Oceanography and Marine Biology: An Annual Review, 2008, 46, 1-23
© R. N. Gibson, R. J. A. Atkinson, and J. D. M. Gordon, Editors
Taylor & Francis
Use, abuse, misconceptions and insights
from quota models — the Droop
cell quota model 40 years on
Kevin J. Flynn
Institute of Environmental Sustainability, Wallace Building, Swansea
University, Singleton Park, Swansea SA2 8PP, Wales, UK
E-mail: [email protected]
Abstract The Droop cell quota model is the most cited model of phytoplankton growth, even
though many pay scant regard to the original description and to its limitations for the description
of the interactions that define phenotypic plasticity. While the mechanistic basis of the concept and
most ecosystem applications of quota models are C based, much experimental work is cell based,
and most theoretical studies ignore the important differences between cell and C nutrient quotas.
The future application of the quota approach would be enhanced by the adoption of a normalised
quota (nQuota) description, employing a dimensionless constant (KQ) to define the response curve,
rather than using the original fixed-curve form. Establishment of the range of these KQ values for
different phytoplankton species would limit the number of free parameters in ecosystem variants of
quota models while recognising the importance of curve shape for phenotypic variation. KQ for N
is typically >3, while for P it is typically <0.2. In addition, appropriate control linkages are required
to regulate nutrient transport to the quotas of limiting and non-limiting nutrients. Together, these
would enable the establishment of a more coherent quota-based description of algal growth more fit
for the development of plankton functional-type models.
Introduction
The Droop quota model was first published in 1968 (Droop 1968), spawning not only applied
applications but extensive theoretical analyses over the following 40 yr. The driving concept of the
model, that organism (specifically phytoplankton) growth is first and foremost a function of internal nutrient availability, contrasts with the Monod (1942, 1949) description of microbial growth,
which relates growth simply to external resource availability. Perhaps ironically, given that both the
works by Monod (1942, 1949) and Droop (1968) were based upon steady-state chemostat studies,
the interface between these contrasting approaches is of most potential value in dynamic situations
(Droop 1975), where organism growth can continue (as allowed by the remaining internal resource)
in the absence of sufficient external resource. More recently, with the increasing appreciation of the
importance of stoichiometric differences between consumer and food (Sterner & Elser 2002), the
quota model has found an additional role for the description of how phytoplankton prey stoichio­
metry varies with nutrient status (e.g., Mitra & Flynn 2006).
Without doubt the work of Droop (the original paper cited now in excess of 330 times) has made
a profound impact on our science. Indeed, sometimes the cell quota model is referred to simply
as the Droop equation (e.g., Borchardt 1994, Oyarzun & Lange 1994, Pascual 1994), just as the
Monod equation carries the name of its originator. The history of the quota model, and of the initial
1
KEVIN J. FLYNN
experiments of Droop, is described by Leadbeater (2006). However, the quota model represents but
a small part of the prodigious research output of Droop and his coworkers. Indeed, the quota model
itself almost seems like an aside in the complex (and well worth reading) story of the vitamin B12
nutrition of the microalga Monochrysis (Droop 1968). Alas, most of the works by Droop still remain
to be digitised and are thus not as readily available to a global audience as they should be.
The aim of this review is not to consider technical aspects or detailed results but to consider the
uses of the quota approach subsequent to Droop’s work, to draw attention to some of the abuses and
misconceptions, and to consider insights from the mathematical analyses of quota-type constructs
as they pertain to the current development of phytoplankton models.
The quota description
The Droop quota model originated as a purely empirical description of the relationship between the
cell quota (an amount of a resource within a cell, hence ‘cell quota’) and the organism’s steady-state
growth rate. The original work was conducted with reference to vitamin B12, later extended to P, and
then even to light (Droop et al. 1982). To differentiate the Droop description type from any others,
it will be accorded the term DQuota from hereon. The original parameter names are not used here
because kQ (etc.), used to signify the subsistence quota by Droop, is confusable with the MichaelisMenten/Monod half-saturation constant k. The DQuota description is thus


µ = µ max′ ⋅ 1 − Qmin  Q


(1)
where µmax′ is the theoretical growth rate at infinite quota, Qmin is the minimum (subsistence) quota,
Q is the current quota, and µ is the resultant growth rate. The curve varies in shape over realistic
values of Q and µ with the values of Qmin and µmax′.
Quotas were originally reported of nutrient per cell, although in later works nutrient:joule quotas were used (Droop et al. 1982), which could be considered as akin to nutrient:C quotas. There
are some important differences between cell- and C-based nutrient quotas, which are considered
further below. However, the concept that growth rate varies with the value of the internal nutrient
quota remains the same.
To normalise the DQuota curve, making the value of µmax′ meaningful, such that when Q attains
the maximum value Qmax then µ = µmax, the Droop description can be rewritten as
µ = µ max ⋅
 Qmin 
1 −

Q

 Qmin

1 −

Q

max 
(2)
The relative growth rate µrel is thus given by
µ rel =
 Qmin 
1 −

Q

µ
=

µ max  Qmin
1 −

Qmax 

(3)
From the form of the DQuota description in Equation 2 it can be seen that the curve becomes increasingly hyperbolic as the ratio Qmax:Qmin increases. From a physiological point of view then, the
2
USE, ABUSE, MISCONCEPTIONS AND INSIGHTS FROM QUOTA MODELs
expectation is that a nutrient that can be accumulated to high relative concentration (such as vitamin
B12 and P) can be essentially amassed in a form that does not affect growth. Excess P, for example,
may be accumulated as polyphosphate in certain microbes (e.g., Rhee 1973, Watanabe et al. 1987) so
that Qmax:Qmin may easily exceed 10. In contrast, N, which cannot be accumulated in an inert state,
does not display such a wide quota range (for N, Qmax:Qmin = ca. 4) and accordingly one may expect a
flatter relationship between the quota and growth rate, as the Droop model indeed describes.
There are various other quota-type descriptions in the literature. The curve attributed to
Caperon & Meyer (1972) contains an additional parameter, Kq, which allows the form of the curve
(Equation 4) to be altered independently of the ratio of Qmax:Qmin. Another constant (µmax″), akin to
µmax′ in the Droop formulation, is used as a scaling constant to describe µ.
µ = µ max′′ ⋅
(Q − Q )
(Q − Q ) + K
min
min
(4)
q
The similarity between Equation 4 and that describing Michaelis-Menten enzyme kinetics is
apparent; the quantity Q − Qmin equates to the concentration of the growth-limiting substrate, with
Kq analogous to the half-saturation constant for the process. However, the shape of this hyperbolic
curve when Q ≤ Qmax, can be varied from being effectively linear (Kq and µmax″ very large), to a true
rectangular hyperbola (Kq small, and µmax″ tending to µmax).
Equation 4 can be normalised (Flynn 2002) to remove µmax″, allowing the direct use of µmax, so
yielding the description given in Equation 5. This normalised quota description is termed nQuota
from hereon.
µ = µ max ⋅
(1 + KQ) ⋅ (Q − Q ) (Q − Q ) + KQ ⋅ (Q − Q )
min
min
max
(5)
min
Constant KQ is dimensionless, unlike the value of Kq in Equation 4, which has the same dimensions
as the quota. KQ must not be confused with Kq (cf. Baklouti et al. 2006). The value of KQ sets the
curve form irrespective of the unit basis (cell or biomass) or of the numeric range of Qmin and Qmax
(Figure 1). Values of KQ exceeding 10 give linear relationships (Figure 1A).
Figure 1B shows the relationship between Qmax:Qmin and the value of KQ (KQequiv) that allows
the original Droop model (DQuota) and nQuota descriptions to yield the same values of µ. The value
of KQ required to obtain equivalence is
−1
KQ
equiv
Q

Qmin
=
=  max − 1 Qmax − Qmin  Qmin 
(6)
While the DQuota description (Equation 2) has a curve form set by the ratio of Qmax:Qmin (and which
becomes linear as Qmax:Qmin approaches unity), nQuota (Equation 5) can describe any curve from
linear to rectangular hyperbolic irrespective of the value of Qmax:Qmin. The simplicity of the DQuota
equation (three constants) versus nQuota (four constants) is thus bought at the cost of a lack of flexibility. However, as considered below, it may be possible to constrain KQ for a given nutrient type
and so remove a free variable.
Cell versus biomass quota descriptions
Original emphasis on quota experimentation centred on the cell quota. It can be argued that the
cell (organism) is the central unit of life and thus warrants its position as the quota base. Although
3
KEVIN J. FLYNN
10
1.0
KQ = 0.05
KQ = 0.1
0.8
1
µrel
0.6
KQequiv
KQ = 0.25
KQ = 1
0.4
KQ = 10
0.1
Qmin:Qmax = 0.2
0.01
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1
1.0
10
100
Qmax:Qmin
Q:Qmax
(B)
(A)
Figure 1 The shape of the normalised quota (nQuota) curve (Equation 5) with different values of KQ (A).
At the value of Qmax:Qmin used (5; Qmin:Qmax = 0.2), the shape of the Droop quota (DQuota) curve is given when
KQ = 0.25 (bold line). Panel B shows the value of KQ required (KQequiv; Equation 6) for the nQuota equation to
give the same shape as a DQuota curve (Equation 2) at different values of Qmax:Qmin.
there is a long history of studies of size-related phytoplankton physiology (Banse 1976, Blasco et al.
1982), and there may be various general trends relating cell size to activity and growth rates, intracellular biochemistry will be most closely related to the concentration of material available within
the cell, and hence to biovolume, which equates to carbon (C). Cell-size-scaled functions are most
closely related to the passage of material and energy into the cell. As the whole basis of the quota
concept is that internal rather than external nutrient concentrations regulate growth, it appears to be
more logical to describe quotas in terms of C.
Cell-based quota relationships are also inevitably skewed by the fact that cell size varies with the
cell cycle and with various environmental and physiological factors. P-deprived cells may be larger
or alternatively of similar size to P-sufficient cells (Lehman 1976, Gotham & Rhee 1981, Elrifi &
Turpin 1985, John & Flynn 2002), while N-deprived cells are typically smaller (e.g., Davidson et al.
1992, Wood & Flynn 1995). This complicates the interpretation of other issues, such as changes in
the quota of non-/lesser limiting nutrients as a function of limiting nutrients (e.g., Elrifi & Turpin
1985). Droop et al. (1982), using calorimetry to derive Joules cell−1 (and assuming here that joules:C
is constant), noted no variation in cell size with light- or indeed with vitamin B12-limited growth,
although they did warn about problems of comparing cell and biomass quotas. Others have noted
changes in cell quota with light (Zevenboom et al. 1980, Falkowski et al. 1985, Healey 1985) as well
as with temperature (Goldman 1979). Zonneveld et al. (1997) modelled changes in cell size in lightlimited algae. Different combinations of factors affecting cell size will then affect the packaging
effect of the photosynthetic apparatus (e.g., for diatoms; Taguchi 1976) while nutrient stress will
further affect C fixation, all of which generates additional problems (Liu et al. 2001).
The term ‘cell quota model’ is ambiguous, which is unfortunate given that quota models based
on different units are not necessarily comparable. Some, like Spijkerman & Coesel (1998), specifically use the term ‘cellular quota’ to indicate that the quota used is on a cell basis. Armstrong (2006)
refers to a ‘nitrogen cell quota’ and then assigns units of N:C. The ‘carbon quota model’ could also
4
USE, ABUSE, MISCONCEPTIONS AND INSIGHTS FROM QUOTA MODELs
be considered ambiguous because it could (and usefully) be referring to the amount of C within a
cell. The simplest solution is perhaps to refer to the units in the title of the model (‘nutrient-C quota
model’; ‘nutrient-cell quota model’).
There are rather few occasions where a model refers to both C- and cell-based quotas. Usually
this is done to accommodate specific requirements, notably simulating changes in cell size (Flynn
& Martin-Jézéquel 2000, Flynn 2001) and changes in cellular concentrations such as toxins (John &
Flynn 2002, Davidson & Fehling 2006). Through the use of X-ray microanalysis it is now possible
to determine C:N:P for individual cells (Heldal et al. 2003). However, while biochemically based
models need quotas in terms of a biochemical base (typically C), in the simulation of lag phases,
where cell-cycle events may be important, operating with a cell quota model may help (Cunningham
& Maas 1978, Davidson et al. 1993, Davidson & Cunningham 1996).
If one accepts that, mechanistically, the quota description is best considered on a C rather than
cell basis, then individual-based models describing the activity of plankton in Lagrangian scenarios
(Woods & Barkmann 1994) should couple cell and C quota descriptions. Relying on a cell quota
description of growth for what may well be synchronised populations of cells could lead to significant errors because cell sizes (and hence for example N:cell) vary 2-fold during their cell cycle; this
2-fold variation over the cell cycle is a large fraction of the 3- to 4-fold variation expected in N:C in
N-starved compared with N-replete cell suspensions.
Ideally, then, a coupled C-cell structure is desirable, the C:cell relationship affecting resource
acquisition, whereas the nutrient:C quota affects internal biochemical dynamics. Ultimately,
however, and perhaps most importantly for the selection of C-based over cell-based formulations,
the vast bulk of ecosystem models use biomass as the main unit, and not organism numbers.
Unfortunately, most traditional quota experimentation was cell based and conversion between
cell and biomass bases is not easy. The result is a large literature of data (little of which involves
marine species of ecological importance) reporting various combinations of Qmin, Qmax, Kq (these
three terms usually in terms of cells), Qmax:Qmin, µmax, and so on, but which are not readily usable for
transforming into other units or quota formats. The form of the quota curve may thus be expected to
vary (and will be shown to do so below) depending on growth conditions and on whether the basis of
the quota is cell or C (or any other unit, such as dry weight). In the following, to differentiate between
different specific values of KQ (as used in nQuota, Equation 5) these will be identified by subscript;
such as KQcell, KQC or KQdryweight for cell, C or dry weight quota-specific values, respectively.
There are few datasets available for making simultaneous comparisons of cell quota and C
quota models, and even fewer for multinutrient applications (e.g., Elrifi & Turpin 1985, Liu et al.
2001, John & Flynn 2002). Figure 2 shows data reconstructed and transformed from figures shown
in Elrifi & Turpin (1985) for the freshwater chlorophyte Selenastrium. Note the variation in cell
size with nutrient status (Figure 2E; a feature that Elrifi & Turpin (1985) played down), that NKQ
> PKQ, and KQcell < KQC. One of the most interesting features is that with P limitation N:C falls
(Figure 2C), while with N limitation P:C increases (Figure 2D); the pattern in cell-specific quotas
(Figure 2A,B) is different because of the variation in cell size (Figure 2E). An explanation for the
variation in C-specific quotas (Figure 2C,D) is sought in the section on nutrient transport regulation.
The similarity between the relationships of growth rate versus Chl:C for N- and P-limited growth
(Figure 2F; see also Liu et al. 2001) is consistent with the demand for C controlling the synthesis of
the photosystems, and hence chlorophyll (Flynn 2001).
Figure 3 shows the fit of cell- and C-based nQuota models of N and P limitation to the batch
culture data for a dinoflagellate. Both models fit the data but, because of the changes in cell size
during P stress (Figure 3E), the shape of the quota curves is different (note the different values of
PKQ
P
cell and KQC). That P-limited cells became larger compensates for changes in growth rate in
P-limited growth (Figure 3E); the C quota relationship covers a greater range of Qmax:Qmin and the
5
KEVIN J. FLYNN
2.0
2.0
NKQ
cell = 0.5
1.6
PKQ
cell = 0.155
N-limited
1.6
µ (d–1)
µ (d–1)
P-limited
1.2
0.8
1.2
0.8
0.4
0.4
0.0
0.0
0.0
0.5
1.0
1.5 2.0 2.5
pgN cell–1
3.0
nQuota fit
3.5
0.0
0.2
0.4
0.6
pgP cell–1
(A)
2.0
1.6
nQuota fit
NKQ = 10
C
µ (d–1)
µ (d–1)
2.0
P-limited
1.2
0.8
1.2
0.8
0.4
0.4
0.0
0.0
0.05
0.10
0.15
1.0
(B)
N-limited
1.6
0.8
0.20
PKQ = 0.44
C
0.00
0.25
0.02
0.04
0.08 0.10 0.12 0.14
N:C
P:C
(C)
(D)
20
0.05
N-limited
Chl:C
pgC cell–1
0.04
P-limited
15
10
5
0.03
0.02
N-limited
0.01
0
P-limited
0.00
0.0
0.5
1.0
µ (d–1)
1.5
0.0
2.0
(E)
0.5
1.0
µ (d–1)
1.5
2.0
(F)
Figure 2 Transformed data digitalised and recompiled from Elrifi & Turpin (1985) for Selenastrum minutum, showing nQuota (Equation 5) fits to cell-specific data (A,B) and C-specific data (C,D), and changes in cell
size (E) and Chl:C (F) with growth rate. Data are for steady state under either nitrate-N or P-limited growth.
Ratios are by mass; Redfield mass N:C = 0.176 and P:C = 0.024.
6
USE, ABUSE, MISCONCEPTIONS AND INSIGHTS FROM QUOTA MODELs
4000
10
C-biomass (mgC L–1)
Cells mL–1
3000
P-replete
2000
P-deplete
1000
8
P-replete
6
P-deplete
4
2
0
0
0
0.00
10
20
30
40
0
10
20
30
40
Time (d)
Time (d)
(A)
(B)
C-quota (gN gC–1)
C-quota (mgP gC–1)
0.05
0.10
0.15
0
0.20
1.0
1.0
0.8
0.8
NKQ = 8.4
C
5
10
15
20
PKQ = 0.7
C
0.6
µrel
µrel
0.6
0.4
0.2
0.4
0.2
NKQ
cell = 10
PKQ
cell = 1.8
0.0
0.0
0.0
0.1
0.2
0.3
0.4
0.5
0
10
20
30
40
50
Cell-quota (pgN cell–1)
Cell-quota (fgP cell–1)
(C)
(D)
60
6
Cell Size (pgC cell–1)
5
4
3
2
P-replete
1
P-deplete
0
0
10
20
30
40
Time (d)
(E)
Figure 3 Fits of cell quota and C quota models to the data of John & Flynn (2002) for P-replete (N-limiting)
and P-limiting batch growth of the dinoflagellate Alexandrium fundyense. The forms of the nQuota curves are
shown in (C) and (D) for N and P, respectively. Also shown is the cell size (E); P-limited cells approach double
the size of non-P-limited cells.
7
KEVIN J. FLYNN
curve is greater (PKQC < PKQcell in Figure 3D; CF PKQC > PKQcell in Figure 2). The DQuota curve
equiv
=
descriptions with reference to the same Qmin and Qmax values shown in Figure 3C,D yield N KQcell
N
equiv
P
equiv
P
equiv
0.38, KQC = 0.38, KQcell = 1.38, KQC = 0.46, which are very different from the KQ values
N
equiv
P
equiv
< KQ
while NKQ > PKQ.
shown in Figure 3 for nQuota, not least because KQ
In order to readily compare quota-based phenotypic descriptions between different organisms
and construct functional group models, a common base is needed. From the above, one can argue
that quota descriptions for comparative purposes are best C-based (avoiding the many orders of
magnitude variation in cell size between organisms and hence in values of Q and Kq as used in
Equation 2 or Equation 4) and are most readily compared between organisms when using the nQuota
formula; the dimensionless curve descriptor KQ allows a comparison between the implications of
quota-µ responses for different nutrients.
Although Qmax:Qmin, and indeed the absolute cell-based value of Qmin, have been used to define
competitive advantage, the shape of the quota curve (which is fixed in DQuota as a function of
Qmax:Qmin; Equation 6) is important in conferring competitive advantage (Flynn 2002). A low value
of KQ (Figure 1A) is advantageous, especially if nutrient stress is not long-lived or too extreme. As
such a condition (i.e., non-extreme limitation) is most likely to occur in nature, the form of the upper
range of the quota-growth curve is the most important for ecosystem models.
Empirical to mechanistic relationships
While the original Quota model was never intended to offer anything other than an empirical
description, a mechanistic basis may be sought for the quota concept. That growth would be related
positively to the amount of substrate within an organism, and that there must be a finite lower
limit below which growth cannot occur (the subsistence quota Qmin), is not an unexpected result.
Consistent with the form of the Michaelis-Menten equation for enzyme kinetics (and even though
growth is a function of myriad enzymic reactions) one may also expect this quota-µ relationship to
be hyperbolic, or perhaps sigmoidal. The availability of internal substrate (i.e., Qmax − Qmin) does
not refer simply to unassimilated material (e.g., nitrate within a vacuole) but also to material that
has been assimilated and that can be recycled and redistributed internally. Clearly the latter requires
more processing effort, and growth (in C and/or cell terms) reliant on internal recycling may be
expected to be slower than using unassimilated material made available in ideal form. The nature
of the nutrient, and the manner in which an organism accumulates, distributes and uses it, will thus
have an impact on the shape of the quota-growth curve, as reflected in the value of KQ.
The shape of the quota-growth curve (KQ) describes an important phenotypic characteristic
(Flynn 2002). Some workers specifically make the relationship between the quota and µ a linear
function. For example, Geider et al. (1998) make µ a linear function of the N:C quota; the form of
this relationship is given by Equation 7, having the same constants as Equation 2.
D
µ = µ max ⋅
Q − Qmin
Qmax − Qmin
(7)
This linear relationship for N appears reasonably robust (NKQC > 5 for over a dozen contrasting algal
species; Flynn unpublished). However, Flynn et al. (2002), using NKQC = 3, commented that even a
shallow curve could have important implications for the behaviour of phytoplankton consuming N
within a light-dark cycle.
As noted, the initial shape of the curve, leading back from Qmax to Qmin as nutrient stress develops, is likely to be all important in nature, where extreme nutrient limitation is not likely due to in
situ nutrient recycling and because loss processes are likely to exceed µ at lower growth rates. By
8
USE, ABUSE, MISCONCEPTIONS AND INSIGHTS FROM QUOTA MODELs
default, the Droop equation assigns a more curved form for the N quota. Thus values of NKQ assumequiv
= 0.2, Elrifi & Turpin 1985; N KQ equiv > 0.3 in various
ing Droop kinetics describe curves ( N KQcell
data collated and presented in Morel 1987) that are more hyperbolic than nQuota fits (NKQ typically
>3). For P, against which classic Droop (DQuota) kinetics appear much better suited than they do for
equiv
< 0.25, Gotham &
N (Caperon & Meyer 1972), the value of KQ is typically less than 0.2 ( P KQcell
equiv
equiv
equiv
< 0.15, Elrifi & Turpin 1985; P KQcell
= 0.052–0.175, Morel 1987; P KQcell
< 0.1,
Rhee 1981; P KQcell
P
equiv
Grover 1991; KQdryweight < 0.075, Ducobu et al. 1998). Whether the much higher values for the dinoflagellate PKQ shown in Figure 3 are typical of these organisms is not known. Certainly it would be
useful to know if different phytoplankton groups expressed different NKQC and PKQC values. From
equiv
of about 0.05.
Figure 5 in Droop (1968) the curve for vitamin B12 is tight, with B12 KQcell
Why should the quota-µ relationship for N typically have a quasi-linear form (i.e., NKQC > 3)?
Nitrogen cannot be accumulated to any significant amount to support another generation in an
inorganic, biochemically inert, form (in total contrast to P; Watanabe et al. 1987). The amount of
inorganic N (as nitrate, for example) that may be accumulated cannot equate to more than a few
per cent of that required for construction of a daughter cell. Nitrogen can only be accumulated at
high densities (gN per cell volume) in organic form, and indeed all the vital components of the cell,
other than the membranes and cell walls, are dominated by proteinaceous and nucleic acid-based
compounds. Accordingly, the concentration of enzymes, photosystems, and so on for a given set of
conditions may be expected to relate more or less directly to the rate of growth. In contrast, nutrients
that can be accumulated to great excess (P, Fe, vitamins, etc.) would be expected to have a very different functional relationship between the quota and growth rate.
Although the linearisation of the quota description for N appears justified, there is no evidence
to support the universal adoption of such simple linear relationships for non-N (P, Si, Fe) quota controls of growth (e.g., Moore et al. 2002), and plenty of evidence to the contrary. While the original
DQuota model by default included a non-linear response curve, the removal of that non-linearity
could be viewed as unjustified. Doing so could be considered as similar to replacing a hyperbolic
description of external resource acquisition (e.g., nutrient uptake) with a linear function (Holling
Type I). An inappropriate choice of quota description has important ramifications beyond the modelling of algal growth dynamics because a proper simulation of changes in the nutrient quota and of
associated changes in behaviour by zooplankton is important in predator-prey simulations (Marra
& Ho 1993, Mitra & Flynn 2005).
Although the quota-growth curve describes growth rate as a function of internal nutrient resource
availability, one may also expect that beyond a certain upper value for Q, no further change in the
growth rate will occur. Thus, in consideration of the P quota, in those groups in which P can be
laid down as polyphosphate (which excludes diatoms and dinoflagellates) this accumulation product
is effectively biochemically inert and further accumulation, while raising Q, cannot increase the
growth rate. Indeed, at the extreme (most likely in consequence of growth limitation by some other
factor) an overabundance of such an inert material occupying the cell volume could conceivably be
counterproductive. Flynn (2003) suggests the inclusion of an absolute maximum value of Q, Qabs,
and a redefinition of Qmax so that this is now the value of Q sufficient (if all else is in excess) to
support µmax. Thus, for example, ammonium-grown phytoplankton may not grow any faster than
nitrate-grown cells but the N:C quota of the former can be higher; both cell lines can attain Qmax
(and hence µ attains µmax) but ammonium-grown cells can attain a value of N:C of Qabs while such a
high value of N:C represses the consumption of nitrate so that these cells cannot attain a N:C much
above Qmax (Flynn et al. 1999). This may explain why Liu et al. (2001) report a curvilinear quota
curve for ammonium-supported growth, while others (e.g., Geider et al. 1998) use a linear function
(note, however, that the original Geider et al. (1998) model was fitted to data (Davidson et al. 1992)
for ammonium-grown cells).
9
KEVIN J. FLYNN
Thus, there would appear to be grounds to assign a mechanistic meaning to the quota concept,
although not to the original DQuota equation itself. A mechanistic basis is easier to consider from
biochemical arguments on a C basis, rather than using the original cell quota description (Droop
1968). That mechanistic basis, and our understanding of it, is important and it should not be treated
lightly by unjustifiably altering the form of the quota-growth curve.
Applicability of the quota approach for different nutrients
Since the original description, other nutrients have been subjected to the quota treatment. These
include, in addition to vitamin B12 and P, N, Fe, Si, and even light (Droop et al. 1982, Baird &
Emsley 1999). The nature of these nutrients, their functional role within phytoplankton, differs
greatly and has a profound impact on the applicability of a quota approach.
From an empirical point of view, a quota application for Si may be justified; a diatom grown in
a Si-limited chemostat (i.e., at steady state) shows a relationship between Si quota and µ (Paasche
1973). However, at a mechanistic level a quota relationship for Si is not acceptable because previously assimilated Si is not available for redistribution within the cell (though it may be redistributed
via dissolution of Si frustules from cells that have lysed; Nelson et al. 1976, Fehling et al. 2004).
There is a quota relationship for Si in a Si-limited chemostat because at steady state, Monod and
quota descriptions fit the same data (Droop 1973). However, while the growth rate can explain the
quota, the Si quota should not be used to deduce the growth rate. The link between Si and µ should
be made directly to external nutrient availability, operating in a Monod-like manner (Flynn &
Martin-Jézéquel 2000).
A rather different issue affects the usefulness of a Fe quota-µ relationship. The need for Fe
varies greatly with light availability because the critical role of Fe in photosynthesis (Raven 1990)
places a variable demand for this element as cells regulate photosystem synthesis during light acclimation. The need for Fe also increases with growth rate in general (via the role of Fe in respiration)
and with the consumption and hence reduction of nitrate as the N source (Raven 1990, Sunda &
Huntsman 1997, Armstrong 1999, Flynn & Hipkin 1999, Kustka et al. 2003). In consequence, a
single quota relationship (on a cellular or C base) is not expected for Fe. Figure 4 shows the output
of the mechanistic model of Flynn & Hipkin (1999) as used by Fasham et al. (2006). Although
there are no data against which to fully tune such models, the model structure costs Fe-mediated
High PFD
Growth rate (d–1)
1
0.1
Low PFD
Ammonium
Nitrate
0.01
0
50
100
150
200
250
Fe:C (µg:g)
Figure 4 Simulated relationship between the Fe:C quota and growth rate for a diatom growing on nitrate
(thin lines) or ammonium (thick lines) at different photon flux densities (PFDs). At any given PFD the ammonium curve is higher than that for nitrate. Output is for the diatom model used by Fasham et al. (2006).
10
USE, ABUSE, MISCONCEPTIONS AND INSIGHTS FROM QUOTA MODELs
growth processes between the three major Fe demands according to biochemical knowledge. Values
of FeKQC for the curves shown in Figure 4 range from 0.05 (ammonium grown, highest light) to 1
(nitrate grown, lowest light). Most likely there is not a single value of KQ for other nutrients either,
the relationship varying depending on other external factors (this being especially likely for KQcell in
cell quota descriptions because of the variation in C cell−1). The quota curves are thus not fixed but
vary with C limitation, consistent with the variation in critical N:P with light (Leonardos & Geider
2004). However, the situation for Fe can be expected to be so extreme it renders a single simple Fe
quota description (fixed FeKQ) for all likely nutrient and light conditions effectively useless.
While vitamins, specifically B12, have enjoyed a recent revival in interest (Croft et al. 2005,
Bertrand et al. 2007, Droop 2007), N and P are the most important nutrients for quota descriptions
in ecosystem models. Of these two, the quota-µ interaction for N is also poorly described using the
DQuota structure. The values of N:C and P:C lend themselves to use not only in quota controls of
growth rate but also in control of nutrient acquisition. Thus N:C can be used to control ammonium
versus nitrate versus amino acids versus N2 fixation (Flynn et al. 1999, Flynn 2003, Stephens et al.
2003). Likewise, P:C could be used to control the use of inorganic versus organic P sources; growth
using dissolved organic P need not be limiting and expression of phosphatase activity indicates sufficient internal stress to derepress enzyme synthesis and not necessarily P-limited growth. In both
instances the stress that biologically derepresses the use of alternative nutrients can be linked to
declining nutrient:C (N:C and P:C) quotas between Qabs and Qmax. The linkage between quotas and
the control of nutrient transport is considered further but first it is necessary to consider the meaning
of high quota values.
Misconceptions and misunderstandings over high quota values
Two misconceptions have commonly been associated with the use of the quota model; one is that a
high quota indicates a high growth rate, and the other is that the maximum growth rate is attainable
only at the highest quota. In part these are functions of a misunderstanding that has arisen over the
physiological explanation of the nutrient quota. They also relate to interpretations of nutrient:C quotas and to the Redfield ratio. The original quota equation (Droop 1968) contained no reference to a
maximum quota; the maximum rate of growth was tied to the rate of nutrient transport and the quota
(as a nutrient:cell quota) had no obvious biochemical boundary as does a nutrient element:C ratio.
The Redfield ratio (Redfield 1958) is an average elemental (stoichiometric) ratio for oceanic
particulate material. It is widely and colloquially used as an assumed value, if not the optimal
value, of C:N:P for phytoplankton. However, algal N:C and P:C can exceed Redfield values (Geider
& La Roche 2002), the highest growth rates often being associated with values exceeding Redfield,
and there is no biological (physiological) basis for such a set ratio (Klausmeier et al. 2004a). The
optimal C:N:P is also expected to vary with cell size (higher N:C in smaller cells) and in eukaryotes
versus prokaryotes (higher P:C versus lower P:C respectively in smaller cells) (Raven 1994). The
qualitative ranges of N:C and P:C that may be expected under N, P or light limitations are shown
in Figure 5.
It has long been known that a high quota for a given nutrient, or indeed for several nutrients
simultaneously, cannot be interpreted to indicate a high growth rate (Donaghay et al. 1978, Tett
et al. 1985). Light and temperature limitations of growth prevent such extrapolations. The quota can
only indicate whether a particular nutrient is non-limiting and not whether growth is occurring at
any particular rate. The more complicated issue involves relating high quotas to high growth rates.
The quota is a ratio, and ratios can be high because the denominator is small, or because the
numerator is large. Thus a high N:C could reflect a relatively high cellular N content (optimal) or a
low C content (suboptimal). From a biochemical standpoint, one expects physiological regulations
to balance the cellular response to these events, to not only increase (up-regulate) acquisition of a
11
KEVIN J. FLYNN
PCabs
P:C
–N
–Light
PCmax
–NP
PCmin
NCmin
N:C
–P
NCmax NCabs
Figure 5 Schematic representation of the C quota range of N and P in relation to N, P or light (i.e., carbon)
limitations assuming that the other factor(s) are non-limiting. Zones indicate N-limitation (-N), P-limitation
(-P), co-NP-limitation (-NP), and light limitation (-Light). Minimum quotas marked NCmin­ and PCmin; quota
required to support maximum growth rates marked NCmax and PCmax; absolute maximum quotas marked NCabs
and PCabs. Scales are only representative, but note the difference between PCabs:PCmax and NCabs:NCmax.
limiting nutrient but, critically, also to decrease (down-regulate) acquisition of non-limiting nutrients. One of the reasons that Flynn (2003) introduced Qabs (absolute maximum possible quota) in
addition to Qmax (quota required to support the maximum growth rate) was in reflection of the fact
that under non-nutrient limiting conditions the quota could exceed Qmax. Values of Q between Qmin
and Qmax would be expected to be associated with up-regulation of nutrient acquisition, and between
Qmax and Qabs with down-regulation. The latter zone would be expected to be especially apparent
for P:C when N or light, or indeed when the intrinsic maximum rate of growth (i.e., cell cycle), is
limiting (e.g., Elrifi & Turpin 1985). It is also noted for N:C with light limitation (Laws & Bannister
1980), with ammonium-grown cells showing higher N:C than nitrate-grown cells (Wood & Flynn
1995, Flynn et al. 1999). Indeed, the variation of N:C over the light-dark diel cycle can be linked
to the control of dark-N assimilation, which is especially important for nitrate assimilation (Clark
et al. 2002, Flynn et al. 2002).
The constant Qabs (Flynn 2003) represents the absolute maximum possible value of Q at which
nutrient transport must be terminated. However, it is apparent that depending on other factors,
transport may be terminated at a lower value of Q, at a value here termed QTcon. Thus in P-limited
cells, NCTcon for N assimilation may be less than NCmax, and hence N:C in P-limited cells declines
ammonium
>
(Figure 2). For different N sources different values of NCTcon are expected such that NCTcon
nitrate
N2
NCTcon > NCTcon . Similarly, different values of QTcon are also expected for the use of dissolved
organic P (requiring the synthesis of the phosphatase enzyme) versus the direct use of dissolved
DIP
DOP
> PCTcon
).
inorganic P (i.e., PCTcon
Armstrong (2006), commenting that the implementation of the linear form of the N:C quota
model by Geider et al. (1998) is inappropriate at low light, offered an optimisation model alternative to describe why the maximum N:C quota is not an optimal quota under such conditions of light
(= C) limitation. The mechanistic basis for this lies in the (de)repression control of cell physiology
that is linked to the cellular concentration of metabolites (Flynn 1991, 2003, Flynn et al. 1997,
2002). The typical algal model does not refer to metabolites and to product-inhibition links (see
Flynn et al. (1997) for ammonium nitrate controls via such links and John & Flynn (2000) for P controls) because of the resultant increase in complexity and decrease in integration step size required
12
USE, ABUSE, MISCONCEPTIONS AND INSIGHTS FROM QUOTA MODELs
to run such models. However, a more empirical association can be made to cellular C:N:P to drive
an active control between quota nutrient acquisition. For N:C and the interactions between light,
ammonium and nitrate acquisition this association is shown in Flynn et al. (2002), with elevated
values of N:C downregulating N-source acquisition. Thus, at very high N:C ratios N-source uptake,
and thus growth rate, is restricted. At the same time, the demand for C (and hence the link to Chl:C)
is heightened, which can be used to modulate the synthesis of Chl:C (Flynn 2001).
The control of nutrient acquisition in models is more of a challenge than the use of quotas to
control growth and the mechanisms by which it is achieved has important implications (Morel 1987,
Flynn 2002). This control is at least as important for the non-limiting nutrient as for the nutrient
that limits growth.
Nutrient acquisition — the quota-transport interface
The great novelty of the Droop approach, in contrast to that of Monod, was to relate growth to
the internal rather than external nutrient pool concentrations. In reality, growth is a function of
both these pools and the methods by which nutrient acquisition (via Monod-like kinetics) into the
quota is described, and the subsequent control of nutrient transport and growth by the quota, are
all important for the final model behaviour. The simplest way to link these functions is by defining
the maximum nutrient transport rate Tmax as equal to µmax·Qmax (Goldman & McCarthy 1978). This
makes the tacit assumption that the maximum growth rate is actually simultaneously co-limited by
both nutrient transport and internal processing. However, this simple link between transport- and
quota-type models fails to describe the development of surge transport capabilities that offer competitive advantage (Turpin & Harrison 1979) and change nutrient uptake ratios. This link also does
not down-regulate nutrient acquisition when another factor limits growth.
Empirically these types of interactions were considered over three decades ago (Droop 1974,
1975) and two quota descriptions were identified that could be viewed as functioning in opposite
directions. For the limiting nutrient, substrate availability controls (limits) nutrient uptake, which
controls the limiting quota QL and subsequently µ via the quota equation. For the non-limiting
nutrient, µ (as set by QL) controls the non-limiting quota QN via a control on the uptake of the nonlimiting nutrient, which in turn affects the remaining non-limiting substrate concentration. Droop
(1975) also considered the special instance of a limiting yet inexhaustible nutrient (namely light).
For the limiting nutrient the quota relationship in Droop’s arguments used the real Qmin, while
control of the acquisition of non-limiting nutrients was by reference to an apparent Qmin; this latter
value equates in some ways to QTcon in the discussion above. Davidson & Gurney (1999) make a version of this regulation using a hyperbolic regulation term rather than using a quota description. The
advance that we make now is to recognise the mechanism behind these interactions and how they
can be manipulated to control transport of different nutrients.
The kinetics of nutrient acquisition are a consequence of changing transport capabilities and
are not fixed; they vary with the type of nutrient and the nutrient status as reflected by the quota
(Smith & Kalff 1982, Ikeya et al. 1997, Flynn et al. 1999). Knowledge of the existence of interactions between external and internal nutrient availability and surge acquisition dates from the 1970s
(Conway et al. 1976). Droop (1973) postulated a linear relationship between cell-specific nutrient
transport and the nutrient:cell quota and ended by questioning whether in non-steady state the transport capacity of the lesser limiting nutrients become controlled (as we now know it to be) by downregulation. Elrifi & Turpin (1985) later concluded that the original DQuota model could not handle
the consumption of non-limiting nutrients correctly over the entire range of nutrient supply ratios
(here N:P) and growth rates because of the lack of fidelity in such controls and the basis of their construction (making reference to external nutrient concentrations). Zonneveld (1996) declares there is
13
KEVIN J. FLYNN
no meaningful basis for the Droop handling of non-limiting nutrients. The point remains, however,
that Droop appreciated that the handling of non-limiting nutrients is important; most experimental
and modelling studies place their emphasis on the assimilation of single nutrients (i.e., that which is
limiting) or upon single nutrient-light interactions.
The ability to perform surge uptake and to modulate uptake of non-limiting nutrients (e.g.,
Conway et al. 1976, Conway & Harrison 1977), especially as they may feature to different extents in
different organisms, has important implications also for the theoretical analyses of DQuota dynamics (e.g., Lange & Oyarzun 1992, Pascual 1994, Bernard & Gouze 1995, Smith 1997). Analyses of
the competitive advantage based solely on transport kinetics (Healey 1980, Button 1991) are also
inadequate. It is a balance of both transport and subsequent internal factors (in part summarised by
the form of the quota curve) that govern the outcome of competition (Flynn 2002) and indeed makes
experimental determinations of transport kinetics such a challenge (Flynn 1998).
There are various examples where the combined kinetics of Monod and quota equations have
been studied, together with empirical data, to drive discussions on nutrient transport kinetics, especially in chemostats at low dilution rates where external nutrient concentrations fall below detection
(e.g., Gotham & Rhee 1981). Nutrient stress, as driven by low chemostat dilution rates relative to
µmax, can have different levels of physiological severity depending on the identity of the limiting
nutrient (thus Si stress may be more severe than N stress at low dilution rates (Harrison et al. 1976)).
In reality, Tmax (rather than being fixed as equal to µmax·Qmax) is itself a variable with Q, typically
initially increasing as Q declines (Gotham & Rhee 1981). Because of this variability, and indeed
because a high Tmax may compensate for a high half-saturation for transport (Kt, nutrient affinity being set by Tmax/Kt), a single set of equations describing Monod-style growth kinetics cannot
describe phytoplankton growth completely. Various approaches have been developed to enhance the
nutrient transport control of the quota model (notably Morel 1987) providing a Droop-based model
for use under dynamic situations (Grover 1991).
There are two issues here. One is the potential surge transport of the limiting nutrient into an
organism previously deprived of that nutrient and the other is of ‘luxury transport’ of non-limiting
nutrients. Even if the transport capacity remains constant (Tmax set by µmax·Qmax), then there is de
facto an increasing capacity for acquisition over that required to satisfy demand as Q declines to
Qmin. Surge transport has been studied for N, P and Si (Conway et al. 1976, Parslow et al. 1984) and
shows different responses for different organisms and different nutrient types. The surge transport
of P (e.g., Smith & Kalff 1982) displays capacities of an order of magnitude above that required to
support the current growth rate (i.e., as Q → Qmin then Tmax → >> µmax ·Qmax). Ammonium displays
similar surge kinetics but nitrate does not (Parslow et al. 1984, Syrett et al. 1986, Flynn et al. 1999).
Hence, Tmax for P can be similar to that for nitrate-N (g element g C−1) despite the 40-fold difference
nitrate
) appears
in minimum quota values. The relationship between the N:C quota and nitrate Tmax (Tmax
nitrate
bell shaped (Flynn et al. 1999). Thus Tmax falls below the demand rate (i.e., µ·(N:C)) at high N:C,
when the N status is so high due to ammonium assimilation or with light limitation that the ability
to use nitrate is repressed. This mechanism prevents nitrate-growing cells from having such a high
nitrate
also falls close to the demand rate at extremely low
N:C as ammonium-growing cells have. Tmax
N:C, when the cell is so starved that it is increasingly physiologically incompetent.
There are various ways in which these changes in dynamics have been explored. Morel (1987)
gives a detailed treatment of simple descriptions of changes in uptake kinetics and how the resultant
value of the apparent half-saturation constant for growth (Kg) also changes. In keeping with experimental data for N-source uptake, Flynn et al. (1997) used linear or curvilinear descriptions, whereas
later Flynn (2003) used sigmoidal functions, which can be readily altered to describe a range of
patterns relating Tmax to the nutrient quota. Although curves are more appropriate as empirical
descriptors (conforming to the way that biochemical feedback processes occur), linear equations
are more tractable for mathematical analysis. Indeed, the simple form of the Droop equations has
14
USE, ABUSE, MISCONCEPTIONS AND INSIGHTS FROM QUOTA MODELs
undoubtedly been instrumental in making them such a popular target for mathematical and theoretical biological studies.
The patterns that are observed between Q and Tmax now provide us with a mechanistic basis for
a description of the nutrient transport component of quota-based models, to enhance the phenotypic
capabilities of the original Droop model. Transport capabilities are under (de)repression regulation,
with synthesis minimised (repressed) when the cell contains a sufficient amount of the incoming
nutrient element and maximised otherwise. The value of Q controlling transport (QTCon) varies with
various factors with interactions that lay behind the form of Figure 5. In reality, rather than being
the whole organism Q that biochemically controls Tmax, control is via metabolites that could be the
nutrients themselves within the cells (transinhibition) or more commonly a downstream product of
nutrient assimilation. Thus for the control of nitrate and ammonium transport, glutamine (Gln) is
considered a likely regulatory metabolite (Flynn 1991) and has been so employed in complex models (Flynn et al. 1997). However, to involve an explicit description of such metabolite pools creates
an overly complicated model for routine use, requiring additional state variables for metabolite
quotas. Even the most demanding of modellers is unlikely to want to consider simulating all of the
phenotypic variation that molecular biology now indicates may be present (with several transporters for each substrate; e.g., Hildebrand 2005). John & Flynn (2000), considering the interactions
between different intracellular P pools, P transport and P:C quota-linked growth, suggest that the
removal of P into inert polyphosphate would enable better decoupling of transport from assimilation. However, they also indicate that it was not necessary to specifically model the presence of
these pools provided that an appropriate variant of the quota model is used. An empirical link back
to the whole organism nutrient quota is thus more appropriate and more desirable (Flynn & Fasham
1997, Flynn 2003, Stephens et al. 2003).
While N stress does not appear to greatly affect P accumulation (although transport is affected
(Conway et al. 1976)), such that the P quota rises to a maximum value and may exceed that attained
during nutrient-replete conditions (Elrifi & Turpin 1985, Liu et al. 2001), P stress appears to have a
greater affect on N assimilation (Healey & Hendzel 1975, Terry 1982, Elrifi & Turpin 1985, Figures 2
and 5). The former event is especially important when one considers marine biogeochemistry because
N limitation will inevitably result in a strong shift away from the Redfield N:P. This is all the more
likely because of the high value of PCabs:PCmax (>1); in comparison NCabs:NCmax is much narrower
(<1). The behaviour of a quota model in which Tmax for the non-limiting nutrient is ultimately brought
to zero as Q → Qabs (Flynn 2003) describes the control of P transport in this situation.
The more interesting event is where P stress is associated with a decline in the N quota (Figures 2
and 5). The significance of this event assumes that N is not exhausted during P-limited growth
(noting, for example, that Elrifi & Turpin 1985 make no mention of measuring residual nitrate
concentrations). This is a potential problem in batch-style experiments; given the great range of
P:C in phytoplankton (Geider & La Roche 2002), and that P can be accumulated to excess during
N limitation, it is not always trivial in a culture system at quasi-natural biomass levels to ensure that
P limitation is not associated with concurrent exhaustion of the N source.
In P-replete cells, the intracellular concentration of glutamine, and the value of the
glutamine:glutamate ratio (Gln:Glu) varies with the N status, being highest in previously N-starved
ammonium-refed cells, lower in nitrate-growing cells, and lowest in N-starved cells (Flynn 1990).
The concentration of Gln is implicated with the regulation of N-source transport (Flynn 1991), but
the impact of P stress upon the intracellular accumulation of Gln is not known. If P stress resulted
in the accumulation of Gln, then this could explain the repression of N transport that must accompany the noted decline in N:C (Figure 2C). From the relationship between N:C and P:C in P-limited
nitrate
nitrate
declines as P:C declines (from data shown in Figure 2; QTcon
= 4.8·P:C
Selenastrum, then QTcon
+0.1125; R2 = 0.91). What is not known is the shape of this relationship in cells grown on ammonium
(Figure 2C shows nitrate-grown P-limited cells). One may expect this relationship to be steeper
15
KEVIN J. FLYNN
than for growth on nitrate (closer to vertical) because ammonium assimilation is repressed at higher
internal concentrations of Gln (and hence of higher N:C; Flynn et al. 1999).
To conclude, the control of nutrient transport is ideally made a function of both the respective
nutrient:C quota and of the quota of other nutrients. There are two components to this: (1) the value
of the quota at which transport halts (QTcon) and (2) the magnitude of Tmax at values of Q < QTcon. In
the simplest form, the combined description is given by Tmax = µmax·Qmax·(Q < QTcon); the Boolean
logic term simply halts transport if Q attains QTcon, with QTcon being a function of other nutrient:C
quotas as required. What can be seen readily is that deviation from Redfield C:N:P may be rapid as
stress is applied and can be strongly divergent between P- versus N-limited cells. Droop recognised,
and modelled, the multinutrient interactions at a simpler level, but appeared (Droop 1975) unsure
regarding whether the resultant model complexity was justified. Today we can be reasonably sure
that the effort is indeed worth it.
Competition, nutrient supply ratios and stoichiometric
predator-prey models
Despite the steady-state origins of the Droop cell quota model, it is still useful in following competition in dynamic systems (Droop 1975, Grover 1991, Ahn et al. 2002). This is so, even though
delays in cell responses are not handled adequately (Davidson & Cunningham 1996), which can
have important implications in competition scenarios (Li et al. 2000). Although links between
nutrient:cell and nutrient:C quotas can be interpreted to explain population growth and resource
availability (Savage et al. 2004), the interplay between nutrient uptake capabilities and the shape
of the quota curve has great capacity for affecting competition between algae (Ikeya et al. 1997,
Vadstein 1998, Flynn 2002, John & Flynn 2002). That diatoms do not accumulate polyphosphate
could be a factor affecting their competitive advantage when under P stress (Egge 1998), especially
if the inability to remove newly assimilated P to an inert form that does not affect further transport
(John & Flynn 2000) prevents diatoms from making best use of P pulses.
The most advantageous quota configuration is a high Qmax:Qmin, low Qmin, high Qabs, coupled
with a low KQ value (Figure 1). This would endow an organism with a capacity to accumulate much
surplus nutrient (Qabs >> Qmax) in times of plenty and to continue growing on that nutrient reserve at
a high relative growth rate for as long as possible in the absence of any new input. The interface with
transport kinetics is important (Klausmeier et al. 2004b), as is the control of non-limiting nutrient
transports (Flynn 2002, 2003) because these processes top-up the quotas and drain the environment
of nutrients required by future generations of potential competitors (Flynn 2005b). Thus the magnitude and control of Tmax is important.
The value of the critical N:P ratio, at which N and P quotas exert equal control of growth (Rhee
& Gotham 1980), varies with growth rate and with light over the diel cycle and with day-integrated
irradiance (Leonardos & Geider 2004). Different algae display different critical N:P ratios at low
growth rates (Ahlgren 1985), as a function of differences in Qmin and in KQ, but these become less
obvious at high µ because of the similarity in values of Qmax. Although competition has been the
subject of many studies since those of Rhee (1974) and others, these studies have been primarily
associated with freshwater phytoplankton (and especially with chlorophytes and cyanophytes, which
accumulate polyphosphate), and considered with nutrient:cell quota formulations. Multinutrient
C-specific studies, especially of marine species, are sorely lacking. The importance of these studies
arises because growth and nutrient consumption must not be made a function only of the most limiting nutrient; otherwise the consumption of the lesser-limiting nutrients is not described correctly
(Sciandra & Ramani 1994, Davidson & Gurney 1999), and competition simulations described using
such models have the potential to give seriously erroneous results (Flynn 2005b).
16
USE, ABUSE, MISCONCEPTIONS AND INSIGHTS FROM QUOTA MODELs
The implications of the quota control of growth are also important for those conducting experiments and ecosystem investigations on the impacts of N:P nutrient supply levels. It is readily
apparent from the manipulation of even the simplest quota models that great care must be taken in
conducting and interpreting batch-style experiments in which the impact of variable nutrient N:P is
studied. The larger Qmax:Qmin, and the smaller the value of KQ, the longer it takes for nutrient stress
to affect growth. Thus it takes very much longer for cultures to become growth limited to the same
extent by P than by N availability. Furthermore, the duration of the period before effective limitation is affected by the extent to which the organisms managed to engage in surge assimilation (i.e.,
the value of Qabs:Qmax and the value of Tmax relative to µ·Qmax) prior to nutrient exhaustion. This is
again far more of a problem for P because the ready accumulation of P (especially in those that
can accumulate polyphosphate) enables Qabs to far exceed Qmax. The nutrient status (quota status)
of the phytoplankton at the start of a given batch study thus becomes an important factor affecting
whether the organisms display characteristics of one or other nutrient limitation during the duration
of the investigation. In steady-state chemostat studies, the importance of the ways in which prior
nutrient history affects the cell’s response to perturbation may be missed, while they can have great
consequences for batch studies and for natural events.
There is increasing interest in modelling the stoichiometric link between planktonic predator and prey and indeed an appreciation of the general importance of stoichiometric differences
between consumers and their food in ecology (Sterner & Elser 2002). It is thus important to get
the quota description of growth rate correct because the description affects the growth not only of
the phytoplankton prey, but also of their predators. The adequacy of the description is especially
important where the interaction is enhanced by the development of antigrazer strategies in nutrientstressed phytoplankton (Mitra & Flynn 2005, 2006). For this purpose, nutrient:C quota relationships are required, although certainly one could argue also for a description of cell size (C cell−1)
because predators capture cells not biomass. In this context, assuming the quota curve is always
linear (Moore et al. 2002) if it is not, or conversely using a DQuota formula to describe the wrong
type of hyperbolic curve, is all the more worrying.
Simulating genetic and phenotypic diversity
Another issue that Droop raised during development of the quota model that is of high relevance to
our current modelling activities is the subject of genetic and phenotypic genetic variation. Droop
(1974) encountered this in the form of ‘fast-adapted’ and ‘slow-adapted’ P- and B12-limited chemostat
cultures of Monochrysis. Harrison et al. (1976) encountered a similar event for Si-limited diatoms.
To what extent such observations reflect evolution within a clone or selection within non-clonal
cultures is not known but for applications of models to field situations the whole topic of diversity is
one that is invariably avoided.
While molecular biology has demonstrated the great diversity of plankton, in total contrast,
modelling typically amalgamates groups, glossing over phenotypic differences that define diversity
and succession (Irigoien 2006). Models assume uniformity within groups, and indeed typically this
is extended to functional groups in the widest possible sense of the term (phytoplankton, zooplankton). We know so little in quantitative terms about integrated algal physiology, including factors such
as how non-limiting nutrients are handled, that we cannot make a fully parameterised model for
even one clone (Flynn 2005a, 2006) and this lack of knowledge makes the whole issue of diversity
even more problematic. Droop (1974) ends by asking whether “the phytoplankton of a region might
be regarded [for modelling purposes] as an envelope, the sum of its component parts, and it would be
interesting to know whether the kinetic properties of this envelope show any coherence”. In fact, it is
now more than a matter of interest because consideration of this matter is becoming a necessity.
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KEVIN J. FLYNN
Conclusions
In one way or another, the development and use of the cell quota model by Droop (1968) has stimulated the research of generations of phytoplankton scientists. It also affects those interested in
predator-prey interactions and those who appreciate the importance of variable stoichiometry in
global-scale ecosystem models. Today the original quota model (Droop 1968) is rarely used, but the
legacy of the work remains powerful. Not infrequently the work of Droop is referenced even though
the methods employed barely follow the original description.
While early researchers appreciated the limitations of the empirical quota descriptions, later
developments have not always maintained a connection with reality as supported by either empirical data fits and/or through a mechanistic basis. The ‘cell quota model’ became the ‘quota model’,
ignoring the differences between C and cell bases that affect the validity of the original empirical
construct. Manipulations of the quota description to describe the competitive advantage of one species over another ignored the vital importance of the nutrient transport process. Furthermore, simplifications of quota descriptions, which have serious implications for simulations of phytoplankton
growth, nutrient transport and trophic dynamics, have been made with little or no justification and/or
no biological validation. There have also been many variations on the theme without a full appreciation of what Droop had achieved by the mid-1970s. In part this is perhaps because the original models, with their cell-based units, did not lend themselves so readily to the C-based models required
in ecosystems. The simplistic elegance of the DQuota equation, while lending itself to ready mathematical investigation, was also inappropriate for one of the main applications, nitrogen.
Assignment of typical values for the normalised, dimensionless curve descriptor KQ in nQuota
(Equation 5) for different nutrient (element) types in different plankton groups would decrease the
number of free variables in models, while better describing the quota-µ relationship than is achievable using DQuota with its fixed curve form. As an approximation, it may be tempting to set the
upper quota values for N:C and P:C to the Redfield ratio (Redfield 1958). However, the maximum
quotas (especially for P:C) readily exceed the Redfield values, and indeed maximum growth rates
may not be attainable at Redfield values (e.g., Figure 2).
One of the greatest challenges still is the correct description of the quota of non- or lesser-limiting nutrients. The call by Elrifi & Turpin (1985) for “much further work … to determine the kinetics
of non-limiting nutrient utilization and their significance to algal competition” has not been met
two decades on (Flynn 2005b). Whether such work is best conducted under steady state (the arena
in which the original quota studies were conducted) or in some form of non-steady-state system
is debatable. That is especially so given the selective pressure that can develop within chemostats
(Dykhuizen & Hartl 1983) affecting the appearance of ‘fast’ and ‘slow’ acclimated populations
(Droop 1974, Harrison et al. 1976, Maske 1982). Furthermore, while a good dynamic model may be
expected to satisfactorily describe steady-state conditions, the opposite need not be so.
The gulf between genetic, phenotypic and modelled diversity needs to be closed. Although we
have a reasonable qualitative understanding of algal physiology, our quantitative understanding is
at best described as incomplete. Likewise, how genetic diversity translates to phenotypic diversity
is unclear. Without an understanding of the breadth of phenotypic diversity we cannot fully explore
the implications for model diversity. However, what we can be sure of is that features such as the
form of the quota-µ curve and of the interactions between quota and nutrient transport will be
important variables in such descriptions.
There is a time and place for all ideas; quite a number of the topics that were explored by Droop
in his early work on the quota concept have recently come back to the forefront of our science.
Whatever the future holds, we can be sure that quota-type models are here to stay.
18
USE, ABUSE, MISCONCEPTIONS AND INSIGHTS FROM QUOTA MODELs
Acknowledgements
I am indebted to Michael Droop, to whom this work is dedicated in his 90th year, because our discussions have added much to the content of this work and to Paul Harrison and to John Raven for
their most useful comments in the later stages of manuscript preparation. Help from John Leftley
and Ian Davies is also much appreciated.
References
Ahlgren, G. 1985. Growth of Oscillatoria agardhii in chemostat culture. 3. Simultaneous limitation of nitrogen
and phosphorus. British Phycological Journal 20, 249–261.
Ahn, C.Y., Chung, A.S. & Oh, H.M. 2002. Diel rhythm of algal phosphate uptake rates in P-limited cyclostats
and simulation of its effect on growth and competition. Journal of Phycology 38, 695–704.
Armstrong, R.A. 1999. An optimization-based model of iron-light-ammonium colimitation of nitrate uptake
and phytoplankton growth. Limnology and Oceanography 44, 1436–1446.
Armstrong, R.A. 2006. Optimality-based modeling of nitrogen allocation and photoacclimation in photosynthesis. Deep-Sea Research II 53, 513–531.
Baird, M.E. & Emsley, S.M. 1999. Towards a mechanistic model of plankton population dynamics. Journal of
Plankton Research 21, 85–126.
Baklouti, M., Diaz, F., Pinazo, C., Faure, V. & Quéguiner, B. 2006. Investigation of mechanistic formulations
depicting phytoplankton dynamics for models of marine pelagic ecosystems and description of a new
model. Progress in Oceanography 71, 1–33.
Banse, K. 1976. Rates of growth, respiration and photosynthesis of unicellular algae as related to cell-size.
Journal of Phycology 12, 135–140.
Bernard, O. & Gouze, J.L. 1995. Transient-behavior of biological loop models with application to the Droop
model. Mathematical Biosciences 127, 19–43.
Bertrand, E.M., Saito, M.A., Rose, J.M., Riesselman, C.R., Lohan, M.C., Noble, A.E., Lee, P.A. & DiTullio,
G.R. 2007. Vitamin B12 and iron colimitation of phytoplankton growth in the Ross Sea. Limnology and
Oceanography 52, 1079–1093.
Blasco, D., Packard, T.T. & Garfield, P.C. 1982. Size dependence of growth-rate, respiratory electron-transport
system activity, and chemical-composition in marine diatoms in the laboratory. Journal of Phycology 18,
58–63.
Borchardt, M.A. 1994. Effects of flowing water on nitrogen-limited and phosphorus-limited photosynthesis
and optimum N/P ratios by Spirogyra fluviatilis (Charophyceae). Journal of Phycology 30, 418–430.
Button, D.K. 1991. Biochemical basis for whole-cell uptake kinetics — specific affinity, oligotrophic capacity,
and the meaning of the Michaelis constant. Applied and Environmental Microbiology 57, 2033–2038.
Caperon, J. & Meyer, J. 1972. Nitrogen-limited growth of marine phytoplankton. I. Changes in population
characteristics with steady-state growth. Deep-Sea Research 19, 601–618.
Clark, D.R., Flynn, K.J. & Owens, N.J.P. 2002. The large capacity for dark nitrate-assimilation in diatoms may
overcome nitrate limitation of growth. New Phytologist 155, 101–108.
Conway, H.L. & Harrison, P.J. 1977. Marine diatoms grown in chemostats under silicate or ammonium limitation. IV. Transient response of Chaetoceros debilis, Skeletonema costatum, and Thalassiosira gravida to
a single addition of the limiting nutrient. Marine Biology 43, 33–43.
Conway, H.L., Harrison, P.J. & Davis, C.O. 1976. Marine diatoms grown in chemostats under silicate or ammonium limitation. I. Transient response of Skeletonema costatum to a single addition of the limiting nutrient. Marine Biology 35, 187–199.
Croft, M.T., Lawrence, A.D., Raux-Deery, E., Warren, M.J. & Smith, A.G. 2005. Algae acquire vitamin B12
through a symbiotic relationship with bacteria. Nature 438, 90–93.
Cunningham, A. & Maas, P. 1978. Time lag and nutrient storage effects in transient growth response of
Chlamydomonas reinhardii in nitrogen-limited batch and continuous culture. Journal of General Micro­
biology 104, 227–231.
Davidson, K. & Cunningham, A. 1996. Accounting for nutrient processing time in mathematical models of
phytoplankton growth. Limnology and Oceanography 41, 779–783.
19
KEVIN J. FLYNN
Davidson, K., Cunningham, A. & Flynn, K.J. 1993. Modelling temporal decoupling between biomass and
numbers during the transient nitrogen-limited growth of a marine phytoflagellate. Journal of Plankton
Research 15, 351–359.
Davidson, K. & Fehling, J. 2006. Modelling the influence of silicon and phosphorus limitation on the growth
and toxicity of Pseudo-nitzschia seriata. African Journal of Marine Science 28, 357–360.
Davidson, K., Flynn, K.J. & Cunningham, A. 1992. Non-steady state ammonium-limited growth of the marine
phytoflagellate, Isochrysis galbana Parke. New Phytologist 122, 433–438.
Davidson, K. & Gurney, W.S.C. 1999. An investigation of non-steady-state algal growth. II. Mathematical
modelling of co-nutrient-limited algal growth. Journal of Plankton Research 21, 839–858.
Donaghay, P.L., Demanche, J.M. & Small, L.F. 1978. Predicting phytoplankton growth-rates from carbonnitrogen ratios. Limnology and Oceanography 23, 359–362.
Droop, M.R. 1968. Vitamin B12 and marine ecology. IV. The kinetics of uptake, growth, and inhibition in
Monochrysis lutheri. Journal of the Marine Biological Association of the United Kingdom 48, 689–733.
Droop, M.R. 1973. Some thoughts on nutrient limitation in algae. Journal of Phycology 9, 264–272.
Droop, M.R. 1974. The nutrient status of algal cells in continuous culture. Journal of the Marine Biological
Association of the United Kingdom 54, 825–855.
Droop, M.R. 1975. The nutrient status of algal cells in batch culture. Journal of the Marine Biological Asso­
ciation of the United Kingdom 55, 541–555.
Droop, M.R. 2007. Vitamins, phytoplankton and bacteria: symbiosis or scavenging. Journal of Plankton
Research 29, 107–113.
Droop, M.R., Mickleson, J.M., Scott, J.M. & Turner, M.F. 1982. Light and nutrient status of algal cells. Journal
of the Marine Biological Association of the United Kingdom 62, 403 – 434.
Ducobu, H., Huisman, J., Jonker, R.R. & Mur, L.R. 1998. Competition between a prochlorophyte and a
cyanobacterium under various phosphorus regimes: comparison with the Droop model. Journal of Phy­
cology 34, 467–476.
Dykhuizen, D.E. & Hartl, D.L. 1983. Selection in chemostats. Microbiology Review 47, 150–168.
Egge, J.K. 1998. Are diatoms poor competitors at low phosphate concentrations? Journal of Marine Systems
16, 191–198.
Elrifi, I.R. & Turpin, D.H. 1985. Steady-state luxury consumption and the concept of optimum nutrient ratios:
a study with phosphate and nitrate limited Selenastrum minutum (Chlorophyta). Journal of Phycology
21, 592–602.
Falkowski, P.G., Dubinsky, Z. & Wyman, K. 1985. Growth-irradiance relationships in phytoplankton. Limnology
and Oceanography 30, 311–321.
Fasham, M.J.R., Flynn, K.J., Pondaven, P., Anderson, T.R. & Boyd, P.W. 2006. Development of a robust marine
ecosystem model to predict the role of iron on biogeochemical cycles: a comparison of results for ironreplete and iron-limited areas, and the SOIREE iron-enrichment experiment. Deep-Sea Research I 53,
333–366.
Fehling, J., Davidson, K., Bolch, C.J. & Bates, S.S. 2004. Growth and domoic acid production by Pseudonitzschia seriata (Bacillariophyceae) under phosphate and silicate limitation. Journal of Phycology 40,
674–683.
Flynn, K.J. 1990. The determination of nitrogen status in microalgae. Marine Ecology Progress Series 61,
297–307.
Flynn, K.J. 1991. Algal carbon-nitrogen metabolism: a biochemical basis for modelling the interactions between
nitrate and ammonium uptake. Journal of Plankton Research 13, 373–387.
Flynn, K.J. 1998. Estimation of kinetic parameters for the transport of nitrate and ammonium into marine
phyto­plankton. Marine Ecology Progress Series 169, 13–28.
Flynn, K.J. 2001. A mechanistic model for describing dynamic multi-nutrient, light, temperature interactions
in phytoplankton. Journal of Plankton Research 23, 977–997.
Flynn, K.J. 2002. How critical is the critical N:P ratio? Journal of Phycology 38, 961–970.
Flynn, K.J. 2003. Modelling multi-nutrient interactions in phytoplankton; balancing simplicity and realism.
Progress in Oceanography, 56, 249–279.
Flynn, K.J. 2005a. Castles built on sand: dysfunctionality in plankton models and the inadequacy of dialogue
between biologists and modellers. Journal of Plankton Research 27, 1205–1210.
20
USE, ABUSE, MISCONCEPTIONS AND INSIGHTS FROM QUOTA MODELs
Flynn, K.J. 2005b. Modelling marine phytoplankton growth under eutrophic conditions. Journal of Sea
Research 54, 92–103.
Flynn, K.J. 2006. Reply to Horizons article ‘Plankton functional type modelling: running before we can walk’
Anderson (2005): II. Putting trophic functionality into plankton functional types. Journal of Plankton
Research 28, 873–875.
Flynn, K.J., Clark, D.R. & Owens, N.J.P. 2002. Modelling suggests that optimization of dark nitrogen-assimilation need not be a critical selective feature in phytoplankton. New Phytologist B 155, 109–119.
Flynn, K.J. & Fasham, M.J.R. 1997. A short version of the ammonium-nitrate interaction model. Journal of
Plankton Research 19, 1881–1897.
Flynn, K.J., Fasham, M.J.R. & Hipkin, C.R. 1997. Modelling the interaction between ammonium and nitrate
uptake in marine phytoplankton. Philosophical Transactions of the Royal Society B 352, 1625–1645.
Flynn, K.J. & Hipkin, C.R. 1999. Interactions between iron, light, ammonium and nitrate; insights from the
construction of a dynamic model of algal physiology. Journal of Phycology 35, 1171–1190.
Flynn, K.J. & Martin-Jézéquel, V. 2000. Modelling Si-N limited growth of diatoms. Journal of Plankton
Research 22, 447–472.
Flynn, K.J., Page, S., Wood, G. & Hipkin, C.R. 1999. Variations in the maximum transport rates for ammonium
and nitrate in the prymnesiophyte Emiliania huxleyi and the raphidophyte Heterosigma carterae. Journal
of Plankton Research 21, 355–371.
Geider, R.J. & La Roche, J. 2002. Redfield revisited: variability of C:N:P in marine microalgae and its biochemical basis. European Journal of Phycology, 37, 1–17.
Geider, R.J., MacIntyre, H.L. & Kana, T.M. 1998. A dynamic regulatory model of phytoplankton acclimation
to light, nutrients and temperature. Limnology and Oceanography 43, 679–694.
Goldman, J.C. 1979. Temperature effects on steady-state growth, phosphorus uptake, and the chemical composition of a marine phytoplankter. Microbial Ecology 5, 153–166.
Goldman, J.C. & McCarthy, J.J. 1978. Steady state growth and ammonium uptake of a fast-growing marine
diatom. Limnology and Oceanography 23, 695–703.
Gotham, I.J. & Rhee, G.-Y. 1981. Comparative kinetic studies of phosphate-limited growth and phosphate
uptake in phytoplankton in continuous culture. Journal of Phycology 17, 257–65.
Grover, J.P. 1991. Nonsteady state dynamics of algal population-growth — experiments with 2 chlorophytes.
Journal of Phycology 27, 70–79.
Harrison, P.J., Conway, H.L. & Dugdale, R.C. 1976. Marine diatoms grown in chemostats under silicate or
ammonium limitation. I. Cellular chemical composition and steady-state growth kinetics of Skeletonema
costatum. Marine Biology 35, 177–186.
Healey, F.P. 1980. Slope of the Monod equation as an indicator of advantage in nutrient competition. Microbial
Ecology 5, 281–286.
Healey, F.P. 1985. Interacting effects of light and nutrient limitation on the growth-rate of Synechococcus linearis (Cyanophyceae). Journal of Phycology 21, 134–146.
Healey, F.P. & Hendzel, L.L. 1975. Effect of phosphorus deficiency on two algae growing in chemostats.
Journal of Phycology 11, 303–309.
Heldal, M., Scanlan, D.J., Norland, S., Thingstad, F. & Mann, N.H. 2003. Elemental composition of single cells
of various strains of marine Prochlorococcus and Synechococcus using X-ray microanalysis. Limnology
and Oceanography 48, 1732–1743.
Hildebrand, M. 2005. Cloning and functional characterization of ammonium transporters from the marine diatom Cylindrotheca fusiformis (Bacillariophyceae). Journal of Phycology 41, 105–113.
Ikeya, T., Ohki, K., Takahashi, M. & Fujita, Y. 1997. Study on phosphate uptake of the marine cyanophyte
Synechococcus sp. NIBB 1071 in relation to oligotrophic environments in the open ocean. Marine
Biology 129, 195–202.
Irigoien, X. 2006. Reply to Horizons Article ‘Castles built on sand: dysfunctionality in plankton models and
the inadequacy of dialogue between biologists and modellers’ Flynn (2005). Shiny mathematical castles
built on grey biological sands. Journal Plankton Research 28, 965–967.
John, E.H. & Flynn, K.J. 2000. Modelling phosphate transport and assimilation in microalgae: how much complexity is warranted? Ecological Modelling 125, 145–157.
John, E.H. & Flynn, K.J. 2002. Modelling changes in paralytic shellfish toxin content of dinoflagellates in
response to nitrogen and phosphorus supply. Marine Ecology Progress Series 225, 147–160.
21
KEVIN J. FLYNN
Klausmeier, C.A., Litchman, E., Daufresne, T. & Levin, S.A. 2004a. Optimal nitrogen-to-phosphorus stoichiometry of phytoplankton. Nature 429, 171–174.
Klausmeier, C.A., Litchman, E. & Levin, S.A. 2004b. Phytoplankton growth and stoichiometry under multiple
nutrient limitation. Limnology and Oceanography 49, 1463–1470.
Kustka, A., Sanudo-Wilhelmy, S., Carpenter, E.J., Capone, D.G. & Raven, J.A. 2003. A revised estimate
of the iron use efficiency of nitrogen fixation, with special reference to the marine cyanobacterium
Trichodesmium spp. (Cyanophyta). Journal of Phycology 39, 12–25.
Lange, K. & Oyarzun, F.J. 1992. The attractiveness of the Droop equations. Mathematical Biosciences 111,
261–278.
Laws, E.A. & Bannister, T.T. 1980. Nutrient- and light-limited growth of Thalassiosira fluviailis in continuous culture, with implications for phytoplankton growth in the ocean. Limnology and Oceanography 25,
457–473.
Leadbeater, B.S.C. 2006. The ‘Droop Equation’ — Michael Droop and the legacy of the ‘Cell-Quota model’ of
phytoplankton growth. Protist 157, 345–358.
Lehman, J.T. 1976. Photosynthetic capacity and luxury uptake of carbon during phosphate limitation in
Pediastrum duplex (Chlorophyceae). Journal of Phycology 12, 190–193.
Leonardos, N. & Geider, R.J. 2004. Responses of elemental and biochemical composition of Chaetoceros
muelleri to growth under varying light and nitrate: phosphate supply ratios and their influence on critical
N:P. Limnology and Oceanography 49, 2105–2114.
Li, B.T., Wolkowicz, G.S.K. & Kuang, Y. 2000. Global asymptotic behavior of a chemostat model with two
perfectly complementary resources and distributed delay. Siam Journal of Applied Mathematics 60,
2058–2086.
Liu, H., Laws, E.A., Villareal, T.A. & Buskey, E.J. 2001. Nutrient-limited growth of Aureoumbra lagunensis
(Pelagophyceae), with implications for its capability to outgrow other phytoplankton species in phosphatelimited environments. Journal of Phycology 37, 500–508.
Marra, J. & Ho, C. 1993. Initiation of the spring bloom in the northeast Atlantic (47° N, 20° W) — a numericalsimulation. Deep-Sea Research II 40, 55–73.
Maske, H. 1982. Ammonium-limited continuous cultures of Skeletonema costatum in steady and transitional
state — experimental results and model simulations. Journal of the Marine Biological Association of the
United Kingdom 62, 919–943.
Mitra, A. & Flynn, K.J. 2005. Predator-prey interactions: is “ecological stoichiometry” sufficient when good
food goes bad? Journal of Plankton Research, 27, 393–399.
Mitra, A. & Flynn, K.J. 2006. Promotion of harmful algal blooms by zooplankton predatory activity. Biology
Letters 2, 194–197.
Monod, J. 1942. Recherches sur la Coissance des Cultures Bactériennes. Paris: Hermann, 2nd edition.
Monod, J. 1949. The growth of bacterial cultures. Annual Review of Microbiology 3, 371–394.
Moore, J.K., Doney, S.C., Kleypas, J.A., Glover, D.M. & Fung, I.Y. 2002. An intermediate complexity marine
ecosystem model for the global domain. Deep-Sea Research II 49, 403–462.
Morel, F.M.M. 1987. Kinetics of nutrient uptake and growth in phytoplankton. Journal of Phycology 23,
137–150.
Nelson, D.M., Goering, J.J., Kilham, S.S., Guillard, R.R.L. 1976. Kinetics of silicic-acid uptake and rates of
silica dissolution in marine diatom Thalassiosira pseudonana. Journal Phycology 12, 246–252.
Oyarzun, F.J. & Lange, K. 1994. The attractiveness of the Droop equations. 2. Generic uptake and growth functions. Mathematical Biosciences 121, 127–139.
Paasche, E. 1973. Silicon and the ecology of marine plankton diatoms. I. Thalassiosira pseudonana (Cyclotella
nana) grown in a chemostat with silicate as the limiting nutrient. Marine Biology 19, 117–126.
Parslow, J.S., Harrison, P.J. & Thompson, P.A. 1984. Saturated uptake kinetics: transient response of the marine
diatom Thalassiosira pseudonana to ammonium, nitrate, nitrite, silicate or phosphate starvation. Marine
Biology 83, 51–59.
Pascual, M. 1994. Periodic-response to periodic forcing of the Droop equations for phytoplankton growth.
Journal of Mathematical Biology 32, 743–759.
Raven, J.A. 1990. Predictions of Mn and Fe use efficiencies of phototrophic growth as a function of light availability for growth and C assimilation pathway. New Phytologist 116, 1–18.
22
USE, ABUSE, MISCONCEPTIONS AND INSIGHTS FROM QUOTA MODELs
Raven, J.A. 1994. Why are there no picoplanktonic O2 evolvers with volumes less than 10−19 m3. Journal of
Plankton Research 16, 565–580.
Redfield, A.C. 1958. The biological control of chemical factors in the environment. American Scientist 46,
205–221.
Rhee, G.-Y. 1973. A continuous culture study of phosphate uptake, growth rate and polyphosphate in Scene­
desmus sp. Journal of Phycology 9, 495–506.
Rhee, G.-Y. 1974. Phosphate uptake under nitrate limitation by Scenedesmus sp. and its ecological implications. Journal of Phycology 10, 470–475.
Rhee, G.-Y. & Gotham, I.J. 1980. Optimum N-P ratios and coexistence of planktonic algae. Journal Phycology
16, 486–489.
Savage, V.M., Gillooly, J.F., Brown, J.H., West, G.B. & Charnov, E.L. 2004. Effects of body size and temperature on population growth. American Naturalist 163, 429–441.
Sciandra, A. & Ramani, P. 1994. The steady-states of continuous cultures with low rates of medium renewal per
cell. Journal of Experimental Marine Biology and Ecology 178, 1–15.
Smith, H.L. 1997. The periodically forced Droop model for phytoplankton growth in a chemostat. Journal of
Mathematical Biology 35, 545–556.
Smith, R.E.H. & Kalff, J. 1982. Size-dependent phosphorus uptake kinetics and cell quota in phytoplankton.
Journal of Phycology 18, 275–284.
Spijkerman, E. & Coesel, P.F.M. 1998. Different response mechanisms of two planktonic desmid species
(Chlorophyceae) to a single saturating addition of phosphate. Journal of Phycology 34, 438–445.
Stephens, N., Flynn, K.J., Gallon, J.R. 2003. Interrelationships between the pathways of inorganic nitrogen
assimilation in the cyanobacterium Gloeothece can be described using a mechanistic mathematical
model. New Phytologist 160, 545–555.
Sterner, R.W. & Elser, J.J. 2002. Ecological Stoichiometry: The Biology of Elements from Molecules to the
Biosphere. Princeton, NJ: Princeton University Press.
Sunda, W.G. & Huntsman, S.A. 1997. Interrelated influence of iron, light and cell size on marine phytoplankton
growth. Nature 390, 389–392.
Syrett, P.J., Flynn, K.J., Molloy, C.J., Dixon, G.K., Peplinska, A.M. & Cresswell, R.C. 1986. Effects of nitrogen
deprivation on rates of uptake of nitrogenous compounds by the marine diatom Phaeodactylum tricornutum Bohlin. New Phytologist 102, 39–44.
Taguchi, S. 1976. Relationship between photosynthesis and cell-size of marine diatoms. Journal of Phycology
12, 185–189.
Terry, K.L. 1982. Nitrate and phosphate uptake interactions in a marine prymnesiophyte. Journal of Phycology
18,79–86.
Tett, P., Heaney, S.I. & Droop, M.R. 1985. The Redfield ratio and phytoplankton growth rate. Journal of the
Marine Biological Association of the United Kingdom 65, 487–504.
Turpin, D.H. & Harrison, P.J. 1979. Limiting nutrient patchness and its role in phytoplankton ecology. Journal
of Experimental Marine Biology and Ecology 39, 151–166.
Vadstein, O. 1998. Evaluation of competitive ability of two heterotrophic planktonic bacteria under phosphorus
limitation. Aquatic Microbial Ecology 14, 119–127.
Watanabe, M., Kohata, K. & Kunugi, M. 1987. 31P nuclear-magnetic-resonance study of intracellular phosphate
pools and polyphosphate metabolism in Heterosigma akashiwo (Hada) Hada (Raphidophyceae). Journal
of Phycology 23, 54–62.
Wood, G. & Flynn, K.J. 1995. Growth of Heterosigma carterae (Raphidophyceae) on nitrate and ammonium
at three photon flux densities: evidence for N-stress in nitrate-growing cells. Journal of Phycology 31,
859–867.
Woods, J. & Barkmann, W. 1994. Simulating plankton ecosystems by the Lagrangian ensemble method.
Philosophical Transactions of the Royal Society B 343, 27–31.
Zevenboom, W., deGroot, G.J. & Mur, L.R. 1980. Effects of light on nitrate-limited Oscillatoria agardhii in
chemostat cultures. Archives of Microbiology 125, 59–65.
Zonneveld, C. 1996. Modelling the kinetics of non-limiting nutrients in microalgae. Journal of Marine Systems
9, 121–136.
Zonneveld, C., van den Berg, H.A. & Kooijman, S.A.L.M. 1997. Modeling carbon cell quota in light-limited
phytoplankton. Journal of Theoretical Biology 188, 215–226.
23