Seascapes: the world of aquatic organisms as determined by their particulate natures

Planktonic organisms are particles. They are discrete objects, suspended in water; they fall or rise; they collide. They concentrate material from a dilute solution and they discharge concentrated material back to the solution. A solitary cell is but one of a mixture of particles, living and dead, small and large. Organisms and other particles structure the physical/chemical environment of the aquatic medium, either by temporarily changing solution concentrations around them or, in an analogous manner, by changing the distributions of organisms. The summed changes from all these interactions give aquatic environments a spatial texture that can be considered a seascape (Fig. 1). This seascape affects an organism’s ability to respond, either feeding passively or hunting actively for prey. In all of this, size is the crucial parameter determining how an organism or other particle interacts with its environment, including other objects. While any organism can be described by a multitude of size measures, we shall use length here, unless otherwise noted.

There is a rich tradition in applied mathematics, physics and engineering describing the interactions between particles and their environments. Most fundamental to understanding how a particle interacts with its surrounding solution is molecular diffusion theory, which has been extensively developed in applied mathematics (e.g. Carslaw and Jaeger, 1959; Crank, 1975). The mathematical descriptions extend to include the role of fluid motions (e.g. Clift et al., 1978; Leal, 1992). The techniques have been applied to describe single cells and the rates at which they take up nutrients from solution (e.g. Munk and Riley, 1952; Dusenbery, 2009). Most such studies emphasize steady-state situations, but there has been interest in the effects of pulsed input (e.g. McCarthy and Goldman, 1979; Jackson, 1980; Jackson, 1987; Lehman and Scavia, 1982).

The first mathematical analysis describing how particles in a fluid interact with each other in what is now known as coagulation theory is usually attributed to Smoluchowski (Smoluchowski, 1917). To describe how particles interact with each other in a fluid, we itemize the possible mechanisms bringing them together and then describe the rates mathematically. Important properties for a particle include its mass, diameter, settling speed and, if it is alive, its mobility and sensory capabilities. Coagulation theory focuses on describing collisions arising from three mechanisms: Brownian motion, differential settling and shear. Brownian motion describes how random fluctuations in particle positions can lead to collisions; differential sedimentation describes how a particle falling faster than another particle can overtake it and collide; shear describes how turbulent water motions can cause particles to collide. These three mechanisms can be extended to describe the rates at which many organisms feed on each other.

We know organisms are also discrete particles, and we know some of their properties, including diameter, density and chemical composition. One important point is that inert particle interactions have analogs in how organisms interact with each other. Early pioneers in developing the correspondence between the two include Gerritsen and Strickler (Gerritsen and Strickler, 1977) and Fenchel (Fenchel, 1984). More recent thinking has been summarized by Kiørboe (Kiørboe, 2008) and Dusenbery (Dusenbery, 2009). Dusenbery has applied these approaches to aquatic organisms, with an emphasis on the factors that affect transport to individuals (Dusenbery, 2009).

As organisms sit or move through the environment, they leave trails, regions depleted in some substances and enhanced in others. They might be consuming oxygen, dissolved organic compounds or plant nutrients; they may be excreting carbon dioxide, ammonia or amino acids. They may be copepods leaving pheromones to attract a mate; they may be predators clearing a path of prey; or they may be aggregates falling, disintegrating while they fall, leaving a trail rich in microbial food. In time, all these trails fade back into the homogeneous background (Fig. 1). While they exist, they provide enhanced resources. For a bacterium looking to colonize a marine snow particle or a male copepod looking for a mate, they increase the chances of success. For a microbe looking to pull in food from the solution, they constrain growth rates. Of particular interest have been the roles that elongated trails, or plumes, behind swimming animals or falling particles can have in helping microbes overcome low nutrient concentrations. These trails provide resources whose extent and lifetime determine how well they can be exploited.

Our understanding of the nature of this environment on the organism scale is not well developed and, as a result, neither is our understanding of how organisms interact with each other. In this paper, I describe the chemical environments of individual microbes and animals; I then describe some of the implications for ecosystem function; lastly, I summarize and extend the mathematics used to describe the temporal and spatial extent of various plumes. In much of this, I emphasize marine bacteria because they are the smallest marine organisms with the simplest physical constraints and because they meet all their nutritional needs by taking in small molecules from solution. Symbols in the text are summarized in Tables 1 and 2. In some cases, the symbols and equations are introduced in the Appendix.

We begin the analysis with the simple bacterial cell, starting with typical values for bacterial size, abundance and environmental nutrient concentrations (Table 2). We compare the bacteria to their environment, both physically and chemically. We express the chemical content of the cell in terms of nitrogen and assume that it is spherical in shape. We assume that the edge of the depletion region resulting from the cell’s uptake is where the concentration reaches half the background concentration. We then examine nutrient sources, including zooplankton excretion and marine snow breakdown, and their implications.

For a marine bacterium with a typical radius of 0.25 μm, the radius of the depleted zone around it is twice its radius, 0.5 μm. The volume of the depleted region is 2³=8 times that of the cell (7 times if the cell’s volume is subtracted). A typical background concentration of the dissolved organic nitrogen (DON) used by the cell is 10 nmol l^–1; with a nitrogen content of 1.4×10^–16 mol per cell, the internal bacterial N concentration is 2.2 mol l^–1, 2×10⁸ greater than the background concentration. This internal content equals all the dissolved material within a distance of 146 μm of the cell.

A typical bacterial abundance is 5×10⁵ cells cm^–3, equivalent to a content of about 70 nmol l^–1 N if distributed equally through the solution. The vastly greater concentration of material within the cells than dissolved in the environment implies that bacterial uptake is relatively efficient and relies on continual resupply to sustain it. Such resupply is part of the regeneration process, in which material present as organisms or other particles is metabolized and released as small molecules available for formation of new particle mass. If the bacterial cells were to be uniformly distributed on a lattice, they would be spaced 126 μm apart. Thus, the diameter of the depletion region around each bacterium (1 μm) is small relative to the spacing between cells (Fig. 2).

Similar calculations can be made for phytoplankton cells, but the range in cell size, geometry and abundance is much greater.

For a typical molecular diffusivity of D=10^–5 cm² s^–1 and a substrate concentration that is effectively 0, the total flow (uptake, F) into the cell can be calculated using values in Table 2 and Eqn A5 (Appendix) as F=3.14×10^–20 mol s^–1=19,000 molecules s^–1. The time to accumulate enough material to double is T=N/F≈4500 s=1.25 h, where N is the amount of nitrogen in the cell. Note that these calculations assume, among other simplifications, that all molecules diffuse at the rate of a small molecule, that all molecules are useful to the cell, and that all are incorporated into particulate material.

The condition for enhancing the flux by swimming is that the Péclet number Pe=vd/D>1, where v is the swimming velocity and d is the diameter (see Eqn A33 in Appendix). For a cell swimming at 20 lengths s^–1, v=10 μm s^–1, Pe=5×10^–3. This small value of Pe implies that the cell does not swim faster than diffusion can establish a gradient, so there is no enhancement of uptake and no elongated depletion plume trailing it. The absence of a depletion plume means that the immediate impact of bacterial passage is limited to a depleted spherical region that moves with it. There is little impact of its passage on the spatial structure of nutrients, except for its adding to the general bacterial consumption of material. To create a plume and leave a chemical signal of its passage, the cell would need to have vd>D=10^–5 cm² s^–1. If v=10 body lengths s^–1=10d s^–1, d²>10^–6 cm², d>10^–3 cm. That is, the minimum size at which bacterial motion has any impact on creating depletion plumes or on enhancing nutrient uptake should be ∼10 μm.

A similar set of analyses can be made for phytoplankton, although the greater range of nutrient and algal concentrations, as well as organism sizes and shapes, makes any situation more complicated and less generalizable.

Blackburn and colleagues (Blackburn et al., 1998) have observed motile bacteria clustering in solution, a response that they suggest is the result of chemotactic movement to the sudden release of cellular contents. Such a release could result from cellular lysis after a viral infection. If we assume that algal carbon content (mol/cell) is given as a function of size by 3.44×10^–5d^2.2 (Mullin et al., 1966), where d is the algal radius in cm, that the carbon to nitrogen ratio (mol/mol) equals the 6.6 Redfield ratio, that the detectable concentration is given by C_r=1 μmol l^–1 N, and that diffusivity D and microbial uptake k are as above, then Eqns A8, A9, A15 and A16 (Appendix) can be used to describe the characteristics of the diffusion cloud (Fig. 5). For a small, d=1 μm, algal cell, the maximum radial extent of the cloud is about 400 μm and the cloud disappears after 70 s. Bacterial consumption has minimal impact on it. For 10 and 100 μm algae, the clouds last almost 40 min and almost 1 day, in the absence of water motions or bacterial uptake. Bacterial uptake significantly decreases these times to about 15 and 52 min. A significant increase in bacterial abundance resulting from chemotactic attraction would further decrease these times. One conclusion is that cell lysis could provide ephemeral patches lasting between 1 min and 1 h, depending on the size of the cell being lysed. The estimated times are probably high because they assume that all cellular material is solubilized instantly.

Benoit-Bird (Benoit-Bird, 2009) found zooplankton layers 0.2–4.6 m thick in Monterey Bay, along the coast of California. The sonar technique she used documented holes of the order of 0.5 m² created when fish passed through one of the layers. These holes took 300–400 s to fill back up. The layer was dominated by 0.9–1.4 mm long copepods (Calanus, Ctenocalanus, Acartia). This provides a highly visible example of a depletion plume caused by a swimming predator.

To test the model (Eqn A31, Appendix), we can use Eqn A35 to estimate an effective copepod diffusivity, D=2.8(0.1)^1.81=0.05 cm² s^–1. This value is low compared with the estimate of D=0.036 cm² s^–1 for d=0.75 mm Temora longicornis and D=0.36 cm² s^–1 for d=1.9 mm Calanus helgolandicus (Visser and Kiørboe, 2006).

The dominant size range where regeneration occurs depends on the size distributions of the animals and aggregates doing the release. While larger objects excrete faster, there are fewer of them. We can examine the importance of size by integrating over the appropriate size ranges.

Most regeneration occurs at the small animal scales, presumably with the few micrometer-sized flagellates (Fig. 6). Increasing the size range of animals covered by increasing d_u has a decreasing effect on the total regeneration. More than 99% of the regeneration is from animals too small to have plumes changed by the bacterial populations (Fig. 3).

How important is microbial consumption of the material in the plume? In Fig. 3, we argued that microbial uptake has an important impact on plume dissipation when the zooplankton forming them are larger than 0.46 mm long or the aggregates are larger than 3.2 mm. The regeneration from either aggregates or animals at least this large is extremely small, 5.6% and 0.5% (Fig. 6). We conclude that microbial consumption localized within plumes is a relatively small fraction of the total consumption.

The total regeneration for both zooplankton and marine snow can be calculated by letting d_u→∞. Assuming that d_l=10 μm, the regeneration rate from the animals calculated here is 5.4×10^–16 mol N cm^–3 s^–1 and that for the marine snow is 5.7×10^–16 mol N cm^–3 s^–1. The two numbers are remarkably close, especially considering the arbitrary pairing of data sets, one from the oligotrophic central North Pacific and the other from nutrient-rich Monterey Bay.

Several factors could change these relationships. Microbial consumption within the plume would be relatively greater if bacterial abundance were greater or if chemotactic responses allowed bacteria to aggregate before a plume dissipated. If their uptake rates were not diffusion limited, however, they would have smaller impacts on the plumes than calculated. In any case, experimental measurements are needed to resolve these issues.

One of the basic questions when considering plumes is just how extensive are they? Are the plumes formed by small or large objects? Again, we can use the size distribution in conjunction with the size-dependent plume volume to answer these questions.

While particle size distributions do not necessarily follow simple power law distributions, such distributions are useful tools in investigating the implications of size and concentration.

For C_r=1 μmol l^–1, the fraction of a volume occupied by plumes is 2.0×10^–7 for those formed by zooplankton and 1.2×10^–8 for those formed by aggregates. These values are equivalent to 200 and 12 mm³ m^–3, respectively, and represent upper estimates, as they do not include shrinkage resulting from bacterial uptake within the plumes. As noted above, such shrinkage is greater for larger plumes.

The integrated cross-sections for both zooplankton and aggregate plumes are quite small. Using 1/Σ to calculate a characteristic distance that an organism would need to swim to find a plume, the distances are 315 and 1000 m. An animal sensing a plume over a finite sensory distance could dramatically increase the contact rate (Jackson and Kiørboe, 2004). Alternatively, a bacterium could be in the path of a falling aggregate and have more frequent contacts (Kiørboe and Jackson, 2001).

When the diffusivity of the incoming food and outgoing metabolic property are the same, a depleting plume and an excretion plume are similar and balance each other out (Fig. 9). This can occur when the diffusivity of diffusing prey and the regenerated nutrients are the same. In particular, the diffusivities associated with swimming bacteria are similar to those of small molecules. As a result, bacterivores such as diffusive-feeding radiolarians or swimming flagellates would make smaller changes to the total N concentration than would an animal feeding on larger prey.

The chemical and particle seascape is a balance between the organisms that texturize it and the diffusive and mixing processes that homogenize it. Understanding organism behavior requires a way to incorporate a range of scales into the description. Using organism size distributions in conjunction with size-based effects provides a means to do this.

The above calculations show that the bulk of nutrient release is from sources too small to be decreased by microbes before the plumes dissipate into the general background. Microbial concentrations would have to be greatly increased by chemotactic microbes in ways that we can calculate if significant amounts of the material were to be consumed in the plume. Consumption of nutrients from the diffuse background is more efficiently done by small, non-motile cells. A system in which non-motile microbes dominate chemical uptake would be consistent with observations by B. Ward (personal communication), who noted that microbial populations in the ocean are dominated by groups with no known motility.

Behavior is difficult to incorporate into these models. There have been efforts to put chemotactic behavior into an analytical framework (e.g. Bearon and Grünbaum, 2008), but most work has involved simulations of individual microbial motions. However, simulations can be used to derive size-specific properties. For example, Kiørboe and Jackson simulated the rates of attachment of both chemotactic and non-chemotactic bacteria to falling particles of different sizes (Kiørboe and Jackson, 2001). The simulations allowed them to describe colonization rates in terms of power law relationships. Such relationships are the basis for the calculations performed above.

The various parameter values given here are not constant in the ocean, varying in time and space as well as by taxa. Because organism and aggregation concentrations vary spatially and seasonally, calculations that are location specific are needed to describe each situation. In addition, size distributions do not always fit Eqn A47 (Appendix) exactly. Using simple power laws can miss important differences in the distributions (e.g. Jackson and Checkley, 2011). More complicated distributions can always be used to make numerical integrations of the various observed system properties, such as the particle size distributions.

In these analyses, we have neglected the chemical nature of the plumes, assuming that simply calculating nitrogen concentration is sufficient. In fact, they are made of many compounds of different desirabilities, detectabilities and affinities. Describing the fates of compounds with properties, including concentrations, would strengthen the results.

Appreciating the role of turbulence is always important, particularly when larger organisms are considered, as the role of shear and organism contacts can dominate as the length scale of the interactions increases (Rothschild and Osborne, 1988; Saiz and Kiørboe, 1995). There are promising approaches to parameterizing the patchiness of larger plumes in the presence of turbulence (e.g. Visser and Jackson, 2004), but further work is needed if the approach developed here is to be extended to larger organisms and scales.

Aquatic organisms exist in an environment that is heavily textured with chemical resources and cues. The production and consumption of chemical substances occurs on a range of spatial and temporal scales, which are constrained by the physics of their movement. The result is a heterogeneous environment (Fig. 1) that we must understand if we are to understand how the organisms interact.

The movement of molecules to and from an organism in solution is ultimately controlled by diffusion. In the absence of water motion, the mathematical description simplifies. The simplest representation of an organism is as a sphere. By calculating the concentrations and material fluxes to and from the sphere, we can understand the world of a cell. The cell’s role in this is to remove molecules that reach its surface, transporting them inward.

For simplicity, we shall assume this spherical symmetry case in subsequent calculations. Dusenbery (Dusenbery, 2009) has an extensive discussion of the effect of non-spherical shapes on uptake. The solution depends on the initial distribution of the substance and the distribution of any sources and sinks.

This relationship is used to describe nutrient uptake by an isolated cell. It states that in the absence of motion and at steady state, diffusion controls the maximum rate at which molecules can approach a cell. By contrast, there is no upper limit on the rate at which material can diffuse away from a cell, if the concentration next to the cell is not constrained.

A relatively simple problem is one in which the sphere suddenly appears in the middle of a solution with uniform solute concentration C₀. A biological example would be a grazer leaving a region with depleted bacterial concentrations to move to one with an untouched population. Before the region around the cell becomes depleted, the concentration of molecules next to the cell is higher and the flux is temporarily greater.

One important situation is the extent of the molecular cloud around a leaking cell or aggregate. The material being leaked could serve as a desirable substrate for bacteria or phytoplankton. The presence of the bacteria taking the material up should reduce the extent of the cloud around the source. If the rate of uptake is proportional to the substrate concentration and a constant bacterial concentration, we can represent the effect as a loss rate –kC, where k is a rate constant.

A moving cell can leave a trail, or plume, of anything it is leaking or consuming. The simplest analysis of this situation focuses on the movement and ignores the complexity of flow around the sphere. If it moves in a straight line and acts as a very tiny (point) source, then the diffusion rate along its path is negligible compared with diffusion along its path because concentration gradients are greater perpendicular to the path. The resulting differential equation is the same as for cylindrical diffusion (Okubo, 1980). We can use an equation of the concentration distribution to calculate different plume properties that may be of ecological interest. One of the situations involves the decrease associated with microbes sitting in the plume.

An animal swimming in search of food sweeps out an area in front of it whose size depends on the distance at which the animal can sense its prey. As it feeds, it leaves behind it a tube emptied of prey, which ultimately fills refills with prey. For a prey species that moves in random walks, changes in its concentration can be described as diffusion, in a situation similar to that of the plume created by a moving point source or sink. In this case, the moving point sink is modified to be a moving disk of radius P, which represents the predator’s sensory distance. While the mathematics are similar for a source and a sink, we will consider the situation of a sink, with a prey concentration of 0 in a moving disk that represents the feeding zone.

We use the same approach as in the previous section, when calculating the profile of a point plume, assuming that this is really a 2-dimensional problem in which t=z/v. If we assume that the plume extends to the region where the concentration is half the background concentration, then we can calculate a plume half-width ρ₂ along the path, a plume length Z₂ and a time T₂. The complete solution to this is given in p. 260 (X–VII) of Carslaw and Jaeger (Carslaw and Jaeger, 1959).

Swimming by an organism or other fluid motions can overcome the transport limitation imposed by diffusion. In some cases, the organism is able to swim fast enough to reach new material faster than diffusion alone can supply it. In addition, external fluid motions driven by turbulence can distort the depleted regions trailing a swimming organism, twisting them and breaking them into pieces.

Visser and Jackson (Visser and Jackson, 2004) examined the effect of turbulence on various metrics of a plume, such as that formed by a leaky swimming animal or a falling marine snow particle. Turbulence had multiple effects on the plume, including stretching and breaking it into pieces as well as wrapping it up into a more compact ball. Such concepts as plume length needed to be altered to distinguish between total length along the plume, the plume length to first break, and extension distance in one direction. The crucial parameter describing the effect of turbulence on the plume is γT₀, where γ is the average turbulent shear rate and T₀=Z₀/v is the length of time the plume lasts in the absence of turbulence (Eqn A24). For further discussion, see Visser and Jackson (Visser and Jackson, 2004).

Animal feeding involves both the depletion of food in the region around an animal and the enhancement of nutrient concentration as food is metabolized and released back into the environment. Animal movement determines the geometry of the regions of depletion and enhancement associated with the animal.

The organic compounds supplying microbial growth have a variety of sources, including leakage from algae, excretion by animals and degradation of particles. While some of these sources are small and sedentary, others move through the water, potentially leaving behind a chemical trail in the form of a higher dissolved nutrient concentration plume. Such a plume creates a finite-sized environment that can be used as a food resource by a microbe or as a sensory trail for a predator. The size of a sensory plume depends on, among other factors, the sensitivity with which the concentration can be detected.

The fastest way for an animal to find food is to swim in a straight line (Visser and Kiørboe, 2006), but this can also be the fastest way to be eaten. How do animals address this dilemma? Bigger animals tend to sense larger distances (R) than the straight paths of their prey. If a small prey moves in a straight line for a ‘short’ distance l, then turns, its larger scale movements are effectively diffusive on the scale at which the larger predators work, but it effectively feeds itself as if moving in a straight line hunting for smaller prey. Random motion like this leads to diffusive behavior, described by diffusion coefficients. What allows this strategy to work is that aquatic food chains tend to be size based, with organisms eating prey 1/10 their size (e.g. Fenchel, 1987).

The molecular diffusion coefficients are typically of the order of D=10^–5 cm² s^–1. Bacteria swim in different patterns, but typical estimates for the bacterial diffusivity are similar to those for molecular diffusivity, 10^–6 to 10^–5 cm² s^–1. Animal diffusivities are larger.

Having searched the region through which it passes, the grazer leaves behind a plume depleted of its prey. If we assume that the prey motions look diffusive on the scale of the grazer’s search distance, then we can use diffusion theory to calculate the plume geometry. The grazer will also leave behind a plume of excreted organic matter and inorganic nutrients, in a shape described by Eqns A20, A21, A22, A23. As noted earlier, turbulent motions are important for larger plumes, as they can distort their shapes (e.g. Visser and Jackson, 2004), but will not be considered further here.

In the absence of turbulent motions, we can estimate the length of an enhancement plume relatively simply from Eqn A19 as Z=leakage rate/(4πDC_r), where C_r is the concentration defining the boundary of the plume. The numerical value of C_r can be chosen as the minimum detectable concentration by the organism trying to detect it (Jackson and Kiørboe, 2004). If we consider a depletion region that is bounded by C_r=0.5C₀, then C_r=ΔC/2, and Z=2aSh. When there is no enhancement of nutrient uptake by motion, Sh=1 and Z=2a=d. This is the same result as we had for the pure diffusion case. The addition of a finite sensory width can increase the effective plume width considerably.

For small turbulence mixing rates, the geometries of the food-depleted and excretion-enhanced regions are described simply by the same diffusion problem (Eqn A1). Thus, the same functions apply for both the inward transport of food and the outward transport of regenerated nutrients, although the relative size of the affected areas depends on the relative diffusivity of the two materials.

Filter feeding is the classical mode for particle feeders and is akin to search feeding. An alternative mode is for animals to sit in one spot and allow particles to fall on them in what has been called flux feeding (Jackson, 1993). While the kernel is similar to β_sw, it substitutes the falling speed v_s of the particles for that of the swimming speed of the predator. Ambush feeding on vertically migrating zooplankton also has the form of the search kernel, although with the swimming speed of the prey substituting for that of the predator.

In both cases, they leave a depleted region below that fills in from lateral movement of particles and water behind the predator. While such a predator forms a depletion plume, it forms a purely diffusive excretion plume.

Most of these effects of organisms on their environments are extremely dependent on the organism size. Thus, if we are to describe the chemical seascape, we need to account for the sizes and abundances of the different organisms generating it. We use the size distribution to describe the size dependence of organism concentration. We integrate rates of interest over relevant size ranges to find the total effect of organisms of multiple sizes, feeding rates and concentrations. We will integrate over all sizes to determine the net effect on the community properties.

Non-organism particles have also been sampled extensively and their distributions fitted to relationships of the form Eqn A47, with values of λ₄ from –2 to –6 (e.g. Sheldon et al., 1972; Guidi et al., 2009). While Eqn A47 is a useful way to summarize a size distribution, it can over-represent the abundance of the largest particles (Jackson and Checkley, 2011).

This work has been inspired by and benefited from conversations with Andy Visser and Thomas Kiørboe of the Danish Technical University. Mark Denny and Thomas Kiørboe provided useful feedback on a draft version of this manuscript. Bess Ward noted the preponderance of non-motile microbes in the ocean.