# 3.1.2 - An example of a thought experiment

This section provides an example of a thought experiment. First, however, some essential background is given on solute transport and factors determining pH in plant cells to allow those who are less familiar with these themes to appreciate this exercise. Independently, students in the biological sciences are likely to find these sections particularly relevant and useful. In my experience, the background is also excellent for biological science teachers who wish to develop their own thought experiments, while the example will illustrate to those from the more fundamental sciences why this approach is rigorous. Even so, it is important to note that these sections are not a comprehensive review. Instead, they give some interesting facets of solute transport and pH regulation that can be used to solve the thought experiment and might be helpful in formulating other thought experiments.

Why choose solute transport and pH regulation? My reasons for choosing these themes were partly because they are particularly suited to foster innovative and independent thought, and partly because I am myself fascinated by them. Another reason is that many principles of solute transport and pH regulation are well defined but understanding how they interact in plants can be fraught. Furthermore, the theme of solute transport and pH regulation is a critical component of the physiology of animal as well as plant cells. As an example, during a personal experience I found criteria that were used to judge recovery of patients in intensive care are remarkably similar to those used by many plant scientists for health of plant cells, e.g. pH, energy-dependent transport and K+/Na+ regulation.

Solute transport is a key process in nearly all biological cells, and pH regulation to a large extent depends on solute fluxes. Further, pH regulation in plant cells is central to metabolism, with nearly all metabolic processes in one way or another regulated by pH. Because they are key processes, some readers are likely to be familiar with the topic and might wish to skip sections 2.1, 2.2 and 2.3 in which we describe the background necessary to participate in the thought experiments. For those not so familiar with ion relations in biological cells I hope they find the topic as fascinating as I did when I came across it during my career in environmental plant biology.

I will now focus on material essential to understanding the thought experiment. This will include some characteristics of the plant cell, and of solute transport across a membrane, including transport along and against a free energy gradient, membrane proton pumps, co-transport systems and very briefly, K+ channels. I will then focus on factors that determine pH in biological cells before turning to a detailed description of a thought experiment.

### 1. Some characteristics of the plant cell

The plant cell is a small volume separated from its environment by a lipid bilayer embedded with proteins, forming a semi-permeable membrane called the plasma membrane. A specific feature of plant cells is that only meristematic cells are similar to animal cells, being mainly cytoplasm. All more mature plant cells are highly vacuolated, i.e. a cytoplasmic shell surrounds a large central vacuole, that contains mainly electrolytes, but can also contain organic acids and sugars. These ‘compartments’, the cytoplasm and vacuole, are separated from each other by a semi-permeable membrane called the tonoplast. At steady-state, the environment and cytoplasm and vacuole are in dynamic equilibrium, that is, in flux equilibrium (Dainty 1962). The cytoplasm in most plant cells has a very high surface area to volume ratio, which helps to explain the very rapid effect that solute transport across its bounding membranes has on concentrations within the cytoplasm.

Plant tissues contain a large number of solutes including inorganic electrolytes, such as K+ and Cl-, and organic metabolites, such as sugars, amino acids and organic acids, some of which carry a charge at physiological pH. The fluid in the cytoplasm of a cell can differ greatly in composition from that in the vacuole, with the pH of the cytoplasm usually ranging between 7.2 - 7.5, while the vacuole is in most cases between 4.5 - 5.5. In most plant cells under optimum conditions, K+ is a major cation, with the concentration of K+ in the cytoplasm typically around 80 mM, while the concentration in the vacuole is more variable depending on the concentration of K+ in the environment (Leigh, 2001). The balancing anions vary, consisting of Cl-, NO3- and, if their sum in equivalents is lower than K+, dissociated anions of weak organic acids make up the deficit. An imbalance between inorganic cations and anions can be met by changes in organic acid metabolism, with organic acids synthesised to provide balancing anions when increases in concentration of inorganic cations exceeds that of inorganic anions, and vice versa, with organic acids catabolised when increases in inorganic anions exceed cations. For example, organic acids increased substantially in excised roots of 6-day old barley plants upon the addition of K2SO4 to the bathing solution containing 0.2 mM CaSO4 (Hiatt 1967). Uptake of cations from this treatment solution would be much larger than of anions, leading to synthesis of organic anions by the root cells. Conversely, organic acids decreased in excised roots when CaCl2 was added to the bathing solution (Hiatt 1967). Here, cation uptake from solution would be less than anion uptake leading to catabolism of organic anions. Table 2 shows that the changes in cations and anions in excised roots were of the same order of magnitude. In this example, the solutions in which plants were raised and the treatments applied contained CaSO4, so providing the calcium ions needed to protect membrane integrity of the plant tissue.

## 3.1-FE-Tab-3.2.png

How do these changes in organic acid metabolism impact on pH? According to Stewart (1983), the H+ concentration is not determined directly by such proton-producing or consuming processes but rather by the strong ion difference (SID) between cations and anions to which the dissociated anions of organic acids contribute (see section 2.3). As such, any changes in pH during organic acid metabolism will reflect the changes in SID. For further insights into pH regulation see section 2.3 of this paper, and Smith and Raven (1979).

Unless stated otherwise, our focus is on the cytoplasm of the plant cell and the transport of solutes across the plasma membrane that separates the cytoplasm from the environment surrounding the cell. Comparable transport mechanisms occur at the tonoplast.

### 2. Solute transport across a membrane

The fluxes of solutes across a membrane can be diffusive, that is along a gradient of free energy, or energy-dependent, that is occurring against a free energy gradient (Nobel 1974).

Before discussing solute fluxes across a membrane, there are two key concepts that need to be understood with regard to fluxes of electrolytes. First, movement of electrolytes will alter and at the same time determine the electrical potential across the membrane. At a steady membrane potential, the flux of positive and negative electrolytes across the membrane is equal. Second, in all solutions bounded by a cell membrane, the number of negative charges is balanced by the number of positive charges (see Plants in Action Chapter 3.6.3, http://plantsinaction.science.uq.edu.au/content/chapter-3-water-movement...). The membrane potential reflects a tiny imbalance in the concentration of cations and anions on either side of the membrane, the membrane potential difference arising from the unequal transport of charge by diverse ions through the membrane (Dainty 1962). In a healthy plant cell, the normal resting potential across the plasma membrane is around –115 mV. The capacitance (C) of biological membranes is very small, at around 1 mFarad cm-2, and so the charge (q) imbalance at the membrane need only have changed by 1 pmol of univalent ion per cm2 of membrane to see a 100 mV change in membrane potential (V), where V = q/C. That is the equivalent of 12 nmol L-1 of univalent ion averaged over a cell volume of about 650 pL (see Plants in Action Chapter 3.6.3). Another way of looking at it, given the cytoplasm is only 10% of the cell volume and assuming a cytoplasmic concentration of ~80 mM, the concentration of a univalent ion in the cytoplasm only needs to change by about 1.5 millionth, a tiny amount, to change membrane potential by 100 mV. The very small membrane capacitance and the tiny change in charge imbalance at the membrane that is associated with a change in membrane potential helps to explain the rapid response of membrane potential to perturbation. Interestingly, a similarly high sensitivity applies to the magnitude of the differences between cations and anion concentrations to enact changes of pH in the physiological range.

#### (a) Transport along a free energy gradient

Consider first the energy-independent transport of a neutral solute along its concentration gradient. For a neutral solute, this can be expressed as:

$da/(dx.R_1)$

where $da/dx$ is the chemical free energy gradient arising from the difference in solute activity, $da$, over distance, $dx$, and $R_1$ is the resistance of the membrane to flow of the solute concerned. In many cases the solutes have close to full activity and therefore the activity coefficient is 1. Then the activity gradient, $da$, becomes the concentration gradient, $dC$, of the solute concerned. As expressed in Fick’s law, where $D$ is the diffusion co-efficient, and, $C$ is the solute concentration:

$net flux = D (dC/dx)\tag{1}$

Electrolyte flux has a similar form but with the free energy gradient including not only a concentration component but also an electrical component, expressed as:

$E_m/R_2$

where $E_m$ is the membrane potential and $R_2$ is the resistance to flow through the membrane for that particular electrolyte. That is, the total potential for net movement of an electrolyte arises from the sum of its chemical potential gradient and its electrical potential gradient; an electrolyte moves under the influence of its electrochemical potential gradient. For both neutral solutes and electrolytes, passive flow is nearly entirely through channel proteins (Dainty 1962; Plants in Action Chapter 3.6.3). The big advantage of such a system versus flow through the lipid part of the membranes is that the proteins confer selectivity, and resistance to transport through channels can be closely regulated.

Assuming a solute activity coefficient of close to 1, the chemical free energy gradient arises from the concentration gradient, $dC$, of the solute concerned. The electro- part of the gradient is related to the electric field, i.e. the gradient in voltage. If the cell is negative relative to the medium, cations will accumulate in the cell and anions will be excluded unless there is energy-dependent transport e.g. pumping Cl- against the electrical gradient (see section 2.2 (b)). The voltage gradient can also drive influx or efflux of ions through ion channels as a passive process, but the voltage gradient itself is developed by an energy-dependent process.

A useful expression here is the Nernst potential, also called the equilibrium potential. If for a particular electrolyte there is no electrochemical potential gradient across the membrane, then there is no driving force on that electrolyte, and it is in passive equilibrium across the membrane. Although passive, this is a dynamic equilibrium in that the electrolyte will be moving across the membrane, but the influx and efflux will be equal. The Nernst equation describes the potential difference (mV) across a membrane that develops at the concentration gradient at which the electrolyte is in passive equilibrium, that is when influx equals efflux.

$E_m = (RT/zF) ln (C_o/C_i) \tag{2}$

where $E$ = membrane potential (mV), $R$ = gas constant (8.314 J.K-1.mol-1), $T$ = temperature (K), $z$ = valency, $F$ = Faraday constant (96500 J.V-1.mol-1), $C_o$ and $C_i$ = outside and inside concentration (mol.L-1) respectively. Solving for constants and substituting log10 for the natural log (ln), this can be rewritten as:

$E_m = (58.2/z) log_{10} (C_o/C_i) \tag{3}$

where 2.303 $RT/F$ at 20°C = 58.2 mV

This form of the Nernst equation illustrates that at an equilibrium membrane potential of –58 mV, the concentration of a monovalent, positively charged electrolyte will be 10-fold higher inside the cell than outside. So, at an $E_m$ of –115 mV, typical of the plasma membrane of a plant cell, an external concentration of 1 mM KCl will be in flux equilibrium with a cytoplasmic concentration of K+ of 100 mM at 20°C. That is, the internal concentration of an electrolyte can be much greater (or smaller) than the concentration outside the cell even though the electrolyte is in passive equilibrium across the membrane. A higher internal concentration of an electrolyte is not evidence of energy-dependent transport of that electrolyte into the cell (Dainty 1962; Plants in Action Chapter 3.6.3). In this example, the equilibrium concentration of Cl- in the cytoplasm would be 0.01 mM (100-fold lower than the external solution).

The Nernst potential gives a useful criterion for determining whether energy-dependent transport is occurring. Taking K+, if the measured resting membrane potential is less negative than the equilibrium potential (Nernst potential) calculated from the observed K+ concentration gradient across the membrane there would be an efflux of K+, so any K+ uptake must require energy (Dainty 1962). As an example, in barley roots the measured potential was –103 mV compared with the calculated equilibrium potential for K+ of –125 mV (Walker et al. 1995), so these authors concluded to maintain flux equilibrium of K+ at –103 mV, there must have been energy-dependent influx of K+.

#### (b) Transport against a free energy gradient

Ions can be transported against an electrochemical potential gradient and neutral solutes against their chemical gradients, provided there is a source of energy.

As an example, we consider the very clear case of low Cl- concentrations outside a cell. In this case, cells as a rule accumulate Cl- against an electrochemical gradient since cells are usually negatively charged relative to their environment. So, with no energy-dependent transport, internal Cl- concentration would be exceedingly low according to the Nernst equation. In fact, at modest external Cl- (say 0.5 - 5.0 mM) the internal concentration usually becomes much higher than the external concentration.

Returning to K+, whether the influx is passive or requires energy depends not only on the free energy gradient arising from the difference in K+ concentration across the membrane but also on the prevailing membrane potential. If the measured membrane potential is less negative than the equilibrium potential predicted by the Nernst equation from the known internal and external K+ concentrations, energy-dependent K+ influx is indicated (Dainty 1962; Walker et al. 1995).

#### (c) Solute/H+ co-transport

In plants, energy-dependent transport is often by solute/H+ co-transport. In this type of transport system, there are primary and secondary transporters. A primary transporter, taking for example the plasma membrane H+- ATPase, uses energy to pump H+ from the cytoplasm to the environment, creating a voltage (electro) and pH (chemical) gradient. The H+ can flow back along its electrochemical gradient e.g. from environment to cytoplasm, and the energy released during the influx of H+ can be used in secondary solute/H+ co-transport to transport solutes against their free energy gradient (Figure 1). Since this type of transport occurs so frequently it is worth drawing a useful analogy. We can say the inward H+ flow along its electrochemical gradient is downhill and the solute co-transport is uphill. Uphill transport can also be referred to as ‘pumping’. Examples are Cl-/H+ transporters (H+ influx - Cl- influx) and K+/H+ transporters (H+ influx - K+ influx) called symports (Figure 1; Ullrich and Novacky 1990), and Na+/H+ transporters (H+ influx - Na+ efflux) called antiports (Atwell et al. 2017). The simplest example is for a neutral solute, such as a sugar. A sugar/H+ symport (H+ influx - sugar influx) can drive sugar uptake against its concentration gradient, that is, where the ratio of the concentration of sugar in the cytoplasm / sugar in medium is greater than 1.0 such as occurs when sugars are ‘loaded’ into the phloem.

## 3.1-FE-Fig-3.1.png

Figure 1 Solute / H+ cotransport at the plasma membrane. The plasma membrane H+ -ATPase pumps H+ against an electrochemical gradient out of the cell. H+ enters the cell along its electrochemical gradient, providing energy for co-transport. For example, Cl-/H+ symport, where Cl- is taken up by the cell against its electrochemical gradient; Na+/H+ antiport, where Na+ is pumped out of the cell against its electrochemical gradient. K+ can be co-transported via a K+/H+ symport (high affinity); uptake can also be substantial through K+ channels (low affinity). The membrane potential across the tonoplast is about 30 mV more positive than that across the plasma membrane (Plants in Action Chapter 3.6.3) because of transport processes across this membrane that are not discussed here.

There is one other membrane proton pump in plant cells, the H+ -PPiase. The energy for the pumping is in this case provided by the high energy compound PPi rather than ATP. In this regard, the ATP and PPi can be considered as two different currencies (Atwell et al. 2017).

#### (d) Fluxes of different ions

Ions like K+ and Cl- are known to be in dynamic equilibrium with their environment. K+ fluxes in particular have a critical role in modifying membrane potentials and presumably pH over very short time periods (Dainty 1962; Plants in Action Chapter 3.6.3 for membrane potentials). Evidence for this includes the short half-times of exchange of the cytoplasm for K+ in barley, being 16 - 25 min for the high-affinity transport system (low external K+ concentrations, 0.1 mM) and only 8 - 9 min for the low-affinity transport system (high external K+ concentrations, > 1.5 mM) (Szczerba et al. 2006). The high-affinity transport system occurs via a K+/H+ symport with the inward flux of K+ against an electrochemical gradient, while the low-affinity transport system, which dominates at a higher external K+ concentration, is via the activity of K+ channels with flow occurring along the electrochemical gradient. Exchange rates of a third phase, presumably the vacuole, were much slower (Szczerba et al. 2006).

Membranes have different permeabilities for different ions because of the transport proteins that are expressed in different cell types. Fluxes of K+ across membranes tend to be very high, as are Na+ fluxes in several halophytes, with much lower fluxes for Ca2+ and Mg2+. For anions, Cl- and NO3- fluxes are much higher than SO42- (Cram 1973). K+ is usually the most permeable ion, with a rate of flux across the membrane typically 10 - 100 fold faster than Cl-.

The dominance of K+ and to a lesser extent of Na+ and Cl- in ion fluxes across biological membranes can be seen in the Goldman equation which predicts the equilibrium membrane potential $E_m$ taking into account all the ions that permeate the membrane. This equation is a function of the permeabilities of all diffusible ions and the concentrations of each of these ions on either side of the membrane. For a detailed description of the Goldman equation see Plants in Action Chapter 3.6.3. Only the permeabilities of K+, Na+ and Cl- tend to be included in the prediction of the equilibrium potential by the Goldman equation (Dainty 1962), which was developed for situations where it was known that the permeabilities of SO42- and Ca2+ were very small relative to K+, Na+ and Cl-. The same has since been assumed for other situations.

As K+ fluxes are usually much larger than Cl- fluxes, longer term K+ uptake is ‘held back’ by the rate of uptake of the balancing ion, Cl-. However, if there is no readily absorbable anion in the external solution (e.g. when K+ is balanced by SO42- or a colloid or non-penetrating organic acid), K+ uptake can still take place by producing organic acids in the cell, which in their dissociated form function as counter ions of K+, providing H+ is pumped out. In general, where cation and anion uptake differ substantially, the biological pH stat comes into play, with either syntheses or catabolism of organic anions (see section 2.1 and 2.3).

### 3. pH in biological cells

Understanding factors determining pH is always useful in view of the widespread importance of pH, for example a basic knowledge of pH is required to understand why, at increased CO2 in the atmosphere, waters of the seas and oceans will acidify.

We chose the regulation of cytoplasmic pH for the present course since changes in pH of biological cells are suited for several ‘thought experiments’.

#### (a) Factors determining pH

Stewart (1983) has stated that pH, an expression of the concentration of protons in solution, is a dependent variable. He then argued that the pH of solutions is determined by:

1. The strong ion difference (SID) between cations and anions, excluding H+ or OH-. Here, we refer to cations as *C+ = (C+ - H+) and to anions as *A- = (A- - OH-);
2. The content of weak acids and bases; these will influence pH since they dissociate, yielding anions or cations respectively;
3. The partial pressure of CO2, which will determine the amount of H2CO3 in aqueous solution. This is just a special case of a weak acid, but is particularly relevant to ecology, photosynthesis and ocean acidification.

That the concentrations of protons and hydroxyl ions in solution are dependent, not independent variables, gives a counter-intuitive dictum. That is, the H+ concentration of a solution is not determined by proton-producing or importing processes directly but via changes in the difference between *C+ = (C+ - H+) and *A- = (A- - OH-), with the pH increasing when *C+ - *A- increases (i.e. *C+ increases relative to *A-) and decreasing when *C+ - *A- decreases (i.e. *A- increases relative to *C+).

pH is a dependent variable because of the very low dissociation constant of water. In pure water at pH 7, the hydrogen ion concentration is 10-7 M = 0.0001 mM. Therefore, according to Stewart (1983), very slight changes in the balance between cations and anions are responsible for changes in pH we see in the physiological range, with H+ concentration fluctuating, depending on the imbalance between cations and anions. Consequently, the same applies to changes in membrane potential, where an increase in membrane potential of 100 mV, i.e. from –200 mV to –100 mV, would require an increase in *C+ = (C+ - H+) relative to *A- = (A- - OH-) of between 10-5 to 10-6 mM. For that to be achieved by H+ concentration alone, the pH of the cytoplasm would have to decrease to ~ 5, a level of acidity too low to allow metabolic reactions to occur.

#### (b) Example of a biochemical pH stat in anoxic rice coleoptiles

In cells growing under optimal conditions, longer term effects of treatments on pH are usually complicated by the many-fold reactions that occur. Fewer complications can be expected for anoxic rice coleoptiles since, while still healthy, as judged by rapid regrowth after return to aeration, their metabolism is very slow (Greenway et al. 2012). At external pH 6.5, coleoptiles elongated and took up K+ slowly, matched by equivalent amounts of malate synthesis. When these cells were put under an additional acid load by transfer to pH 3.5, malate and succinate decreased over 18 h by 0.3 and 0.1 µmol g-1 (fresh weight) h-1 respectively to just below 50% of their previous levels, with levels of both declining more slowly thereafter to near exhaustion. These losses were matched by leakage of K+, initially 2.5 µmol g-1 (fresh weight) h-1 in the period immediately following transfer to pH 3.5, dropping to 0.5 µmol g-1 (fresh weight) h-1 in the period 10 - 18 h after transfer (Greenway et al. 2012). In interpretation, at low external pH, H+ leaked into the cell and was incorporated into the organic anions, which were catabolised to ethanol and CO2 and lost to the medium, while the balancing cation, in this case K+, leaked to the medium as the organic anions declined (Greenway et al. 2012). The long term data confirmed organic acids played a major role in the biochemical pH stat (see section 2.1).

To sum up, biological cells need a very close balance of cations and anions. We have seen that a substantial electrical potential gradient across the plasma membrane results from a very slight imbalance between the concentrations of cations and anions. Likewise, a change in difference between strong cations and strong anions in a solution need only be minute to generate substantial changes of pH within the physiological range. Otherwise the balance would have to be provided by H+ or OH- and pHs would develop that can’t be tolerated by metabolic reactions.

### 4. A thought experiment

Experiments by Ullrich and Novacky (1990) are used here to illustrate a successful thought experiment, which in this particular case was actually tested in the laboratory. With knowledge gained from reading through sections 2.1, 2.2 and 2.3, readers may wish to try their hand at this thought experiment, after reading the information in the next two paragraphs.

Ullrich and Novacky‘s experiments concerned a key biological issue, namely, in what way and by what mechanism does the ubiquitous proton-pumping at the plasma membrane affect pHcyt. As background to this question, the research literature describes the effects on pHcyt of fusicoccin, a stimulator of the plasma membrane H+-ATPase that uses energy to transport protons out of the cell. Some experiments supported the plausible hypothesis that increased proton-pumping would, by removing H+ from the cytoplasm, increase cytoplasmic pHcyt. However other investigators reported an ‘unexpected’ result; rather than increasing pHcyt, stimulation of proton-pumping decreased pHcyt.

Ullrich and Novacky then used their knowledge of solute transport and pH regulation to hypothesise that the changes in pHcyt were related to changes in the strong ion difference (SID; section 2.3 (a)) and that stimulation of H+ export from a cell stimulated a solute/H+ co-transport system, i.e. there were two processes happening in sequence over short periods (likely to be in seconds). So, they devised a cunning thought experiment that extended a methodology used by Hiatt (1967) and involved placing plant roots in a basal solution of CaSO4 then applying two treatments plus and minus fusicoccin. At the start of the experiment, the first treatment received K2SO4 resulting in a bathing solution from which the cation (K+) was taken up by the tissue faster than the anion (SO42-), while the second treatment received CaCl2, from which the anion (Cl-) was absorbed faster than the cation (Ca2+). They predicted that pHcyt would increase when cation uptake exceeded anion uptake and, vice versa, pHcyt would decrease when anion uptake exceeded cation uptake. This is a testable hypothesis and typically, would end the thought experiment.

However, in fact, Ullrich and Novacky were fortunate in having the equipment and skill to test their hypothesis, the outcomes of which supported their predictions (Table 3).

## 3.1-FE-Tab-3.3.png

#### (a) Resolving the thought experiment

The thought experiment is key to our goal of stimulating and fostering independence in students, so here the Ullrich and Novacky narrative is reiterated with a different emphasis to show how the ‘unexpected’ result was resolved.

The original hypothesis, that H+ export from the cell reduces internal H+ concentration, i.e. increases pHcyt, cannot explain the result observed with CaCl2 where pHcyt decreases. So, we need to develop a new hypothesis that accommodates both the results observed with K2SO4 and with CaCl2, as follows:

1. In CaCl2, extra proton pumping stimulated by fusicoccin in turn stimulates influx by the H+/Cl- co-transporter;
2. Little Ca2+ is taken up;
3. So the SID changes, with an increase in *A- relative to *C+. This means that the concentration of the dependent variable (H+) increases, and pHcyt decreases even though proton extrusion was increased by the fusicoccin.

Thus, we can formulate a new hypothesis. That H+ export from the cell leads to changes in the difference between cations and anions with consequent changes in proton concentration.

This hypothesis also accommodates the K2SO4 result. K+ is much more readily taken up than SO42-, and so the SID changes with an increase in *C+ relative to *A-, which means that the H+ concentration will decrease, so in this case the pHcyt increases.

The students can then be asked to formulate a reasonable hypothesis as a further thought experiment. One suggestion might be the transport of a neutral solute when one would predict there would be no change in pHcyt. To check this hypothesis, apply a non-metabolised sugar in the form of 3-0-methyl glucose and there should be no effect on SID, that is, on pH. Note that glucose itself would add a new complexity, because as a metabolite it may stimulate uptake metabolism and hence uptake of a cation or anion.

Once the concept of thought experiments is adopted, most science disciplines abound with examples. Two such, using themes very different from those used here, can be found in Dixon and Grace (1984) on the impact of wind speed on evaporation from a leaf, and at a whole plant level, in Turner et al. (2020) on suckering and plant size in plantains.