales. Such innovative modelling approaches are proving very promising, revealing new mechanistic insights about how auxin functions within a multicellular context to control plant growth and development. In this review we initially describe examples of models capturing auxin transport and response pathways, and then discuss increasingly complex models that integrate multiple hormone response pathways, tissues and/or scales.

Modelling hormonal response and development

As our knowledge of the complexity of hormone homeostasis, transport, perception, and response increases, and their outputs become less intuitive, modelling is set to become more important. Initial modelling efforts have focused on hormone transport and response pathways. However, we now need to move beyond the network scales and use multicellular and multiscale modelling approaches to predict emergent properties at different scales. Here we review some examples where such approaches have been successful, for example, auxin-cytokinin crosstalk regulating root vascular development or a study of lateral root emergence where an iterative cycle of modelling and experiments lead to the identification of an overlooked role for PIN3. Finally, we discuss some of the remaining biological and technical challenges.