A random encounter model for the microbial planktonic community

Transcription

1 OLOGY May 1992 LIMN AND OCEANOGRAPHY / Volume 37 Number 3 Limnol. Oceanogr., 37(3), 1992, , by the American Society of Limnology and Oceanography, Inc. A random encounter model for the microbial planktonic community Dale A. Kiefer and Juli Berwald Department of Biological Sciences, University of Southern California, University Park, Los Angeles Abstract The numeric concentration of procaryotic and eucaryotic cells that form the planktonic microbial community decreases monotonically and rapidly with increasing size. This size distribution has been interpreted to be the result of allometric control of growth and respiration rates, but we propose an alternative random encounter hypothesis to explain such a distribution. According to this hypothesis, the size distribution of cells that range from 0.3 to 100 pm results from sizeselective predation by protozoans. A phagotrophic population will swim randomly through the water encountering both prey and predators. At steady state, such a population must ingest smaller prey at a sufficient rate to balance its rate of loss to larger predators. The size distribution of cells must, therefore, satisfy this condition. In particular, a mathematical statement of the hypothesis is based on the following relationships: volume of a cell varies as the cube of its diameter; clearance rate (L3T-I) by a predator varies as the square of its diameter; and, for a given population, its range of possible prey sizes varies with its prey diameter at the same time that the range of predator sizes able to capture the given population varies with its predator diameter. Measurements at sea of the distribution of particle sizes by electronic detection (Sheldon et al. 1972, 1973) and by phase contrast microscopy have shown that there is a characteristic distribution for particles in the size range between 2 and 100 pm. Samples collected from diverse regions of the open ocean are all characterized by a steep and continuous decrease in the particle concentration with increasing particle size. This size distribution has been mathematically characterized as a hyperbolic distribution in which the the concentration of particles decreases as roughly the 4th power of the particle s size (e.g. McCave 1975). Acknowledgments This work was supported by both the Oceanic Biology Program of ONR (grant NO J-308 1) and the Processes Branch of NASA (grant NAGW-3 17). We thank Dariusz Stramski, Richard Emlet, Rick Reynolds, and Nick Appelmans for their interest and suggestions. Recent studies using not only electronic particle counters and phase contrast microscopes, but also flow cytometers (Chisholm et al. 1988) and epifluorescent microscopes (Caron 1983; Cho and Azam 1990) have increased our knowledge of both the size distribution of particles that are <2+m diameter and the living and detrital components of the distribution. After reviewing such information, Stramski and Kiefer (199 1) have concluded that the steep decrease in particle concentration with increasing size is a feature of particles in the size range between 0.3 and 2 pm and is a feature of living cells as well as total particles. Thus, most importantly, the 4th power dependency appears to be a feature of the diverse taxa that constitute the socalled microbial food web. Included in the food web are heterotrophic and autotrophic procaryotic cells, which most commonly are < 1 pm in diameter, and phagotrophic, heterotrophic, mixotrophic, and autotrophic 457

2 458 Kiefer and Berwald A D 0, D,, D, D2, J (D) h N(D) S V(o) ~(0, 6) CL2 u erfz Notation Steady state assimilation efficiency Diameter of the cell under consideration, L Diameters of cells of trophic levels, 1 being the smallest and 3 the largest, L Diameter of autotrophic population of diameter D2, L Concn of particles within an infinitesimal band at diameter D, L -4 Specific rate of loss of cell volume from population 2 by egestion and excretion, T-l Specific rate of ingestion of cell volume by population 2, T-I Specific rate of cell volume lost from population 2 by predation, T-l Ratio of the diameter of the predator to the diameter of the prey at which clearance is maximal Concn of cells in a bandwidth from D to D + dd, L- L),.. F(D) dd s amn Biochemical sensitivity of a predator to its prey Volume of a cell of diameter D, L3 Mean clearance rate of an individual predator of diameter II2 that interccpts a cell of diameter D,, T-l Specific growth rate of population 2, T-1 Standard deviation of the distribution of capture efficiencies Specific mortality rate of population 2, T-1 eucaryotic cells, which are > 1 pm in diameter. Because the spectrum of cell size distribution is an important manifestation of the trophic interaction of populations composing the microbial food web (Fenchel 1987), we have begun to examine the factors that may influence the distribution of cell sizes. This examination has been guided and aided by recent laboratory and field research on the population dynamics of species and taxonomic groups composing the microbial food web. We have found that the hypotheses of Kerr (1974), Platt and Denman (1977), and Silvert and Platt (1978) that have attempted to explain particle size distributions are very general in their description and thus do not provide much insight into the dynamics of microbial populations. These hypotheses are based on the princi- ples of energy conservation, the conditions of steady state, and empirical allometric rules that describe the metabolic rates of organisms and the turnover times of natural populations as functions of body or cell size. In this paper we present an alternative hypothesis which states that the steady state flow of material tlhrough the microbial food web and the associated spectrum of size distribution is determined by the random encounter of predator and prey populations. According to this hypothesis, the spectrum of particle sizes is the result principally of the constancy of chemical composition of the taxonomically distinct populations in the food web and the physical constraints on food capture in a viscous fluid. Before discussing the random encounter hypothesis, we present examples of cell size distributions in the open ocean and then derive a mathematical model of random encounter that attempts to explain the 4th power dependency of the frequency of cell size. Cell size distribution The differential size distribution of particles, F(D), is dehned in the list of notation. The concentration of particles within a given bandwidth, Dmax-Dmin, that is centered at a given diameter D can be determined from F(D) by integrating the function between the limits: Dmax N(Drnin < D -c D,,,,) = r F(D) dd. (1) J Dmin We further define N(D) as the numeric concentration of cells in a bandwidth from D to D + dd. In the case of marine particles, the differential size distribution generally has been described as a hyperbolic function (Eq. 2) in which the exponent b commonly has a value of 4. Such a 4th power dependency has been noted by numerous scientists sampling geographically diverse waters (Sheldon et al. 1972). F(D) = a D-k (2) In fact, the value of b may vary between the 3rd and the 5th power (McCave 1975; J.C. Kitchen pers. comm.) for differing regions. In addition, Gaussian features are sometimes superimposed on the hyperbolic distribution (Jonasz 1983).

3 Random encounter model 459 Fig. 2. Schematic illustration of trophic tions in the random encounter model. interac ; lb 100 cell diameter. D Cm) Fig. 1. Cumulative size distribution of cells in the central Pacific. The ordinate shows the number of cells that exceed the equivalent diameter (adapted from Stramski and Kiefcr 199 1). In our presentation (Fig. l), the concentration of cells on the ordinate represents the total concentration of cells that exceeds the equivalent diameter on the abscissa. Data for cells >2-pm diameter were obtained both by microscopic measurements and counts of settled water samples (Takahashi and Bienfang 1983). Data include eucaryotic photosynthetic cells. Data for the smaller procaryotes were obtained by counts and measurements of free-living bacteria stained with DAPI. The slope of the logtransformed data of cumulative size classes is -3; thus the differential size distribution of the cells conforms to a 4th power dependency. It is likely that such a pattern of distribution is an important feature of the planktonic community and that it is a manifestation of factors that control the steady state populations of diverse species. Mathematical derivation We propose that a set of interactions (Figs. 2, 3) are the primary sources of the sizefrequency distribution. Consider a population of swimming, phagotrophic protozoans of diameter D2, whose differential concentration is F(D,). As these cells swim randomly through the water they encounter and consume prey of size class 1 whose diameters range from D, min to D1 max (Fig. 3). The range of cell sizes in this prey class is determined by the ability of population 2 to ingest these cells. Less frequently but inevitably these consumers will also encounter predators of size class 3 whose diameters range from D3 min to D3 max. Smaller or larger cells cannot ingest the consumer. The range of cell sizes in class 3 is determined by their ability to ingest cells in population 2. Although the prey of size class 1 may be phagotrophic, autotrophic, or heterotrophic cells, the two higher trophic levels must of course be phagotrophic. The rate of ingestion of prey biomass by population 2 will be simply the product of the total rate of encounter of population 2 with its prey and the mass of an individual prey item (Fig. 2). Likewise, the rate of biomass loss of population 2 to its predators, represented by the arrow pointing from compartment 2 to 3, is simply the product of the total rate of encounter of population 2 with its predator and the mass of an individual within the population. Population 2 will also lose mass by excretion and egestion; the arrow pointing upward from compartment 2 represents the sum of the rates of these two processes. At steady state, gains in biomass of population 2 by ingestion must equal losses in biomass by predation, excretion, and egestion. If one assumes that the density of cells (weight per unit of cell volume) within the three size classes is the same, then the fluxes of biomass into and out of a population are proportional to the fluxes of cell volume. Thus (Fig. 2), we characterize the lower and upper size classes, 1 and 3, by their differential cell size distributions, F(D) (Eq. 1) and by their volume, V(D), where we make the simplifying assumption that microbial cells are spherical. The entity fz1 is the flux of cell volume that is ingested by population 2 in a unit volume of seawater, fze is the flux of cell volume that is lost to egestion and excre-

4 460 Kiefer and Berwald ,00 A ul s h , Dlmin Dl max D3mln D3max cell diameter (urn> Fig. 3. Illustration of interactions between components of the random enocunter model. The central population can encounter and ingest prey of the bandwidth D, min to D, max. The central population will also encounter, with less frequency, predators from bandwidth D, min to D, max. h, and h, represent the ratios of optimal clearance. See text for discussion. tion, and the arrow labeledfzp is the flux of cell volume that is lost to predation by size class 3. At steady state conditions, fzr = fze + f-p. If we further assume that fze is a constant fraction, (1 - A), offzl, the steady state condition is Af, =fzp. (4) A is thus the steady state assimilation efficiency. Since fzz is the total encounter rate of population 2 with size class 1, the lefthand side of the steady state equation can be written S Dlmax D~mm Af,l= A N(D,) HDlMm -xul 02) dd,. (5) We define x(dl, D2) as the mean clearance rate of an individual predator of diameter D2 that intercepts a cell of diameter D,, The right-hand side of Eq. 3 can be written tip = WWW2) D3UlliX. s WUG 2, 03) dd3. (6) D3min By dividing Eq. 5 and 6 by the concentration of cell volume of population 2 one expresses steady state in terms of the population s specific rates of growth, p2, and mortality, ti2. P2 = =w2= DlIMiX A W,Y(n,)x(D,, 02) dd, s Dl?lUIl s D smax D3min VP2) F(Dh(D2, D3) dd,. (7) Equation 7 is key in our random encounter model. x, the individual clearance rate, is the product of two terms: ZP, the encounter rate and s, the biochemical selectivity coefficient. The selectivity coefficient, will depend on the nutritional state of the predator (Verity 1985) and its biochemical preference for food. There are several published studies (e.g. Verity 199 1) that suggest differential behavior by microorganisms in the presence of chemical cues. Because information on chemical selection of prey is very limited, we set the value of s constant and equal to 1. Gerritsen and Strickler ( 1977) have derived a formulation of the rate of encounter of prey by an individual predator as follows: zp = ; N2N(D,) (VI2 + 3V22) (8) V2 where V, < v2, 12 is the radius of a sphere of encounter, N(D,) is the prey cell concentration, and v1 and v2 are the swimming

5 Random encounter model 461 speeds of prey and predator. Gerritsen and Strickler did not present a formulation for R, although in the case of protozoans, the function must include terms for the size of prey and predator cells. We chose to call the product rr2 the effective cross-sectional area for cell capture and represent this area as the product of 7r, the square of the radius of the predator, (D2/ D), and the hydrodynamic capture efficiency coefficient, q(dl, D2). It has been suggested that capture of prey by protozoans is best described by the physical term for direct interception. This hydrodynamic capture efficiency is defined as the ratio of the cell diameter of the prey to the cell diameter of the predator, q(d1, D2) = D,lD2 (Rubenstein and Koehl 1977; Fenchel 1984; Shimeta and Jumars 1991). The predicted clearance rate then becomes x(hd2) = ~rd~d~s~(v~~ + 3V22) V2. (9) We have examined published studies of food size selectivity by protozoans (Bernard and Rassoulzadegan 1990; Chrzanowski and Simek 1990) and have found such a description unsatisfactory. Such a function predicts that the clearance by a given size predator is a linear function of the diameter of the prey and that the maximum clearance per prey item will occur when the prey is the same size as the predator. Feeding studies like those discussed below indicate that clearance rates are maximal when the prey is many times smaller than the predator but that very small and very large prey are not ingested. Equation 9 also predicts that the clearance rates of predators of variable sizes feeding on prey of a given size increase linearly with predator size. This result seems unlikely and is incompatible with the 4th power dependency of cell size. The feeding apparatus of protozoa is an elaborate structure specifically designed to facilitate effective capture and subsequent phagocytosis. Effective cross section of particle capture and ingestion will depend on the number, size, and arrangement of cilia, flagella, and pseudopodia as well as the flow of water in and around such a structure. We suggest that the structure and the associated flow field limit the range of particle sizes that can be captured and ingested. There is a theoretical optimum in prey size, and cells that are larger or smaller will be captured and ingested with diminished efficiency. The normal distribution is an appropriate formulation of such a relationship (Bernard and Rassoulzadegan 1990; Chrzanowski and Simek 1990). In the case of a predator of constant diameter D2 feeding on prey of variable diameter D we write rs2d22 (v12 + 3v22) x(q 02) = -yj- V2 eexp- By analogy, the clearance rate of the predator s predator with a variable diameter of D and feeding on prey of constant diameter D2 is ns,(hd,)2 (v22 + 3~~~) xp2,d> = 12 V3 sexp- (11) The exponential terms are the capture efficiencies for the central population on its prey, ~(0, D2), in Eq. 10 and for the predator on the central population, 7(D2 D), in Eq. 11. Values vary between 0 and 1. Note that differences in size between prey and predator are parameterized as the ratio of the cell diameter of the prey to predator. This term is the direct interception term discussed previously; h is the ratio of the diameter of the predator to the prey at which the clearance rate is maximal. Therefore, when the ratio of prey diameter to predator is optimal, efficiency will be 1. The constant, g, is the standard deviation of the distribution of capture efficiencies with variations in the ratio of the cell diameter of the prey to that of the predator. Although observed distributions of clear-

6 Kiefer and Berwald prey Ulameter: predator dfameter OS Prey diameter, predator diameter 3 so 1 Fig. 4. A. Strombidium sulcatum feeding on bacteria and phytoplankton. Measured values from Bernard and Rassoulzadegan (1990). Note the similarity of the shape of the predicted clearance curve to measured values. B. Clearance rates of the flagellate Ochromorzas feeding on various sizes of the bacteria Pseudomonas ji uowscens. The predicted and measured curves show similarity in shape. Measured values from Chrzanowski and Simek (1990). ante are not as symmetric and smooth as predicted by Eq. 10, the shape is similar (Fig. 4). Our interpretation of such data is that they are dominated by variations in capture efficiency, 7, rather than variations in the selection coefficient. At the large end of the prey size spectrum however, selection may have greater relative importance. When we substitute Eq. 10 and 11 into Eq. 7, we obtain two equations that allow us to calculate the value of b in the hyperbolic distribution. For such a prediction we introduce values for the swimming speeds of the flagellates, v2, and ciliates, v3, of 0.2 and 1 mm s-l (Fenchel 1987). The first trophic level is assumed to be nonmotile. A is given a value of 0.5 (Verity 1985). Because Eq. 7 is solved numerically, we choose values for the cell diameter of the predator, D2 14 = 10 pm, for the optimal ratio of cell diameter of predator to prey, h = 10, and for the standard deviation of hydrodynamic clearance efficiency, (T = The exponent of the hyperbolic function of particle size (Eq. 2) equals - 4.2, which satisfies the measurements of cell distribution in the sea. Most importantly the solution does not depend on the value of D2, h, or Q. In fact, inspection of Eq. 7, 10, and 11 indicates that the 4th power dependency results from the fact that the volume of a cell varies as the 3rd power of its diameter, the maximum clearance rate of a phagotroph varies as the 2nd power of its diameter, and the size of prey that a predator ingests is proportional to i.ts own size. Discussion The food chain diagramed in Fig. 2 is obviously extremely simplified. The marine microbial food web is characterized by phagotrophic, photoautotrophic, mixotrophic, and heterotrophic cells in the same size class, so a hypothesis of microbial community structure should, at a minimum, accommodate such metabolic diversity. SigniJicance of swimming speed-it is obvious from our mathematical description of the random encounter model that swimming speed is an important parameter of community structure. Fenchel (1987) has presented evidence that the swimming speeds of protozoans are quite variable. This variation may be due to biomechanical and physiological factors. Additionally, velocities may be influenced by small- and large-scale turbulence in the aqueous environment. Conversely, this variance may be attenuated by biased or directional swimming on local scales. Therefore, variance of swimming speed for an organism of a given size may vary by a factor of about six and in general swimming speeds are independent of cell size. Table 1 illustrates the interactions of various predators with different prey. Of particular interest in this table are the predicted and measured specific clearance rates for various types of predation. The predicted specific clearance rate of a predator is calculated by dividing the predator s clearance rates (Eq. 10) by the cellular volume of the predator. (The cap-

7 Random encounter model 463 Table 1. Specific rates of clearance for various microbial encounters. Predicted values assume maximal efficiency and selectivity. Predator Flagellate Flagellate Flagellate assemblage Flagellate assemblage Flagellate assemblage Ciliate Ciliate Tintinnopsis vasculum Tintinnopsis acuminata Ciliate Strombidium sp. Strombidium sp. Strombidium sp. Cyclidium sp. Ciliate Specific clearance rate Predator (s-9 Mea- CP% Prey sured Predicted Reference 15 Flagellate Nonswimmer 4-24 Nannochloris atomis 1 Sherr et al Chlorella capsulata 23 Sherr et al Thalassiosira 12 Sherr et al pseudonana 30 Flagellate Nonswimmer 3-18 Dicrateria inornata 2 Verity 1985 Isocrysis galbana 1 Verity Nonswimmer N. atomis 12 Sherr et al C. capsulata 27 Sherr et al T. pseudonana 19 Sherr et al Microcystis sp. 10 Gerritsen et al Ciliate 8-49 ture efficiency coefficient is assumed to be unity.) The swimming speeds for flagellates and ciliates have been summarized by Fenchel as a range of values. The calculations in Table 1 include the clearance rates for flagellates and ciliates of specified sizes feeding on nonswimming prey as well as feeding on other flagellates and ciliates that are swimming at comparable speeds. Two features of Table 1 are noteworthy. First, the clearance rates measured by Gerritsen et al. (1987), Verity (1985), and Sherr et al. (199 1) most often fall within the range of predicted rates. Thus, our simple description of ingestion is supported. Second, variations in the swimming speeds of prey cells may have a small but significant effect on their encounter rates with predators (Gerritsen and Strickler 1977). For example, the clearance rate of a ciliate predator swimming at a typical speed of 1 mm s-l is predicted to be 25% lower when feeding on a nonswimming prey than on a comparably sized prey swimming at 1 mm s-l. If the ciliate s prey is swimming at a speed typical of a flagellate, 0.2 mm s-l, then its clearance rate is 24% lower than for a prey of comparable size that swims at the same speed of 1 mm s-l. Thus, the difference between the clearance rates of a ciliate feeding on a slowly swimming flagellate and a nonswimmer is very small. Such differences may be important to the structure of the microbial food chain. One can speculate that swimming speeds in natural populations are variable and could be an adapted or selected feature of a population. According to Eq. 7, for a given phagotrophic population there is a unique swimming speed that is determined not only by the ratio of cellular concentrations of its prey and predator populations, but also by the swimming speeds of its prey and predator. If perturbations occur to the cellular concentrations of a population s prey or predator, a different swimming speed for individuals in the population might be favored. If, for example, the local concentration of a predator swims at the same speed as its prey, the fitness of slower swimming individuals in the prey population would increase relative to the faster swimmers. Signifxance of size- Within a population of a given size class, there will be a distribution of cell sizes. Within a given species, population cell size will vary because of phenotypic differences and because size changes are inherent features of the cell cycle. In addition, several species in a trophic level may be preyed on by the same pred-

8 464 Kiefer and Berwald ator( The specific mortality rate of a population (Eq. 7 and 11) is the product of the hydrodynamic capture efficiency, q(d2, D3), the biochemical selectivity coefficient, s2, and the rate of volume swept by the predator population: - FUL d&j. (12) The capture efficiency, v(dz, DJ, is the o.nly term in Eq. 12 that explicitly relates mortality rate to the size of the cells in the prey and predator populations. In the random encounter model, capture efficiency is described as a normal distribution in which the larger cells and the smaller cells are less effectively captured by a predator of a given size. Populations may, in effect, reduce rates of mortality to a specific predator by maintaining genotypic and phenotypic variability in cell size. Such a process may help to stabilize the community and maintain species diversity. It may be a particularly important process for the smallest prey cells that might effectively escape capture by the smallest phagotrophs. As pointed out by Chrzanowski and Simek (1990), bacteria growing slowly because of nutrient limitation, may find a refuge from predation because of associated decreases in cell size. Nonphagotrophic populations -Let us consider the criterion for survival of a population of photoautotrophic or heterotrophic cells of the size class 2j that are of the same size as the cells of class 2i, where the subscripts i and j designate heterotrophic and autotrophic populations (Fig. 5). For nonphagotrophic cells, encounter with cells of another species is never beneficial. In particular, this population will experience losses by predation from size class 3,. Unlike the phagotrophic cells of its size class, the rate of replacement of individuals in population 2j is not directly dependent on the concentration of cells in size class li. Instead, the rate of replacement of population 2, will depend on the concentration of a limiting nutrient or perhaps, in the case of a photoautotrophic population, light intensity. Before discussmg the criterion of survival for the nonphagotrophic population, we wish to point out that the existence of a nonphagotrophic population 2 in the same size class as the phagotrophic population 2, does not appear to violate the 4th power dependence of cell concentration on cell diameter. Although the presence or absence of such a population will affect the size distribution of phagotrophic cells, it will have little effect on the concentration dependence of all cells in size class 2i phagotrophic and nonphagotrophic. The result of introducing nonphagotrophic populations into the model is to shift the intercept of the 4th power function vertically up the concentration axis while preserving, the slope of the line. Mixotrophs can thus be incorporated into the model as either heterotrophs or autotrophs. Survival of a nonphagotrophic population (Fig. 4) simply requires that it satisfy the steady state conditions as presented in Eq. 7. That is, its specific growth rate, a function of nutrient availability, must equal its specific mortality rate: P2j = W2j = s Dz,max Dz,min E (D3j)X(D2j, DJj) dd,* ( 13) If we assume for simplicity that there is no chemical selection by the predator between phagotrophic and nonphagotrophic populations of the same-sized individuals, the ratio of the growth rates of nonphagotrophic to phagotrophic populations is simply a function of the capture efficiencies and the swimming speed of the two prey populations:

9 Random encounter model 465 P2j -= q(d2j, D3i) (v2j2 + 3v3i2) P2i dd2i, D3i) (~2; + 3v3i2) = V2j2 + 3Y3i2 V2i2 + 3V3i2 ' (14) The efficiencies cancel because D2i = D,. If the nonphagotrophic cells swim at the same speed as the phagotrophic cells of the population, the specific rates of loss by the two populations will be the same, and thus, the two populations must have the same net specific growth rates. If, as is often the case, the nonphagotrophic cells swim more slowly or not at all, the specific rates of predation loss and specific rates of net growth will be smaller than the phagotrophic population. Values for volume-specific rates of clearance (Table 1) provide rough estimates of the relative differences in growth rates expected for populations in the same size class that have different swimming speeds. Note that the microbial food chain differences in steady state growth rate that can be ascribed solely to differences in swimming speed are relatively small. The largest within-size difference in growth rates between a nonswimming population and a swimming population occurs in the size classes of the ciliates. In such a case the growth rate of the nonswimming, nonphagotrophic population is 75% of the rate of the corresponding swimming population. Comparison with previous explanations of size distributions-kerr ( 1974), Platt and Denman (1977), and Silver-t and Platt ( 1978) have proposed models of cell size distribution. Because these models are generally similar, we will consider only one: Kerr s derivation is the simplest and its mathematical approach most resembles ours. He assumes a steady state in which net growth rate of the population at trophic level 1 equals the sum of the net growth rate and respiration rate of the population at trophic level 2. He then invokes a relationship between specific rates of growth, here u (Sheldon et al. 1972, 1973) and respiration, r, of a population and the size of individuals within the population. Specifically: u=kw-fl,rfcx WY. (15) Wis the body weight of an individual within the population. He further assumes, as we do, that the assimilation efficiency of the predators, A, is constant, and that the size of individuals in population 2 is a multiple, h, of its prey. Given the assumption of constant cell density, Kerr found that such relationships yield a predicted ratio of the concentration of total cell volume for adjacent trophic levels: WA) VW ND21 W2) 1. (16) Kerr refers to typical values for y of 0.8 and for p of 0.2; thus, the exponent of the power function is close to 0. Additionally, he uses an h of 20 and an A of 0.7. The ratio a2 : k is calculated to be -4. Therefore Kerr concludes that the total biomass of a given trophic level will be about 1.2 times the standing stock of its predators. According to Kerr this value agrees well with the estimates of Sheldon et al. (1972). Perhaps the most important difference between the random encounter hypothesis and these earlier hypotheses involves the treatment of the growth rate of populations. As indicated by Eq. 15, the earlier hypothesis is based on the assumption that the specific growth rates of natural populations are uniquely determined by cell size. The random encounter hypothesis ignores such an empirical allometry; the growth rate of a population of phagotrophic cells is a function not only of the size of the cell but also the assimilation rate (Eq. 7). The random encounter model predicts that the growth rates of nonphagotrophic cells are at least loosely coupled to the growth rates of similarly sized phagotrophic cells (Eq. 14). Unlike the earlier hypothesis, the random encounter model strongly implies that steady state growth rates will tend to increase with increases in both prey and predator cell concentration. Although estimates of growth rates of nat-

10 466 Kiefvr and Berwald ural microbial populations are difficult to measure and few, evidence favors the random encounter hypothesis. The growth rate of microorganisms cultured continuously in the laboratory indicates that specific rates of growth and respiration are strongly dependent on the rate of supply of food or nutrients (e.g. Verity 1985). Even in the absence of nutrient limitation when growth rates are maximal, there is little evidence that microbial growth rates are closely tied to size. For phytoplankton and photosynthetic bacteria, the growth rates of many larger species are comparable or even faster than that of smaller species. Phagotrophic protozoans grow as fast or faster than phytoplankton and strict size dependency is not evident (Banse 1982). As mentioned earlier, the growth rate of heterotrophic bacteria tends to increase rather than decrease.with cell size within a given strain. Furthermore, measurements of growth rates of natural microbial populations have provided no evidence of strong size dependence on growth rate. In oligotrophic waters the doubling time of the smallest cells, the heterotrophic bacteria, is thought to be in the range of 1 d to several days (Fuhrman et al. 1989). The photosynthetic bacteria, which consist of the prochlorophytes and cyanobacteria, are larger than the heterotrophic bacteria and grow at least this fast (Iturriaga and Marra 1988). The same may be said of many of the protists that are much larger than the procaryotes. This apparent decoupling of growth rate and size is consistent with predictions in the random encounter hypothesis. Another important difference between the two hypotheses involves the effects of respiration on community structure. In the earlier model loss of cell volume by respiration is treated explicitly as an important loss of biovolume, while in the random encounter model respiration is treated implicitly and considered to be less important than loss by predation, In fact, the specific respiration rates of most marine microorganisms are much smaller than their specific rates of mortality by predation. Specific respiration rates are commonly < 0.2 d-l while specific rates of mortality are commonly > 0.5 d-l. Furthermore, in the earlier hypothesis, rates of respiration depend only on cell size. In the random encounter model, the assimilation efficiency, which is the ratio of the rate of growth to the sum of the rates of growth, respiration, and egestion, is assumed constant. As a consequence, the rate of loss by both respiration and egestion is proportional to the growth rate. This relationship is generally supported by measurements of the growth and respiration of phytoplankton (Laws and Bannister 1980) and protozoa (Verity 1985) maintained in semicontinuous culture. Finally, the hypothesis of Platt and Denman and Silvert and Platt is based on a mathematical description of a single trophic and size class of the biomass spectrum and that of Kerr is based on a description of two adjacent trophic levels. These workers treated the size distributions of unicellular and multicellular organisms and these size distributions are affected by both the growth of individuals within a population and trophic exchange. As pointed out by Silvert and Platt, the mathematical description of size changes affected both by growth of individuals and by predation is difficult. Because the random encounter model is specifically designed to examine the size distribution of the single-celled organisms that compose the planktonic microbial food web, changes in the size of an individual can be ignored and attention centered on trophic exchange. Qur approach does not necessarily conflict with their arguments. The derivation presented in this paper is based on three trophic levels. This conceptual framework is appropriate to the microbial planktonic ecosystem because the material transformations within this system are dominated by predator-prey interactions. More specifically, in the absence of physical refuges, the search for food is inevitably accompanied by exposure to predation. In order to survive, a population of phagotrophic cells must encounter a sufficient number of smaller prey to counter its losses to larger predators. From this simple concept, we propose that the size distribution of single cells in the sea is not the result of allometric constraints on metabolism but rather a con-

11 Random encounter model 467 sequence of random encounter in an environment that lacks physical refuges. References BANSE, K Cell volumes, maximal growth rate of unicellular algae and ciliates, and the role of ciliates in the marine pelagial. Limnol, Oceanogr. 27: BERNARD,~.,AND F. RASSOULZADEGAN Bacteria or microflagellates as a major food source for marine ciliates: Possible implication for the microzooplankton. Mar. Ecol. Prog. Ser. 64: CARON, D. A Technique for enumeration of heterotrophic and phototrophic nanoplankton, using epifluorescent microscopy, and comparison with other procedures. Appl. Environ. Microbial. 46: 49 l-498. CHISHOLM, S. W., AND OTHERS A novel freeliving prochlorophyte abundant in the oceanic euphotic zone. Nature 334: CHO, B. C., AND F. AZAM Biogeochemical significance of bacterial biomass in the ocean s euphotic zone. Mar. Ecol. Prog. Ser. 63: CHRZANOWSW, T.H., AND K. SIMEK Prey-size selection by freshwater flagellated protozoa. Limnol. Oceanogr. 35: 1429-l 436. FENCHEL, T Suspended marine bacteria as a food source, p. 30 l In Flows of energy and materials in marine ecosystems. NATO Conf. Ser. 4, Mar. Sci. V. 13. Plenum Ecology of protozoa. Sci. Tech. FUHRMAN, J. A.,T. D. SLEETER,~. A. CARLSON, AND L. M. PROCTOR Dominance of bacterial biomass in the Sargasso Sea and its ecological implication. Mar. Ecol. Prog. Ser. 57: GERRITSEN, J., R. W. SANDERS, S. W. BRADLEY, AND K. G. PORTER Individual feeding variability of protozoan and crustacean zooplankton analyzed with flow cytometry. Limnol. Oceanogr. 32: 69 l , AND J. R. STRICKLER Encounter probabilities and community structure in zooplankton: A mathematical model. J. Fish. Res. Bd. Can. 34: ITURRIAGA, R., AND J. MARRA Temporal and spatial variability of chroococcoid and cyanobac- teria Synechococcus spp. specific growth rates and their contribution to primary production in the Sargasso Sea. Mar. Ecol. Prog. Ser. 44: 175-l 8 1. JONASZ, M Particle-size distributions in the Baltic. Tellus 35B: KERR, S. R Theory of size distribution in ecological communities. J. Fish. Res. Bd. Can. 31: LAWS, E. A., AND T. T. BANNISTER Nutrientand light-limited growth of Thalassiosira fluviatih in continuous culture, with implications for phytoplankton growth in the ocean. Limnol. Oceanogr. g5: MCCAVE, I. N Vertical flux of particles in the ocean. Deep-Sea Res. 22: PLATT, T., AND K. L. DENMAN Organization in the pelagic ecosystem. Helgol. Wiss. Mecresunters. 30: RUBENSTEIN, D.I., AND M. A.R. KOEHL The mechanisms of filter feeding: Some theoretical considerations. Am. Nat. 111: 98 l-994. SHELDON, R. W., A. PRAKASH,AND W.H. SUTCLIFFE, JR The size distribution of particles in the ocean. Limnol. Oceanogr. 17: ,AND The production of particles in the surface waters of the ocean with particular reference to the Sargasso Sea. Limnol. Oceanogr. 18: SHERR, E. B., B. F. SHERR, AND J. MCDANIEL Clearance rates for ~6 pm fluorescently labeled algae (FLA) by estuarine protozoa: Potential grazing impact of flagellates and ciliates. Mar. Ecol. Prog. Ser. 69: SHIMETA, J., AND P. A. JUMARS Physical mechanisms and rates of particle capture by suspcnsionfeeders. Oceanogr. Mar. Biol. Annu. Rev. 29: 19 l SILVERT, W., AND T. PLATT Energy flux in the pelagic ecosystem: A time-dependent equation. Limnol. Oceanogr. 23: STRAMSKI, D., AND D. A. KIEFER Light scattering by microorganisms in the open ocean. Prog. Oceanogr. 28: TAKAHASHI, M., AND P. K. BIENFANG. 1983, Size structure of phytoplankton biomass and photosynthesis in subtropical Hawaiian waters. Mar. Biol. 76: VERITY, P. G Grazing, respiration, excretion, and growth rates of tintinnids. Limnol. Oceanogr. 30: Feeding in planktonic protozoans: Evidence for non-random acquisition of prey. J. Protozool. 38: Submitted: I7 June 1991 Accepted: I9 September I991 Revised: 8 March I992