Modeling Auxin Transport and Plant Development

Abstract

The plant hormone auxin plays a critical role in plant development. Central to its function is its distribution in plant tissues, which is, in turn, largely shaped by intercellular polar transport processes. Auxin transport relies on diffusive uptake as well as carrier-mediated transport via influx and efflux carriers. Mathematical models have been used to both refine our theoretical understanding of these processes and to test new hypotheses regarding the localization of efflux carriers to understand auxin patterning at the tissue level. Here we review models for auxin transport and how they have been applied to patterning processes, including the elaboration of plant vasculature and primordium positioning. Second, we investigate the possible role of auxin influx carriers such as AUX1 in patterning auxin in the shoot meristem. We find that AUX1 and its relatives are likely to play a crucial role in maintaining high auxin levels in the meristem epidermis. We also show that auxin influx carriers may play an important role in stabilizing auxin distribution patterns generated by auxin-gradient type models for phyllotaxis.

Appendices

APPENDIX 1. DETAILED MODEL DESCRIPTION

In the model tissue is built up by cytoplasmic and wall compartments separated by plasma membranes. The auxin transport model is based on the detailed chemiosmotic description for transport across membranes partly dependent on saturable efflux and influx mediators. In addition to cross-membrane transport also apoplastic diffusion is modeled. Transport mediators cycle between cytoplasm and plasma membrane compartments. In the present study it is assumed that the cycling is fast such that transport mediators are positioned in equilibrium. In each compartment auxin is divided into pH-dependent fractions of the protonated and anion forms (assuming fast dynamics and equilibrium for the reversible reaction a^H ↔ a⁻ + H). The auxin flux J_a from a cytoplasm compartment into a wall compartment is described by

$$ J_{a} = p_{a}^{H} (f_{a^H}^{cell}a_{i} - f_{a^H}^{wall} a_{ij})\,+\,p_{PIN} P_{ij} {\left(N(\Phi){f_{a^-}^{cell} a_{i} \over K_{p} + f_{a^-}^{cell} a_{i}} - N(-\Phi){f_{a^-}^{wall} a_{ij} \over K_{p} + f_{a^-}^{wall} a_{ij}}\right)}\,+\,p_{AUX} A_{ij} {\left(N(-\Phi){f_{a^-}^{cell} a_{i} \over K_{A} + f_{a^-}^{cell} a_{i}} - N(\Phi){f_{a^-}^{wall} a_{ij} \over K_{A} + f_{a^-}^{wall} a_{ij}}\right)} $$

where a_i, a_ij are the auxin concentrations in the cytoplasm and wall compartment respectively. P_ij and A_ij are the PIN1 and AUX1 concentrations in the membrane, p _a , p _PIN , and p _AUX are the permeabilities, $f^{cell}_{a^H}$, $f^{wall}_{a}$, $f^{cell}_{a^-}$, $f^{wall}_{a^-}$ are the fractions of protonated and anion forms of auxin within the cell and wall compartments. N(Φ), and N(–Φ) are factors coming from the carrier mediated transport across the membrane potential given by

$$ N(\Phi) ={\Phi e^{\Phi} \over e^{\Phi} - 1},\, \Phi = {zV_{m}F \over RT} $$

where Φ = ±4.65 has been used assuming a membrane potential V _m = –120 mV (negative inside). z is the valence, F is the Faraday constant, R is the gas constant, and T is the absolute temperature.

In addition to the membrane transport there is diffusion between neighboring walls with a diffusion constant D _a. In the equations describing the transport also spatial factors are included (Jönsson and others 2006), where we have simplified by using constant values for these assuming in the two-dimensional simulations a cell “volume” of 25 μm², a cell-wall crossing “area” of 5 μm, and a wall thickness of 50 nm.

The PIN1 and AUX1 cycling determines how much of the proteins are within the cell membranes toward different neighbors. Although we use a symmetric AUX1 polarization, the PIN1 cycling is also dependent on the auxin in the neighboring cells. In the simulations we use the equilibrium calculated from the cycling rate (from cytosol to membrane compartment)

$$ {\eqalign {J_{A} &= k_{A1} A_i - k_{A2} A_{ij} \cr J_{P} &= k_{P1} {\left((1 - c_{P}) + c_{P}{a_{j}^{n} \over K_{H} + a_{j}^{n}}\right)}P_{i} - k_{P2} P_{ij}}} $$

where a _j is the auxin in the neighboring cell, A _i , (P _i ) is the AUX1 (PIN1) in the cytoplasm compartment, and A _ij (P _ij ) is the AUX1 (PIN1) in the membrane. The cytoplasm concentrations are measured as molecules per volume, whereas the membrane concentrations are measured as molecules per area. For a more thorough description we refer to Jönsson and others (2006), where a minor difference is that we here also include a symmetric term for PIN1 endocytosis (the k _P1(1–c) term in J _P ).

Finally we include production and degradation for the molecules, which for auxin also could be interpreted as transport in and out of the simulated tissue at the boundary. These processes are described by

$$ {\eqalign {{da_{i} \over dt} &= c_{a} - d_{a} a_{i} \cr {dA_{i}^{total} \over dt} &= (c_{A1} + c_{A2} L_{1}){\left((1 - k_{A}) + k_{A}{a_{i} \over K_{M} + a_{i}}\right)} - {d_{A} A_{i}^{total}} \cr {dP_{i}^{total} \over dt} &= (c_{P1} + c_{P2} L_{1}){\left((1 - k_{P}) + k_{P}{a_{i} \over K_{M} + a_{i}}\right)} - {d_{P} P_{i}^{total}}}} $$

where a _i , A _i , and P _i are the concentrations of auxin, AUX1, and PIN1. All molecules have a simple degradation proportional to its concentration. Production is allowed to be different in the epidermis (L₁ = 1) compared to inside the tissue (L ₁ = 0). The proteins are produced only within the cytoplasmic compartments and have one constant term and one Michaelis-Menten term for the auxin-induced production. These protein amounts are then redistributed between the cytoplasmic and membrane compartments according to the previous cycling equations (see Jönsson and others 2006).

The parameter values used in the simulations are presented in Table 1, and are mostly estimated from experiments. All simulations are done using a C++ software based on a 5th order Runge-Kutta solver with adaptive step length for the numerical integration.

Table A1. Model Parameters

APPENDIX 2. EXPERIMENTAL METHODS

Auxin treatments were carried out by either applying auxin paste made from 5 mM IAA, 1% DMSO in lanolin (Sigma) or a mock treatment of 1% DMSO in lanolin to pin1-1 apices. The apical 2 mm of tissue was then collected for RNA extraction after 30 min. RNA extraction and quantification was carried out according to Heisler and others (2005). For amplifying AUX1 we used the primers 5′ GTCCAATCAATTCCGCTGTC 3′ and 5′ GCATAAAGAACGGTGGCTTC 3′. We used both the ACTIN2 and ACTIN8 genes as internal controls as described in Heisler and others 2005.

In situ hybridizations was carried out according to Long and Barton (1998). Preparation of tissue for confocal imaging was carried out as described in Heisler and others (2005).