FACTORES QUE INFLUYEN EN LA FILTRACIÓN EN MEDIO GRANULAR

In canalisation models, the formation of auxin canals due to the feedback between auxin transport and auxin transporter allocation is examined. Early models have been

established by Mitchison [140, 141, 142], based on biological experiments of Tsvi

Sachs [189]. Sachs showed in a series of experiments with pea stems that vascu-

lar tissue formation occurs between auxin sources (for example, applied exogenous auxin) and sinks (for example, existing vasculature which removes auxin from a tis- sue). Mitchison also considered the predictions of the chemiosmotic hypothesis of auxin transport, namely that transporter polarity in cells accounts for polar auxin trans-

port, since otherwise auxin would be trapped in cells [70]. Two possible mechanisms

were employed in Mitchisons models: facilitated diffusion [141] and transporter po-

larisation. In the facilitated diffusion model, auxin diffusion increases with flux, which remains controversial because a mechanistic explanation of this diffusion enhance-

ment remains elusive [185]. However, Mitchison makes it clear in his article that this

increase in diffusion might be an emergent property of underlying mechanisms in a similar manner as the increase of conductance of electrical currents in gas as the temperature rises. It has also to be noted that ’facilitated diffusion’ does not mean diffusion in the strict sense of Brownian motion of molecules, eventually leading to the dissipation of concentration gradients in a given fluid, but this term also includes trans-membrane movement of substance by carriers, as long as the substance is not transported against a chemical gradient.

In his polar transport model [142], auxin fluxes directed the insertion of transporters

into the membrane, such that higher fluxes between two cells resulted in a stronger abundance of transporters between them. Cell wall space was not explicitly consid- ered in this model, instead transport occurred directly from cell to cell, which is a com-

mon abstraction in auxin models [102]. This model was able to reproduce canal for-

mation between auxin sources and sinks, and even could account for the formation

of more complex loops in leaf venation [142]. Both models, facilitated diffusion and

polar transport, were dependent on the square of auxin flux, such that a non-linear re- lationship between auxin flux and the flux enhancement mechanism is needed for the formation of stable canals. An example of canalisation in a 2D grid of cells is shown in figure 3.1.

3.2. Auxin transport models

Figure 3.1: Auxin transport canalisation. A schematic tissue is drawn as a two dimen- sional grid of square cells. Cells acting as auxin sources are black; auxin sinks are white. PIN localisation in the membrane (green) shows the direction of auxin efflux and thus the direction of overall auxin transport through the tissue. Initial fluxes of auxin (upper left) are reinforced and lead to the development of auxin transport canals between source and sink (upper right). Lower left and lower right: The situation with two source cells. One possible result could be a branching pattern where both canals drain into the same sink. In both situations, auxin is drained from surrounding tissues

(grey cells). This figure is reproduced from [63].

A prediction of Mitchison’s canalisation model was that the canals exhibit high auxin flux and low auxin concentration, contrary to the results obtained by indirect visuali-

sation techniques using the DR5 auxin responsive element [194], indicating high con-

centration in auxin canals. In a more recent model, Kramer could show that these canals with high auxin concentration could be reproduced when including auxin influx carriers in canal-forming cells, which were able to deplete the surrounding tissue from

auxin [101]. Similarly, if parts of the membrane compete for the allocation of auxin

efflux transporters, such canals with high flux and high concentration of auxin could also be established, with the result of reproduction of branched vein patterns, but no

closed loops [54]. A more recently presented canalisation model [239] has been able

to reproduce both high flux, high concentration canals, and several leaf venation pat- terns, such as closed loops, with the underlying mechanism of auxin flux enhancement depending on the inhibition of PIN endocytosis (relocation of PINs towards endoso- mal compartments) by a signal from a slowly diffusing apoplastic auxin-receptor. The

canalisation mechanism relies on the competition between cell membranes for free diffusing auxin receptors, however it requires the sudden immobilisation of receptors once they are bound to auxin. While it is indeed unlikely that a receptor molecule such as auxin binding protein 1 (ABP1, the putative candidate for this receptor molecule as suggested by the authors) would change its diffusion coefficient in such a dras- tic way by binding to auxin, which has only around 1.5 % of the molecular weight of

ABP1 [50,99], this is in fact an approximation to the receptor-auxin complex binding

at the cell membrane which is nearest to where the complex formed. This model was also able to reproduce canalisation in the context of bud outgrowth competition due to apical dominance with the same mechanistic assumptions of cells competing for an extracellular auxin receptor.

A minimal canalisation model for PIN polarisation in a file of cells has been proposed

by Alim and Frey [4], predicting an excitable polarisation front which is able to trigger

PIN polarisation within cells. The model is implemented in 1D, it does not consider apoplastic space and it is notably solved analytically, thus it is a pure mathematical model not relying on computer simulation. It is able to reproduce bipolar cells, which could account for the occurrence of closed vein loops in leaf vein patterning. The feedback between auxin flux and auxin transport involves a mechanism by which the endocytosis rate of PINs from the membrane into the endosomal compartments is reduced in membrane parts exhibiting high flux.