How Far Can a Molecule of Weak Acid Travel in the Apoplast or Xylem?^,

The plant hormones auxin, abscisic acid (ABA), and the gibberellins (GAs) are all weak acids subject to the ion-trapping mechanism that tends to remove them from the extracellular space and concentrate them in the cytoplasm of plant cells. If a molecule of one of these compounds enters the extracellular space, it can therefore travel only a limited distance before reentering a cell. Influx carriers can only shorten this distance. Here I present a simple but quantitative estimate of this distance, and discuss its relevance for various models of short- and long-range signaling in plants.

To review, the weak acids of interest all have one or more carboxyl groups, with dissociation constants between 4 and 5 (Table I). In the weakly acidic apoplast, a fraction of each hormone will be protonated and thus membrane permeable. However, once in the approximately neutral cytoplasm, the molecules dissociate and become membrane-impermeable anions. In the absence of transmembrane efflux carriers, the molecules will accumulate in the cytoplasm—the so-called ion-trapping mechanism. Although the principle of ion trapping has been known for decades (Rubery and Sheldrake, 1973), its contribution to the overall hormone economy has remained vague. The actual accumulation of a weakly acidic plant hormone in any cell is dominated by the activity of influx and efflux carriers, if present, with additional contributions from biosynthesis and metabolism pathways (Davies, 2004). However, the fact remains that a weak acid, upon entering the extracellular space, will tend to be trapped by adjacent cells. How far can we expect a molecule of hormone to travel?

Table I.

Decay length for extracellular movement of some plant hormones

Estimates for the decay length of travel through the apoplast (pH = 5.5) and the xylem (pH = 5.5) are given. The diffusion coefficient for all molecules is estimated to be 10% of the aqueous value for auxin, D = D_aq/10 = 0.0024 cm² h⁻¹. Decay lengths in the xylem assume a vessel radius of R = 100 μm. All values for the GAs, except pK_a, should be regarded as order-of-magnitude estimates only. The bottom row, an estimate for any hormone subject to the activity of an influx carrier, assumes P_eff = 1.0 cm h⁻¹.

	pKaa	PAHb	Lapo		Lxylem
					If h = 0.1 μm	If h = 1.0 μm	If v = 1 m h−1	If v = 10 m h−1
		cm/h	μm		m
Auxin	4.8	0.2	13	42	0.31	3.1
ABA	4.7	0.04	35	110	2.2	22
GA3	4.0	0.008	150	500	50	500
Early-hydroxylation pathway GAs
GA20	4.2	0.08	40	150	3	30
GA1	4.0	0.006	200	500	50	500
Late-hydroxylation pathway GAs
GA9	4.3	4	5	15	0.04	0.4
GA4	4.2	0.4	20	60	0.6	6.0
With influx carrier	NA	NA	2.5	8.0	0.01	0.1

^apK_a values are provided by the LogD software suite, version 9.0 (ACD Labs). Values generally agree with experiment within ±0.1 (Tidd, 1964; Rubery and Sheldrake, 1973; Kaiser and Hartung, 1981; Tomlin, 2000).

^bThe permeability of protonated auxin has been discussed previously (Swarup et al., 2005). The permeability of protonated ABA comes from Astle and Rubery (1980) and Baier et al. (1990). GA permeabilities are estimates based on the octanol-water partition coefficient predictions of the LogD software suite. The value for GA₁ falls between the experimental values of Drake and Carr (1981) and Nour and Rubery (1984).

Consider the idealized situation shown in Figure 1A. A transmitter cell secretes a pulse of hormone into the apoplast. The hormone then moves through the apoplast between two sink cells. What fraction of the excreted hormone reaches the receiver cell, a distance x away? The answer (derived in the supplemental data) is 10^∧(−x/L_apo), where L_apo is a characteristic decay length

(1)

where h is the thickness of the wall, D is the diffusion coefficient of the hormone in the wall, and P_eff is the effective permeability of the sink cell membranes. The latter depends on a variety of factors, but in the absence of influx carriers it reduces to

(2)

where P_AH is the membrane permeability of the protonated form of the hormone and the term in parenthesis is the fraction of the hormone that is protonated in the apoplast.

Figure 1.

Sketch of the movement of a weak acid (blue circles) in the apoplast. A, Transmitting cell (T) is separated from the receiving cell (R) by a cell wall of length x. B, Transmitting and receiving cells share a common wall of width w. S, Parenchyma cells that act as sinks for the hormone.

There are several important caveats to this discussion. First, Equation 1 is exact only for the simplified geometry shown in Figure 1. Cell walls seldom have uniform thickness, so the parameter h is an approximate width (see supplemental data for additional discussion). Second, arrangements with more than one layer of sink cells between the transmitter and the receiver will have a shorter decay length due to the presence of more cell surface area available for import. Third, eukaryotic cell membranes are complex structures—inhomogeneous in composition, crowded with membrane proteins, and subject to rapid turnover (Engelman, 2005). The diffusive permeability of protonated hormones may therefore be expected to vary from cell to cell and even between different microdomains of the same membrane. Last, auxin and other hormones can trigger proton secretion, leading to changes in apoplastic pH on a time scale of minutes (Davies, 2004). A decrease in apoplastic pH by 0.5 will decrease L_apo by about one-half. For all these reasons, a value for the decay length in plant tissues should be regarded as approximate. However, it should also be noted that the square root in Equation 1 tends to limit the impact of parameter variations.

In Table I, I estimate typical values for the decay length of auxin, ABA, and several GAs. These data constrain proposed models of hormone action. The decay length is the distance over which the hormone concentration decreases by a factor of 10. Thus, a distance of 3L_apo can be taken as a practical upper bound on the distance between the transmitting and receiving cell. A pulse of hormone can travel much farther than 3L_apo only if sink cells export the hormone back into the apoplast via efflux carriers or into the cytoplasm of adjacent cells via the plasmodesmata. Table I thus implies that, in thin-walled meristematic tissue, an apoplastic pulse of auxin or late-hydroxylation pathway GAs can travel just a few cell diameters. In mature tissues with well-developed cell walls, the decay lengths are uniformly larger by a factor of about 3. Note, in particular, GA₁ and GA₃. These are sufficiently membrane impermeable that an apoplastic signal can travel farther than 1 mm. Experiments applying radiolabeled GA₁ or GA₃ to sectioned plant tissues often find transport over distances >10 mm (Phillips and Hartung, 1976; Drake and Carr, 1979), which suggests a role for both apoplastic transport and efflux carriers.

The decay length also provides a useful bound on the efficiency of signaling between adjacent cells. Figure 1B shows a sketch of the apoplastic interface between a transmitter cell and a receiver cell—analogous to the synapse between two neurons. If the width w of the interface is large compared to L_apo, then most of the hormone secreted into the interface enters the receiver cell (or reenters the transmitting cell). If w = L_apo, less than one-half of the hormone secreted by the transmitting cell remains at the interface. The rest diffuses into the adjacent cell walls. For w = 0.1L_apo, 98% leaves the interface via the cell walls (see proof in supplemental data). Table I indicates that, for thin-walled cells with no influx carriers, only the late-hydroxylation GAs and possibly auxin would be efficient paracrine signals. Of course, specific influx carriers in the receiver cell membrane would permit efficient paracrine signaling with any hormone. There is good evidence for influx carriers specific for ABA and some GAs (Astle and Rubery, 1983; Nour and Rubery, 1984; Perras et al., 1994; Yamaguchi et al., 2001). For auxin, at least one gene family of influx facilitators is known (Parry et al., 2001; Terasaka et al., 2005).

This analysis of interface efficiency is relevant for models of auxin transport and auxin-mediated morphogenesis. Auxin transport is transcellular, which means auxin moving through a file of cells traverses the cell wall between each pair of neighbors in the file (Goldsmith et al., 1981). In addition, auxin transport is often concentrated in a narrow file or a uniserrate layer of cells (Swarup et al., 2005; de Reuille et al., 2006; Jonsson et al., 2006; Smith et al., 2006). Efficient transport thus requires that the cell interfaces not be too leaky; in other words, w > L_apo. Swarup et al. (2005) estimate that influx facilitators in the root epidermis of Arabidopsis thaliana increase membrane permeability by a factor of 15, giving L_apo = 3 μm for thin-walled cells. Apoplastic diffusion is not negligible in this system.

Regarding auxin-mediated morphogenesis, consider the case of spiral phyllotaxis in the shoot apical meristem. Three recently published computer models of phyllotaxis all couple the auxin flux between cells with cell differentiation triggered by high cytoplasmic auxin concentration (de Reuille et al., 2006; Jonsson et al., 2006; Smith et al., 2006). The results are patterns of primordia initiation that match observations. However, two of these models assume that cell interfaces are 100% efficient (i.e. no apoplastic diffusion; de Reuille et al., 2006; Smith et al., 2006), whereas the third uses a value for the membrane permeability of protonated auxin that is 60 times larger than the measured value in plant cells (Delbarre et al., 1996; Jonsson et al., 2006). From the previous paragraph, it is clear that apoplastic diffusion can be neglected only if the width of a cell interface is large compared to L_apo. Even if auxin influx carriers increase the effective membrane permeability by a factor of 15, L_apo will still be comparable to the typical interfacial width of 5 μm.

Similar considerations apply to the movement of a weak acid in the xylem (Fig. 2). In this case, the decay length (derived in supplemental data) is

(3)

where v is the speed of the xylem sap and R is the radius of the xylem vessel. Values for L_xylem are estimated in Table I. In the absence of carriers, most of the weak acids have a decay length of approximately 2 m or greater. The only exceptions are the late-hydroxylation GAs and (at low transpiration rates) auxin, due to their relatively high membrane permeabilities. Conversely, GA₁ and GA₃ again have the longest range, due to their low membrane permeability. This is consistent with autoradiography studies that show considerable movement of exogenously applied GAs in the xylem (Zweig et al., 1961; Couillerot and Bonnemain, 1975).

Figure 2.

Sketch of the movement of a weak acid (blue circles) in a xylem vessel (Xy). Arrow indicates the direction of water movement. All other labels are as in Figure 1. Note that the calculation of L_xylem in the text only considers the loss of hormone across plasma membranes that face the xylem vessel. The additional loss of hormone due to diffusion along the radial walls between sink cells is expected to be small.

The most thoroughly studied hormone in the xylem is ABA, which is a drought stress signal that moves from the roots to the leaves in the transpiration stream (Sauter et al., 2001). The decay length for ABA is on the order of 10 m, consistent with a role in long-range signaling. Note also that L_xylem is proportional to the speed of flow in the xylem. At low transpiration rates, losses to the xylem parenchyma cells are proportionately higher. In this way, the concentration of ABA in the xylem may provide the plant with a local measure of transpiration rates.

In this letter, I have discussed the kinetics of acid trapping. There is a general tendency in the literature to regard membrane permeability as an all-or-nothing phenomenon. However, it is clear from the above discussion that the relative degree of membrane permeability has important consequences for models of hormone transport and signaling.