A ‘simplest’ steady-state Munch-like model of phloem translocation, with source and pathway and sink

We thank Matthew V. Thompson for reading our manuscript, making helpful suggestions and diligently provoking us to greater transparency for the benefit non-mathematical readers.

Batchelor GK (1967) ‘An introduction to fluid dynamics.’ (Cambridge University Press: Cambridge)

Behnke HD (1989) Structure of the phloem. In ‘Transport of photoassimilates’. (Eds DA Baker, JA Milburn) pp. 79–136. (Longman Scientific & Technical: Harlow, UK)

Bluman GW, Cole JD (1969) The general similarity solution of the heat equation. Journal of Mathematics and Mechanics 18, 1025–1042. to Peters et al. 2007, 2008) and a biophysical background in both the ascent of sap (Pickard 1981) and osmotic swelling (Pickard 2008).

³The logic of this assertion is not a priori obvious, especially since the anatomy of the phloem might encourage one to favor a fully three dimensional formulation. Having initially attempted some small generalisation in this direction, we feel that neglecting intra-tube variations in the transverse plane really is necessary and will greatly simplify matters for the reader. This is basically an application of the Variation of the Problem technique favored by mathematicians confronted with stubborn problems (e.g. Polya 1957, p. 209). Besides, the nature of the generalisation from one dimension is not a priori obvious; and, physiologically, phloem experiments normally report only axial variations along a phloem bundle.

⁴In science and engineering, a ‘dimensionless group’ is ‘the product of a subset of a problem’s dimensioned parameters which is itself dimensionless’. Generally, a problem will have far fewer useful dimensionless groups than it has parameters. Also, as a rule of thumb, exploring a dependent variable’s behavior is much simpler using its dimensionless groups than its more numerous parameters. A standard and still useful treatise on the topic is the monograph of Langhaar (1951).

⁵‘Analytical’, as it will be used in this paper, refers to a function or system of equations which is treated by the methods of algebra and calculus and yields results as formulas expressed in computable algebraic functions which have neither discontinuities nor infinities over the range of interest. In contrast, a ‘numerical’ method yields no formulas but rather numerical predictions. Philosophically, an analytic method is generally preferred if one is seeking thorough understanding of an idealised problem. Whereas a numerical method may be more appropriate when seeking usable predictions about a practical situation.

⁶Sieve tube sap is a complex substance which varies widely from season to season and species to species (cf. Ziegler 1975). It contains, among others, inorganic ions, amino acids, and raffinose family oligosaccharides. As a simplification, we shall speak only of ‘an osmolyte’ and pick sucrose as illustrative of this substance.

⁷The parameter Ξ will turn out to be linear in the product (source strength) × (plasmalemmal water permeability). Intuitively, non-zero values of both are required for phloem transport to take place by way of a Münch-like mechanism. Since the boundary conditions (Eqn 20) involve neither, the only way either source strength or plasmalemma permeability enters is by way of Ξ. Hence, phloem transport can be said to be ‘driven’ or ‘forced’ by Ξ.

⁸We argue that copious output from a computer program, although desirable in an agronomic prediction, is less desirable in a setting which requires deep inward digestion of recondite concepts to appreciate the qualitative variations of a complicated process. For example, a numerical model can show that some quantity may increase rapidly with time: but we believe that achieving an intuitive grasp of the process would be facilitated by additional knowledge that the variation is exponential with an algebraically expressible rise time.

⁹We note that what makes a sink a sink is the proclivity of nearby tissue to take up photosynthate. To a ‘pure’ applied mathematician looking for a boundary condition which would simplify analysis, this at once suggests taking this uptake to the limit υ_arb = 0. Since the sieve tube is not a storage tissue with an imperative to accumulate photosynthate, and since our interest is more qualitative understanding of mechanism than quantitative prediction, we would concur with such a choice.

Initially, we note that the physically expected downward-concavity of u(z) requires u″ < 0; and, therefore, even if the ψ′ term on the right hand side of Eqn (13) does not swamp the u term, it must at least outweigh it.

By Eqn (14) the condition for neglecting the ‘pressure’ terms of Eqns (8) and (13) is

where a typical Table 1 value for the term in curly braces is roughly 1/7. In all ω(ζ) computations successfully conducted, ω^## was non-positive and ω non-negative, as expected. For Ξ = 1000, the range of the ratio of Eqn (A1) was (0,0.1427); for Ξ = 100, 10 and 1, it was (0,0.0339), (0,0.0206) and (0,0.0186), respectively. This trend is not unexpected because study of a progression of ω(ζ) plots shows that, with increasing Ξ, the mid-regions of the plots flatten to a plateau while the edges of the plateau become markedly steeper; this can be seen in Fig. 4.

We conclude that neglect of the p′(z) term is justified. Moreover, for this qualitative study, its neglect enormously simplifies the problem.

As caveats for the reader, we note two further numerical observations. First, for Ξ = 10 000, the range of the ratio is (0,14.9548); this could be interpreted as an artifact of computational roundoff. Second, if one began the integration at ζ = 1 and shot backwards towards the origin, the problem of meeting the two boundary conditions at ζ = 1 of course went away: but ω^## developed a small non-negative region for Ξ > 400, which conceivably could be computational error manifesting itself differently; and, again for Ξ > 400, both computational stability and desired behaviour near the shot-at-boundary became more serious issues than for the preferred method of shooting forward from ζ = 0.

The conspicuous linearity displayed in Fig. 3 by the dimensionless osmolality [ω^# − κ] coupled with the boundary conditions Eqn (20) leads one to approximate the dimensionless speed as

Back-substituting this equation into Eqn (19) yields

Therefore, the approximation should be qualitatively useful for Ξ  1 and implies that

and

The actual numerical values for Ξ = 1 are κ = −0.4844, ω_max = 0.1189 and [ω^#(0) − κ] = 0.9615. For Ξ = 0.1 they are κ = −0.0498, ω_max = 0.0124 and [ω^#(0) − κ] = 0.0996.

These considerations for unrealistically small Ξ reveal that, for sufficiently large D (or small ϒ), the expected diffusion-like osmolality profile does appear. They also give us confidence that our numerical integration does indeed work properly when Ξ is non-large and validate its usefulness for guiding our development of asymptotic solutions.

A series of the form ΣA_nexp(–α_nχ), where the α_n are real non-negative constants and χ is a real variable, is a particular case of what are called Dirichlet series (Mandelbrojt 1972); it is known that, if the series converges for χ = 0, then it converges for all χ > 0. Figure 5 and the developments of the Accessory Publication to this paper lead one to believe that ω(χ) displays a prominently exponential behavior and might readily be represented using a Dirichlet series. To examine this intuition, let α_n = nΛ (n = 0, 1, 2, …), where Λ > 0 is a finite real decay parameter not yet specified and express the speed as

The differential equation for the speed is (19) which may be written as

where * denotes d/dχ and χ = ξ = 1 − ζ when the + sign is selected but χ = ζ when the − sign is chosen. The + sign is appropriate near the boundary at ζ = 1 where the independent variable ξ = 1 − ζ is preferable; the −; sign is appropriate near the boundary at ζ = 0 where the preferable independent variable is ζ itself. Next define

so that 1 ≧ τ ≧ e^−Λ > 0. Then, defining ω⁺ = dω/dτ, rewrite (C2) as the first order system:

Because this system satisfies Lipschitz conditions over 1 ≧ τ ≧ e^−Λ > 0, both ω and ω⁺ possess unique continuous solutions over this interval and will satisfy arbitrary initial conditions at τ = 1 (Ince 1956, s. 3.3). If (C4b) is differentiated repeatedly, it will be observed that the higher order derivatives are well defined ratios of polynomials over the envisioned range and able to meet the Lipschitz conditions. Thus ω⁺⁺, ω⁺⁺⁺, etc. will also be well defined. Therefore, over our range of interest, we can utilise Taylor-Dirichlet expansions of the form

although the rate of convergence of the series is not guaranteed.

The structure of the system will become clearer if extensive substitutions are made:

and

Substitution of (C5) and (C6) into (C2) then yields, upon grouping, a series of terms, each of the form

where the summation extends over all combinations of p and q such that and where . In the abstract, the problem has now been solved to any order of approximation N. However, solving for the N + 1 unknown coefficients _n plus λ and is harder than might be thought because it turns out not to be possible to do a simple recursion up the chain of coefficients and then apply boundary conditions as an afterthought to determine λ and . One will always need N+3 independent relations among the unknowns, but they need not include all of the first N+1 instances of (C7).

Consider first n = 0. In Eqn (C7) setting to zero the term in brackets yields

as might have been expected from Eqn (S6) of the Accessory Publication to this paper. Since the ± sign does not appear in (C8), it follows that both the ξ-expansion and the χ-expansion yield the same plateau region far from the ends of the sieve tube. This is encouraging.

Next consider n = 1. Here (C7) yields , which, being homogeneous, cannot determine ₁ and instead constrains λ:

Now consider n = 2. Here (C7) yields

If ₁ were known, one could efficiently recurse using (C7) to find higher order coefficients. However, it is not. Therefore, we shall proceed to the 2nd order not by invoking (C10) but rather (i) by focussing on the ξ-solution and (ii) by imposing three boundary conditions at  = 0:

and

(C11a) expresses (20b) in the present notation and (C11b) expresses (20c) and (C11c) expresses (C2). Since, to order 2, ,

(C12b) and (C12c) combine to yield

(C8), (C13a) and (C13b) then combine with (C12a) to yield

(C9) and (C14) can then be combined to show that

Moreover, this value of can be back-substituted into (C9) to show that

From (C15) it can be predicted that κ(1000) = −114.04; the value derived by numerical integration is −113.83. Moreover, the plot of ω(ζ) given in Fig. 4 predicts that the distance-of-rise near ζ = 0 is roughly twice the distance-of-fall near z = 1; and this is in accord with the derivable from Eqns (C16). However, the value of Λ(10 000) measured from Fig. 5 is 17.6 in contrast to the 15.4 predicted from (C16b).

Using (C13) and the calculated values of and λ yields, for the expansion near ζ = 1, the coefficients

and

The value of ₂ predicted from Eqn (C10) using (C17a) and (C17b) is

This is sufficient for predictions of (C17) to be used along with (C7) for qualitative studies of convergence of Taylor-Dirichlet series (C5). As a practical matter, it seems probable that only terms of orders 0 and 1 will be needed throughout the transport phloem, because term-1 dies off exponentially quickly and the total contribution of the other terms should be negligible.

Finally, the simplification from Eqn (18f) to Eqn (19) depended upon assumed inequality

which (using |κ| ∼ Ξ^2/3 and the values of Table 1) requires υ_arb ≪ 4. Thus, the simplification leading to Eqn (19) is probably justified; or, at the least, it is a good enough approximation to have justified the analysis lavished upon it, because it is almost always the case that υ_arb  1 (Zimmermann and Ziegler 1975).

Symbol table