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Plant Physiology 141:1233-1236 (2006) © 2006 American Society of Plant Biologists How Far Can a Molecule of Weak Acid Travel in the Apoplast or Xylem?1,[W]Biology Department, University of Massachusetts, Amherst, Massachusetts 01003
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
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/Lapo), where Lapo is a characteristic decay length
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 Lapo 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 3Lapo 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 3Lapo 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
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 Lapo, then most of the hormone secreted into the interface enters the receiver cell (or reenters the transmitting cell). If w = Lapo, 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 = 0.1Lapo, 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
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
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
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
; Couillerot and Bonnemain, 1975).
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 Lxylem 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.
This article was written while the author was on sabbatical in the Biology Department of the University of Massachusetts, Amherst. Received May 17, 2006; returned for revision May 17, 2006; accepted June 16, 2006.
1 This work was supported in part by the U.S. National Science Foundation (grant no. 0316876) and by the hospitality of the labs of Tobias I. Baskin and Peter Hepler. Software was purchased with funds provided by the Biotechnology and Biological Sciences Research Council (UK). 
2 Present address: Physics Department, Simon's Rock College, 84 Alford Rd., Great Barrington, MA 01230. The author responsible for distribution of materials integral to the findings presented in this article in accordance with the policy described in the Instructions for Authors (www.plantphysiol.org) is: Eric M. Kramer (ekramer{at}simons-rock.edu).
[W] The online version of this article contains Web-only data. www.plantphysiol.org/cgi/doi/10.1104/pp.106.083790. * E-mail ekramer{at}simons-rock.edu; fax 413–528–7365.
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