Self-Organization of Plant Vascular Systems: Claims and Counter-Claims about the Flux-Based Auxin Transport Model

Abstract

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.

Auxin-mediated regulation of vascular patterning in Arabidopsis thaliana leaves

“…2b VI). This is interesting from the perspective of two proposed mechanisms for the polarization of PIN1 proteins (Bayer et al 2009; Feller et al 2015). The first mechanism, “up the gradient”, assumes that polarization of PIN1 proteins occurs toward the highest auxin concentration, and it was suggested for the organogenesis at SAM and the formation of convergence points during leaf development (Reinhardt et al 2003; Scarpella et al 2006; Heisler et al 2005; Jönsson et al 2006; Smith et al 2006; Bayer et al 2009; Marcos and Berleth 2014).…”

“…The first mechanism, “up the gradient”, assumes that polarization of PIN1 proteins occurs toward the highest auxin concentration, and it was suggested for the organogenesis at SAM and the formation of convergence points during leaf development (Reinhardt et al 2003; Scarpella et al 2006; Heisler et al 2005; Jönsson et al 2006; Smith et al 2006; Bayer et al 2009; Marcos and Berleth 2014). The second mechanism, “with the flux”, assumes polarization of PIN1 proteins in accordance with the direction of auxin transport but toward low auxin concentrations, and it was proposed for the continuous vascular pattern formation (Sachs 1969; Mitchison et al 1981; Bayer et al 2009; Feller et al 2015). It is likely that two primarily separated processes, organogenesis, and vascularization, became regulated in Arabidopsis by the same PIN1 proteins due to evolutionary loss of one from two functionally distinct PIN1 clades, which probably occur in the most angiosperms.…”

Mathematical Modelling of Auxin Transport in Plant Tissues: Flux Meets Signalling and Growth

“…Until recently, one of the main criticisms of the canalisation hypothesis (auxin-flux related localisation of PIN to cell membrane) was its inability to produce spot patterns in auxin distribution in a plant tissues without any additional assumptions on cell types (e.g. source/sink cells) [50], despite its accurate capturing of passage patterns [12]. Recent results [8,20] have shown that it is possible to obtain both spot and passage patterns in auxin distribution considering the canalisation hypothesis, provided an extra mechanism of either auxin-mediated PIN degradation or auxin self-induced production is considered.…”

Shaping leaf vein pattern by auxin and mechanical feedback

“…On the other hand, theoretical models predict that high auxin flux can lead to low auxin concentration in pre-existing veins ( Rolland-Lagan and Prusinkiewicz, 2005 ). However, high auxin concentration in pre-existing veins is made plausible by the implementation of additional factors, such as dynamics of auxin flux and efflux carriers ( Feugier et al , 2005 ), or a modulation of the source strength ( Feller et al , 2015 ). High auxin concentration in pre-existing veins may also be maintained by in situ auxin biosynthesis.…”