ARTICLE IN PRESS
Deep-Sea Research I 55 (2008) 1311– 1317
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Deep-Sea Research I
journal homepage: www.elsevier.com/locate/dsri
Nutrient uptake rate as a function of cell size and surface
transporter density: A Michaelis-like approximation to the
model of Pasciak and Gavis
Robert A. Armstrong
Marine Sciences Research Center, Stony Brook University, Stony Brook, NY 11794-5000, USA
a r t i c l e in fo
abstract
Article history:
Received 2 November 2006
Received in revised form
6 May 2008
Accepted 9 May 2008
Available online 16 May 2008
Pasciak and Gavis were first to propose a model of nutrient uptake that includes both
physical transport by diffusion and active biological transport across the cell membrane.
While the Pasciak–Gavis model is not complicated mathematically (it can be expressed in
closed form as a quadratic equation), its parameters are not so easily interpretable
biologically as are the parameters of the Michaelis–Menten uptake model; this lack of
transparency is probably the main reason the Pasciak–Gavis model has not been adopted
by ecologically oriented modelers. Here I derive a Michaelis-like approximation to the
Pasciak–Gavis model, and show how the parameters of the latter map to those of the
Michaelis-like model. The derived approximation differs from a pure Michaelis–Menten
model in a subtle but potentially critical way: in a pure Michaelis–Menten model, the
half-saturation constant for nutrient uptake is independent of the density of transporter
(or ‘‘porter’’) proteins on the cell surface, while in the Pasciak–Gavis model and its
Michaelis-like approximation, the half-saturation constant does depend on the density of
porter proteins. The Pasciak–Gavis model predicts a unique relationship between cell size,
nutrient concentration in the medium, the half-saturation constant of porter-limited
nutrient uptake, and the resulting rate of uptake; the Michaelis-like approximation
preserves the most important feature of that relationship, the size at which porter
limitation gives way to diffusion limitation. Finally I discuss the implications for
community structure that are implied by the Pasciak–Gavis model and its Michaelis-like
approximation.
& 2008 Published by Elsevier Ltd.
Keywords:
Nutrient uptake kinetics
Diffusion limitation
Porter limitation
Cell size
Pasciak–Gavis
Michaelis–Menten
1. Introduction
Biophysical reaction–diffusion models have been used
for many years to explore the effects of size, shape,
swimming mode, and flow regime on rates of nutrient
uptake by phytoplankton (e.g., Munk and Riley, 1952; Berg
and Purcell, 1977; Lazier and Mann, 1989; Karp-Boss et al.,
1996; Pahlow et al., 1997; Karp-Boss and Jumars, 1998;
Völker and Wolf-Gladrow, 1999). These studies usually
make use of the mathematically convenient approximation that cells immediately absorb all nutrients reaching
E-mail address: [email protected]
0967-0637/$ - see front matter & 2008 Published by Elsevier Ltd.
doi:10.1016/j.dsr.2008.05.004
the cell surface (Munk and Riley, 1952); this approximation implies a boundary condition that the nutrient
concentration at the cell surface must be zero. In a
departure from this view, Pasciak and Gavis (1974, 1975)
recognized that nutrient transport across cell membranes
may also be limited by the finite processing times of
uptake (‘‘porter’’) units. In this case, and near steady state,
diffusive flux to the cell surface must equal transport flux
across it; the boundary condition of Pasciak and Gavis
(1974, 1975) is therefore a condition on flux, not on
concentration.
Unfortunately, the Pasciak–Gavis (PG) approach does
not yield a mathematical form that is familiar to
ecosystem modelers. In even the simplest of their models
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(spherical cells not moving relative to the medium),
uptake flux is an inverse quadratic function of nutrient
concentration; when compared to Michaelis–Menten
hyperbolas, the behavior of such functions are difficult
to visualize in terms of underlying parameters. Perhaps
more important, quadratic (or more complicated) descriptions do not fit neatly within the paradigmatic Droop
family of descriptors of nutrient-limited growth, where
the concatenation of a Michaelis–Menten model of
nutrient uptake with a Droop model of growth regulation
by internal cell quota yields a Monod model for nutrientlimited growth (Droop, 1968; Goldman, 1977; Burmaster,
1979; Morel, 1987). Largely for these reasons, biophysicsbased descriptions of nutrient uptake are seldom used by
modelers of phytoplankton interaction in ecology, biological oceanography, and biogeochemistry.
Here I derive a Michaelis-like approximation to the PG
model, and show how the parameters of the latter model
map to those of the Michaelis-like model. The derived
approximation differs from a pure Michaelis–Menten
model in a subtle but potentially critical way: in a pure
Michaelis–Menten model, the half-saturation constant for
nutrient uptake is independent of the density of transporter (or ‘‘porter’’) proteins on the cell surface, while in
the PG model and its Michaelis-like approximation, the
half-saturation constant does depend on the density of
porter proteins.
The PG model predicts a unique relationship between
size, nutrient concentration in the medium, the halfsaturation constant for porter-limited uptake, and the
resulting rate of nutrient uptake; I show that the
Michaelis-like approximation preserves the most important feature of this relationship, the location of the size at
which porter limitation gives way to diffusion limitation. I
next fit the Michaelis-like approximation to uptake data
from Sunda and Huntsman (1997); the conclusion here is
that it will be difficult to estimate parameters needed for
either model from laboratory cultures alone. Finally I
discuss the implications for community structure that are
implied by the PG model and its Michaelis-like approximation. I conclude that the predictions of the present
paper, when combined with theoretical calculations for
the influences of size, shape, and movement on uptake
rates, may allow prediction of the largest size of
phytoplankton species that can occur at steady state
under given environmental conditions; this last prediction
may provide the most accessible, and also most important, test of this model.
2. Foraging models of substrate acquisition
Michaelis–Menten uptake kinetics can easily be derived from foraging theory (Stephens and Krebs, 1986), an
ecological application of queuing theory. Let n be the
number of transport (‘‘porter’’) units per unit cell surface
(mol transport units m2), and let f(t) be the fraction of
these units engaged in transporting nutrient molecules
into the cell at time t. If unoccupied transport units
(a fraction 1f(t) of the total) find new substrate
molecules at rate aS0, where S0 is substrate concentration
(mol m3) near the cell surface and where a is an
‘‘acquisition’’ rate (the rate at which a single unoccupied
porter unit acquires ions for transport across the cell
membrane ((mol m3)1 s1), unoccupied transport units
will become occupied at rate naS0(1f) (s1). Similarly, if
occupied transport units take handling time h (s) to
transport nutrient molecules across the cell membrane,
occupied transport units will be freed at rate nf/h. The
resulting differential equation for f(t) is
df =dt ¼ aS0 ð1 f Þ f =h.
(1)
At steady state df/dt ¼ 0, f ¼ aS0h/(1+S0h), so that the
total porter-mediated transport flux per unit cell surface
area fP (mol m2 s1) is {number of occupied porter units
nf per unit area} {processing rate 1/h per porter unit}, or
n
S0
h ð1=ahÞ þ S0
S0
fmax
.
kP þ S0
fP ¼
(2)
In Eq. (2), fmaxn/h (mol m2 s1) is the maximum
uptake rate (again per unit surface area), and kP(ah)1
(mol m3) is the ‘‘half-saturation’’ constant for porterlimited transport, the nutrient concentration at which
porter-limited transport reaches half its maximum
value.
Note here that the definition of half-saturation constant in Eq. (2) does not include the density n of porter
units on the cell surface. This is an important difference
from the model that will be derived below, where n will be
shown to be integral to determining the half-saturation
constant, and so will be critical for setting the boundary
between porter-limited and diffusion-limited uptake.
3. The model of Pasciak and Gavis
Pasciak and Gavis (1974, 1975) began their derivation
by assuming that Eq. (2) describes substrate flux across
the surface of a cell of radius r0 (m). To relate the substrate
concentration S0 at the cell surface to the concentration in
the bulk medium SN, they next noted that in the simplest
case of a totally quiescent medium with only molecular
diffusion, and with no motion of the cell relative to the
medium, diffusive transport of nutrient from the bulk
medium to the cell surface is given by
F D ðrÞ ¼ 4pDr 2 qSðrÞ=qr,
(3)
where S(r) is substrate concentration (mol m3) at distance r from the center of the cell, and where D is
molecular diffusivity (m2 s1). The quantity FD(r) is the
diffusive flux (mol s1) of nutrient toward the origin (r ¼ 0)
through a spherical surface of area 4pr2. At steady state,
the flux of nutrient through a series of concentric
spherical shells must be independent of r; this assumption
may be written as FD(r) ¼ FD(r0) for all r. With this steadystate assumption, for a spherical cell, and assuming that
the bulk concentration SN is reached only as r-N, Eq. (3)
can be integrated as
Z S1
Z 1
dS ¼ ðF D ðr 0 Þ=4pDÞ
r 2 dr,
(4)
S0
r0
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model as
which on rearrangement becomes
F D ðr 0 Þ ¼ 4pDr 0 ðS1 S0 Þ.
(5)
The flux per unit area is then
fD ðr 0 Þ ¼ F D ðr 0 Þ=4pr 20 ¼ ðD=r 0 ÞðS1 S0 Þ.
(6)
Since Eq. (6) is a limiting case for spherical cells in
cases where molecular diffusion is the only diffusive
element, it excludes the effects of shape and of transport
due to swimming, sinking, shear, and ‘‘eddy diffusivity,’’
all of which can be modeled as advective shear at the
scale of single cells (Karp-Boss et al., 1996). Here I
denote by F the multiplicative enhancement of diffusive
transport associated with non-spherical shapes (e.g.,
Pasciak and Gavis, 1975; Pahlow et al., 1997). Next,
the further enhancement (over pure diffusion) due to
movement-associated processes can be represented by
the dimensionless Sherwood number Sh (Karp-Boss et al.,
1996). The value of Sh is a function f(Pe) of the
dimensionless Péclet number Pe, which is in turn defined
as the ratio of advective to diffusive transport and is
calculated as Ur0/D, where U is a characteristic flow
velocity. The mathematical form of f(Pe) depends on
shape, flow regime, and other factors (Karp-Boss et al.,
1996).
Using these definitions, Eq. (6) can be generalized to
fD ðr 0 Þ ¼ FShðD=r 0 ÞðS1 S0 Þ.
(7)
Setting porter-limited flux (Eq. (2)) equal to diffusionlimited flux (Eq. (7)) at the cell surface, we arrive at the
(slightly generalized) PG model
fPG ðr 0 Þ ¼ fmax
S1 FShD=r 0 fPG ðr 0 Þ
,
ðkP þ S1 ÞFShD=r 0 fPG ðr 0 Þ
(8)
where fPG(r0) ¼ fD(r0) ¼ fP(r0) is the rate of transport
into a cell per unit surface area.
For further analysis, it is convenient to rewrite Eq. (8)
in the nondimensional form
r 0 fPG ðr 0 Þ 2
r 0 S1 fPG ðr 0 Þ S1
1
þ
þ
þ
¼ 0,
fmax
fmax
rn
r n kP
kP
(9)
where
r n kP FShD=fmax
(10)
is the characteristic length scale of the reaction–
diffusion process. Eq. (9) can be solved using the quadratic
formula:
fPG ðr 0 Þ
1 þ ðr 0 =r n Þ þ ðS1 =kP Þ
¼
fmax
2ðr 0 =r n Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
4ðr 0 =r n ÞðS1 =kP Þ
1 1
2 .
1 þ ðr 0 =r n Þ þ ðS1 =kP Þ
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(11)
Remarkably, Eq. (11) can be approximated by a
Michaelis-like equation both when cells are very small
(r0/r51) or very large (r0/rb1). In both cases the term
y ¼ 4(r0/r)(SN/kP)(1+r0/r+SN/kP)2 is 51, so that one
pffiffiffiffiffiffiffiffiffiffiffiffi
can use the relationship
1 þ y 1 þ y=2 to derive a
Michaelis–Menten-like (PG/MM) approximation to the PG
S1 =kP
1 þ r 0 =r n þ S1 =kP
S1
¼ fmax
kP ð1 þ r 0 =r n Þ þ S1
S1
¼ fmax
.
kP þ fmax =ðFShDÞr 0 þ S1
fPG=MM ðr 0 Þ ¼ fmax
(12)
Since Eq. (12) is an excellent approximation in these two
critical limits, it seems reasonable to posit that it could
serve as a general approximation to Eq. (11).
Note particularly that Eq. (12) is of strict Michaelis–
Menten form only in the limit r0-0, because when r040
the parameter fmax participates in determining the halfsaturation constant. Since fmaxn/h, both the maximum
uptake rate and the half-saturation constant are sensitive
to n. Because the density of porters on the cell surface
varies with the nutrient environment (Morel, 1987;
Hudson and Morel, 1990), including n in the halfsaturation constant may change significantly theoretical
predictions of how nutrient adaptation may affect fmax
(Pahlow, 2005; Smith and Yamanaka, 2007).
In Fig. 1, PG and PG/MM uptake rates (normalized to
maximum uptake rate fmax) for three normalized nutrient
concentrations SN/kP are plotted as functions of normalized cell radius r0/r*. Three things in particular should be
noted: (1) that at each nutrient concentration SN/kP, the
PG and PG/MM curves are virtually identical both at small
cell sizes (r0/r*51) and at large cell sizes (r0/r*b1); (2)
that the PG and PG/MM curves are virtually identical at
low nutrients (e.g., SN/kP ¼ 0.1; see Fig. 1) for all values of
r0/r*, because in this limit all size classes are diffusion
limited; and (3) that even at high nutrients (e.g., SN/
kP ¼ 30; see Fig. 1), where the PG curve can be
substantially higher than the PG/MM curve, the ‘‘transition size’’ of the two models, the size at which porter
limitation gives way to diffusion limitation (Hudson and
Morel, 1990), is identical for PG and PG/MM; this last fact
follows directly from point (1) above. In both the PG and
PG/MM models, the asymptotic uptake rate of small
cells (r0/r*51) is f(r0)/fmax ¼ (SN/kP)/(1+SN/kP)o1,
and the asymptote at large sizes (r0/r*b1+SN/kP) is
f(r0)/fmax ¼ (SN/kP)/(r0/r*); these two lines intersect at
the transition size
ðr 0 =r n Þc ¼ 1 þ S1 =kP .
(13)
Cells smaller than (r0/r*)c will be porter limited, while
cells larger than (r0/r*)c will be diffusion limited.
The importance of this last point cannot be overemphasized, because the transition size may be critical in setting
the size of the largest phytoplankton species in a steady-state
community; it is therefore critical that the PG/MM model
represent this property properly (see Discussion).
4. Discussion
I have here presented an approximation (Eq. (12)) to
the PG model (Eq. (11)) of size-dependent nutrient
uptake; this form should appeal to ecosystem modelers
because of its PG/MM form. The PG/MM prediction of
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PG, Sinf / kp = 30
PG, Sinf / kp = 1
PG, Sinf / kp = 0.1
locus of corners of PG model
locus of corners of MM model
phi / phimax
1
0.1
0.01
0.001
0.01
0.1
1
10
100
r0 / r*
Fig. 1. Predicted uptake rates per unit area f scaled to maximum uptake rates fmax as a function of size (equivalent radius r0) divided by environmental
size scale (r*). The three highlighted curves are for the PG model; the dotted curves below each are the PG/MM approximation to each PG curve. Also
noted by dotted lines are the asymptotes to both the porter-limited and diffusion-limited sections of each curve; where they intersect is the ‘‘turnover
point’’, where porter limitation gives way to diffusion limitation. Note particularly that the turnover points for the PG/MM model are the same as the
turnover points for the PG model for each value of SN/kP.
uptake rate is virtually identical to the PG prediction over
large regions of parameter space; if more accurate
predictions are needed near the transition sizes shown
in Fig. 1, Eq. (11) can be used directly, with no increase in
the number of parameters needed (relative to Eq. (12)).
Where such increased accuracy is desired, I would hope
that the above exposition of the connection between the
PG and PG/MM models might give ecosystem modelers
the courage to use Eq. (11) directly.
Empirical assessments of phytoplankton community
structure (Raimbault et al., 1998; Chisholm, 1992) suggest
that as nutrient availability increases, chlorophyll concentrations in smaller size classes do not increase
significantly; instead, larger phytoplankton size classes
are added. This response has been modeled by assuming a
combination of almost-uniform grazing rate across size
classes, coupled with decreasing phytoplankton growth
rate due to nutrient limitation of phytoplankton growth
(Armstrong, 1994, 1999, 2003a, b; Hurtt and Armstrong,
1996, 1999; Kriest and Oschlies, 2007).
In the most recent of these publications (Armstrong,
2003a, b; Kriest and Oschlies, 2007) the model of Aksnes
and Egge (1991) has been used to model size dependence
of nutrient uptake (and, by extension, growth; see Morel,
1987) as an allometric function of the half-saturation
constant for nutrient uptake (Moloney and Field, 1989,
1991). The model of Aksnes and Egge (1991) is a
Michaelis-form model based on the work of Holling
(1959, 1966); it is identical in concept to the queuingbased uptake model presented in Section 2. The Aksnes–
Egge (AE) model (1991; their Eq. (3)) is
fðr 0 Þ ¼
n
S
,
h 1=ðhAvÞ þ S
(14)
where A is the effective collection area of a porter unit,
and where v is a characteristic velocity near the cell
surface. Their variable S is defined (their p. 67) as the
‘‘nutrient concentration in the medium,’’ which in the
notation of the present paper is SN. Eq. (14) is quite
general; the problem lies in predicting the value of the
characteristic velocity v under porter-limited and diffusion-limited conditions.
The contrast between the AE and PG/MM models can
best be appreciated by recasting the PG/MM model in AE
notation. First, in the case of porter limitation, Berg and
Purcell (1977, p. 208, last line) note that the acquisition
rate of nutrient molecules by an individual porter unit
(Eq. (2) ff) is a ¼ 4Da(A/p)1/2, where a is the probability
that a nutrient ion that enters area A is indeed captured.
Since kP ¼ 1/(ah), this result implies that in porter-limited
cases
kP ¼
p 1 pffiffiffiffiffiffiffiffi
A=p,
4a hAD
(15)
which in turn implies that in the limit of porter limitation
pffiffiffiffiffiffi
v ¼ 4aD= pA.
(16)
There is nothing particularly surprising about this result.
However, in the limit of diffusion limitation, the PG/
MM model predicts that the half-saturation constant
should be
kD;PG=MM !
fmax
nA 1
r0 ,
r0 ¼
FSh hAD
FShD
(17)
which in turn implies a characteristic velocity in the AE
model of
v ¼ FSh=ðnAr 0 Þ.
(18)
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This result is very strange, because it is difficult to see how
the velocity of nutrient molecules near the cell surface
could possibly depend on the density of porter proteins on
the cell surface.
The form of Eq. (17) is not difficult to rationalize,
however, when one views it in the light of PG/MM:
combining Eqs. (15) and (17), the half-saturation constant
of the present model, rewritten in the notation of Aksnes
and Egge (1991), becomes
1
p pffiffiffiffiffiffiffiffi
nA
kS;PG=MM ¼
r0 ,
(19)
A=p þ
hAD 4a
FSh
which is a linear combination of the limiting results of
porter limitation and diffusion limitation. In Eq. (19), an
increase in porter density n would shift the half-saturation
constant away from porter limitation and toward diffusion
limitation, which is what one would intuitively expect.
In short, the model of Aksnes and Egge (1991) is not so
general as one would guess at first glance; it is based on
the work of Holling (1959, 1966), who did not include the
effects of diffusion or other types of movement in his
models.
Regardless of whether Eq. (11) or Eq. (12) is used, the
parameters kP and r* must be estimated if the PG/MM
model is to be implemented. Therefore, I fitted the data of
Sunda and Huntsman (1997) on iron uptake kinetics in
four species of phytoplankton that differed in size (and
also taxon, which complicates the interpretation, as noted
below). By normalizing data on uptake rates to cell surface
area, Sunda and Huntsman (1997) showed that to a good
approximation the uptake kinetics of all four species could
be described by the same curve: fFe ¼ fFe, max[Fe0 ]/
[Fe0 ]+kFe), where fFe, max ¼ 1.276 mmol m2 d1 and kFe ¼
0.51 mmol m3.
1315
Here I used exactly the same data (all data points
having total inorganic iron concentrations [Fe0 ] less than
0.75 mmol m3, where iron hydroxides start to form) to
estimate key parameters in Eq. (12): the maximum uptake
rate fmax ¼ n/h, the half-saturation constant for porterlimited uptake kP, and the uptake length scale r*.
(Remember that r* is not determined solely by physical
processes, but rather by the ratio of physical transport to
the porter-limited half-saturation constant kP; in Sunda
and Huntsman (1995, 1997), the effective diffusivity D is
that of total inorganic carbon Fe0 chelated by EDTA.) My
aim was to test whether Eq. (12) fit these data significantly better than the model of Sunda and Huntsman
(1997), which did not include possible size-dependent
influences on diffusivity.
Eq. (12) was fit to the Sunda and Huntsman (1997) data
by minimizing likelihood (Edwards, 1992; Hilborn and
Mangel, 1997; see Fig. 2). Values estimated for fmax, kP,
and ShD were assumed to apply to all species. Results
from these fits are plotted in Fig. 2.
The major result is that the fits to Eq. (12) were not
significantly better than the size-independent fit of Sunda
and Huntsman (1997). Inspection of Fig. 2 gives a hint why
they were not better, and also gives one pause about how
these parameters might be measured. First, the data
points for Thalassiosira pseudonana, a diatom that that is
smaller (r0E1 mm) than the dinoflagellate Prorocentrum
mimimum (r0E3 mm), lie below those for P. minimum at
low iron concentrations, suggesting that kP may vary
among species. Second, the data points for these two
species near [Fe0 ] ¼ 0.75 mmol m3also suggest that they
may have different fmax, raising the possibility (even
probability) of strong taxonomic effects (cf. the corresponding conclusions of Chisholm, 1992, on maximum
10
phi (umol m-2 d-1)
T. pseudonana
T. weissflogii
P. minimum
P. micans
Sunda and Huntsman (1997)
Eq. (12), r* = 70 um
1
0.1
0.01
0.01
0.1
1
-3
[Fe′] (umol m ]
Fig. 2. Fits of the PG/MM model to the data of Sunda and Huntsman (1997). The solid line is the fit made by Sunda and Huntsman (1997). The dashed lines
are the fits of the present model (Eq. (12)) to the same data. The two dashed lines are predictions for the largest size class (Procentrum micans; r0 ¼ 15 mm)
and the smallest size class (Thalassiosira pseudonana; r0 ¼ 2 mm). It is clear that with the estimated environmental size scale (r* ¼ 70 mm), model
predictions do not include most data points.
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Table 1
Parameters used in this paper
n
r
R0
r*
S(r)
SN
S0
a
h
kP
fP
fmax
fD
D
F
Sh
v
A
m
c
Number of transport units per unit cell surface (mol m2)
Distance from the center of a cell (m)
Cell radius (equivalent spherical radius) (m)
Scale length for determining the transition from porter limitation
to diffusion limitation (m)
Nutrient concentration at distance r from the
Center of a cell (mol m3)
Nutrient concentration in the ambient medium (mol m3)
Nutrient concentration at the cell surface (mol m3)
Acquisition rate: the rate at which an unoccupied
Porter unit captures a new nutrient molecule ((mol m3)1 s1)
Handling (transport) time for a single nutrient molecule (s)
Half-saturation constant for porter-limited uptake (mol m3)
Porter-limited nutrient flux (mol m2 s1)
Maximum porter-limited nutrient flux (mol m2 s1)
Diffusion-limited nutrient flux (mol m2 s1)
Molecular diffusivity (m2 s1)
Shape coefficient (dimensionless)
Sherwood number (motion coefficient) (dimensionless)
Characteristic velocity in the model of Aksnes and Egge (m s1)
Collection area of a porter unit in Aksnes and Egge (m2)
Growth rate (d1)
Susceptibility to predation (dimensionless)
growth rate). Taken together, these observations suggest
that it may be very difficult to test Eq. (12) empirically in
laboratory experiments.
Third, the estimated length scale r*, 70 mm, suggests a
diffusivity of roughly 24 105 m2 d1. For comparison,
values of D given by Völker and Wolf-Gladrow (1999) are
15.5 105 m2 d1 for Fe III, 12.4 105 m2 d1 for Fe II, and
1.9 105 m2 d1 for Fe–ligand complex (Fe-L). Given that
the results of Sunda and Huntsman (1995, 1997) and
Völker and Wolf-Gladrow (1999) are comparable in detail
(since Völker and Wolf-Gladrow (1999) made extensive
use of the data of Sunda and Huntsman (1995, 1997), one
presumes that these diffusivities are at 20 1C, the
temperature used by Sunda and Huntsman, 1995), we
would expect D for Fe-EDTA to lie between those for Fe II
and Fe-L, and probably to lie closer to Fe-L. The fact that it
does not again suggests the difficulty of estimating
parameter values from laboratory culture experiments
(Table 1).
Another important assumption in this fit was that the
Sherwood number is the same for all species. However,
since the Péclet number is a function of cell size, this
assumption may not be valid. To assess whether this
assumption was warranted, I followed the procedure used
by Karp-Boss et al. (1996) to construct the relationship
between Sh and algal size for nonmotile phytoplankton in
decaying turbulence (their Table 2). I substituted 15 mm,
the approximate radius of the largest cells in Sunda
and Huntsman (1997), into their formulas, calculating a
Sherwood number of 1.04. Since the other species are
smaller, their Sherwood numbers would be even closer
to 1. I therefore conclude that all the species used by
Sunda and Huntsman (1997) are small enough that
assumption that their Sherwood numbers are all near 1
is an acceptable approximation.
To use these results to predict community structure,
predictions of nutrient uptake rate must be converted to
predictions of growth rates, and the effects of grazing
must be estimated. In general this will be a daunting task.
However, Armstrong (1979) showed that ranking species
in terms of competitive ability should be done not simply
by their growth rates m, but instead by the ratio m/c of
their growth rates to their susceptibility to grazing. Since
growth rate and grazing rate are likely to follow the same
allometric relationship with size (Moloney and Field,
1989), use of the ratio m/c may help overcome complications arising from surface:volume relationships. In addition, the fact that slower-growing phytoplankton taxa,
such as dinoflagellates, can coexist with faster-growing
taxa, such as diatoms, suggests that taxonomic differences
in competitive ability may also largely disappear when
m/c is used as the metric of competitive ability.
If both these speculations prove correct, it then follows
the transition point between porter limitation and diffusion limitation may be the most important predictor of
phytoplankton growth, and that community structure—or
at the very least, the size of the largest phytoplankton
species under steady-state condiditions—may be more
directly determined by the location of this point than by
the intricacies of growth–herbivore interactions.
In its strongest form, this quantitative hypothesis
concerning community structure is that the size of the
largest species under steady-state conditions is given by
Eq. (13). Given this prediction, a new perspective for
biophysical investigations of diffusion limitation may be
that species are adapted not to cope with diffusion
limitation, but to avoid it. The implication is that under
natural conditions, one may find diffusion-limited species
only in transient conditions, such as those following a
bloom, where there has not been enough time for species
replacement to have occurred.
Acknowledgments
I thank Bill Sunda for providing the original data (on
spreadsheet) from the Sunda and Huntsman (1997) Nature
paper. This work was supported in part by NSF Grant OCE
0049009 to R.A. Armstrong and by NOAA Grant DE-FG0200ER63009 to J.L. Sarmiento. This is contribution #1372
from the Marine Sciences Research Center, Stony Brook
University.
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