Eur. Phys. J. Special Topics 225, 669–683 (2016) © EDP Sciences, Springer-Verlag 2016 DOI: 10.1140/epjst/e2015-50261-1
THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS
Review
Fluid dynamic constraints on resource acquisition in small pelagic organisms T. Kiørboea Centre for Ocean Life, DTU Aqua, Technical University of Denmark, Kavalerg˚ arden 6, 2920 Charlottenlund, Denmark Received 17 September 2015 / Received in final form 3 May 2016 Published online 15 July 2016 Abstract. Physicists have long examined the fluid dynamics of swimming at low Reynolds number, but the main scope has rarely been to understand the behavior and ecology of microorganisms. However, many ecological questions about the functioning of small aquatic organisms can only be addressed by the application of formal fluid physics. Here, I examine resource acquisition mechanisms in small aquatic organisms, ranging from uptake of dissolved molecules to feeding on suspended particulate prey, and examine how organism behaviors and morphologies may be shaped by the often non-intuitive small-scale fluid physics.
1 Introduction Feeding and swimming are often intimately related in microscopic plankton organisms. Not only may swimming bring the organisms to regions of high food but swimming and feeding currents are often also required to encounter prey. Fluid physicists have a long tradition for examining the fluid dynamics of swimming – not feeding – in micro-organisms, that is, swimming in a viscous environment at low Reynolds numbers (Re1 ). That tradition was intensified by the publication of the classical paper by Purcell [1] on “Life at low Reynolds numbers” and the field was recently reviewed by several authors [2,3]. The main scope of most of this literature has been fluid dynamics problems, and many of the problems studied, ranging from Purcell’s 3-hinge swimmer [4] to the study of gyrotactic bioconvection and “dense suspensions” [5] relate only remotely to the ecology and biology of “real” microorganisms in their natural environments. Yet, these studies have demonstrated that the morphology, behavior, and evolution of microorganisms are constrained by fluid physics in often non-intuitive ways [6]. And, equally important, these studies have developed analytical tools and models that may be used to address also ecologically relevant questions. The starting point for this brief tutorial is from the point of view of a biologist: how can formal a
e-mail:
[email protected] The dimensionless Reynolds number (Re) is defined as the product of the velocity (U ) and size (a) of a moving object, divided by the kinematic viscosity of the medium in which it is moving: Re = aU . It expresses the relative significance of inertial and viscous forces. ν 1
670
The European Physical Journal Special Topics
fluid physics help us understand the ecology of small plankton in the ocean? In particular we shall here examine the implications of a low Re environment for resource acquisition in small aquatic organisms, since resource acquisition is probably the most fundamental activity in an organism and a prerequisite for the ultimate Darwinian mission of reproduction. So, the focus is on feeding rather than on swimming. We shall start by noting that most of the biology of the oceans is due to small organisms. Quantified in terms of carbon turnover, maybe 95% or more of the oceans’ carbon metabolism is due to organisms smaller than a few millimeters. The reason for this is that the plants in the ocean are (almost) consistently small, single-celled phytoplankton. This is because the plants (primary producers) in the ocean are limited by the availability of inorganic nutrients, unlike terrestrial plants that are limited mainly by water. And small size is at a competitive advantage in a nutrient limited environment (see below). The main plants in the ocean are therefore microscopic phytoplankton that typically measures less or much less than 100 μm in diameter. In fact, the dominating group, the cyanobacteria, measures less than 1 μm in diameter, which is probably the smallest size possible for an intact cell with a cell wall and room for organelles [7]. Small plants are eaten by small grazers that, in turn, are eaten by larger predators – that are still pretty small – and so on (Fig. 1). As a rule of thumb (with many exceptions), the size ratio between grazer and phytoplankton prey, or predator and prey in plankton food chains is about 10:1 [8], and so the typical size of zooplankton predators on, say, the 3rd trophic level still only measures less than a mm in size. Because the ‘trophic efficiency’ due to metabolic and other losses is maybe only about 20%, the carbon turnover at one trophic level is only 20% of that at the level above. The integrated carbon turnover of organisms smaller than 1 mm therefore absolutely dominates the carbon turnover and entire biogeochemistry of the ocean, even though the total biomass of organisms at each trophic level is approximately constant ([9]; see also legend of Fig. 1). This small-scale life is also the basis for life at higher trophic levels, including fish and other living resources that, directly or indirectly, support more than 10% of the global human population [10]. This implies that the most significant (in terms of carbon turnover) oceanic life unfolds in a low Reynolds number, counterintuitive, sticky environment. Marine ecologists therefore want to understand how the life at small scales is constrained by this particular physical environment as this is a prerequisite to understand the biology of the ocean.
2 Feeding on solutes Organisms can either feed on dissolved or particulate material. Organisms feeding on dissolved molecules are called “osmotrophs”. In the ocean they encompass two groups of organisms, namely autotrophic phytoplankton that require inorganic nutrients and dissolved inorganic carbon as substrates for photosynthesis fueled by light energy, and heterotrophic bacteria that use dissolved organic material as their energy resource. The concentration of inorganic nutrients is typically low and, hence, limiting for growth rate in phytoplankton, and similarly, the concentration of degradable organic molecules are low. When the delivery of substrate or nutrient molecules are limited by diffusion, which is most often the case, then the diffusive delivery of molecules to a spherical collector (the cell) and, hence, the uptake rate, Q, is given by [12, 13], Q = 4πDaShC
(1)
where D is the diffusivity of the substrate molecules, a is the radius of the cell, C the bulk concentration of molecules, and Sh the so-called Sherwood number. In the absence of advection, Sh = 1. Because the absolute nutrient uptake rate scales with the
Cooperative Particles: Patchy Colloids, Active Matter and Nanofluids
0.001 cm
Phytoplankton
0.01 cm
Ciliate
0.1 cm
Copepod
671
1 cm
Fish larva
Fig. 1. Schematic of pelagic food chain. The organisms at each trophic level increase in size (length) by a factor of about 10 and, hence, in body mass by a factor of about 1000. The energy turnover at successive trophic levels decreases by about 80%, whereas the biomass (B) is approximately constant at successive levels. To see this, assume (1) steady state, i.e., that the production at the i’ the trophic level equals the consumption by the next level i+1; (2) that the trophic efficiency, e, is 0.2; and (3) that the specific growth of an individual scales with its body weight (W ) to a power of −1/4 (“Kleibers rule”; [11]). Then the production −0.25 at level i, aBi Wi−0.25 , equals the consumption at the next level (1/e)aBi+1 Wi+1 , where 0.25 T). Equating consumption and production a is a proportionality constant (dimensions M and noting that Wi+1 /Wi ∼ 1000, one finds that the ratio of total biomasses at successive trophic levels (Bi /Bi+1 ) is about unity. That is, we find approximately the same biomass at each tropic level in the ocean. The “real” world is better described as a food web than as a food chain, but the arguments can be generalized. Image courtesy Torkel Gissel Nielsen.
radius of the cell, it follows that the uptake rate relative to the mass or volume of the cell scales inversely with the cell radius squared; that is, by doubling the radius of a cell, the diffusive nutrient delivery relative to its volume is decreased by a factor of 4. As noted above, that is the reason why small size is a strong competitive advantage in osmotrophic phytoplankton as well as bacteria. Larger organisms cannot efficiently harvest dissolved molecules in the ocean, and so osmotrophs are generally very small. While small, osmotrophs are not all equally small. Phytoplankton cells, for example, may range in size from less than a micron to nearly a millimeter, which implies a nearly 9 order of magnitude variation in cell volume (and mass). The advantage of being small is countered by a higher risk of being eaten, and so there is also a selective pressure for larger size. When the availability of nutrient molecules is higher and the pressure for small size is relaxed, cells may grow bigger. Thus, bacteria attached to surfaces, where nutrient conditions may be better, are typically larger than those in suspension [14], and phytoplankton cells growing under eutrophic (nutrient rich) conditions are often much larger than those growing in oligotrophic water [15, 16]. The variable nutrient conditions and predation pressure in natural communities may account for some of the size-diversity found among unicellular organisms in the plankton. Natural selection has also favored the evolution of mechanisms for relaxing the constraints of nutrient limitation in osmotrophs, and we shall briefly mention a few here. First, cells may “inflate” by having a high water content. This way they can increase their radius and, hence, the diffusive delivery of nutrient molecules (cf. Eq. (1)); this strategy is found both among phytoplankton and bacteria [17]. Second, they can swim to regions of higher nutrient availability by being chemotactic, i.e., the ability to sense the concentration of nutrient molecules and through a
672
The European Physical Journal Special Topics
biased random walk on average swim up nutrient gradients [13]. Many bacteria and phytoplankton cells are motile, and most motile bacteria are chemotactic and able to efficiently utilize small ephemeral patches of nutrients that are characteristic of the apparently homogenous aquatic environment [18]. Some larger phytoplankton cells, best described for the group of dinoflagellates, conduct diel vertical migration: they swim downwards during night to harvest inorganic nutrients at depth where nutrients are typically plentiful, and swim towards the nutrient-poor surface layer during day to harvest light [19]. Swimming may finally also increase nutrient uptake by replenishing nutrientdepleted water in the immediate vicinity of the cell and thus steepen concentration gradients and diffusive fluxes. Ambient turbulence may have the same effect. In the absence of advection, nutrient transport to the cell is solely through diffusion. The Sherwood number2 (cf. Eq. (1)) quantifies the enhancement of nutrient transport due to advection (and is therefore equal to 1 in the absence of advection). The Sherwood number depends on the magnitude of the kinematic viscosity of the medium relative to the magnitude of the diffusivity of the molecule as well as on the Reynolds number. An organism moving at low Reynolds numbers is surrounded by a viscous boundary layer that moves together with the organism and through which nutrient molecules have to diffuse. At Reynolds number 0 – obviously an idealized situation – the thickness of the boundary layer is infinite, but the higher the Reynolds number, the thinner the boundary layer and, hence, the steeper the concentration gradients and the higher the Sherwood number. Theoretical studies based on calculations of mass transport to a sinking sphere suggest that for unicellular micro-organisms smaller than about 10–100 μm and swimming at typical speeds, the Sherwood number is very close to 1 and the enhancement thus negligible [2, 8]. However, real organisms are self-propelled and are not moved by a body force (like gravity), i.e., they swim by means of cilia or flagella. The general effect of self-propulsion is that streamlines moves closer to the cell body compared to those for a sinking cell and thus further increases concentration gradients and nutrient fluxes [21]. Several theoretical models, such as the “squirmer” model3 ([22–24], CFD models [25], or other idealized models [21] have been developed to examine that effect and have shown that Sherwood numbers may be significant even for rather small organism, e.g., a 25% enhancement for ap ) is 1–10 μm, i.e., similar to that for chemical detection. This size threshold is consistent with observed size threshold in rheotactic copepods [35]. From the above it follows that prey smaller than 1–10 μm cannot be perceived and captured individually. The encounter with small prey therefore rely on direct interception (e.g., filtration) or some other automated process.
4
A stresslet consists of two oppositely directed forces of equal magnitude, working at two points in the fluid that are separated by a short distance. This model is often used to describe a steadily cruising micro swimmer.
680
The European Physical Journal Special Topics
C
B
A
20 µm
25 µm
5µm
Fig. 8. Unicellular zooplankton with different positions of the propulsion apparatus: the ciliate Mesodinium rubrum has cilia arranged in a band around the equator of the cell (A); the flagellate Fibrocapsa japonica has a single flagellum that pulls the cell through the water (B); and the flagellate Ochromonas moestupii also has a single flagellum that pushes the cell through the water (C). Image courtesy Lasse Tor Nielsen.
5 Eat, not eaten We noted above the design conflict between requirements for efficient swimming and requirements for efficient acquisition of prey. There is yet another important constraint posed by the risk of being perceived by a predator: the fluid disturbance that is an unavoidable implication of feeding and swimming may signal the presence of the organism to its rheotactic predators, as described above from point of view of the consumer. However, any consumer is also a prey to larger predators, and is therefore sandwiched between the need to eat and the need to not being eaten. The extension of the fluid signal generated by a moving or feeding zooplankton is an estimate of the predator encounter cross section, and is therefore proportional to the risk of encountering a rheotactic predator. The spatial attenuation of a fluid signal and, hence, its spatial extension depends strongly on both the kinematics of the force production as well as on the position of the propulsion apparatus on the body of the organism (Fig. 8). For example, propulsion forces positioned near the “equator” of an organism appears to be optimal for quiet propulsion: the flow velocity attenuates with the cube of the distance to the organism and the extension of the flow field is minimized. This has been demonstrated by means of CFD models [25], highly idealized point-force models [44], and experimentally for a variety sub-mm sized zooplankton. This position may also be optimal for propulsion efficiency [25]. Similarly, intermittent impulsive force production, resulting in erratic swimming-byjumping, the generation of viscous vortex rings, and a spatial attenuation scaling with distance to power 4, appears also to be superior to continuous cruising at the same average speed in terms of “noise” production. Again, this has been demonstrated by simple point force models (impulsive stokeslet and stresslet models5 ; [64, 65]), CFD simulations [65] and observations in a variety of zooplankton organisms ([26, 67–69]). The conflict between feeding and predator avoidance becomes evident in a taxatranscending comparison between zooplankton in which swimming and feeding are intimately related with those in which swimming and feeding are separate processes (as in ambush feeders): the former produces a signal with an order of magnitude larger spatial extension (Fig. 9). The swimmers can minimize their disturbance, while the feeders need to examine – and, hence, disturb – a large volume of water in order to acquire food. 5
A stokeslet consists of a force working in a point in the water. The impulsive versions of the stresslet and stokeslet models refer to the forces working “impulsively”, i.e., for a very short time.
log(S0.5 mm/s, mm2)
Cooperative Particles: Patchy Colloids, Active Matter and Nanofluids
3 2 1 0 -1 -2 -3 -4 -5 -6
Active
Ambush -3
-2
-1
0
1
2
log (Reynolds number)
3
681
Podon intermedius Temora longicornis naupl Acatia tonsa naupl Oithona davisae fem Oithona davisae male Acartia tonsa cop Mesodinium rubrum Acatia tonsa cop Metridia longa Dinoflagellates Temora longicornis naupl Brachionus pliciatilis Temora longicornis cop Swimmers Feeders
Fig. 9. The spatial extension (area, S) of imposed fluid velocities exceeding a threshold velocity of 0.5 mm s−1 by a large variety of swimming zooplankton here plotted as a function of the Reynolds number. The different species can be broadly divided among those in which feeding and swimming are intimately related (Active) and those in which swimming and feeding are separate processes (Ambush feeders). Modified from [14].
6 Conclusion A classical discipline in biology and ecology is to relate form with function in order to achieve a mechanistic understanding of why organisms are shaped the way they are and behave as they do. Such insight is necessary to understand how organisms interact through competition and predation and thereby shape population dynamics and community and ecosystem structure and function. As terrestrial animals living in a high Reynolds number world, we can intuitively relate differences in form to differences in function in many of our fellow terrestrial animals: The beating wings of the birds, the jumping legs of a kangaroo, and the undulating body of a snake, etc. While we can observe the microscopic organisms living in the ocean – although only in microscope preparations and not in the complex 3-dimensional environment in which they live – we cannot always understand the relation between form and function intuitively. We can only relate the beating appendages, cilia or flagella of small aquatic organisms to the general purpose of propulsion and feeding, but we need formal physics to understand the implications of differences in design and kinematics. In the past we have made some progress but it has been slow because it has been led either by biologist with insufficient analytical power or by physicists with limited biological insights. By combining the insights of the biologist with the analytical power of the physicist, we may be able to speed up much needed progress, and the main purpose of this tutorial has been to encourage such collaboration. As noted, small organisms account for most of the biogeochemical processes in the oceans including carbon sequestration and thus have major impact on the global climate. They are also the basis for the higher trophic levels in the ocean that support fisheries. All biological process ultimately depend on individual level process, i.e., how individuals interact with one another and with their environment.
References 1. E.M. Purcell, Am. J. Phys. 45, 3 (1977) 2. J.S. Guasto, R. Rusconi, R. Stocker, Ann. Rev. Fluid Mech. 44, 373 (2012)
682 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.
The European Physical Journal Special Topics R. Goldstein, Ann. Rev. Fluid Mech. 47, 343 (2015) D. Tam, A.E. Hosoi, Phys. Rev. Lett. 98, 068105 (2007) T.J. Pedley, J.O. Kessler, Ann. Rev. Fluid Mech. 24, 313 (1992) D.B. Dusenbery, Living at Micro Scale: The Unexpected Physics of Being Small (Princeton University Press, Princeton, 2009) J.A. Raven, J. Plankton Res. 16, 565 (1994) T. Kiørboe, A Mechanistic Approach to Plankton Ecology (Princeton University Press, Princeton, 2008) R.W. Sheldon, W.H. Sutcliffe Jr., M.A. Paranjape, J. Fish. Res. Bd. Can. 34, 2344 (1977) FAO, The State of World Fisheries and Aquaculture (FAO, Rome, 2014) J.H Brown, J.F. Gillooly, A.P. Allen, V.M. Savage, G.B. West, Ecology 85, 1771 (2004) H.S. Carslaw, J.C. Jaeger, Conduction of heat in solids, 2nd edn. (Clarendon Press, Oxford, 1959) H. Berg, Random walks in biology, 2nd edn. (Princeton University Press, Princeton, 1992) M. Simon, Limnol. Oceanogr. 32, 591 (1987) S.W Chisholm in Primary Productivity and Biogeochemical Cycles in the Sea, edited by P.G. Falkowski, A.D. Woodhead, K. Vivirito, Vol. 43 (Environmental Science Research, 1992) T. Kiørboe, Adv. Mar. Biol. 29, 1 (1993) T.F. Thingstad, L. Øvre˚ as, J.K. Egge, T. Løvdal, M. Heldal, Ecol. Lett. 8, 675 (2005) R. Stocker, Science, 338, 628 (2012) J.A. Raven, K. Richardson, New Phytol. 98, 259 (1984) R. Clift, J.R. Grace, M.E. Weber. Bubbles, drops and particles (Academic Press, New York, 1978) V. Langlois, A. Andersen, T. Bohr, A. Visser, T. Kiørboe, Aquat. Microbiol. 54, 35 (2009) V. Magar, T. Goto, T.J. Pedley. Q. J. Mech. Appl. Math. 56, 65 (2003) R.N Bearon, V. Magar, J. Plankton Res. 32, 1599 (2010) D. Tam, A.E. Hosoi, Proc. Natl. Acad. Sci. USA 108, 1001 (2011) H. Jiang, J. Plankton Res. 33, 998 (2011) T. Kiørboe, H. Jiang, R.J. Goncalves, L.T. Nielsen, N. Wadhwa, Proc. Natl. Acad. Sci. USA 111, 11738 (2014) L.T. Nielsen, T. Kiørboe, The ISME Journal 9, 2117 (2015) V. Smetacek, J. Biosci. 37, 589 (2012) T. Kiørboe, Biol. Rev. 86, 311 (2011) T. Fenchel, Adv. Microb. Ecol. 9, 57 (1986) T. Kiørboe, J. Titelman, J. Plankton Res. 20, 1615 (1998) T. Kiørboe, H. Jiang, J. Roy. Soc. Int. 10, 20120693 (2013) T. Kiørboe, A. Andersen, V. Langlois, H.H. Jakobsen, T. Bohr, Proc. Natl. Acad. Sci. 106, 12394 (2009) V. China, R. Holzman, Proc. Natl. Acad. Sci. USA 111, 8083 (2014) E. Bruno, M.B. Andersen, T. Kiørboe, PLosOne 7, e47906 (2012) A.J. Calado, Ø. Moestrup, Phycologia 36, 47 (1997) H.H. Jakobsen, L.M. Everett, S.L. Strom, S. L. Aquat. Microb. Ecol. 44, 197 (2006) T. Fenchel, Protist 152, 329 (2001) J.T. Berdach, J. Phycol. 13, 243 (1977) H. Nguyen, R. Ortiz, R. Cortez, L. Fauci, J. Fluid Mech. 671, 574 (2011) T. Nakai, I. Shihira-Ishikawa, A. Miyawaki, T. Goto. J. Aero Aqua Bio-Mech. 3, 79 (2013) M.A. Sleigh, BioSystems 14, 423 (1981) I. Inouye, T. Hori, Protoplasma 164, 54 (1991) Andersen, N. Wadhwa, T. Kiørboe, Phys. Rev. E 91, 042712 (2015) J.R. Blake, J. Fluid Mech. 46, 199 (1971)
Cooperative Particles: Patchy Colloids, Active Matter and Nanofluids
683
46. K. Drescher, J. Dunkel, L.H. Cisneros, S. Ganguly, R.E. Goldstein, Proc. Natl. Acad. Sci. USA 108, 10940 (2011) 47. K. Drescher, R.E. Goldstein, N. Michel, M. Polin, I. Tuval, Phys. Rev. Lett. 105, 168101 (2010) 48. T. Fenchel, in Flows of Energy and Materials in Marine Ecosystems, edited by M. Fasham (Plenum Press, New York, 1984), p. 301 49. P. Andersen, Mar. Microb. Food Webs 3, 35 (1989) 50. P. Flood, Mar. Biol. 111, 95 (1991) 51. M.A.R. Koehl, R. Strickler, Limnol. Oceanogr. 26, 1062 (1981) 52. E.A. Martens, N. Wadhwa, N.S. Jacobsen, C. Lindemann, K.H. Andersen, A. Visser, Proc. Roy. Soc. B 282 (in press) (2015) 53. P.K. Bjørnsen, Limnol. Oceanogr. 33, 151 (1988) 54. Z.M. Fuzessery, J.J. Childress, Biol. Bull. 149, 522 (1975) 55. F. Lombard, M. Koski, T. Kiørboe, Limnol. Oceanogr. 58, 185 (2013) 56. J.R. Strickler, Science 218, 158 (1982) 57. J.C. Andrews, Can. J. Fish. Aquat. Sci. 40, 1293 (1983) 58. H.H. Jiang, T.R. Osborn, C. Meneveau, J. Plankton Res. 24, 495 (2002) 59. R.J. Gon¸calves, T. Kiørboe, Limnol. Oceanogr. 60, 1286 (2015) 60. T. Kiørboe, A.W. Visser, Mar. Ecol. Prog. Ser. 179, 81 (1999) 61. C. Svensen, T. Kiørboe, J. Plankton Res. 22, 1155 (2000) 62. H.S. Jiang, G.-A. Paffenh¨ ofer, Mar. Ecol. Prog. Ser. 373, 37 (2008) 63. J. Yen, P.H. Lenz, D.V. Gassie, D.K. Hartline, J. Plankton Res. 14, 459 (1992) 64. T. Kiørboe, A. Andersen, V. Langlois, H.H. Jakobsen, J. Roy. Soc. Interface 7, 1591 (2010) 65. H. Jiang, T. Kiørboe, J. Exp. Biol. 214, 476 (2011) 66. H. Jiang, T. Kiørboe, J. Roy. Soc. Interface 8, 1090 (2011) 67. D.W. Murphy, D.R. Webster, J. Yen, Limnol. Oceanogr. Meth. 10, 1096 (2012) 68. K. Katija, S.P. Colin, J.H. Costello, H. Jiang, J. Exp. Biol. 218, 2333 (2015) 69. T. Kiørboe, H. Jiang, S.P. Colin, Proc. Roy. Soc. B. 277, 3229 (2010)
