AMER. ZOOL., 32:566-582 (1992)
A Chemical Engineering View of Cnidarian Symbioses1
MARK R. PATTERSON2
Division of Environmental Studies, University of California, Davis, California 95616
SYNOPSIS. Chemical engineering theory proves useful in predicting selective constraints on algal-cnidarian symbioses. Scleractinian corals are used
as a model for understanding how the architecture of polyps and colonies
affects the transport of material between the host and symbiotic algae,
and the environment. Transport properties of the symbiosis are described
by mathematical functions involving molecular diffusion and forced convection (water motion). Metabolic rates (photosynthesis and aerobic respiration) measured under manipulated regimes of water motion provide
evidence that the physical state of the boundary layer (laminar or turbulent) surrounding the symbiosis directly affects both symbiotic algae
and coral. However, the algae's photosynthetic rate is affected by changes
in the ambient flow regime to a lesser degree than the symbiotic association's aerobic respiration rate, indicating a buffering effect of the host
tissue. Two mathematical models presented explore the relationship of
size and shape of contracted and expanded polyps on maximal rates of
gas or nutrient exchange: the size/shape spectrum of scleractinian polyps
is understandable in terms of how diffusion limits delivery of metabolites
to coral and algae. Polyps of differing size are not geometrically similar;
the shape changes observed are consistent with keeping fluxes of dissolved
substances to these symbiotic associations "diffusionally similar."
Expanded polyps possess diffusive boundary layers of considerable depth
which limit delivery of metabolites to the algae. The mass transfer characteristics and size range of scleractinian polyps lie within the range where
theory predicts an optimal polyp wall thickness should occur.
can survive for awhile without their symbionts but do not appear to grow (Williams
and
Bunkley-Williams, 1990). However,
wnile some
benefits of the association to the
host
scleractinian are easy to demonstrate,
the
benefit to the symbiotic dinoflagellates
(=zooxanthellae), Symbiodinium spp. is not
wel1
demonstrated (Douglas and Smith,
19
89).
Scleractinian-dinoflagellate associations
forming reefs are confined to tropical and
subtropical regions of the Earth's oceans,
where the m e a n annual surface water
temperature remains above 20°C (Kozloff,
1990). Most are colonial in form and usually
• From the Workshop on The Impact of Symbiosis exhMi
contraction of the polyps during
INTRODUCTION
Reef-building scleractinian corals are wellknown examples of symbioses involving
invertebrates as habitats for endosymbiotic
algae (Muscatine and Porter, 1977). Symbioses between cnidaria and algae are traditionally considered to be mutualisms, at
least in the sense of Muscatine and McNeil
(1989, p. 371): "Bilateral exchange of
metabolites, particularly between heterotroph and autotroph, is usually interpreted
as mutually beneficial. . . ." The association
appears obligatory: bleached scleractinians
on Invertebrate Physiology, Ecology, and Evolution
presented at the Centennial Meeting of the American
Society of Zoologists, 27-30 December 1989, at Bost0
"^en a tSe t s S s:Schoo.ofMarineScien Ce ,vi^nia
institute of Marine Science, College of William and
Mary, Gloucester Point, Virginia 23062.
daylight w h e n p h o t o s y n t h e s i s b y t h e Sym-
bionts is at its peak. At night, the expanded
i y p s e n g a g e i n heterotrophic feeding on
zooplankton. The degree to which corals
may depend on autotrophy for their energy
requirements varies considerably, with some
po
566
MASS TRANSFER IN SYMBIOTIC CNIDARIANS
species presumably able to satisfy all their
carbon requirements by algal photosynthesis (McCloskey and Muscatine, 1984). There
is increasing evidence that much of the
translocated photosynthate is unaccounted
for, and may be lost from the association as
mucus (Edmunds and Davies, 1986).
Translocation products may constitute
"junk food" for the host in that holozoic
feeding may be needed to obtain nitrogen
and phosphorus necessary to synthesize new
protein (Steen, 1988).
Biotic reefs are built by symbiotic scleractinians and coralline algae. Photosynthesis by zooxanthellae is thought to be necessary for the accretion of coral limestone
skeletons composing the reef framework.
Zooxanthellae provide the coral with energy
needed to calcify, probably prevent phosphate poisoning of calcification, and enhance
the conditions under which aragonite can
precipitate by stripping carbon dioxide from
the site of calcification (review in Jaap and
Hallock, 1990).
Although in many ways, scleractiniandinoflagellate symbioses are self-sufficient
ecosystems, with tight nutrient cycling
between host and algae (Hallock, 1981),
fluxes of materials to and from the external
environment are also important (D'Elia,
1988; Steen, 1988). How might the physical
environment affect exchange rates of carbon
dioxide, oxygen, and other nutrients important for the growth of the symbiosis? Does
the architecture of the host affect the internal environment experienced by the symbionts?
Fluxes of materials within a symbiotic
coral and between the coral and its environment are complex. The delivery of
metabolites from the external environment
can potentially be modified by water motion.
In a hermatypic scleractinian, carbon fixed
by photosynthesis is translocated from the
zooxanthellae to the coral (Goreau and
Goreau, 1960). Some of the oxygen produced as a photosynthetic byproduct is used
in maintenance respiration by both alga and
coral, but under conditions of high irradiance, there is a net efflux of oxygen from
the symbiosis to the environment (McCloskey et al., 1978). Conversely, at night, oxygen diffuses in from the surrounding sea-
567
water through the boundary layer
surrounding the symbiosis. Carbon dioxide
taken up by photosynthetic symbionts
comes from the host coral as a byproduct
of the coral's respiration, and also from the
external environment under conditions of
enhanced photosynthesis (Muscatine et al.,
1989). Though some nitrogen taken up by
the algae may arise from coral metabolism
(Muscatine and Porter, 1977), symbiotic
corals also take up significant amounts of
inorganic nitrogen from the external environment (Muscatine et al., 1979; D'Elia,
1988). Inasmuch as there are many nitrogen-fixing bacteria in reef systems (Johannes
et al., 1972), phosphate, rather than nitrogen, may actually be the limiting nutrient
affecting growth rate of corals (Littler and
Littler, 1984), although there is growing evidence that zooxanthellae may be nitrogenstarved when occurring at high density in
the host (Cook and D'Elia, 1987). Fixed
nitrogen, phosphate, and other nutrients
necessary for the maintenance of coral tissue are also acquired by mucus feeding
(Lewis and Price, 1975) and particulate
(zooplankton) feeding (Johannes et al.,
1970); these fluxes (capture rates) are also
influenced by the flow of water around the
coral. Yet few studies have investigated
whether water motion influences the metabolic transactions of coral-algal symbioses
(Dennison and Barnes, 1988; Patterson et
al., 1991).
Physiological studies of symbiotic cnidarians have focussed on carbon budgets
(Muscatine et al., 1983; Muscatine et al.,
1984), nitrogen budgets (Zamer and Shick,
1989), energy budgets (Edmunds and
Davies, 1986, 1989), the process of photoadaptation and controls on limiting rates
of primary production (Porter et al., 1984),
the role played by excess oxygen in stimulating production of anti-oxidants to reduce
cellular and genetic damage to host (Dykens
and Shick, 1982) and zooxanthellae (Lesser
and Shick, 1989), and how hyperoxia resulting from symbiont photosynthesis overcomes diffusion limitation through the
boundary layer of cnidarians (Shick, 1990).
All of the above phenomena involve the
transport process of diffusion, either between
animal and plant (molecular diffusion) or
568
MARK R. PATTERSON
between the symbiosis and the environment
(molecular plus turbulent diffusion).
Chemical engineering theory predicts
fluxes of mass, heat, and momentum
between generalized "exchange surfaces"
and moving or still fluids. Mass transport
theory is a subdiscipline of chemical engineering; it was developed to characterize and
predict the rates of chemical reactions in
man-made systems such as fermenters, furnaces, jets, etc. (Brodkey and Hershey, 1988),
but it can be applied to living systems as
well. Industrial operations are concerned
with "scaling-up" prototype chemical reactors to full-scale production reactors: thus
chemical engineers have a longstanding
interest in deriving correlations that allow
predictions of the production rate from
changes in the size, the shape, and the internal mixing rate of the reactor. Analogous
problems concern biologists interested in
symbiotic associations as "reactors" of sorts.
Mass transfer and fluid motion are intimately intertwined, and thus principles of
fluid mechanics are invoked. Thus, this theory overlaps with the domain of biomechanics in analyzing organismic form and
function (Vogel, 1983).
I present some mathematical models
showing how fluxes of dissolved gases and
nutrients in cnidarian symbioses can be
directly affected by the physical environment and the morphology of algal-coral
associations. New avenues of analysis are
illustrated, complementing the approaches
more familiar to physiologists working with
algal/invertebrate mutualisms.
MASS TRANSFER IN SYMBIOTIC
SCLERACTINIANS
Diffusion through boundary layers
Delivery of materials such as carbon
dioxide (as bicarbonate ion), oxygen, dissolved organic matter, and nitrogenous
compounds, from the environment to a
symbiotic scleractinian occurs through the
combined processes of diffusion and convection (water motion). Scleractinians can
be viewed, to a first approximation, as having a planar geometry in terms of exchange
with the environment, i.e., they can be considered sheets of tissue wrapped over an
aragonitic skeleton. Thus, the dynamics of
one-dimensional diffusion are appropriate
for calculations, especially of contracted
polyps.
The diffusive pathway into a sessile
aquatic invertebrate includes part of the
momentum boundary layer (Patterson and
Sebens, 1989), that is, the layer of water
over the organism where the flow speed is
changing from zero at the animal surface to
99% of some "freestream" value obtained
some distance above the animal (Vogel,
1983).
That section of the momentum boundary
layer where the concentration of a dissolved
substance is increasing or decreasing from
the exchange surface {i.e., the scleractinianseawater interface) is defined as the diffusive, or mass transfer, boundary layer
(White, 1988). Its properties bear some further investigation.
In seawater, the momentum boundary
layer and diffusive boundary layer are not
of identical thickness. The dimensionless
index, Schmidt number (Sc) describes the
ratio of the diffusivity of momentum in a
fluid to the diffusivity of a dissolved substance (Brodkey and Hershey, 1988). Sc =
y/D, where v = the kinematic viscosity of
the seawater (ca. 10~2 cmVsec), and D is
the diffusion coefficient of the dissolved species (ca. 2 x 1O~5 cmVsec for most small
molecules and ions). Sc does not vary greatly
for small molecules or ions such as bicarbonate, dissolved oxygen, calcium, ammonium, etc.; it is approximately 500 for all
these cases. For metabolically important
molecules and ions dissolved in the surrounding seawater (or in the coelenteron
fluid), the diffusive boundary layer surrounding the symbiosis (dd) is related to the
momentum boundary layer (dm), by dd/dm
= Sc~'/j). Thus, the ratio of momentum to
diffusive boundary layer thickness around
corals is approximately eight (5001/j).
Flow regime
Because of its effects on the thickness of
the momentum and diffusive boundary layers, water motion near the surface of a symbiotic scleractinian can affect metabolism
by altering the delivery rate of molecules
(e.g., oxygen or bicarbonate ion). The relationship of the diffusive boundary layer
thickness to changes in size, shape, or flow
569
MASS TRANSFER IN SYMBIOTIC CNIDARIANS
speed depends on the kinds of flow (hydraulically smooth, rough or transitional in
nature; Jumars and Nowell, 1984). In
hydraulically smooth flow, a layer of water,
the laminar sublayer lies immediately adjacent to the scleractinian's surface. In this
region, turbulent eddies are damped, and
there is a linear gradient in the flow speed
as a function of height above the coral. In
a transitional flow, this laminar sublayer
exists intermittently, both spatially and
temporally, above the coral surface. In rough
flow, there is no organized laminar sublayer,
and turbulent eddies penetrate all the way
to the surface (Denny, 1988), causing
extreme changes in flow speed and direction
(Praturi and Brodkey, 1978), and greatly
increasing the transport rate of dissolved
materials (Okubo, 1980), compared to a
smooth flow.
Diffusion mechanics
Any decrease in the diffusive path length
from the environment into the cnidarian
symbiosis has two important consequences:
(i) it increases the dissolved species flux
(FJ by linearly increasing the concentration
gradient
Fick's first law: Fx = - D —
dx
where D = the diffusion coefficient (cm2/
sec), C = concentration (moles/liter), and x
= distance coordinate perpendicular to the
tissue (cm), and (ii) it parabolically decreases
the time (t) needed to equilibrate changes
in concentration
ir
i.
A^
d C
T^2C
Fick s second law: — = D —-2
dt
dx
resulting from changes in the aquatic environment or within the organism (Berg,
1983).
Diffusive and momentum boundary layer
thicknesses also vary as an inverse power
function of the flood speed; the form of the
function depends on whether the boundary
layer is laminar or turbulent (Schlichting,
1979), and this in turn depends on the
Reynolds number (Vogel, 1983). Re is the
ratio of inertial to viscous forces acting on
the fluid as it moves past the organism, and
hence it serves as an index of the gross character of the flow past a symbiotic coral
(Schlichting, 1979). For scleractinians,
Reynolds number (Re) is calculated from
the flow speed (U; cm/sec), colony head
width or polyp diameter (W; cm), and
momentum diffusivity (kinematic viscosity) of the fluid (c; cm Vsec):
Re =
UW
(eq. 1)
Diffusive boundary layer thickness over
some benthic substrates and organisms can
be on the order of 103—104 nm in thickness
(Revsbech and Jergensen, 1986; Patterson,
unpublished data), which is equal to or
greater than tissue thickness (gastroderm +
mesogleal layer + ectoderm) for most scleractinian species. If the diffusive boundary
layer thickness is a substantial part of the
total distance for diffusion, it is easy to see
how changes in the thickness of the diffusive
boundary layer can alter the rate at which
material is exchanged between the symbiotic coral and the environment. The gradient, dC/dx, changes with dx, and fluxes
into the symbiosis are altered accordingly.
Effects of colony size and shape on
mass transfer
Is it possible to predict a priori how
changes in colony size or shape might affect
metabolic rates of host and symbiont? Two
dimensionless indices relevant to size and
shape in mass transfer studies are the Sherwood number (Sh), which can be used as a
measure of the metabolic rate (e.g., respiration rate, photosynthetic rate) of the symbiotic scleractinian, and the Reynolds number, a measure of the strength of water
motion past the symbiosis.
Sherwood number is the ratio of the measured mass flux (moles cm" 2 sec~') assisted
by fluid motion (Fconv, eq. 2) to the average
flux (moles cm" 2 sec~') that would occur if
molecular diffusion through a diffusive
boundary layer proportional to organism
size (Fdiff, eq. 3) was the sole mechanism for
mass transport (Brodkey and Hershey,
1988). Since Sherwood number is a flux
divided by a flux, it is dimensionless (eq.
4). Sherwood number can be thought of as
the mass transfer coefficient (hm) made
dimensionless. hm (cm/sec) is determined
570
MARK R. PATTERSON
empirically from the ratio of the mass flux and Re is used to investigate how the overall
per unit area assisted by convection (¥„,„„) geometry of the symbiosis affects the degree
to the dissolved nutrient or gas concentra- to which convective loss or gain of material
tion difference (eq. 2) between the environ- augments diffusive fluxes (Brodkey and
ment (Ce) and the site of metabolism inside Hershey, 1988). In other words, for a given
the symbiosis (Q). The value of hm will increase in flow speed and/or colony size,
change with the flow speed around a coral how much increase in flux of the dissolved
colony, as well as the size and shape of the substance into the symbiosis occurs? For
colony. In general, hm is positively corre- constant kinematic viscosity, v, and conlated with flow speed, negatively correlated stant diffusion coefficient, D, the relationwith size, and indeterminately correlated ship between the two dimensionless variables can be simply given as
with shape changes (White, 1988).
Fconv = -h m (C e - Q)
-D(C e - Q)
Sh =
hm<9d(Ce - Q) _ hmdd
D(Ce - Q)
D
(eq. 2)
Sh = a(Re)b
(eq. 3)
This formula provides insight into the
qualitative and quantitative nature of a given
problem in mass transfer, independent of
the effects of size or initial concentrations
of dissolved substances in the organism or
the environment. If b = 0.5, metabolic rate
(photosynthesis or respiration) increases at
a rate identical to the rate of decrease in
laminar boundary layer thickness around
the symbiosis. If b > %, then metabolic
rates of the symbiosis increase at a rate identical to the rate of decrease in boundary layer
thickness seen in a turbulent boundary layer
(White, 1988).
Turbulent boundary layers are absolutely
thicker than laminar boundary layers by the
99% definition rule (Denny, 1988). We
might be tempted to conclude erroneously
that the flux of metabolites through a turbulent boundary layer around a scleractinian-algal association will thus be lower than
the laminar case since the concentration
gradient is now less steep (Fick's first law).
However, in a turbulent boundary layer,
most of the change in flow speed and dissolved substance concentration will occur
very close to the surface and thus the flux
of metabolites can be much higher than the
laminar case.
The degree of local turbulence will change
with location around a cnidarian colony and
this can locally alter the metabolic rate of
individual polyps. In octocorals, flow patterns at the level of the polyp are subject to
drastic change with increasing Re (Patterson, 1991a) and similar patterns are
observed for scleractinians (W. S. Price, personal communication). Thus the physical
(eq. 4)
In practice, the diffusive boundary layer
thickness, dd, is not used to compute Sh; the
characteristic dimension of the symbiosis
(coral head or polyp diameter, W) is used
instead. This is an accepted practice inasmuch as the boundary layer thickness (dd)
is proportional to the size of an object in a
moving fluid (White, 1988). Thus, a working expression for Sherwood number
becomes:
Sh =
hmW
D
(eq. 5)
For Sh calculations, it is helpful to know
the concentration of the dissolved material
at the site of metabolism, Q. For a given
organism, this information may be available from the literature, or it may be possible to measure it directly, e.g., oxygen
microelectrode measurements inside coral
tissue are possible (R. Carlton, personal
communication; Patterson, unpublished
data). Ce is usually much easier to determine
experimentally (by sampling the water
around the organism), and often does not
vary much during a given experiment,
although it can vary between experiments,
and thus have a direct effect on the "driving
pressure" for diffusion. Using this approach,
it is not necessary to know the distance over
which the concentrations are changing.
The functional relationship between Sh
(eq. 6)
571
MASS TRANSFER IN SYMBIOTIC CNIDARIANS
thickness of the diffusive boundary layer
over scleractinian surfaces will change as the
flow increases, but to a differing degree,
depending on exact location on the colony
and the local direction of the ambient flow.
However, in metabolic chamber measurements, since Sh is calculated for the entire
colony rather than small patches of surface
tissue, the Sh/Re relationship will indicate
the average behavior of all exchange sites
involved in mass transfer.
Using Sh, rather than metabolic rate, to
examine theflowdependent behavior nicely
takes into account changes in the "driving
pressure" for diffusion present during an
experiment (eq. 2). For example, it is known
that the amount of dissolved oxygen available in the surrounding seawater can strongly
affect respiration in invertebrates (cf. Sebens,
1987; Shick, 1990), yet many investigators
do not report the initial oxygen concentrations present when determining respiration
rate. Sh/Re plots offer a convenient way to
graph simultaneously data from many
experiments which may have used organisms of different size, different initial concentration of metabolite, or different flow
speeds on one graph.
Figure 1 shows Sh/Re relationships for
oxygen transfer during dark respiration and
maximum photosynthesis in the symbiotic
scleractinian Montastrea annularis (Patterson et ai, 1991). It is instructive to investigate the effect of shape on the transfer process: included are respiration rates in two
cnidarians without symbionts with different
shapes, the actiniarian Metridium senile and
the alcyonarian Alcyonium siderium (Patterson and Sebens, 1989) and theoretical
curves for oxygen uptake by a flat plate (data
of Pohlhausen in Brodkey and Hershey,
1988; Holman, 1986), a sphere (Whitaker,
1972), and a right circular cylinder (Zukauskas and Ziugzda, 1985). The metabolic data
were collected using a microcomputer-controlled recirculating metabolism chamber
that is capable of generating a uni-directional flow of known speed and turbulence
intensity (described in Patterson and Sebens,
1989; Patterson *tf ai, 1991).
The slope of the plots for the geometric
shapes is about 0.5, until a significant degree
of flow separation has occurred in the wake
4
3.6
3
Plate
2.5
©
2
1.5
1
0.5
1
2
3
4
5
6
Log Re
FIG. 1. Sherwood (Sh) number/Reynolds (Re) number plots for mass transfer of oxygen in a symbiotic
scleractinian (Montastrea annularis; dark respiration
and maximum photosynthesis), a non-symbiotic sea
anemone (Metridium senile), a non-symbiotic octocoral (Alcyonium siderium), and theoretical uptake
curves for three geometries (plate, cylinder, sphere)
derived from the engineering literature.
of the objects (sphere, cylinder), or the surface boundary layer has become completely
turbulent (flat plate). Then the slope of the
Sh/Re increases, indicating that further
increases in flow speed are having a greater
effect on enhancing the delivery of dissolved
material through the boundary layer. This
breakpoint in mass transfer behavior generally occurs at Re « 105 where the boundary layer usually becomes completely turbulent for these shapes.
In contrast, the slopes for mass transfer
of oxygen during respiration in the three
cnidarians are considerably higher over the
same range of Re. For the scleractinian colonies tested, this convection-enhanced mass
transport is related to their morphology. The
heads used in these experiments were hemispherical to globular in shape. The exponent
(b) in the allometric equation is significantly
greater than 0.5 (2-tailed Mest; P < 0.05),
indicating that mass transfer is governed by
a turbulent boundary layer with the size of
the wake and the location of the separation
point both important in determining the
mass transfer as a function of flow speed in
this species. The boundary layer over the
572
MARK R. PATTERSON
colony becomes turbulent at a lower Re than
would be expected for a spherical shape,
since the surface of colony is covered by
polyps which promote a transition to a turbulent boundary layer. This turbulent
boundary layer allows the transport of
nutrients and dissolved gases to increase at
a faster rate as the flow speed increases (and
the boundary layer becomes thinner), than
if the boundary layer had remained laminar.
A laboratory study investigating the effect
of stirred and unstirred conditions on the
photosynthesis/irradiance curve for an
acroporid also found increased primary
production and respiration under stirred
conditions (Dennison and Barnes, 1988).
The effect of morphology on how quickly
changes in flow speed or size can augment
respiration rates are shown by the data for
two other cnidarians. While not symbiotic
species, the results are probably applicable
to symbiotic species with similar morphologies. Metridium has a finely divided tentacular crown located far from the substrate
(Shick et ai, 1979). Alcyonium, a colonial
form, is studded with polyps (ca. 1 cm
height) that greatly retard flow from the surface of the colony (Patterson, 1984, 19916).
Colonies used were ellipsoidal. The allometric exponent relating Sh to Re in Metridium is significantly greater than 0.5 (2-tailed
Mest; P < 0.05). In contrast, exchange of
gas in Alcyonium conforms with laminar
boundary layer theory since the exponent is
not statistically larger than 0.5 (although the
mean value is 0.92); the reduced flow in the
layer of water lowing in and around the polyps results in a functionally thicker diffusive
boundary layer than in Metridium at comparable Re. Note both species had calculated slopes that were higher than slopes
measured empirically for the analogous
property of heat transfer in similar geometric shapes (Fig. 1).
If we assess the relative ranking of transport ability, for the same biomass arranged
in different geometries (plate, cylinder,
sphere) at an Re large enough for the boundary layer to be completely turbulent (e.g.,
Re > 5 x 10s, we find that the cylinder >
sphere > flat plate. The ratio of transport
abilities is about 2.2 to 2.1 to 1.0, respectively. Under laminar boundary layer con-
ditions (e.g., Re = 103), the situation reverses
itself, and the relative ranking is sphere >
(cylinder and flat plate). The flat plate and
cylinder have almost equal Sh, and the ratio
of transport abilities is about 1.5 to 1.0 to
1.0. The implications for a direct coupling
between morphology and gas or nutrient
transfer ability are profound, and are only
beginning to be examined by coral physiologists.
A study on gas exchange in the scleractinian Pocillopora damicornis found that
both maximum photosynthetic rate and respiration rate positively correlated with size
(Jokiel and Morrissey, 1986). This supports
the above mass transfer arguments, since
increased size will lead to increased Re, all
else being equal, and the authors were careful to keep the relative degree of water
motion similar in their treatments. However, no studies have as yet examined the
implications of colony shape per se in affecting the physiological rates at which symbiotic corals can fix carbon and hence grow
and compete for space on a coral reef. The
local microenvironment should play an
important role in determining the success
of different morphologies, since the degree
to which the local boundary layer over the
coral is laminar or turbulent will determine
maximal rates of gas and nutrient exchange.
The interaction of size and shape of symbiotic scleractinians with flows in various
microhabitats can thus affect the productivity of coral reefs (Patterson et ai, 1991).
Available evidence for two species of
symbiotic scleractinian strongly suggests that
oxygen limitation occurs at night (Patterson
et ai, 1991; Dennison and Barnes, 1988).
Oxygen limitation probably also occurs in
symbiotic sea anemones (Fredericks, 1976;
cf. Steen, 1988). Symbiotic sea anemones
which become anoxic can generally undergo
anaerobiosis but at greater risk of depleting
valuable energy reserves (Steen, 1988) which
may reduce fitness (Sebens, 1981).
An interesting exception to the above pattern of augmented, turbulent boundary
layer-like transfer, is the curve for photosynthesis in Montastrea. The slope of the
Sh/Re plot (0.63) is statistically indistinguishable from 0.5 (r-test; P < 0.05). Because
the zooxanthellae are found only in the
MASS TRANSFER IN SYMBIOTIC CNIDARIANS
endoderm, and are surrounded by coral tissue which is producing carbon dioxide
(which equilibrates to bicarbonate ion
quickly), less sensitivity to forced convection during maximum rates of photosynthesis is consistent with the mass transfer
analysis. The diffusive path includes many
tissue layers and carbon is available from
the host (short diffusive path length). At high
levels of productivity, which can occur in
well-lit subtidal environments on coral reefs,
it does appear necessary for bicarbonate ion
to be drawn from sources external to the
symbiosis, leading to the probable build-up
of a significant diffusive boundary layer
around the coral (Muscatine et al, 1989).
Hence the primary production of an
intracellular photosynthetic symbiont can
be affected by the overall shape of the host
in which it is embedded. For symbiotic
associations, the interaction described above
between flow regime, colony shape, and size
in affecting diffusive transport becomes
extremely relevant during both day (when
maximum photosynthetic rate may be limited) and night (when aerobic respiration
rate can decrease potentially to the point
that anaerobiosis may occur). Overall form
of the scleractinian is important to the symbiont in that it determines probability of
the boundary layer becoming turbulent for
a given flow regime and size, which will
augment the delivery of dissolved materials
to the symbionts. Increased colony size leads
to increased Re for a given flow speed, which
will shift the mass transfer process (aerobic
respiration, photosynthesis) further up the
Sh/Re curve. Is it possible that the form of
individual polyps composing a symbiotic
scleractinian colony can also affect the
delivery of dissolved substances to symbiont as well as host?
MASS TRANSFER AND SCLERACTINIAN
POLYP MORPHOLOGY
Contracted polyps
Polyps are the major site of zooxanthellar
populations. In most species the polyps are
contracted during the hours when photosynthesis occurs. At night, polyps expand to
feed on zooplankton, thus changing their
mass transfer characteristics. Can the diver-
573
sity of polyp size and shape seen in scleractinians be explained by examining diffusion mechanics?
Porter (1976) proposed that scleractinian
species have polyps of different sizes because
even though most species possess symbiotic
algae, they nonetheless differ in their degree
of reliance on autotrophy for supply of
organic carbon. Porter interpreted largepolyped species with mounding morphologies living in deeper water {e.g., Montastrea cavernosa) as plankton-feeding
specialists and small-polyped species with
a branching form {e.g., Acropora cervicornis) as specialized for the capture of photons. The crux of Porter's argument rested
on an examination of surface area/volume
ratios for different species. Species with small
polyps living in areas where light is saturating the photosynthetic machinery of the
symbionts have a high surface area/volume
(S/V) ratio; this morphological attribute
should maximize the efficiency of light capture (defined as irradiance intercepted per
unit volume). Larger-polyped species have
smaller S/V ratios. Inasmuch as plankton
are brought to the scleractinian colony by
water currents coming from a single direction at a time, the projected surface area (the
product of the height and width of the polyp)
normal to the flow is more important than
the S/V ratio in determining encounter rate
with prey items. Projected surface area will
be absolutely larger for a bigger polyp compared to a smaller polyp. Hence largerpolyped species will be better at capturing
zooplanktonic prey.
Porter's (1976) hypothesis makes intuitive sense as a way of organizing observations of polyp size, though there has been
little direct test of it (D'Elia, 1988). A potential drawback to the use of the S/V ratio in
analyzing morphological features is that S/V
is not dimensionless, and thus the effects of
shape are confounded with size (Vogel,
1983). Both large- and small-polyped species harbor zooxanthellae at similar concentrations in the endoderm layer (Sebens,
1987). Given that diverse scleractinians
have comparable rates of nighttime respiration and maximum photosynthetic rates,
I ask: Is the internal diffusional environment experienced by the zooxanthellae in
574
MARK R. PATTERSON
Araeonitic skeleton?
FIG. 2. Diagrammatic section of a portion of a generalized scleractinian colony. Boxes below polyps indicate
previous growth stages. Wavy lines indicate seawater in coelenteron of polyp. (A) Schematic of the geometry
used to model diffusion into a contracted polyp. Note that coelenteron is filled with tissue in the contracted
state, x is the distance coordinate into the polyp. (B) Expanded polyp showing the radial coordinate (r) and
dimensions used in the model given in the text. Coelenteron is now filled with a larger volume of seawater.
some sense approximately the same between
species? How might changes in size and
shape in coral polyps be effected to keep the
diffusion rate for carbon dioxide (in the form
of bicarbonate), oxygen, free amino acids,
etc., similar?
To a first approximation, a contracted
polyp is similar to a partitioned (septate)
cylindrical pore (Fig. 2A). The contracted
polyp is largely surrounded by an aragonitic
skeleton which is several orders of magnitude less permeable to dissolved oxygen and
carbon dioxide than the overlying water and
tissue (Constantz, 1986). Materials diffusing
into a contracted polyp encounter a tissue
and water-filled tube. The solution for the
diffusive flux (FJ into this tubular geometry
is straightforward:
F =
- c D dM
(1 - M) dx
(eq. 7)
where Fx = flux of the dissolved species
(moles cm" 2 sec-')> c = the molar concentration of the dissolved species outside the
polyp (moles/ml), M = the local molar fraction of the dissolved gas (dimensionless), D
= the binary diffusion coefficient assumed
to be similar for water and tissue (cmVsec),
and x = the direction coordinate into the
contracted polyp (cm).
Imagine that the polyp has achieved a
steady state concentration difference inside
and outside the polyp. Conservation of mass
of the diffusing material requires that over
an infinitesimal depth in the polyp, dx, that
SFx(x) - SFx(x + dx) = 0 (eq. 8)
where S = the cross-sectional area of the
polyp (cm2). Dividing through by Sdx and
taking the limit, gives a differential equation:
575
MASS TRANSFER IN SYMBIOTIC CNIDARIANS
dx
= 0.
(eq. 9)
Substituting eq. 7 into eq. 9, yields:
cD dM
dx|_(l - M) dx
Taking c and D to be constants and integrating once with respect to the distance into
the polyp, x, produces:
1
dM
Separating variables and integrating again
yields:
-ln(l - M) = k,x + k2 (eq. 12)
The boundary conditions at the ends of the
polyp are:
M = M, at x = x,, near the
bottom of the polyp.
(eq. 13)
M = M2 at x = x2, near the
top of the polyp.
(eq. 14)
Solving for the constants of integration
yields:
- M.) - ln(l - M2)
x, - x.
~
7
t 6.5
.2 5.5-1
0.2
O
0.4
0.6
0 . 8
Distance into polyp (x/L)
FIG. 3. Diffusion into a contracted coral polyp.
Numerical solution of the mathematical model given
as a function of dimensionless depth (x/L) into the
polyp, for oxygen concentrations typical of those found
on a coral reef and inside the polyp (Patterson, unpublished observations). Note the curvilinear nature of the
solution, even though the mass flux down the polyp
remains constant.
diffusion of a substance (in this case, dissolved oxygen) inside a contracted scleractinian polyp. While polyps are usually
expanded at night, contraction does occur
periodically (Sweeney, 1976) and in any case
the form of the curve will be qualitatively
similar for fluxes of other dissolved substances. During the day, the flux of oxygen
will be in the opposite direction due to photosynthesis, but the form of the solution will
(eq. 15) be identical. What is important to note about
the steady-state solution is that the slope of
- M2) - x2ln(l - M.)
the
solution (dM/dx) is not constant, as one
k,=
proceeds deeper inside the polyp; however,
x, - x,
the molar flux, Fx, per unit cross-sectional
(eq. 16) area is constant. The total mass transfer rate
in moles (T) is just:
where x2 — x, = L = polyp contracted depth.
Substituting the constants k,, k2 into eq.
(eq. 19)
T = F,S.
12 gives the solution to the flux equation
given above in eq. 7. The local molar conNoting that Fx oc l/L, it is clear molar
centration of the dissolved substance in the mass transfer into (or out oj) the polyp is
polyp (M) into the coelenteron is thus:
proportional to the ratio S/L, not the absolute dimensions per se. Contracted polyps
from different species possessing the same
value for S/L can be thought of as being
diffusively similar. Diffusively similar polThe flux FK (the metabolic uptake or loss yps will have comparable steady-state
rate) is obtained by substituting k, into eq. transfer characteristics for dissolved sub11 and then substituting eq. 11 into eq. 7: stances to the host tissue and symbionts.
Are symbiotic scleractinian polyps diffusively similar?
Figure 4 gives the calculated surface area/
Figure 3 gives a steady-state solution for volume curve for a size series of diffusively
576
MARK R. PATTERSON
1
2
3
4
Polyp radius (mm)
FIG. 4. Surface area/volume ratio plot vs. polyp radius
for polyps with a constant ratio of cross-sectional area
(S)/length (L), showing Porter's (1976) data and upper
95% confidence intervals for 43 species of Caribbean
corals. The model proposed in the text (diffusive similarity) falls within the error of the data, as does a
geometrically similar series of polyps (not shown). Distinguishing between geometric similarity and diffusive
similarity can only be done using the axes shown in
Figure 5.
similar polyps along with data for 43 Caribbean coral species (Porter, 1976). Diffusive
similarity falls within the error of Porter's
measurements. However, a curve for geometric similarity as polyp size increases also
falls within the 95% confidence interval. To
reject geometric similarity, it is necessary to
use the axes shown in Figure 5, which exploit
the predictions of the above model.
Figure 5 shows the expected morphological scaling for a diffusively and geometrically similar series of polyps. The theoretical curve for the diffusively similar set
assumes that the polyp cross-sectional area/
polyp height equals 7.24, the mean for ten
species (Acropora cervicornis, A. palmata,
Montastrea annularis, M. cavernosa, Porites
astreoides, P. porites, Madracis decactis,
Dichocoenia stokesii, Tubastrea coccinea,
and Eusmilia fastigiata) that were studied
extensively during a photographic survey
conducted from the NOAA Aquarius underwater habitat in St. Croix, U.S. Virgin
Islands. Measurements were taken from
expanded polyps as a proxy for contracted
polyp dimensions. Contracted and expanded
polyp dimensions show good correlation (W.
S. Price, personal communication), but using
expanded dimensions avoids destructive
sampling which is discouraged at the Aquarius site. The data indicate that geometric
scaling has not occurred in these species.
Polyp height (mm)
FIG. 5. Polyp height (a proxy for retracted polyp depth)
vs. polyp radius/height for 10 species of Caribbean corals (104 polyps, 8-11 polyps/species), along with the
predicted curve for a geometrically similar series of
polyps (horizontal line), and the theoretical curve
expected by the diffusive similarity model for the mean
polyp cross-sectional area (S)/polyp height (L) for all
species measured (S/L = 7.24).
Polyp shape appears constrained by the
necessity of achieving comparable delivery
of diffusing substances to the symbiosis in
polyps of differing size.
Expanded polyps
In an expanded polyp, two important
constraints are removed which make the
mass transfer analysis more complicated.
Diffusion of dissolved gases and nutrients
occurs laterally through the walls of the
cylindrical polyp and the tentacles, which
are now exposed more directly to moving
seawater.
Within the past decade, micro-sensor
techniques have been developed which allow
measurements of very small gradients of
dissolved gases using miniature polarographic or galvanic probes (Revsbech and
Jergensen, 1986). While most measurements of oxygen gradients have been made
in the laboratory, some attempts to use these
devices in situ have succeeded. Figure 6
shows the diffusional boundary layer for
oxygen measured in situ immediately adjacent to a scleractinian colony (Diploria clivosa), at night, in a gentle (3 cm sec~' maximum speed) bidirectional current.
The gradient of oxygen concentration is
linear in the laminar sublayer immediately
adjacent to the respiring coral surface. As
the probe is moved past the level of the
tentacles, which sprout from the coelen-
577
MASS TRANSFER IN SYMBIOTIC CNIDARIANS
teron openings in the grooves of the exoskeleton, forming a "canopy" of sorts, the
oxygen gradient shows a dip and then a sharp
increase, rising to mainstream values typical of the water over the reef quite a distance
above the colony. Thus the diffusive boundary layer for oxygen concentration around
D. clivosa is quite large compared to tissue
thickness. Thinning of the diffusive boundary thickness by increased flow can occur
which in turn alters the aerobic respiration
rate of the symbiosis at night. Are there
morphological attributes of expanded scleractinian polyps which might enhance the
diffusive delivery of molecules such as oxygen?
Again, to a first approximation, I consider
an expanded polyp to be a cylinder (Fig. 2B).
Fick's second equation for diffusion in a cylinder, using cylindrical polar coordinates (r,
0, z) is:
dt
x\dr
dr
3d r 30
3.75
c
en
2.25
2000
4000
6000
Height (microns)
8000
FIG. 6. Oxygen microelectrode measurements of the
diffusional boundary layer collected in situ over a
roughly hemispherical colony of the reef coral Diploria
clivosa in an oscillatory flow, where the maximum Re
(based on colony diameter) = 3.7 x 103. Probe transect
was run perpendicular to a groove. Arrow indicates the
level of the bottom of tentacle clusters. Depth = 16 m
adjacent to the NOAA Aquarius mobile saturation
facility, Salt River Canyon, St. Croix, U.S. Virgin
Islands.
with the constants of integration, K and K',
determined from the boundary conditions
at r = i and r = o. Let C(o) = Ce and C(i) =
Q. The specific solution is thus:
where D is the diffusion coefficient (cm2/
sec), C is the local concentration of the diffusing species (moles/ml), and t is time (sec).
I assume that diffusion everywhere into
the cylinder is predominantly radial, i.e., the
polyp cylinder is much longer than it is wide
(by at least a factor of 5), a condition met
by some, but not all scleractinian species.
Now the diffusion equation becomes much
simpler, since only r and t are the relevant
variables:
—=- - r D —
.
a
at
r [dr[
-"
4.00
v(eq.
M
21)
'
The polyp has an inner and an outer wall
radius of i and o, respectively. The steadystate condition for diffusive flux is thus,
A r ^ 4 = 0, with i < r < o. (eq. 22)
dr I drj
Integrating, the general solution of this
equation is then:
C(r) = K + K' In r,
(eq.23)
C(r) =
Q ln(o/r) + Ce
(eq. 24)
Consider the situation for oxygen diffusing into an expanded polyp at night. For
values of o/i = 2 or less (similar to that of
many scleractinian polyps), and assuming
that the concentration of the oxygen is less
in the coelenteron, this function predicts an
almost linear gradient of oxygen concentration in the wall of the polyp (Fig. 7 A).
Now if there is a gradient of diffusing substance, such as oxygen, immediately adjacent to the outer layer of the ectodermis (as
the data in Figure 6 seem to indicate), with
the concentration just inside the surface of
the outer layer of the polyp wall (the ectodermis) given by Ce, then the boundary condition is best described as:
^
= h m (Q - C), at r = o,
(eq. 25)
with constant of proportionality given by
hm (cm"1). hm is similar (but not identical
578
MARK R. PATTERSON
A.
C(r) =
hmoln(o/r)} + hmoCe
l+h m oln(r/i)
(eq. 26)
The diffusion current {moles (M) of oxygen diffusing per unit length (dl) of the polyp
per unit time (dt)} is found by differentiating
the previous equation with repsect to r, and
multiplying by - D :
1.0
1.2
M(dldt)- 1
27rh m 0
(eq. 27)
hmo
The factor 2ir appears since results are
expressed per unit length of polyp, rather
than per unit surface area.
B.
I have plotted this function in dimen0.60
sionless form in Figure 7B, assuming ihm =
0.5. The curve has a maximum of o/i = 2.
2 0.58"
This is a very non-intuitive result, since it
predicts that below a certain size, making a
polyp's walls (gastrodermis + mesogleal
layer + ectodermis) thicker will actually
increase the diffusion current to (or from)
the polyp. Above the critical size, determined by the local maximum of the func0.48
tion, the diffusion current will drop, i.e.,
adding "insulation" by making the walls
FIG. 7. (A) Steady state oxygen concentration (y-axis) thicker will decrease the diffusion current as
through the polyp wall (x-axis), predicted by the model our intuition tells us it should.
for diffusion in an expanded coral polyp and plotted
If however, ihm is greater than 1.0, it can
in dimensionless form. Note quasi-linearity of concentration profile. (B) Effect of changing wall thickness (x- be seen by differentiating eq. 27 with respect
axis) on the diffusion current to or from an expanded to o that the sign of the expression is always
coral polyp (y-axis) plotted in dimensionless form; note
that an optimum ratio of inner to outer wall thickness negative, i.e., there is no local maximum.
exists for maximum exchange. See text for explanation This means making thicker walls always
of symbols.
lowers the diffusion current. The implications of the model results are thus two-fold:
(1) Are real polyps characterized by ihm <
in this formulation) to the mass transfer 1 and thus subject to a solution possessing
coefficient, which appeared above in our a maximum (Fig. 7B)? (2) If so, has selection
considerations of the effects of flow on acted on scleractinian polyp dimensions to
metabolism of entire coral colonies. hm can, maximize the diffusion current of dissolved
in principle, be determined empirically with species to and from the polyp?
an oxygen microelectrode, and the values
Oxygen microelectrode data show typical
of o and i can be determined from the lit- values for hm are about 0.33 mm" 1 . Morerature or direct measurement using histo- phological data from fixed coral specimens
gives a range for i of 0.5-3.0 mm. It appears
logical techniques.
If the concentration of the metabolite at that ihm < 1.0, and thus it is possible for a
the inner gastrodermis is constant, the spe- polyp to have a wall thickness which maxcific solution of the concentration as a func- imizes fluxes with the environment. Data
to test the second question are not yet availtion of radius is:
= D(Q - Ce)-
MASS TRANSFER IN SYMBIOTIC CNIDARIANS
able. A comparative study of diffusional
boundary layers around expanded coral polyps of different species is worth pursuing to
answer this intriguing problem in diffusional morphology. Such data might provide additional evidence that diffusive constraints have been important selective agents
in the evolution of symbiotic scleractinian
polyp size and shape.
579
The size and shape of the host have been
shown to directly affect the metabolic rates
of the symbiont and host tissue. From the
symbiont's perspective of needing to achieve
suitably high rates of mass transfer with the
environment, how does being embedded in
the endoderm compare with a free-living
existence? A dinoflagellate swimming in the
water column over a reef will encounter
extremely low concentrations of dissolved
NEW DIRECTIONS
nutrients. Nitrogenous and phosphatic
Microsensors capable of spatially resolv- compounds have concentrations orders of
ing dissolved gases and ions on the scale of magnitude higher inside the host, although
micrometers permit direct test of allometric there is some question whether nutrients
relationships important in mass transfer in might still be limiting inside the symbiosis
symbiotic associations with complex mor- because of large population size of symbiphologies. Diffusive boundary layer map- onts (Cook and D'Elia, 1987). A free-living
ping using these tools promises to dissect cell might deplete a microzone around itself.
the effects of form on metabolic rate between Any such discontinuity in the concentration
and within colonies of symbiotic scleractin- around a cell would be only partially ameliorated by turbulence in the oceanic surface
ians.
The Sh/Re relationship offers more pow- layer (Okubo, 1987). Dinoflagellate cells are
erful insights into how the external envi- on the order of 10 Mm in diameter while the
ronment affects metabolic rates in a sym- length scale of the smallest eddies in the
biosis than the traditional measures of ocean is 1,000 ^m-10,000 nm, depending
metabolic rate and flow speed. Local pat- on depth (Mitchell et ai, 1985), and eviterns of mass transfer within a given shape dence suggests phytoplankton search out and
have been extensively investigated by utilize enriched nutrient microzones near
chemical engineers (Brodkey and Hershey, zooplankton in oligotrophic waters (McCar1988). Generally, areas of any shape exposed thy and Goldman, 1979; Lehman and Scato greater velocity gradients experience the via, 1982).
greatest mass transfer. In this paper, emphaRespiration rates of freshly isolated zoosis has been placed previously on calcula- xanthellae are an order of magnitude higher
tion of average Sh for the entire colony, but than those calculated for algae resident in
Sh can be calculated for single polyps from the host (McCloskey and Muscatine, 1984;
microelectrode measurements. Because Smith and Muscatine, 1986), but the values
boundary layer thickness can change locally compare favorably to those measured for
within a symbiotic scleractinian colony, Sh free-living phytoplankton (Laws, 1975). This
should also vary among polyps. Intracolony supports the hypothesis that diffusion limdifferences in fluxes of dissolved materials itation for oxygen occurs at night in the
to polyps should result in intracolony dif- symbiosis. Of course, during the day oxygen
ferences in polyp metabolic rate. For exam- produced by photosynthesis is readily availple, polyps on the upstream side of a colony able to the symbiosis and can enhance calwill be more active metabolically compared cification rate (Rinkevich and Loya, 1984),
to those on the downstream side, located in and offer an energetic advantage to symbithe more slowly moving wake. Little is otic cnidarians by keeping the coelenteron
known about the role of ciliated surfaces in and endoderm oxic (Shick and Brown,
disrupting the laminar sublayer and locally 1977). Host produced carbon dioxide will
increasing the velocity gradient immedi- be readily available to the symbiotic algae,
ately adjacent to the mucus-covered ecto- but only if photosynthetic rates are not
derm of symbiotic cnidarians, and what the approaching maximum values at which
consequences might be for uptake of dis- point available evidence points to diffusion
solved substances (Sebens, 1987).
limitation for bicarbonate ion (Muscatine et
580
MARK R. PATTERSON
ai, 1989; Pattersons a/., 1991). Free-living
cells can thus be at a mass transfer advantage relative to symbiotic forms for oxygen
transfer by night and carbon dioxide transfer by day. Not only is the diffusive path
length smaller by a factor of 5-10 for free
living cells, but competition between algae
for bicarbonate (Dubinsky et ai, 1990) and
oxygen seems likely given their dense packing in the endoderm (Smith and Douglas,
1987).
Is there a mass transfer advantage for cells
to be physically attached to a substrate over
which fluid is being pumped, relative to living free in the water column? A free-living
dinoflagellate will have such a small relative
velocity between cell and seawater that the
Re will be several orders of magnitude less
than 1. In this convective regime, viscosity
dominates over inertia in determining the
pattern of fluid flow around the cell (Berg,
1983). Sh for a spherical cell will approach
2.0 (Csanady, 1986), indicating that the fluid
motion is only slightly more effective than
diffusion in delivering material. Calculation
of Re for an algal cell bound to a substrate
in a shear velocity typical of that near the
ocean bottom gives a value of order 1.0,
several orders of magnitude higher than the
free-living case. However, zooxanthellae are
held endodermally, not ectodermally. The
importance of irrigation of the coelenteron
in allowing zooxanthellae to benefit by being
immobilized near a moving fluid, and transport of nutrients between polyps via the gastrovascular system remains largely unexplored (Gladfelter, 1983), and will probably
prove the key in understanding upper limits
to rates of aerobic respiration and photosynthesis achievable by the symbiosis.
This analysis is more applicable to ascidians that hold their symbiotic cyanobacteria
extracellularly along the common cloacae
(Olson, 1986) or to tridacnids that harbor
zooxanthellae extracellularly in the hemal
sinuses of the mantle tissue (Trench et at,
1981). In these situations, immobilization
of algal cells near a moving fluid is a possible
benefit to the symbionts, in that it allows
them to ride higher up on the Sh/Re plot
compared to free-living cells at much lower
Re.
Application of principles of mass transfer
from the chemical engineering literature to
analysis of other symbiotic associations may
help provide an improved understanding of
the functional ecology and physiology of
host-symbiont interactions.
ACKNOWLEDGMENTS
The material presented in this paper has
benefited from discussions with Peter Basser, Robert Carpenter, Tom Daniel, Mark
Denny, Betsy Gladfelter, Ric Grosberg,
Todd Hopkins, Alen Kaiser, Joel Kingsolver, Shani Kleinhaus, Mimi Koehl, Simon
Levin, Steven Miller, Len Muscatine, Randy
Olson, Tom Powell, Wm. Stephen Price,
Mary Beth Saffo, S. Laurie Sanderson,
Michael Savarese, Malcolm Shick, Ken
Sebens, and Stephen Wing. Two anonymous reviewers provided useful suggestions
for improvement. Christine Alexander,
Andrea Alfaro, Todd Hopkins, Brita Otteson, Wm. Stephen Price, S. Laurie Sanderson, Michael Savarese, and Stephen Wing
provided expert technical assistance. Doug
Nelson, Ric Carlton, and B. B. Jorgensen
gave helpful hints on successful oxygen
microelectrode fabrication. I am grateful to
the National Oceanic and Atmospheric
Administration
National
Undersea
Research Program (Aquarius Mission 886), the National Science Foundation (OCE8716427 and OCE-9016721) and the UC
Davis Committee on Research for financial
support.
REFERENCES
Berg, H. C. 1983. Random walks in biology. Princeton University Press, Princeton.
Brodkey, R. S. and H. C. Hershey. 1988. Transport
phenomena. McGraw-Hill, New York.
Constantz, B. 1986. Coral skeleton construction: A
physicochemically dominated process. Palaios 1:
152-157.
Cook, C. B. and C. F. D'Elia. 1987. Are populations
of zooxanthellae ever nutrient-limited? Symbiosis
4:199-212.
Csanady, G. T. 1986. Mass transfer to and from small
particles in the sea. Limnol. Oceanogr. 31(2):237248.
D'Elia, C. F. 1988. The cycling of essential elements
in coral reefs. In L. R. Pomeroy and J. J. Alberts
(eds.), Concepts ofecosystem ecology, pp. 195-230.
Springer-Verlag, New York.
Dennison, W. C. and D. J. Barnes. 1988. Effect of
water motion on coral photosynthesis and calcification. J. Exp. Mar. Biol. Ecol. 115:67-77.
MASS TRANSFER IN SYMBIOTIC CNIDARIANS
Denny, M. W. 1988. Biology and the mechanics of
the wave-swept environment. Princeton University
Press, Princeton.
Douglas, A. E. and D. C. Smith. 1989. Are endosymbioses mutualistic? Trends Ecol. Evol. 4(1 l):35O352.
Dubinsky, Z., N. Stambler, M. Ben-Zion, L. R.
McCloskey, L. Muscatine, and P. G. Falkowski.
1990. The effect of external nutrient resources on
the optical properties and photosynthetic efficiency of Stylophora pistillata. Proc. R. Soc. Lond.
B 239:231-246.
Dykens, J. A. and J. M. Shick. 1982. Oxygen production by endosymbiotic algae controls superoxide dismutase in their animal host. Nature (London) 297:579-580.
Edmunds, P. J. and P. S. Davies. 1986. An energy
budget for Porites porites (Scleractinia). Mar. Biol.
92:339-347.
Edmunds, P. J. and P. S. Davies. 1989. An energy
budget for Porites porites (Scleractinia), growing
in a stressed environment. Coral Reefs 8(1):3743.
Fredericks, C. A. 1976. Oxygen as limiting factor in
phototaxis and in intraclonal spacing of the sea
anemone Anthopleura elegantissima. Mar. Biol.
38:25-28.
Gladfelter, E. H. 1983. Circulation of fluids in the
gastrovascular system of the reef coral Acropora
cervicornis. Biol. Bull. 165:619-636.
Goreau, T. F. and N. Goreau. 1960. The uptake and
distribution of labelled carbon in reef building corals with and without zooxanthellae. Science 131:
668-669.
Hallock, P. 1981. Algal symbiosis: A mathematical
analysis. Mar. Biol. 62:249-255.
Holman, J. P. 1986. Heat transfer, 6th ed. McGrawHill, New York.
Jaap, W. C. and P. Hallock. 1990. Coral reefs. In R.
L. Myers and J. J. Ewel (eds.), Ecosystems ofFlorida, pp. 574-616. University of Central Florida
Press, Orlando.
Johannes, R. E., S. L. Coles, and N. T. Keunzel. 1970.
The role of zooplankton in the nutrition of some
scleractinian corals. Limnol. Oceanogr. 15:579586.
Johannes, R. E. and the Project Symbios Team. 1972.
The metabolism of some coral reef communities:
A team study of nutrient and energy flux at Eniwetok. BioScience 22:541-543.
Jokiel, P. L. and J. I. Morrissey. 1986. Influence of
size on primary production in the reef coral Pocillopora damicornis and the macroalga Acanthophora spicifera. Mar. Biol. 91:15-26.
Jumars, P. A. and A. R. M. Nowell. 1984. Fluid and
sediment dynamic effects on marine benthic community structure. Amer. Zool. 24:45-55.
Kozloff, E. N. 1990. Invertebrates. Saunders College
Publishing, Philadelphia.
Laws, E. A. 1975. The importance of respiration losses
in controlling the size distribution of marine phytoplankton. Ecology 56:419-426.
Lehman, J. T. and D. Scavia. 1982. Microscale nutrient patches produced by zooplankton. Proc. Nat.
Acad. Sci. (USA) 79:5001-5005.
581
Lesser, M. P. and J. M. Shick. 1989. Effects of irradiance and ultraviolet radiation on photoadaptation in the zooxanthellae of Aiptasia pallida: Primary production, photoinhibition, and enzymic
defences against oxygen toxicity. Mar. Biol. 102:
243-255.
Lewis, J. B. and W. S. Price. 1975. Feeding mechanisms and feeding strategies of Atlantic reef corals.
J. Zool., London 178:77-89.
Littler, M. M. and D. S. Littler. 1984. Models of
tropical reef biogenesis: The contribution of algae.
Prog. Phycol. Res. 3:323-364.
McCarthy, J. J. and J. C. Goldman. 1979. Nitrogenous nutrition of marine phytoplankton in nutrient-depleted waters. Science 203:670-672.
McCloskey, L. R. and L. Muscatine. 1984. Production and respiration in the Red Sea coral Stylophora pistillata as a function of depth. Proc. R.
Soc. London B 222:215-230.
McCloskey, L. R., D. S. Wethey, and J. W. Porter.
1978. Measurement and interpretation of photosynthesis and respiration in reef corals. In D. R.
Stoddart and R. E. Johannes (eds.), Coral reefs:
Research methods, pp. 379-396. Unesco, Paris.
Mitchell, J. G., A. Okubo, and J. A. Fuhrman. 1985.
Microzones surrounding phytoplankton form the
basis for a stratified marine microbial ecosystem.
Nature 316:58-59.
Muscatine, L. and P. L. McNeil. 1989. Endosymbiosis in Hydra and the evolution of internal
defense systems. Amer. Zool. 29:371-386.
Muscatine, L. and J. W. Porter. 1977. Reef corals:
Mutualistic symbioses adapted to nutrient poor
environments. BioScience 27:454-460.
Muscatine, L., P. G. Falkowski, and Z. Dubinsky.
1983. Carbon budgets in symbiotic associations.
Endocytobiology 11:649-658.
Muscatine, L., P. G. Falkowski, J. W. Porter, and Z.
Dubinsky. 1984. Fate of photosynthetic fixed
carbon in light- and shade-adapted colonies of the
symbiotic coral Stylophora pistillata. Proc. R. Soc.
London B 222:181-202.
Muscatine, L., H. Masada, and R. Burnap. 1979.
Ammonium uptake by symbiotic and aposymbiotic reef corals. Bull. Mar. Sci. 29:572-575.
Muscatine, L., J. W. Porter, and I. R. Kaplan. 1989.
Resource partitioning by reef corals as determined
from stable isotope composition. I. d'3C of zooxanthellae and animal tissue vs depth. Mar. Biol.
100:185-193.
Okubo, A. 1980. Diffusion and ecological problems:
Mathematical models. Springer-Verlag, Berlin.
Okubo, A. 1987. Fantastic voyage into the deep:
Marine biofluid mechanics. In E. Teramoto and
M. Yamaguti (eds.), Mathematical topics in population biology, morphogenesis, and neurosciences: Lecture notes in biomathematics, Vol. 71,
pp. 32-47. Springer-Verlag, Berlin.
Olson, R. R. 1986. Light enhanced growth of the
ascidian Prochloron symbiosis Didemnum molle.
Mar. Biol. 93:437^42.
Patterson, M. R. 1984. Patterns of whole colony prey
capture in the octocoral Alcyonium siderium. Biol.
Bull. 167:613-629.
Patterson, M. R. 1991a. The effects offlow on polyp-
582
MARK R. PATTERSON
level prey capture in an octocoral, Alcyonium siderium. Biol. Bull. 180:93-102.
Patterson, M. R. 19916. Passive suspension feeding
by an octocoral in plankton patches: Empirical test
of a mathematical model. Biol. Bull. 180:81-92.
Patterson, M. R. and K. P. Sebens. 1989. Forced
convection modulates gas exchange in cnidarians.
Proc. Nat. Acad. Sci. (U.S.A.) 86:8833-8836.
Patterson,M.R.,K.P.Sebens,andR. R.Olson. 1991.
In situ measurements of the effect of forced convection on primary production and dark respiration in reef corals. Limnol. Oceanogr. 36:936-948.
Porter, J. W. 1976. Autotrophy, heterotrophy, and
resource partitioning in Caribbean reef-building
corals. Amer. Natur. 110:731-742.
Porter, J. W., L. Muscatine, Z. Dubinsky, and P. G.
Falkowski. 1984. Primary production and photoadaptation in light- and shade-adapted colonies
of the symbiotic coral Stylophora pistillata. Proc.
Roy. Soc. London Biol. Sci. B 222:161-180.
Praturi, A. K. and R. S. Brodkey. 1978. A stereoscopic visual study of coherent structures in turbulent shear flow. J. Fluid. Mech. 89:251-272.
Revsbech, N. P. and B. B. Jergensen. 1986. Microelectrodes: Their use in microbial ecology. In K.
C. Marshall (ed.), Advances in microbial ecology,
Vol. 9, pp. 293-352. Plenum, New York.
Rinkevich, B. and Y. Loya. 1984. Does light enhance
calcification in hermatypic corals? Mar. Biol. 80:
1-6.
Schlichting, H. 1979. Boundary layer theory. McGrawHill, New York.
Sebens, K. P. 1981. The allometry of feeding, energetics, and body size in three sea anemone species.
Biol. Bull. 161:152-171.
Sebens, K. P. 1987. Coelenterate energetics. In F. J.
Vernberg and T. J. Pandian (eds.), Animal energetics, pp. 55-120. Academic Press, New York.
Shick, J.M. 1990. Diffusion limitation and hyperoxic
enhancement of oxygen consumption in zooxanthellate sea anemones, zoanthids, and corals. Biol.
Bull. 179:148-158.
Shick, J. M. and W. I. Brown. 1977. Zooxanthellaeproduced O2 promotes sea anemone expansion and
eliminates oxygen debt under environmental hypoxia. J. Exp. Zool. 201:149-155.
Shick, J. M., W. I. Brown, E. G. Dolliver, and S. R.
Kayar. 1979. Oxygen uptake in sea anemones:
Effects of expansion, contraction, and exposure to
air and the limitations of diffusion. Physiol. Zool.
52:50-62.
Smith, D. C. and A. E. Douglas. 1987. The biology
of symbiosis. Edward Arnold, London.
Smith, G. J. and L. Muscatine. 1986. Carbon budgets
and regulation of population density of symbiotic
algae. Endocyt. C. Res. 3:213-238.
Steen, R. G. 1988. The bioenergetics of symbiotic
sea anemones (Anthozoa: Actiniaria). Symbiosis
5:103-142.
Sweeney, B. M. 1976. Circadian rhythms in corals,
particularly Fungiidae. Biol. Bull. 151:236-246.
Trench, R. K., D. S. Wethey, and J. W. Porter. 1981.
Some observations on the symbiosis with zooxanthellae among the tridacnidae. Biol. Bull. 161:
180-198.
Vogel, S. 1983. Life in moving fluids: The physical
biology of flow. Princeton University Press,
Princeton.
Whitaker, S. 1972. Force convection heat transfer
correlations forflowin pipes, past flat plates, single
cylinders, single spheres, and flow in packed beds
and tube bundles. Amer. Inst. Chem. Eng. Jour.
18:361-371.
White, F. M. 1988. Heat and mass transfer. AddisonWesley, New York.
Williams, E. H., Jr. and L. Bunkley-Williams. 1990.
The world-wide coral reef bleaching cycle and
related sources of coral mortality. Atoll Res. Bull.
335:1-71.
Zamer, W. E. and J. M. Shick. 1989. Physiological
energetics of the intertidal sea anemone Anthopleura elegantissima. III. Biochemical composition of body tissues, substrate specific absorption,
and carbon and nitrogen budgets. Oecologia 79:
117-127.
Zukauskas, A. and J. Ziugzda. 1985. Heat transfer of
a cylinder in crossflow. Hemisphere Publishing
Corp., Washington.