How can large-celled diatoms rapidly modulate sinking rates episodically?

rate of large-celled diatoms a periodicity of s their basis in cell density the we the cost lower for in the cell of different

and Bessarab, 2010). Therefore multiple repetitive fast action potentials would be needed in order to significantly increase and decrease Na+ and K+ cellular concentrations (in the mol m-3 range), respectively. However, decreasing overall cellular K+ concentration would require maintenance of a high K+ in the cytosol, nucleoplasm, chloroplast and mitochondria for enzyme activity perhaps via transport of vacuolar K+ transport out of the cell (Raven, 2018). By contrast, re-establishment of the initial low Na+ and high K+ concentration, and hence the original sinking rate, would need cellular energy, probably via the use of a plasmalemma Na+ K+ ATPase, which apparently occurs in diatoms (Bhattacharya and Volcani, 1980;Flynn et al., 1987;Rees, 1984). The use of a Na+-H+ antiporter in parallel with a H+ efflux ATPase is less likely, in view of the small H+ electrochemical potential difference across the plasmalemma of Coscinodiscus (Melkikh and Bessarab, 2010).
What would be the quantitative effect of Na+/K+ exchange on fast cyclical changes in sinking rate of large celled diatoms? Would it be energetically costly? Using the model of Lavoie et al. (2015) adapted for C. wailesii, here we computed the decrease in sinking rate due to Na+ influx and K+ efflux coupled to action potential-like downhill fluxes through ion channels, and estimate the energy cost related to the use of a Na+ K+ ATPase to restore the initial Na+ and K+ cellular composition. We used the modeling methodology described in Lavoie et al. (2015Lavoie et al. ( , 2016 with further details below in order to predict C. wailesii sinking rate or cell density as a function of cell biochemical composition and morphology. We find that the investment in cellular energy require for this strategy is probably high in order to sustain fast oscillations in sinking rates. We ran our model assuming a high cellular osmolarity, i.e.18% higher than that of seawater (1195 osmole m-3) as assumed in previous work for large celled diatoms (Boyd and Gradmann, 2002) and a low osmolarity (859.4 osmole m-3, i.e., 15 % lower than that of seawater) to cover a range of possible osmolarities that should minimize or maximize, respectively, the sensitivity of sinking rate to changes in the cell osmolarity while keeping the cell negatively buoyant (i.e., cell density > seawater density). Note that for the low osmolarity case, we assume active water influx (Raven and Doblin 2014) to prevent plasmolysis. Osmolarity was adjusted with addition of the putative organic osmolyte glycine betaine (GBT) and by modulation of intracellular ions concentrations. The concentration of GBT was 105 mole m-3 or 0 mole m-3 for the high and low osmolarity case, respectively. The concentrations of structural components (proteins, lipids, carbohydrates) were first approximate with the empirical equations of Hitchcock (1982) and then adjusted so that the total sum of cellular organic carbon, including the osmolyte GBT and carbon from protein, lipid and carbohydrate equal the cellular C concentration modeled with the empirical equation of Strathmann (1967) (2.82 x 10-4 pg C / cell) (mass balance within 1%). We consider that the mass of carbon accounts for 52, 40 and 85% of the protein, carbohydrate and lipid mass, respectively (Boyd and Gradmann, 2002).
For both osmolarity cases, to compute total cell density, we consider a model C.
The model cell sap density of 1036.90 and 1022.2 kg m-3 for the high and low osmolarity, respectively, is close to ( 0.8%) or within the range of measured cell sap density in C. wailesii, i.e., between 1015.0 and 1028.7 kg m-3 (Villareal, 1988). The total cell density was 1039.52 and 1024.92 kg m-3 for the high and low osmolarity case, respectively. Under those conditions, the rate of sinking of the cells is 0.33 mm s-1 (high osmolarity) and 1.3 x 10-4 mm s-1 (low osmolarity).
For the high osmolarity case, the model predicts that a 100 mol m-3 increase in Na+ coupled to a 100 mol m-3 decrease in K+ only decreased the sinking rate of C. wailesii by 4% while a 200 mol m-3 or 400 mol m-3 Na+ / K+ swap only decreased cell sinking rate by 8% or 19%, respectively. Such a very high ion exchange on a time scale of around 10 seconds translate into a Na+ uptake flux of up to 2.5 mol m2 s-1, which would approach diffusive limit of Na+ uptake by the cell (around 6 mol m-2 s-1 for an equivalent cell radius of 105 m and a Na+ diffusion coefficient of 1.33 x 10-9 m-2 s-1 measured by Li and Gregory (1974) ). The cellular Na+ concentration would also approach the seawater Na+ concentration, which would decrease the concentration component of the driving force for uptake. For the low osmolarity case, a much smaller swap of only 0.9 mol m-3 Na+ for 0.9 mol m-3 K+ decreased sinking rate by as much as 8.8-fold, which is a trend consistent with the results of Gemmell et al. (2016).
Is restoring the initial low Na+ and high K+ through the use of a Na+ K+ ATPase costly in terms of energy? Taking a total cellular C of 2.8 x 10-8 mol C cell-1 calculated above and the need for 24 mol absorbed photons per mol particulate organic C as measured in Chlamydomonas reinhardtii by Kliphuis et al. (2012) and predicted by mechanistic modeling of algal bioenergetics with some allowance for lack of waste minimization (Raven et al., 2014;Raven and Ralph, 2015), we calculate that 6.72 x 10-7 mol photons are needed per cell doubling. For the low osmolarity case, a Na+ release flux of 0.9 mol m-3 to re-establish baseline low Na+ cellular concentration using an electrogenic Na+ K+ ATPase (1 mole ATP per 3 mole Na+ release and 2 mole K+ taken up, with 1 mole K+ entering energetically downhill by K+ channels giving electroneutrality) would cost 5.03 x 10-12 mol ATP per cell per around 10 seconds.
Taking into account that the cell will oscillate between period of Na+ uptake / K+ efflux and period of Na+ efflux / K+ uptake of 10 seconds each, it follows that 4.35 x 10-8 mole ATP per cell will be required per 24 h. Assuming 10 mole photon (400-700 nm) per mol CO2 converted to 1 mol carbohydrate, including the energy costs of the CO2 concentrating mechanism (Raven et al. 2014), and 2.7 mol ATP produced in complete oxidation of 1 mole carbohydrate (Raven and Beardall, 2017), 3.7 mole photons (10/2.7 = 3.7) are needed to produce 1 mole ATP. 4.35 x 10-8 mole ATP per cell per 24 h thus requires 1.6 x 10-7 mole absorbed photon per cell per 24 h. For a cell doubling of 24 h in C. wailesii (Olson et al 1986) or longer under N limitation, an increase of 24% (16 x 100 / 67.2) of the absorbed photon cost for growth would be required just to fuel the Na+ K+ ATPase for the low osmolarity case. The additional energy cost is greater for the high osmolarity case. Furthermore, energy is required for moving K+ and Na+ between cytosol and vacuole, where most of the cellular K+ and Na+ occurs. Less is known of the energetics of these fluxes than of those at the plasmalemma (Bussard and Lopez 2014;Raven and Doblin 2014;Schreiber et al. 2017), but they could be as high as those at the plasmalemma, giving an additional cost of 48% of the energy cost of organic matter production in cell growth. These calculations suggest that this strategy involving fast Na+ / K+ exchange at the plasmalemma and tonoplast for episodic sinking rate modulation in large celled diatoms is possible, but might not be energetically sustainable. Our calculations do not rule out however the use of Na+ / K+ action potentials be use as a signal triggering the mechanisms that rapidly modulate buoyancy changes.

S2: Metabolism interconverting organic cations forming low density solutions and organic cations forming higher density solutions
This possibility is based on a role for organic cations, and especially tetramethylammonium (= tetramine), yielding low density solutions (Boyd and Gradmann, 2002;Lavoie et al., 2016;Raven and Doblin, 2014) and decreasing cell density, and conversion of this cation into a cation forming higher-density solutions, increase cell density; reconversion to tetramethylammonium once more decreases cell density. Tetramethylammonium has been reported from the terrestrial flowering plant Courbonia virgata (Henry 1948) and in some sea anemones and predatory marine gastropods (Mathews et al. 1960;Anthoni et al. 1989;Turner et al. 2018). In addition to the absence of information on the occurrence of tetramethylamonium (or related compounds yielding low-density solutions) in diatoms, the pathway of catabolism of tetramethylammonium to, (ex hypothesis), a denser product from which tetramethylammonium can be resynthesized, is not clear (Urakami et al., 1990). Even assuming tetramethylammonium occurs in algae, its synthesis in parallel with cell growth in the giant-celled centric diatom Ethmodiscus rex at a concentration required for positive buoyancy consumes over half as much energy (expressed as absorbed photons used in photosynthesis to synthesise the particulate organic matter of the cell) (Lavoie et al. 2016), i.e. a total energy cost of more than 1.5 times that of particulate organic matter synthesis. The variations in tetramethylammonium concentrations needed to cause the alterations in cell density and explain episodic sinking would presumably require an even greater energy input, especially considering the high frequency of changes in sinking rate. This would be especially the case if tetramethylammonium efflux to seawater occurs, with replacement by a denser inorganic anion, as a means of increasing density, with a subsequent synthesis of more tetramethylammonium and efflux of the dense inorganic cation, is the explanation of the density changes needed for episodic changes in sinking rate.

S3: Fast cyclical modulation of the rate of cell expansion and water uptake, with active water influx
An alternative to the previously discussed ion exchange mechanism would be to exploit periodic cell and protoplasm expansion at a rate faster than ion uptake in order to explain the enigmatic high frequency episodic variations in sinking of large celled diatoms. The cell volume of Coscinodiscus sp. indeed expands episodically due to an increase in the length of the centric diatom cells by decreased overlap of the frustules while the cell radius stay constant (Olson et al., 1986). However, small episodic changes in cell volume at high frequency (on the order of seconds) in order to explain the fast sinking rate modulation recorded by Gemmell et al. (2016) andDu Clos et al. (2019) have not been demonstrated. The occurrence of high frequency variations in the rate of increase of length of the protoplasm and frustule could modulate cell density and account for episodic sinking, provided the increase in cell volume is not driven by turgor, which depends on cell osmolarity in excess of that in the medium, but rather by cytoskeletal motors (Harold et al., 1996;Pickett-Heaps and Klein, 1998;Raven and Doblin, 2014). Starting with a turgid cell with a greater density (density for the low and high osmolarity case is 1024.92 kg m-3 or 1039.52, respectively), than that of seawater (seawater density = 1024.91 kg m-3), increase in cell volume at a rate faster than ions are accumulated but not faster than water can enter (down the (decreased) water potential gradient resulting from the smaller difference in osmolarity between cell contents and the seawater medium) decreases the density of the cell. This decrease in cell osmolarity can potentially continue until reaching seawater osmolarity, at this point, no further net water uptake can occur and hence dilution of intracellular ions is no longer possible. After the rapid cell expansion phase, cell expansion rate could become slower than ion uptake. In this case, as continuing ion influx gradually increases again the intracellular osmolarity and hence the cell density to maximum values, the sinking rate increases too until the next cycle of protoplast and/or cell expansion starts again.
For the energy cost of cell expansion, the involvement of the actomyosin system mentioned above is supported by the demonstration in Figure 4 of Gemmell et al.
(2016) that episodic sinking of Coscinodiscus wailesii was eliminated by the myosin ATPase inhibitor 2,3 butanedione monoxine (BDM) and by the actin inhibitor latrunculin A, applied separately, and that the episodic sinking was restored after rinsing the cells in filtered seawater, consistent with involvement of the actomyosin mechanochemical motor in the episodic sinking. If the actomyosin system is involved in decreasing the overlap of the frustules, and the mean rate of frustule overlap is the same with and without episodic extension means identical energy costs of episodic and non-episodic extension, although episodic elongation sinking may need more actomyosin than a constant rate of elongation and hence sinking. More plausibly, rapid but episodic elongation could require a more rapid specific reaction rate of myosin ATP hydrolysis and/or longer stroket length in the actomyosin system during the rapid cell elongation phase. However, the mean sinking rate is increased (reversibly) by the two actomyosin inhibitors relative to controls, so the inhibitors may act on other sinking-related processes, either by acting on actomyosin or in other ways. The occurrence of actomyosin-based cytoplasmic streaming in characaean algae is controlled by action potentials (Beilby and Casanova MT 2014); as indicated in the main text, action potentials are known in diatoms. Overall, the discussion in this paragraph suggests that there need not be additional energy input to account for episodic rather than continuous cell expansion.
Here we used the C. wailesii sinking rate model (see details in the first section of Supplementary Information) in order to explore the sensitivity of sinking rate to the aforementioned differential cell expansion hypothesis. For a cell doubling time of 24 h (Olson et al 1986), here we speculate that a cyclical cell expansion of 0.023 % occurs each 10 sec followed by a 10 sec. period of no growth. We computed the effect of the associated 0.023% decrease in cell osmolarity on sinking rate due to episodic cell expansion for a rapid small increase in cell length by 0.023%. For the high osmolarity case, we found that decreasing cell osmolarity by 0.023% coupled to an equivalent increase in cell length did not decrease the sinking rate significantly.
However, for the low osmolarity case, this cell expansion effect could decrease by 25fold the cell sinking rate. In this condition, the cell density is similar to the density of seawater. Since those two densities are subtracted in Stokes law, changes in cell sinking rate in response to each incremental change in cell osmolarity (and thus cell density) becomes disproportionately greater when the cell density is similar to seawater density (Lavoie et al., 2015). Therefore, it appears that a rapid increase in the volume enclosed by the frustules and correspondingly increase in protoplast volume, combined with a constant rate of K+, Cl-and Na+ (and other ions) influx, would decrease the sinking rate. After the first 10 s. phase of cell expansion, ion uptake rate could continue at the same rate than in the first 10 seconds phase while the cell volume stays constant (no expansion in the second 10 s phase) and density and sinking rate increase.
The above differential cell expansion strategy must be performed at a significant energy cost, which is potentially low (or at least manageable by the cell) according to the following reasoning. Here, it is assumed that active water transport occurs at the tonoplast, as in Raven and Doblin (2014). Here passive (downhill) water movement from the seawater medium to the cytosol using aquaporins in the plasmalemma, is assumed, requiring a higher osmolarity in the cytosol (and the compartments therein) than in seawater and the vacuole. Active water transport equates the water potential in the vacuole and the cytosol, maintaining the ratio of the volume of the cytoplasm relative to that of the vacuole.
The assumed volume expansion of 0.023% in a 10s period corresponds to an absolute cell volume increase of 1.13.10-15 m3 (0.023% multiplied by total cell volume of 4.91.10-12 m3 , equals 1.13.10-15 m3). With 55.55 kmole water per m3, the water flux is 6.27.10-11 mole water. The cation-chloride transporter moves 500 mole water per mole (K-Na-2Cl), and restoring the gradient of one mole (K-Na-2Cl) gradient requires 2 mole ATP (Raven and Doblin 2014) or 250 mole water per mole ATP. The influx of 6.27.10-11 mole water requires 6.27.10-11/250 = 2.5.10-13 mole ATP per cell. This calculation ignores the additional energy cost of compensating for any water backflux down the water free energy (water potential) gradient from the medium into the cytoplasm. 6.72.10-7 mole absorbed photo per cell. Therefore, the minimum computed cost of active water transport (1.08.10-7 mole absorbed photon per cell) represents a16% increase of the total energy cost for growth of 6.72.10-7 mole absorbed photon per cell.