Diffusional Interactions among Marine Phytoplankton and Bacterioplankton: Modelling H₂O₂ as a Case Study

Abstract

Marine phytoplankton vary widely in size across taxa, and in cell suspension densities across habitats and growth states. Cell suspension density and total biovolume determine the bulk influence of a phytoplankton community upon its environment. Cell suspension density also determines the intercellular spacings separating phytoplankton cells from each other, or from co-occurring bacterioplankton. Intercellular spacing then determines the mean diffusion paths for exchanges of solutes among co-occurring cells. Marine phytoplankton and bacterioplankton both produce and scavenge reactive oxygen species (ROS), to maintain intracellular ROS homeostasis to support their cellular processes, while limiting damaging reactions. Among ROS, hydrogen peroxide (H₂O₂) has relatively low reactivity, long intracellular and extracellular lifetimes, and readily crosses cell membranes. Our objective was to quantify how cells can influence other cells via diffusional interactions, using H₂O₂ as a case study. To visualize and constrain potentials for cell-to-cell exchanges of H₂O₂, we simulated the decrease of [H₂O₂] outwards from representative phytoplankton taxa maintaining internal [H₂O₂] above representative seawater [H₂O₂]. [H₂O₂] gradients outwards from static cell surfaces were dominated by volumetric dilution, with only a negligible influence from decay. The simulated [H₂O₂] fell to background [H₂O₂] within ~3.1 µm from a Prochlorococcus cell surface, but extended outwards 90 µm from a diatom cell surface. More rapid decays of other, less stable ROS, would lower these threshold distances. Bacterioplankton lowered simulated local [H₂O₂] below background only out to 1.₂ µm from the surface of a static cell, even though bacterioplankton collectively act to influence seawater ROS. These small diffusional spheres around cells mean that direct cell-to-cell exchange of H₂O₂ is unlikely in oligotrophic habits with widely spaced, small cells; moderate in eutrophic habits with shorter cell-to-cell spacing; but extensive within phytoplankton colonies.

1. Introduction

1.1. Marine Phytoplankton

Phytoplankton are a polyphyletic functional grouping of ~275,000 species of oxygenic photosynthetic microorganisms that grow while suspended in marine or freshwater habitats [1,2]. Phytoplankton interact with their aqueous environment [3], and with co-occurring bacterioplankton cells, through diffusion of nutrients, toxic compounds and signaling molecules, notably including reactive oxygen species (ROS). Phytoplankton span a wide size range [4] (Table 1), which directly affects their diffusional interactions. Larger cells have longer paths for intracellular diffusion or transport paths for solutes, and can therefore potentially impose larger concentration gradients between their internal cell contents and the local microenvironment. Larger cells also have lower surface to volume ratios which impose limits on their trans-membrane transport per biovolume of the cell [5]. On the other hand, some larger cells have dynamic control over their sinking through water, and can thereby refresh their surface boundary layers more effectively than can smaller cells [3,6].

1.2. Phytoplankton Cell Spacing in Oceans

Phytoplankton and bacterioplankton cell numbers in the ocean vary widely, and therefore so does intercellular distance (Table 1). Direct colonization of phytoplankton by bacteria [7], or colony-forming phytoplankters, bring multiple cells into yet closer local proximities. For example, within Phaeocystis colonies, cell densities reach 2.4 × 10⁷ cells mL⁻¹ [8] (Table 1).

Table 1. Phytoplankton and bacterioplankton cell sizes and representative suspension densities.

Habitat	Taxa	Metabolism	Cell m−3	Cell Radius µm	Citation
Oligotrophic	Bacterioplankton	Hetero	4.7 × 1010	0.5	[9]
Oligotrophic	Bacterioplankton	Hetero	2.0 × 1011	0.5	[10]
Oligotrophic	Diatoms	Phyto	5.6 × 107	10.0	[11]
Oligotrophic	Dinoflagellates	Phyto	6.5 × 106	10.0	[11]
Oligotrophic	PicoEukaryotes	Phyto	6.7 × 108	2.0	[12]
Oligotrophic	PicoEukaryotes	Phyto	2.0 × 109	2.0	[10]
Oligotrophic	Prochlorococcus	Phyto	4.7 × 1010	0.3	[12]
Oligotrophic	Prochlorococcus	Phyto	3.8 × 1010	0.3	[10]
Oligotrophic	Silicoflagellates	Phyto	1.3 × 107	10.0	[11]
Oligotrophic	Synechococcus	Phyto	3.0 × 109	1.0	[12]
Oligotrophic	Synechococcus	Phyto	2.0 × 109	1.0	[10]
Mesotrophic	Bacterioplankton	Hetero	3.3 × 1011	0.5	[10]
Mesotrophic	PicoEukaryotes	Phyto	3.0 × 109	2.0	[10]
Mesotrophic	Prochlorococcus	Phyto	3.1 × 1010	0.3	[10]
Mesotrophic	Synechococcus	Phyto	6.0 × 109	1.0	[10]
Eutrophic	Bacterioplankton	Hetero	4.6 × 1011	0.5	[10]
Eutrophic	Bacterioplankton	Hetero	4.6 × 1013	0.5	[13]
Eutrophic	Chlorophytes	Phyto	8.2 × 1010	2.5	[14]
Eutrophic	Chroococcales	Phyto	8.2 × 1010	1.0	[14]
Eutrophic	Cyanobium	Phyto	4.0 × 1012	1.0	[14]
Eutrophic	PicoEukaryotes	Phyto	3.0 × 109	2.0	[10]
Eutrophic	PicoEukaryotes	Phyto	1.5 × 1013	2.0	[13]
Eutrophic	Synechococcus	Phyto	4.5 × 109	1.0	[10]
Colony	Cyanobium	Phyto	1.0 × 1013	1.0	[14]
Colony	Phaeocystis	Phyto	2.4 × 1013	2.2	[8]

Table 1. Phytoplankton and bacterioplankton cell sizes and representative suspension densities.

Habitat	Taxa	Metabolism	Cell m−3	Cell Radius µm	Citation
Oligotrophic	Bacterioplankton	Hetero	4.7 × 1010	0.5	[9]
Oligotrophic	Bacterioplankton	Hetero	2.0 × 1011	0.5	[10]
Oligotrophic	Diatoms	Phyto	5.6 × 107	10.0	[11]
Oligotrophic	Dinoflagellates	Phyto	6.5 × 106	10.0	[11]
Oligotrophic	PicoEukaryotes	Phyto	6.7 × 108	2.0	[12]
Oligotrophic	PicoEukaryotes	Phyto	2.0 × 109	2.0	[10]
Oligotrophic	Prochlorococcus	Phyto	4.7 × 1010	0.3	[12]
Oligotrophic	Prochlorococcus	Phyto	3.8 × 1010	0.3	[10]
Oligotrophic	Silicoflagellates	Phyto	1.3 × 107	10.0	[11]
Oligotrophic	Synechococcus	Phyto	3.0 × 109	1.0	[12]
Oligotrophic	Synechococcus	Phyto	2.0 × 109	1.0	[10]
Mesotrophic	Bacterioplankton	Hetero	3.3 × 1011	0.5	[10]
Mesotrophic	PicoEukaryotes	Phyto	3.0 × 109	2.0	[10]
Mesotrophic	Prochlorococcus	Phyto	3.1 × 1010	0.3	[10]
Mesotrophic	Synechococcus	Phyto	6.0 × 109	1.0	[10]
Eutrophic	Bacterioplankton	Hetero	4.6 × 1011	0.5	[10]
Eutrophic	Bacterioplankton	Hetero	4.6 × 1013	0.5	[13]
Eutrophic	Chlorophytes	Phyto	8.2 × 1010	2.5	[14]
Eutrophic	Chroococcales	Phyto	8.2 × 1010	1.0	[14]
Eutrophic	Cyanobium	Phyto	4.0 × 1012	1.0	[14]
Eutrophic	PicoEukaryotes	Phyto	3.0 × 109	2.0	[10]
Eutrophic	PicoEukaryotes	Phyto	1.5 × 1013	2.0	[13]
Eutrophic	Synechococcus	Phyto	4.5 × 109	1.0	[10]
Colony	Cyanobium	Phyto	1.0 × 1013	1.0	[14]
Colony	Phaeocystis	Phyto	2.4 × 1013	2.2	[8]

These wide ranges of cell suspension densities for phytoplankton and bacterioplankton across habitats strongly influence the extent to which the growth and activity of a phytoplankton population or community alters, or even governs, solute concentrations in the local habitat [15,16,17,18]. Beyond the bulk activities of the phytoplankton population or community, differences in cell suspension densities mean the diffusional path lengths for solutes separating individual cells vary by orders of magnitude across habitats and taxa. Intercellular distances then influence whether the activity of an individual cell can directly influence the microenvironment of neighboring cells [19].

Our understanding of the taxa- and habitat-specific effects of cell-to-cell separation is then influenced by schematic visualizations of phytoplankton communities with implied cell suspension densities far higher than is realistic, or with cell symbol sizes scaled differently than the scaling of the spatial axes [20]. Symbol size scaling exaggerated relative to the scaling of the spatial axes allows schematic visualizations of cells, but can give a visual impression of a community with fairly close cell-to-cell interactions. In our visualizations, we maintain cell symbol size at the same size scaling as the spatial axes (Figure 1).

1.3. Reactive Oxygen Species

Electrons can transfer from photosynthetic or respiratory electron transport to reduce O₂ to form the superoxide radical, a reactive oxygen species (ROS) [21,22] (Reaction (1)).

ROS include both radicals (superoxide, nitric oxide and hydroxyl radicals) and non-radicals (hydrogen peroxide, singlet oxygen and peroxynitrite), which are more reactive than ground state di-oxygen molecules [23,24]. Reduction of O₂ to H₂O requires the sequential addition of four electrons, and therefore progresses through three ROS intermediates: superoxide (O₂^•−), hydrogen peroxide (H₂O₂) and hydroxyl radical (OH^•) (Reaction (1)) [25]. ROS are by-products of both photosynthetic and heterotrophic metabolism, but are also generated abiotically in seawater [26].

$$\begin{array}{c}{\mathrm{O}}_2+{\mathrm{e}}^{-}\to {\mathrm{O}}_2^{•-}\\ {\mathrm{O}}_2^{•-}+2{\mathrm{H}}^{+}+{\mathrm{e}}^{-}\to {\mathrm{H}}_2{\mathrm{O}}_2\\ {\mathrm{H}}_2{\mathrm{O}}_2+{\mathrm{H}}^{+}+{\mathrm{e}}^{-}\to {\mathrm{H}}_2\mathrm{O}+{\mathrm{OH}}^{•}\\ {\mathrm{OH}}^{•}+{\mathrm{e}}^{-}+{\mathrm{H}}^{+}\to {\mathrm{H}}_2\mathrm{O}\end{array}$$

(1)

Biogenic production of extracellular ROS mediated by microorganisms is a significant contributor to marine environments [27,28], mediated by taxa ranging from heterotrophic bacteria to diatoms including those from the Thalassiosira and Coscinodiscus genera [28,29,30,31,32,33,34,35].

ROS are important at low concentrations due to their role in cell signaling [36,37,38,39,40,41,42] and lipid accumulation [43]. In order to benefit from ROS, organisms must maintain ROS levels above the cytostatic threshold, the minimum concentration of ROS required to maintain normal cellular processes, as well as below the cytotoxic threshold, the level at which the negative effects of ROS outweigh the positives [44].

1.4. Concentrations and Properties of H₂O₂

H₂O₂ is an uncharged compound documented in the ocean at concentrations of 10⁻⁹ to 10⁻⁶ mol L⁻¹ [29,45] depending upon location and conditions. These [H₂O₂] are high enough to significantly contribute to the redox cycling of copper [46,47]. [H₂O₂] in seawater increases after precipitation, including rain and snowfall [18,26,48]. [H₂O₂] in seawater also follows a diurnal cycle with a peak at mid-day [18,49,50], which suggests significant direct or indirect photochemical or photobiological generation of H₂O₂. [H₂O₂] also exhibits latitudinal variation at the surface, higher in the brighter, warmer, mid to low latitudes compared to the colder, darker, higher latitudes [50]. The authors of [50] suggest that regional variation in [H₂O₂] may be caused by the depth of the mixed layer and the concentration of total dissolved organic carbon. Abiotic production rates of H₂O₂ vary from 0.28 to 4.2 pM s⁻¹ in near surface open ocean waters vs. 1.0 to 84.4 pM s⁻¹ in coastal sites [26].

The diffusion coefficients (a measure of how quickly a molecule travels via diffusion), and therefore the diffusional mobilities, of ROS through seawater vary widely. Diffusion coefficients in water decrease with increasing molecular mass, and from uncharged to charged compounds. Diffusion coefficients are strongly influenced by temperature and slightly influenced by pressure. For H₂O₂ at ~20 °C the diffusion coefficient is ~1500 µm² µs⁻¹ (note conversion of units from more typical reporting of cm² s⁻¹) [51]. In aquatic systems, H₂O₂ has extracellular lifetimes of hours to days [29,52] before decay.

H₂O₂ is acutely toxic to most cells in the range of 10⁻⁵ to 10⁻⁴ mol L⁻¹ range [53], about 10-fold higher than seawater concentrations [29,45]. We found few quantitative estimates of the intracellular concentrations of H₂O₂ within cells [54,55,56,57,58,59,60], with a particular lack of measurements from phytoplankton. Limited data, from mammalian cells, support an intracellular H₂O₂ concentration of 10⁻⁶ M [61]. Although only weakly directly reactive, H₂O₂ can react with thiols and methionine [62] and thereby modulate gene expression and transcription [63,64]. Meanwhile, limited experimental data indicate that heterotrophic bacteria maintain an internal [H₂O₂] of 1/10 of the external [H₂O₂] when extracellular H₂O₂ is present [58,60], as it is in seawater. The authors of [58] also predicted an intracellular steady-state [H₂O₂] of 20 nM in the absence of extracellular H₂O₂, which is far below the toxic threshold of intracellular H₂O₂ [60], and is likely for the maintenance of physiological roles of H₂O₂. Cytotoxic effects of H₂O₂, including lipid damage, are primarily caused by H₂O₂ converting to the hydroxyl radical, which is strongly oxidative [52]. Interestingly, H₂O₂ may be used as a terminal electron acceptor in the absence of oxygen in Escherichia coli [65].

H₂O₂ readily crosses the cell membrane [53] primarily through aquaporins [66,67,68,69]. As a result, intracellular H₂O₂ exchanges with extracellular H₂O₂, but cells can nevertheless maintain intracellular [H₂O₂] significantly different than extracellular [H₂O₂] [58]. Other ROS rarely cross cell membranes and therefore the intra- and extracellular pools are not directly connected, although they may influence each other indirectly through transformations to other ROS, for example dismutation of intracellular O₂^•- to form intracellular H₂O₂ which can then exchange with extracellular H₂O₂, and vice versa.

A significant fraction of H₂O₂ destruction in aqueous systems may be attributed to the activities of heterotrophic microbes [70], although heterotrophs are minor contributors to bulk [H₂O₂]. Microbial scavenging of H₂O₂ indeed increases the survival of Prochlorococcus through extended periods of darkness, as demonstrated through co-culture studies [71] with ‘helper’ bacteria which carry genes for catalase. In the open ocean, bulk [H₂O₂] levels generally remain below the cytotoxic threshold, partly because heterotrophic organisms provide catalase [72].

Changes in [H₂O₂], or other ROS, around a cell will depend upon net production, or consumption by the cell, which generate net release (or uptake) of [H₂O₂] from the cell after any intracellular or cell-associated scavenging. These inputs can interact to generate a diffusion-generated gradient of [H₂O₂] around the cell, above or below seawater [H₂O₂]. Whether or not a cell establishes a zone of [H₂O₂] different from the [H₂O₂] background also depends upon extracellular destruction and diffusion rates. The extent of this altered zone will then influence downstream processes. For specific cell–cell interactions or responses, a cell has to change the [H₂O₂] in the local microenvironment sufficiently to provoke a response in neighboring cells. Beyond this local sphere of influence around an individual cell, the collective activity of cells in a community contributes to generating a general seawater [H₂O₂], which in turn influences cellular gene expression and metabolism.

1.5. Goals

Microbes consume and produce H₂O₂, thereby potentially helping, or at least influencing, other microbes. We sought to quantify the influences of cell density and cell size on these potential microbial interactions. H₂O₂ presents a good case study for diffusional interactions among microbes. H₂O₂ readily traverses cell membranes, and there is information on background concentrations of H₂O₂ in seawater, H₂O₂ diffusion coefficients in seawater and some estimates of cellular concentrations of H₂O₂, as either absolute values or ratios compared to external [H₂O₂]. We can therefore approximate concentration gradients generated by [H₂O₂] diluting outwards from a cell maintaining a homeostatic intracellular [H₂O₂] higher than seawater [H₂O₂], or the concentration gradient inwards towards a hypothetical cell maintaining a homeostatic intracellular [H₂O₂] lower than seawater [H₂O₂]. We can use these simple approximations to generate quantitative thresholds for when H₂O₂ released from a cell can directly influence neighboring cells, i.e., how much, and how far, a phytoplankton or bacterioplankton cell can change the local [H₂O₂].

2. Results and Discussion

2.1. Cell to Cell Spacing across Habitats and Taxa

Possible cell-to-cell exchanges of H₂O₂ are mediated by diffusions through intercellular path lengths, varying by orders of magnitude across habitats and taxa. To understand such diffusional exchanges we need realistic visualizations of cell spacings, achieved by maintaining cell symbol size at the same size scaling as the spatial axes (Figure 1). In the visualization of randomized cell spacing under oligotrophic conditions, we see wide spacing among cells, because the small cell symbols are on the same size scaling as the spatial axes. In the visualization of randomized cell spacing under eutrophic conditions, some cells are in close proximity. Note that this standardized scaling visualization approach is only feasible with highly expanded spatial axes such as the 1 mm axis used in Figure 1, equivalent to simulation of only 1 µL. Attempting to visualize 10 mm axes, equivalent to a 1 mL volume, causes the cell symbol size to shrink below visibility (data not presented). A further nuance not captured in these visualizations and subsequent simulations is non-random clustering of cells, as perhaps around particles of marine snow.

2.2. H₂O₂ Concentration Gradients around Cells

Figure 2 shows a light blue background for the approximate range of natural [H₂O₂] in seawater, with the simulated profiles of [H₂O₂] moving outward from a ‘Bacterioplankton’ cell with intracellular [H₂O₂] maintained below seawater [H₂O₂]; or outward from small phytoplankton cells of different sizes with intracellular [H₂O₂] maintained at 1 µmol L⁻¹ [57], near the highest concentrations expected for seawater [ROS]. For [H₂O₂] generators (nominally, phytoplankton of different sizes), [H₂O₂] concentrations are highest near the cell surface and decrease with distance outwards. For [H₂O₂] consumers (nominally, bacterioplankton), [H₂O₂] concentrations are lowest near the cell surface.

Decay is a key influence on the pseudo-steady state bulk [H₂O₂] in seawater, but for our conceptual estimates of [H₂O₂] changing within the local region of the cell, the dilution term (black points, Figure 2) dominates the concentration gradient moving out from the cell. Including the influence of the decay term (red points, Figure 2) has only a negligible influence on [H₂O₂] moving out from the phytoplankton cell over simulated timescales of <100 µs and distances of ~100 µm. For other, less stable ROS, decay would become a significant influence, acting to narrow the local concentration gradients around cells.

Seawater contains H₂O₂ which varies significantly depending upon conditions and region. At some threshold distance out from the cell-specific sphere of influence upon local [ROS], it reaches the seawater [H₂O₂] (Figure 2), beyond which distance net [H₂O₂] production or consumption by the specific cell has no local influence on [H₂O₂], but is simply one input into the wider seawater [H₂O₂]. All other factors being equal, in our estimations the region over which a cell changes the local [H₂O₂] away from seawater [H₂O₂] increases with cell radius. This wider influence of cells with larger cell radii results primarily from the volume of the intracellular homeostatic [H₂O₂] increasing as the cube of cellular radius (Figure 2).

In this simple estimate, a heterotrophic bacterioplankton cell maintaining intracellular [H₂O₂] at 1/10 of the external seawater [H₂O₂] locally drops [H₂O₂] below the lower end of the seawater [H₂O₂] for a radius of only 1.2 µm. If the bacterioplankton cell were to maintain an intracellular [H₂O₂] of <1/10 of the external seawater [H₂O₂], the threshold distances would be greater. The phytoplankton cells maintaining intracellular [H₂O₂] at 1 µmol L⁻¹ influence local [H₂O₂] above a low background level of seawater [H₂O₂] for radii of 3.1 µm for Prochlorococcus up to 90 µm for diatoms, or even more for yet larger phytoplankters not simulated. Furthermore, these threshold distances apply to the lowest published estimates for seawater [H₂O₂]. Under environments with higher seawater [H₂O₂], the threshold distances would be smaller.

2.3. Visualizing Cell to Cell Exchange of [H₂O₂]

We next combine our visualizations of cell suspensions from eutrophic, oligotrophic, or colonial habitats, overlaid with the mathematical expressions of cell-specific spheres of [H₂O₂] above or below seawater levels, to visualize the extent to which cells directly influence the [H₂O₂] environments of their neighbors.

In eutrophic habitats, bacterioplankters are numerous, but their small cell size, and correspondingly small cell-specific spheres of upon local [H₂O₂], mean that their individual contributions towards local [H₂O₂] are negligible (Figure 3), even though their collective influence upon background [H₂O₂] may be considerable [71,72]. Phytoplankton cells, particularly the larger taxa, can however influence local [H₂O₂] over wider distances, leading to overlaps in their cell-specific spheres of influence (Figure 3) with bacterioplankton and with other phytoplankton, and opening the possibility of direct cell-to-cell signaling or cell-to-cell metabolic influences through [H₂O₂], beyond contributions to background seawater [H₂O₂]. In oligotrophic habitats, wider cell-to-cell spacings, and generally smaller cell sizes, make direct cell-to-cell reciprocal influences upon local [H₂O₂] unlikely, although the phytoplankton and bacterioplankton as a whole indeed contribute to establishment of the background [H₂O₂].

For phytoplankters growing colonially, the [H₂O₂] leaving specific cells within the colony strongly influence the local [H₂O₂] environments of neighboring cells in the colony (Figure 4). Such a colony could also be usefully simulated as a single homeostatic sphere of [H₂O₂], with a significant local sphere of influence beyond the colony. Although we did not include bacteria in Figure 4, in fact Phaeocystis antarctica colonies have rich bacterial components [73], which may greatly alter the local H₂O₂ environment within the colony, beyond the activities of the Phaeocystis cells.

2.4. Threshold Distances for Cell to Cell Exchange of [H₂O₂]

We find that cell size, cell suspension density, cellular [H₂O₂] and seawater [H₂O₂] interact to influence the potential for direct cell-to-cell diffusional interactions. We sought to generalize the estimations presented earlier for combinations of taxa x habitat (Figure 3) to summarize how these variables potentially affect cell-to-cell interactions via H₂O₂ (Figure 5).

Figure 5 plots combinations of cell suspension density (X axis, governing cell-to-cell distance), and the radii of the cell-specific sphere of influence upon [H₂O₂] exceeding seawater [H₂O₂] (Y axis), for homogeneous populations. The threshold lines mark the transition for cell-to-cell interactions, for four different ratios of cellular [H₂O₂] to seawater [H₂O₂]. The areas above the threshold lines show the regions of cell radius and cell suspension density where the cell-specific spheres of influence upon [H₂O₂] exceed the average cell-to-cell spacing, for a given ratio of cellular [H₂O₂] to seawater [H₂O₂]. For the purpose of presentation, we collapsed cellular [H₂O₂] and seawater [H₂O₂] to the ratio of cellular [H₂O₂] to seawater [H₂O₂] since that ratio determines the position of the threshold line, rather than the absolute values for cellular [H₂O₂] and seawater [H₂O₂]. We see that a homogeneous population of large cells of 40 µm radius would maintain overlapping cell-specific spheres of influence down to cell suspension densities of 2.7 × 10⁹ cells m³ (Figure 5). In contrast, a homogeneous population of cells of 0.5 µm radius would only achieve directly interacting cell-specific spheres of influence at a hypothetical 1.9 × 10¹⁵ cells m³, far above cell suspension densities for most marine habitats (Table 1). For comparison, in Figure 5 we overlay symbols for representative taxa from different habitats (Table 1) showing that for hypothetical homogeneous populations the cell-specific spheres of influence are generally far below the expected cell-to-cell spacing.

Figure 6 shows that for natural population densities of bacterioplankton (Figure 6; Points), the spheres of lowered local [H₂O₂] would overlap when [H₂O₂] intracellular:extracellular ratio is less than or equal to 0.01 (Figure 6, short dashed line). However, the limited available information [58] suggests that heterotrophic bacteria maintain an [H₂O₂] intracellular:extracellular ratio of about 0.1 (Figure 6, dotted line). It is possible that cells may not be able to maintain an intracellular:extracellular [H₂O₂] ratio of 0.1 at the lowest seawater [H₂O₂] given the likely roles of H₂O₂ in cell signaling [36], and the need to maintain a basal [H₂O₂] [44]. The authors of [58] predict an intracellular steady-state [H₂O₂] of 20 nM in heterotrophs in the absence of extracellular H₂O₂, which would suggest that at low extracellular [H₂O₂], the intracellular:extracellular [H₂O₂] ratio would be >0.1, and the spheres of lowered local [H₂O₂] would not overlap (Figure 6). Thus, the cell-specific effects of bacterioplankton upon local [H₂O₂] are unlikely to overlap at reasonable cell suspension densities.

3. Conclusions

Our approximations of cell-to-cell spacing are applied to each taxa x habitat combination separately, based upon published measures of cell suspension densities in marine systems. We then overlaid the taxa-specific simulations from a given habitat to give a visualization of community level cell-to-cell spacings, across taxa within a habitat (Figure 1). As a next step we aim to implement probability distributions of cell-to-cell spacings across all cells from a mixed community in the habitat, to more accurately capture the likelihood of cell-to-cell interactions. Our work used available cell suspension density data exclusively from marine habitats. In general, we would assume that if cell densities and [H₂O₂] in freshwater systems are similar to that of marine systems, the diffusional patterns described for marine habitats would be comparable to freshwater systems, but future studies would need to confirm this.

Our simple approximations of diffusion gradients out from cells are limited by a dearth of published information on the intracellular concentrations of H₂O₂, and other ROS, in phytoplankton or bacterioplankton cells. There are published estimates of intracellular and extracellular H₂O₂ production and decay rates in seawater, which could support more sophisticated reaction/diffusion simulation approaches.

The threshold lines for interacting cell-specific spheres of influence as functions of cell radii and cell suspension densities are generated for individual ‘taxa’ from marine environments with uniform intracellular [H₂O₂] (Figure 5), rather than for a mixed community of different cell sizes and physiologies. These fixed threshold lines also apply to a fixed ratio of internal to external [H₂O₂], whereas this ratio could vary rapidly with changes in cell physiology and external conditions, causing the threshold for direct cell-to-cell interactions to shift. Our models suggest that it is unlikely that H₂O₂ diffusing out of a phytoplankton cell directly interacts with another phytoplankton cell at natural seawater cell densities (Figure 5), with the exception of Phaeocystis, whereby H₂O₂ diffusing out of a cell would directly interact with other cells if the intracellular:extracellular [H₂O₂] ratio is greater than 100. A future approach aims to generate probability distributions for direct cell-to-cell interactions, since even populations with widely spaced cells occasionally generate close cell-to-cell distances. Analyses show higher rates of cell-normalized H₂O₂ production in bloom-forming phytoplankton taxa, providing support for the selective influence of cell suspension density upon ROS dynamics [29].

It is likely that extracellular ROS play a role in the growth and development of phytoplankton, given that [74] found that the addition of superoxide dismutase and catalase inhibited the growth of Chattonella marina. Our models suggest that H₂O₂ diffusing out of phytoplankton cells reaches external [H₂O₂] before interacting with other cells (Figure 3 and Figure 5), and are primarily driven by the influence of diffusion, and with negligible contribution from the decay of H₂O₂ (Figure 2). Indirect interaction of H₂O₂ diffusing out of cells is also possible as H₂O₂ can affect biogeochemical cycles and bioavailability of nutrients [75].

The simple estimates presented herein using H₂O₂ as a case study are generalizable to other, less stable ROS where decay will be a more immediate influence. For example, O₂^•− itself is not cell membrane permeable and so forms separate intra- and extracellular pools. However, O₂^•− is metabolized to H₂O₂, both inside and outside cells, and so could indirectly influence [H₂O₂], and thereby alter cell-to-cell interactions beyond the simple diffusion-dominated simulations we present. For example, extracellular production of O₂^•− by heterotrophic bacteria can significantly alter O₂^•− concentrations in the dark ocean [28]. Heterotrophs might also produce enough O₂^•− to indirectly influence concentrations of O₂^•− in the photic zone of the ocean. The phytoplankter T. weissflogii [34] generates ranges of ~8 × 10⁻¹⁶ mol O₂^•− cell⁻¹ hr⁻¹, while four raphidophytes show production ranges of 0.45 to 4 × 10⁻¹² mol O₂^•− cell⁻¹ hr⁻¹ [76], illustrating the wide range of extracellular production which can differentially influence extracellular [ROS] across communities or conditions.

The approaches presented here are a step towards visualizing and constraining estimates of cell-to-cell interactions in plankton communities. We next need to validate these approaches with quantitative estimates of intracellular [H₂O₂], and other [ROS], within phytoplankton and bacterioplankton cells.

4. Methods, Simplifying Assumptions, and Limitations

4.1. Estimation of Cell to Cell Spacing

A simplistic estimator of cell-to-cell spacing is the reciprocal of the cube root of the cell suspension density. For example, if cell suspension density is expressed as cells µL⁻¹ (equivalent to cells mm⁻³), the reciprocal of the cube root gives a cell-to-cell spacing estimate in mm. This cube root estimate assumes equally spaced cells, which somewhat overestimates average cell-to-cell spacing, since equally spaced cells give a maximum spacing, not the average spacing of randomly located cells. An arithmetic correction based upon the gamma function of 4/3 generates a numeric correction to multiply the reciprocal of the cube root of the cell suspension density by 0.55 to approximate average spacing of cells [77] (Equation (2)).

$$CellSpace\mathrm{mm}=\sqrt[3]{\frac{1}{CellCount{\mathsf{\mu }\mathrm{L}}^{-1}}}\ast 0.55$$

(2)

We generated simulated cell suspension densities for each taxa and habitat by generating points separated by the average cell spacing between cells, along each of the X, Y, Z spatial axes. We then added random variation to the values of the evenly spaced points (R jitter, factor = 2), to generate a XYZ coordinate cloud. In a parallel approach (data not presented) we directly generated normally distributed random distributions around the average cell spacings, but that approach was computationally slow for denser cell suspension densities. Overlaying the results from co-occurring taxa then gives a visualization of the community in a habitat (Figure 1). Although overlaying data from separately simulated taxa is adequate for visualization, a more sophisticated model would be required to explicitly generate and retrieve the distributions of cell-to-cell distances within, or across, taxa. Furthermore, even in oligotrophic habitats with low average cell suspension densities the few cells present may be clustered non-randomly, an ecological nuance not captured in our approach.

4.2. Simulating Concentration Gradients

To simulate H₂O₂ concentration gradients generated around a given cell, we consider the cell as a fixed spherical volume which maintains a steady, homeostatic intracellular [H₂O₂]. For simulation of phytoplankton cells, we set a homeostatic intracellular [H₂O₂] of 10⁻⁶ M, based upon limited literature from other eukaryotes [61], and as a level below the cytotoxic threshold of 10⁻⁵ M. For simulation of bacterioplankton cells, we set a homeostatic intracellular [H₂O₂] of 1/10 the external seawater [H₂O₂] [58,60].

Using the H₂O₂ diffusion co-efficient in water, and time increments from 0 µs with a log₁₀ series upwards to 100 µs, we calculate the progressive net diffusional displacement of H₂O₂ outward from the cell surface. We then estimate dilution of [H₂O₂] with diffusion outwards from a source ‘Phytoplankton’ cell or diffusion inwards towards a sink ‘Heterotrophic’ cell. In addition to the simple volumetric dilution as ROS diffuses outwards from ‘Phytoplankton’ cells, we add a time-dependent decay term based upon measured lifetimes of H₂O₂ in seawater, which also vary widely. We use nominal time increments to generate the diffusion distances of H₂O₂ outward from the cell and to simulate the concurrent effect of pseudo-first order decay as H₂O₂ moves outward from the cell with time, but this simple simulation assumes a steady state, without considering local mixing or fluctuations in cellular or environmental H₂O₂.

4.3. Simulating Threshold Radii for [H₂O₂] above or below Seawater [H₂O₂]

On a visual plot, the intersect of the [H₂O₂] outwards from a ‘Phytoplankton’ source cell, with the lowest seawater [H₂O₂] is obvious, but it proved challenging to estimate this threshold distance when outwardly diffusing [H₂O₂] declines to the lowest seawater [H₂O₂]. The time for [H₂O₂] to decline to seawater [H₂O₂] depends upon the ratio of intracellular:extracellular [H₂O₂]; the diffusion coefficient for ROS, which determines how far [H₂O₂] moves out from the cell radius in a given time interval; and thus the [H₂O₂] volumetric dilution factor over a given time interval. For [H₂O₂], concomitant decay was only a negligible factor, but to retain capacity for generalizations to other, less stable [ROS] we nevertheless included an extracellular decay rate constant in the estimations. After algebraic transforms we found that a LambertW function was appropriate [78] to estimate the threshold radius at which [H₂O₂] around a cell falls to the lowest seawater [H₂O₂] (Equation (3)).

In Equation (3), [H₂O₂]_seawater is the seawater [H₂O₂] in the simulated habitat in mol L⁻¹; [H₂O₂]_cell is the intracellular homeostatic [H₂O₂] in mol L⁻¹; DiffusionRad_threshold in µm is the radius of the diffusion sphere when [H₂O₂] drops to [H₂O₂]_seawater; CellRadius is the radius of the cell in µm; µ is the pseudo-first order decay constant for H₂O₂ in seawater; us_threshold is the time in µs at which [H₂O₂] drops to [H₂O₂]_seawater; Diffusion Coefficient (D) is expressed in µm² µs⁻¹; W₀ is the Lambert W function, also known as the Omega function, used to solve for f(x) = xe^x.

$$\begin{array}{c}{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{seawater}}={\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{cell}}\ast \frac{\frac{4}{3}\ast \mathsf{\pi }\ast {\left(\mathrm{CellRadius}\right)}^3}{\frac{4}{3}{\ast \mathsf{\pi }\ast \mathrm{DiffusionRad}}_{\mathrm{threshold}}^3}{\ast \mathrm{e}}^{-{\mathsf{\mu }\ast \mathrm{us}}_{\mathrm{threshold}}}\\ {\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{seawater}}={\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{cell}}\ast \frac{{\left(\mathrm{CellRadius}\right)}^3}{{\mathrm{DiffusionRad}}_{\mathrm{threshold}}^3}{\ast \mathrm{e}}^{-{\mathsf{\mu }\ast \mathrm{us}}_{\mathrm{threshold}}}\\ {\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{seawater}}={\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{cell}}\ast \frac{{\left(\mathrm{CellRadius}\right)}^3}{{\left(\mathrm{CellRadius}+\sqrt[2]{{\mathrm{D}\ast \mathrm{us}}_{\mathrm{threshold}}}\right)}^3}{\ast \mathrm{e}}^{-{\mathsf{\mu }\ast \mathrm{us}}_{\mathrm{threshold}}}\\ \frac{{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{seawater}}}{{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{cell}}\ast {\left(\mathrm{CellRadius}\right)}^3}=\frac{1}{{\left(\mathrm{CellRadius}+\sqrt[2]{{\mathrm{D}\ast \mathrm{us}}_{\mathrm{threshold}}}\right)}^3}{\ast \mathrm{e}}^{-{\mathsf{\mu }\ast \mathrm{us}}_{\mathrm{threshold}}}\\ \frac{{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{seawater}}}{{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{cell}}\ast {\left(\mathrm{CellRadius}\right)}^3}=\mathrm{k}\\ \sqrt[3]{\mathrm{k}}=\frac{1}{\left(\mathrm{CellRadius}+\sqrt[2]{{\mathrm{D}\ast \mathrm{us}}_{\mathrm{threshold}}}\right)}{\ast \mathrm{e}}^{\frac{-{\mathsf{\mu }\ast \mathrm{us}}_{\mathrm{threshold}}}{3}}\\ \mathrm{CellRadius}+\sqrt[2]{{\mathrm{D}\ast \mathrm{us}}_{\mathrm{threshold}}}=\frac{{\mathrm{e}}^{\frac{-{\mathsf{\mu }\ast \mathrm{us}}_{\mathrm{threshold}}}{3}}}{\sqrt[3]{\mathrm{k}}}\\ \sqrt[2]{{\mathrm{D}\ast \mathrm{us}}_{\mathrm{threshold}}}=\frac{{\mathrm{e}}^{\frac{-{\mathsf{\mu }\ast \mathrm{us}}_{\mathrm{threshold}}}{3}}}{\sqrt[3]{\mathrm{k}}}-\mathrm{CellRadius}\\ \frac{\sqrt[2]{{\mathrm{D}\ast \mathrm{us}}_{\mathrm{threshold}}}}{{\mathrm{e}}^{\frac{-{\mathsf{\mu }\ast \mathrm{us}}_{\mathrm{threshold}}}{3}}}=\frac{1}{\sqrt[3]{\mathrm{k}}}-\mathrm{CellRadius}\\ \frac{{\mathrm{D}\ast \mathrm{us}}_{\mathrm{threshold}}}{{\mathrm{e}}^{\frac{-{\mathsf{\mu }\ast \mathrm{us}}_{\mathrm{threshold}}\ast 2}{3}}}={\left(\frac{1}{\sqrt[3]{\mathrm{k}}}-\mathrm{CellRadius}\right)}^2\\ {\mathrm{D}\ast \mathrm{us}}_{\mathrm{threshold}}{\ast \mathrm{e}}^{\frac{{\mathsf{\mu }\ast \mathrm{us}}_{\mathrm{threshold}}\ast 2}{3}}={\left(\frac{1}{\sqrt[3]{\mathrm{k}}}-\mathrm{CellRadius}\right)}^2\\ {\mathrm{us}}_{\mathrm{threshold}}{\ast \mathrm{e}}^{\frac{{\mathsf{\mu }\ast \mathrm{us}}_{\mathrm{threshold}}\ast 2}{3}}=\frac{{\left(\frac{1}{\sqrt[3]{\mathrm{k}}}-\mathrm{CellRadius}\right)}^2}{\mathrm{D}}\\ \frac{{\mathsf{\mu }\ast \mathrm{us}}_{\mathrm{threshold}}\ast 2}{3}=\mathrm{Y}\\ \frac{\mathrm{Y}}{\mathsf{\mu }\ast \frac{2}{3}}{\ast \mathrm{e}}^{\mathrm{Y}}=\frac{{\left(\frac{1}{\sqrt[3]{\mathrm{k}}}-\mathrm{CellRadius}\right)}^2}{\mathrm{D}}\\ {\mathrm{Y}\ast \mathrm{e}}^{\mathrm{Y}}=\frac{{\left(\frac{1}{\sqrt[3]{\mathrm{k}}}-\mathrm{CellRadius}\right)}^2}{\mathrm{D}}\ast \mathsf{\mu }\ast \frac{2}{3}\\ \mathrm{Y}={\mathrm{W}}_0\left(\frac{{\left(\frac{1}{\sqrt[3]{\frac{{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{seawater}}}{{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{cell}}\ast {\left(\mathrm{CellRadius}\right)}^3}}}-\mathrm{CellRadius}\right)}^2}{\mathrm{D}}\ast \mathsf{\mu }\ast \frac{2}{3}\right)\\ {\mathrm{us}}_{\mathrm{threshold}}=\frac{{\mathrm{W}}_0\left(\frac{{\left(\frac{1}{\sqrt[3]{\frac{{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{seawater}}}{{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{cell}}\ast {\left(\mathrm{CellRadius}\right)}^3}}}-\mathrm{CellRadius}\right)}^2}{\mathrm{D}}\ast \mathsf{\mu }\ast \frac{2}{3}\right)}{\left(\mathsf{\mu }\ast \frac{2}{3}\right)}\end{array}$$

(3)

For ‘Heterotrophic’ sink cells we simulated the sphere of [H₂O₂] below seawater [H₂O₂] accounting only for diffusion around the cell without including a rate constant for time-dependent bulk decay because the cell is acting, on top of any decay term, to further lower [H₂O₂] below seawater [H₂O₂] (Reaction (4)).

$${\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{seawater}}={\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{cell}}\ast \left({\mathrm{DiffusionSphere}}_{\mathrm{threshold}}\right)/\left(\left(4/3\right)\ast \mathsf{\pi }\ast \mathrm{CellRadius}\right)$$

where [H₂O₂]_seawater is again the seawater [H₂O₂] in the simulated habitat in mol L⁻¹; [H₂O₂]_cell is the intracellular homeostatic [H₂O₂] in mol L⁻¹; and DiffusionRad_threshold in µm is the radius of the diffusion sphere when [H₂O₂] drops to [H₂O₂]_seawater; CellRadius is the radius of the cell in µm.

Alter substitutions and re-arrangements:

Where µs_thresh is the elapsed time in µs to reach the threshold [H₂O₂]; Diffusion Coefficient (D) is expressed in µm² µs⁻¹; µs [1] is the first simulated time step. We then recover the threshold distance as:

$$\begin{array}{c}{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{seawater}}={\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{cell}}\ast \frac{\left(4/3\right){\ast \mathsf{\pi }\ast \mathrm{DiffusionRad}}_{\mathrm{threshold}}^3}{\left(4/3\right){\ast \mathsf{\pi }\ast \mathrm{CellRadius}}^3}\\ {\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{seawater}}={\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{cell}}\ast \frac{{\mathrm{DiffuseRad}}_{\mathrm{threshold}}^3}{{\mathrm{CellRadius}}^3}\\ \frac{{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{seawater}}}{{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{cell}}}={\left(\frac{{\mathrm{DiffuseRad}}_{\mathrm{threshold}}}{\mathrm{CellRadius}}\right)}^3\\ {\mathrm{DiffuseRad}}_{\mathrm{threshold}}=\sqrt[3]{\frac{{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{seawater}}}{{\left[{\mathrm{H}}_2{\mathrm{O}}_2\right]}_{\mathrm{cell}}}}\ast \mathrm{CellRadius}\end{array}$$

(4)

4.4. Simulations and Plotting

We used R [79] running under RStudio [80], using packages ‘tidyverse’ [81], ‘pracma’ [82], ‘ggplot2′ [83], ‘cowplot’ [84], ‘ggpubr’ [85], ‘glue’ [86], ‘kableExtra’ [87], ‘LambertW’ [88], ‘ggforce’ [89], ‘googledrive’ [90], and ‘googlesheets4’ [91]. Citations were managed using Zotero [92] open access reference manager connected to RStudio using the ‘citr’ [93] package, with output generated using the ‘knitr’ [94,95,96] and ‘bookdown’ [97] packages.