Journal of Plankton Research Vol.23 no.9 pp.977-997, 2001
© Oxford University Press 2001
A mechanistic model for describing dynamic multi-nutrient, light, temperature interactions in phytoplankton
Kevin J. Flynn
Ecology Research Unit, School Of Biological Sciences, University Of Wales Swansea, Singleton Park, Swansea Sa2 8pp, Uk
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Abstract
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Developed by combining components simplified from previously
developed mechanistic models, acomplete dynamic model is described
for simulating the growth of phytoplankton as functions of ammonium,
nitrate, light, iron, silicon, phosphorus and temperature. Components
may be safely added or deleted to the base model, describing
ammonium–nitrate–light interactions, to suit particular
modelling scenarios. Biomass is described in terms of C and
cells, while chlorophyll is also a state variable enabling the
simulation of changes in Chl a:C with photoacclimation. The
model is capable of simulating variable silicon deposition (diatoms)
and C cell
–1 with Si, Fe, or P limitations. Mechanisms
for inclusion of temperature control of nutrient transport,
growth rate and cell size are given. The model is suitable for
placement in ecosystem models, containing various components
that can be readily modified to tune the simulation to mimic
the behaviour of specific algal groups or species. Most of those
components have biological significance and can be estimated
from experiments or by analysis of existing data.
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INTRODUCTION
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Phytoplankton are responsible for a significant proportion of
global primary production as well as supporting the bulk of
marine food webs. Not surprisingly, considerable effort is expended
in attempts to model these processes [e.g. (Evans and Garçon,
1997
)]. However, submodels of phytoplankton placed within existing
ecosystem simulators are invariably simple. They are often restricted
to one or two state variables, describing all phytoplankton
as one group or dividing them into two groups [e.g. diatoms
and non-diatoms (Fasham and Evans, 2000
)]. Models of different
phytoplankton groups capable of simulating multi-nutrient interactions
would enable a more realistic consideration of production (growth
rates, acclimation to nutrient limitation), transfer to the
benthos (sinking rates, C:N:P:Si elemental composition) and
support of secondary production (cell size and value as feed).
The chemical composition of algal cells varies with nutrient availability, having an impact on trophic transfer both for microbial predators (Davidson, 1996; Sommer, 1998) and mesozooplankton (e.g. Miralto et al., 1999). Models of these transfers into predators [e.g. (Anderson, 1992; Anderson, 1994; Davidson, 1996)] rely on phytoplankton models giving an adequate description of the chemical composition. Microbial loop activity is supported not only by smaller species but by dissolved organic carbon released by phytoplankton (Anderson and Williams, 1998) as functions of nutrient status and algal group [e.g. (Penna et al., 1999)]. Modelling the operation of microbial planktonic systems, such as that described by Priddle et al. (Priddle et al., 1995), thus requires much more than a single nutrient/single phytoplankton group implementation.
The varied chemical composition of the phytoplankton has important implications for biogeochemical fluxes. Calculations of anoxia in ocean waters by Hotinski et al. (Hotinski et al., 2000) and Lenton and Watson (Lenton and Watson, 2000) may be criticized (Hoppema and Goeyens, 1999; Thomas et al., 1999; Pahlow and Riebesell, 2000) for their assumption of fixed Redfield ratios (Redfield, 1934) of phytoplankton C:N:P. It would be logical to employ more realistic phytoplankton models reflecting differences in population composition and nutrient status. Studies of opal flux (Wong and Matear, 1999) require models that handle variable C:N:P:Si rather than relying on Redfield ratios because the transfer of Si as biosilicate is a function of the level of silicification of diatoms and the N:Si assimilation ratio (Treguer and Jacques, 1992; Dunne et al., 1999). Iron availability also interacts with other nutrient assimilations and with silicification (Hutchins and Bruland, 1998; Takeda, 1998). Discussions of the NP-limitation of marine primary production (Downing, 1997; Falkowski, 1997; Benitez-Nelson and Buesseler, 1999; Tyrrell, 1999) as functions of algal size, type and physiology would all be better supported with multi-nutrient multi-group phytoplankton models. In addition, the N:P nutrient concentration ratio is important for phytoplankton species succession in freshwater systems (Reynolds, 1999) and also in some marine systems; e.g. Emiliania huxleyi blooms with high irradiance, high nitrate and low phosphate availability (Tyrrell and Taylor, 1996).
While the potential utility of multi-nutrient, multi-group phytoplankton models in ecosystem simulators is clear, the task of constructing a suitable model is non-trivial. Traditionally, mathematical models for algal growth have employed the equations of Monod or the quota models of Caperon and Droop (Monod, 1942; Caperon, 1968; Droop, 1968). Such models are essentially curve-fitting exercises in which most biological detail has been lost or combined within single equations. They have the advantage of simplicity but the disadvantages that they cannot handle multi-nutrient interactions or transients very well (Davidson and Cunningham, 1996; Andersen, 1997); when exposed to different external forcing such models can fail badly. An alternative strategy is to formulate models with mechanistic components that more closely mimic physiological interactions. The behaviour of such models should mirror that of the target system over a wide range of external forcing and hence be of more general use. The down side is the inevitable increase in complexity.
Previously we have described individual mechanistic models for various aspects of nutrient physiology in phytoplankton. The original models describing ammonium–nitrate interactions (Flynn et al., 1997; Flynn and Fasham, 1997) were followed by models describing light acclimation and nitrite release (Flynn and Flynn, 1998), iron stress (Flynn and Hipkin, 1999), silicate (Flynn and Martin-Jézéquel, 2000) and phosphate nutrition (John and Flynn, 2000). The aim of this paper is to present a single model structure developed from these former models as an important step in promoting the use of mechanistic multi-nutrient multi-group models of phytoplankton in ecosystem models. Because the model includes both carbon (C) and cells as state variables, cell-size (affecting size-dependent predatory activity) and cell density [affecting sedimentation rates (Bienfang et al., 1982)] can also be included within the ecosystem model.
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DESCRIPTION OF THE MODEL
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It is important to appreciate at the outset that the user may
readily customize the model described here. The particular requirements
of a given modelling scenario will dictate the level of detail
needed in the phytoplankton model. To this end the user may
wish, or need, to delete (such as Si for non-diatoms) or modify
(such as light : dark controls if the model is being run with
a daily photon dose) parts of the structure. Information on
how to customize the model is given below. That said, the model
is presented as a generic phytoplankter and may be used as is.
The full model is available as a Powersim Constructor (Powersim
AS, Isdalstø, Norway) file from the author or from
http://www.swan.ac.uk/biosci/research/kjf.htm.
The full reasoning behind the construction of the model equations has been given in previous papers (referenced with the equations given below). Common features include the widespread use of feedback processes employing rectangular hyperbolic and sigmoidal functions normalized to maximum pool sizes [discussed in (Flynn et al., 1997)], and the indexing of rate processes to the maximum growth rate. A schematic of the model is given in Figure 1; the model contains nine state variables describing C biomass, cell number, C-quotas of glutamine, nitrogen, iron, silicon, inorganic and organic pools of phosphorus, and chlorophyll. Because all nutrient C-quotas vary not only with input and output of nutrient but also with changes in the amount of C (due to respiration and photosynthesis), all differential equations for C-quotas include a term that corrects for the C-specific growth rate, Cu. Thus, for nutrient X with C-quota XC (where XC is the mass ratio X:C),
Steps taken
to simplify the total structure have resulted in the combination
of some equations given in the original papers. Parameter names
have been shortened and simplified to facilitate equation construction
in modelling software. All parameters are described in Tables
I to III
, with the external parameters (i.e. nutrients) given
in Table IV
. Boolean conditional tests (within { } in equations)
take the value 1 if true and 0 if false. The model is typically
operated with a time step of 1/128th of a day (= 11.25 min)
with the Euler integration method. This time step gives a good
balance between correct integration and computing effort; like
all models the suitability of the time step should be verified
for a given scenario.
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NITROGEN
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The maximum value of ammonium transport varies with the N-status
of the cells (Flynn
et al., 1997
, 1999
), indexed to the maximum
growth rate, Um [equation (2)
].
Ammonium
transport [equation (3)
], limited by Atq, is a hyperbolic function
of the availability of external am-monium and is repressed by
the intracellular concentration of an early product of inorganic
nitrogen assimilation, glutamine (GC), using a sigmoidal function.
The
maximum value of nitrate transport is given by equation (4)
[Cf. equation (2)
].
There is evidence
that nitrate transport in phytoplankton may be biphasic (Collos
et al., 1997
; Lomas and Glibert, 1999
; Flynn, 1999
), as it is
in higher plants (Nissen, 1991
). While this situation needs
clarification, it is useful to consider how such a capability
can be included within a mechanistic model. Accordingly, nitrate
transport as described in equation (5)
occurs through two systems,
one a high-affinity low-rate porter and the second with a lower
affinity (here with a half saturation constant set as 200 times
the high-affinity constant) and higher transport rate (here
with the maximum 10 times higher) [see also Figure 1
in (Flynn,
1999
)]. The short-term interaction between ammonium and nitrate
assimilations is controlled by the value of GC; GC is itself
synthesized using N entering from the inorganic N-sources, providing
the feedback control (Figure 1
). Nitrate transport is repressed,
and ultimately halted, by lower GC values than those controlling
ammonium use. For the control of the dual nitrate porters, repression
halts transport at two different levels of GC (NmG1 and NmG2;
compare with that for AmG in Table III
). Nitrate is still transported
by the second (low-affinity) system even when the simultaneous
assimilation of ammonium elevates GC to a level at which the
highaffinity system is halted. Nitrate transport is further
modulated by the availability of reductant, Nred [equation (19)
]
(Flynn
et al., 1997
; Flynn and Fasham, 1997
).
Nitrogen
from ammonium and/or nitrate leads to changes in the concentration
of glutamine, GC [equation (6)
], in turn providing a feedback
regulation of transport (Flynn
et al., 1997
). Nitrogen from
GC supports synthesis of all other nitrogenous components, NC.
The rate of synthesis of NC, a hyperbolic function of GC, is
also a function of the current nitrogen-status [indicated by
NCu, equation (8)
], thus nitrogen-starved cells exhibit a lag
in being able to process new nitrogen (Flynn
et al., 1997
).
The availability of C for this process is indexed by CAAs [equation
(18)
]. In recognition that net protein synthesis can be performed
faster than the maximum growth rate, the maximum NC synthesis
rate is double Um; this is a modification over previous models
and ensures that the growth rate can indeed attain Um under
optimal conditions. GC is a ratio of nitrogen : carbon and so
declines with Cu [equation (1)
].
Changes
in NC are given by equation (7)
. NC declines with Cu.
A
normalized quota equation (Flynn
et al., 1999
) is used to describe
the relative growth rate at a given NC, giving the quotient
NCu [equation (8)
]. In some circumstances NC can briefly exceed
NCm (when growth is limited by a nutrient other than nitrogen)
but the logic statements ensure that NCu cannot exceed 1.
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SILICON ASSIMILATION IN DIATOMS
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The simplified silicon model of Flynn and Martin-Jézéquel
(Flynn and Martin-Jézéquel, 2000
) has been used.
Silicate transport [equation (9)
], indexed to Um, is a hyperbolic
function of the availability of external silicate and related
to cell size, and only occurs if diat is set at 1 (diatoms).
The larger the relative cell size [RS, equation (23)
] the closer
it is to undertaking cell division and hence the greater the
need for silicon. The value of the constant StP affects the
level of silicon deposition when silicon is non-limiting; the
lower StP the greater the deposition.
Changes
in the silicon content per cell [equation (10)
] occur due to
the balance of transport and division of silicon between daughter
cells with cell growth rate, cellu [the latter by analogy with
equation (1)
]. Note the silicon content is a cell, not C, quota.
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IRON
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Details for the basis of these equations are given in Flynn
and Hipkin (Flynn and Hipkin, 1999
). Transport of iron is indexed
to Um, to hyperbolic functions of the availability of external
iron and to the current normalized iron-quota given by FC/FCm
[equation (11)
]. Changes in FC occur as a balance between transport
and an increase in C with Cu.
The level
of cellular iron that can be accounted for, assuming known requirements
of biochemical processes is given by equation (12)
[see (Flynn
and Hipkin, 1999
) for details]. For respiration, 1.2065
x 10
–5 g Fe in cytochromes is required to support a respiration rate
of 1 g C day
–1, while 0.1146
x 10
–3 g Fe in nitrate
and nitrite reductases is required to support a reduction rate
of 1 g of nitrate-N through to ammonium-N per day. The 24 h
moving average of the activities of respiration and nitrate
reduction (aRes and aNNiR, respectively) are used to compute
the iron costs for these processes. If the model is not being
run within a light–dark cycle these average values may
be replaced with their current values given by (At+Nt)•1.5+basres,
and Nt, respectively. This assumes 1.5 g C g
–1 N for N-assimilation
and growth [see (Flynn and Hipkin, 1999
)]. The photosynthesis
subunit (PSU) may contain 500–1300 chlorophyll
a (Chl
a) molecules (Falkowski
et al., 1981
) and each PSU contains
23 Fe atoms (Raven, 1990
). At 500 Chl a PSU
–1, this gives
0.003 g Fe g
–1 Chl
a, as used in equation (12)
. At the
other extreme, 1300 Chl
a PSU
–1 equates to 0.001156 g
Fe g
–1 Chl
a. The implications for employing different
PSU sizes are discussed in Flynn and Hipkin (Flynn and Hipkin,
1999
).
As a simplification, only the
synthesis of the most iron-resource-expensive component, the
photosystems [see Figure 7
in (Flynn and Hipkin, 1999
)], is
made a function of iron availability. Effects of Fe-stress on
nitrate assimilation are primarily indirect through the limitation
of photosystem synthesis; Flynn and Hipkin describe these effects
and how more subtle interactions (through the modification of
the rates of synthesis of the enzymes nitrate and/or nitrite
reductases) may be incorporated for more detailed physiological
models (Flynn and Hipkin, 1999
). Fcon [equation (13)
] controls
synthesis of iron-dependent components (in this simplified model,
just ChlC) as a hyperbolic function of the availability of un-accounted
FC (i.e. FC–Ftot).
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Fig. 7. Changes in cell size for a diatom (a) when growth is limited by the indicated nutrient or by light (see Figure 6 for abbreviations), and for a non-diatom (b). For the latter, other than S (which is not used), other limitations follow the pattern shown with Fe-stress.
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PHOTOSYNTHESIS, CARBON AND RESPIRATION
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Changes in chlorophyll content, ChlC, are described by equation
(14)
. This has been modified from equations given for photoacclimation
(Flynn and Flynn, 1998
), for cell size control (Flynn and Martin-Jézéquel,
2000
), and for iron control (Flynn and Hipkin, 1999
). Synthesis
is a function of the N-status [NCu, equation (8)
], scaled to
the maximum growth rate Um and indexed to the relative demand
for carbon (ratio of photosynthesis rate to the maximum attainable
rate at the current nutrient status, given by PS Pqm
–1).
Synthesis is controlled in darkness by the availability of C
supporting biosynthetic processes, through quotient CAAs [equation
(18)
]. Fine-tuning of the rate of photoacclimation may be achieved
using scalar M. This value is typically between 1 (Flynn and
Flynn, 1998
) and 3 (Flynn and Hipkin, 1999
), and is set at 2
here. Synthesis is limited by hyperbolic functions to the maximum
possible value (ChlCm), and also by cell size (RS) so that synthesis
halts when size is maximal. Iron availability [Fcon, equation
(13)
] limits the rate of ChlC synthesis so that ChlC cannot
increase unless Fe is available to support synthesis. ChlC declines
with increases in Cu and also due to a decay process enhanced
by a low NC [indicated by NCu, equation (8)
] and a maximum cell
size. This decay process may be deleted or modified as required;
some phytoplankton, such as dinoflagellates, ‘bleach’
rapidly when exhausted of nitrogen, while species of other taxa
may not do so. Flynn, Marshall and Geider (in preparation) describe
the performance of the different nitrogen-photoacclimative models
developed from the models of Flynn and Flynn (Flynn and Flynn,
1998
) and by Geider
et al. (Geider
et al., 1998
).
Note
in both equations (14) and (15)
NCu (as indicated by *) is replaced
by NPCu [see equation (29)
] if phosphorus interactions are to
be included.
The absolute maximum rate of photosynthesis enables the maximum growth rate to be attained when using nitrate, accounting for respiratory and reductant costs (Flynn and Flynn, 1998). The current maximum rate of photosynthesis [Pqm, equation (15)] is a function of the NC status, described by NCu. The value of (redco+1.5) accounts for the total cost of assimilating nitrate-N, for both reduction and synthesis of organics [see (Flynn and Hipkin, 1999)].
Basal respiration [basres,
equation (16)
] is indexed to Um assuming a rate of 5%; the value
varies greatly, being much higher in dinoflagellates for example
(Harris, 1978
). The function with NC ensures that cells cease
respiration at a high N:C status (when there is no C to respire).
A value of basres close to zero could be used to trigger a death
function, if required. The logic statement ensures that basres
does not fall below 0 if/when NC exceeds NCm.
Cresv
[equation (17)
] describes the level of reserve C available for
metabolic processes in darkness or at very low light when photosynthesis
is inadequate (Flynn
et al., 1997
).
The
value of Cresv is then used in the calculation of the quotients
CAAs [equation (18)
] and Nred [equation (19)
] describing, respectively,
the availability of C for amino acid and ChlC synthesis and
the regulation of nitrate reduction to ammonium [see (Flynn
et al., 1997
)].
Equation
(20)
[modified after (Jassby and Platt, 1976
)] accounts for
photosynthesis with the current values of ChlC and Pqm and photon
flux density (PFD). The time unit for PFD in the model must
be the same as for Um, in this instance d
–1.
The
C-specific growth rate, Cu [equation (21)
], represents a balance
between photosynthesis and respiration. Respiration accounts
for reduction of nitrate [assuming concurrency with nitrate
transport, with no loss of nitrite (Flynn and Flynn, 1998
)],
synthesis of NC and basal respiration (Flynn and Hipkin, 1999
).
A hyperbolic function prevents cell size exceeding a maximum
value as described by RS [equation (23)
]; implications for the
release of DOC are mentioned by Flynn and Martin-Jézéquel
(Flynn and Martin-Jézéquel, 2000
). The increase
inC-biomass is then given by equation (22)
.
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CELL DIVISION IN DIATOMS
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Silicon metabolism and diatom cell division are inextricably
linked. The value of the relative cell size (RS, where 1 indicates
the maximum) used in the regulation of cell division [equation
(24)
] and silicate transport [equation (9)
], is given by equation
(23)
. Cell size, Ccell, is described in terms of C mass per
cell. [Note that the definition of RS, and hence its use elsewhere,
is opposite to that in (Flynn and Martin-Jézéquel,
2000
) where RS=1 indicated the minimum rather than the maximum
size.]
Changes in cell number [equation
(25)
] occur with cell division [equation (24)
], which in turn
is affected by the current silicate transport rate [equation
(9)
]. Cell division becomes increasingly more likely as cell
size becomes larger, controlled by the sigmoidal function to
RS in equation (24)
[the derivation of this equation is explained
in (Flynn and Martin-Jézéquel, 2000
)]. The value
of celluP is important for controlling enhanced silicon-deposition
where factors other than silicon limit growth; the higher celluP
the greater the deposition. Simulating a variable cell size
for non-diatoms is described below; here cellu for non-diatoms
(diat=0) simply equals Cu.
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INTRODUCING PHOSPHORUS
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For the simulations presented here the multiple pool phosphorus
model, iPIM (John and Flynn, 2000
), was chosen. This contains
pools of inorganic P (IPC, including polyphosphate) and organic
P (OPC), with the total P:C (PC) thus IPC+OPC. Values assigned
to constants come from John and Flynn (John and Flynn, 2001
).
Transport of P in the model can occur at a (surge) rate up to four times that required to support the maximum growth rate (Um) at the maximum organic P quota (OPCm). Transport is a hyperbolic function of the external phosphate concentration, limited by a hyperbolic function to the maximum size of IPC (IPCm). Removal of P from IPC to OPC can occur at twice the rate required to match maximal growth requirements [cf. synthesis of NC in equations (6) and (7)] as a hyperbolic function of the availability of IPC and repressed by OPC to halt synthesis when OPC attains OPCm. Both IPC and OPC are also corrected for increases in C with Cu.
The total
P:C (PC) is used to compute a normalized P-quota regulatory
parameter analogous to NCu in equation (8)
, giving PCu [equation
(28)
]. Here the constants given in John and Flynn (John and
Flynn, 2001
) have been used.
The interactions
between nitrogen and phosphorus nutrition are complex and numerous;
almost all biochemical processes involve enzymes (which are
nitrogenous) and phosphorylated intermediaries. At present,
and in the absence of knowledge indicating an alternative approach,
the interaction is considered by reference to the quotients
describing the nutrient C-quotas (i.e. NCu and PCu). Traditionally,
interaction terms between nutrient quotas, here giving NPCu,
have employed a threshold (NPCu set by the nutrient with the
lowest quota) or multiplicative (product of NCu and PCu) mechanism.
Although synergistic (hence multiplicative) interactions may
be expected, the multiplicative approach is invariably unacceptable
because the growth rate decreases too rapidly as the quotas
decline. The threshold approach is more often the accepted mechanism
(Andersen, 1997
). There are no biochemical bases for either
approach.
Logic would dictate that when two nutrients are limiting, the consequences of the most limiting nutrient would moderate the biochemical implications of limitation by the other nutrient. The threshold approach takes this to extremes, with a total compensation and hence deletion of the effect of the less limiting nutrient. Now consider a situation where NCu is 0.4 and PCu is 0.8; NCu is thus more limiting. The restriction by PCu alone is (1–0.8) = 0.2 (i.e. 20%), but suppose this is now down-played by the effect of NCu giving a modified restriction by PCu equal to NCu x 0.2. The operational value of PCu is now [1–(0.2 x NCu)] = 0.92 and NPCu thus equals (0.4 x 0.92) = 0.368; cf. a threshold value of 0.4, and a multiplicative value of 0.32. This interaction is described by equation (29).
Figure 2
shows a comparison of the output of the
three different forms of handling the dual-nutrient interaction.
The resultant controlling value of NPCu from equation (29)
is
closer to the threshold value than the simple multiplicative
value, while still retaining a multiplicative interaction so
that if either of the individual quotas are altered the resultant
NPCu changes.
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Fig. 2. Comparison between the quota control achieved by interacting two separate nutrient quotas (NCu and PCu) according to the threshold, multiplicative and NPCu formulations as explained in the text.
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The cellular mechanisms for the N–P interaction are not
understood in sufficient detail to enable a definitive choice
between NCu, PCu and NPCu (however computed) as regulators for
individual processes. Here NPCu replaces NCu in equations (14)
and (15)
, placing the control of the N–P interaction at
the level of photosystem (ChlC) synthesis and C-growth. Whether
this is appropriate for the control of ChlC, in particular,
is not yet clear. Either way, the equations are readily changed.
Attempts to involve NPCu with the synthesis of NC [replacing
NCu with NPCu in equations (6) and (7)
] resulted in occasional
oscillatory model behaviour as a consequence of feedback processes
within the model.
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CELL SIZE IN NON-DIATOMS
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Phosphorus-stressed cells are often considerably larger than
P-replete cells, with a higher C cell
–1, while many microalgae
are smaller when stressed of light, iron, or nitrogen (Raven,
1990
; Falkowski and Raven, 1997
; Raven, 1998
). The easiest way
to incorporate P-stress with cell division is via PCu [equation
(28)
]. When PCu is 0, there is no growth and cell size will
be maximal (i.e. RS, from equation (23)
, tends to 1). When PCu
is 1, with other nutrients in excess, the cell can divide rapidly.
Equation (30)
describes cellu as a sigmoidal function of RS.
The
power in the equation [analogous to the constant celluP in equation
(24)
] is now a variable with the relative growth rate (given
by Cu/Um) and P-status (PCu). With non-P limitation the power
tends to 0.5, with optimal conditions it has a value of 3 (Cf.
celluP), while with P-limitation the power tends to 20.5. These
limits, together with the constants 0.001 and 1.001 (which further
affect the shape of the sigmoidal curve), can be altered as
required to optimize the relationship between cell size and
growth rate. Equation (30)
now replaces the final (non-diatom)
section in equation (24)
.
It should be noted that this approach does not simulate a diel/diurnal division cycle and that this control of cell division is for a population not for individuals. Flynn and Martin-Jézéquel describe an individually based diatom model that also includes cell cycle processes (Flynn and Martin-Jézéquel, 2000).
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TEMPERATURE
|
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All rate processes have been indexed to the maximum growth rate,
Um. If one assumes all cellular functions alter
pro rata with
temperature then Um can become a variable referenced to a temperature–growth
rate curve. This capability is explicitly included in the model
of Geider
et al. (Geider
et al., 1998
) using the Arrhenius equation
(Goldman, 1979
; Raven and Geider, 1988
). Haefner (Haefner, 1996
)
gives equations that are more appropriate for describing the
full temperature–growth curve. As nutrient stress increases
a larger decrease in temperature is required for growth to become
temperature controlled (Rhee and Gotham, 1981
); the inference
is that transport is affected to a lesser extent by temperature
than other metabolic processes. Shifts in the half saturation
constants for growth with temperature (Rhee and Gotham, 1981
)
also reflect primarily post-nutrient transport processes. Flynn
(Flynn, 1998
) discusses the difference between half saturation
constants for transport (as used in the model here) and those
for assimilation (transport + incorporation, as in Monod and
quota models). Different temperature correction factors may
be employed at different stages in models (Geider
et al., 1998
),
as in Flynn's (1999) explorations of the results of Lomas and
Glibert (Flynn, 1999
; Lomas and Glibert, 1999
).
For the temperature-based simulations shown here, the constant Um is replaced by the growth rate, Uref, at the reference temperature, Tref. Temperature-rate relationships in biology are often assigned Q10 values indicating the change in rate per 10°C [e.g. (Raven and Geider, 1988)]; typically these are around 2 (i.e. doubling the rate per 10°C increase). Tref would typically be at the upper end of the temperature range for which Q10 is constant. As transport processes are less susceptible to temperature, these are assigned a lower Q10 than growth processes. Here Q10 values for all transport processes are designated Q10t, and for all assimilatory growth processes, Q10g, yielding Umt and Umg for maximum rates for these processes [substituting ‘t’ or ‘g’ for x in equation (31), respectively].
T is the operational temperature. Umt is then used
to replace Um in equations (2), (4), (9), (11)
, and the first
part of (26)
. Umg replaces Um in all other equations.
Low temperature may decrease growth rates while enhancing the size of the individual cell. The data of Rhee and Gotham (Rhee and Gotham, 1981) for a non-diatom show a linear relationship between the reciprocal of C cell–1 and growth temperature. This can be implemented within the model structure by making Ccellm and Ccello (maximum and minimum values of C cell–1) variables indexed to temperature. For a given temperature T, Ccelly (where y is either ‘o’ or ‘m’) is given by equation (32) where Ccellrefy is the minimum or maximum cell size at the reference temperature Tref, with slope Zy.
There appears to be no simple temperature–size
relationship for diatoms; any relationship there may be is confused
by the decline in cell size during successive generations (Durbin,
1977
; Furnas, 1978
). An increase in silicon per cell with growth
at low temperature (Durbin, 1977
) will be handled by the model
in the same way as any other non-silicon limitation of growth
(Flynn and Martin-Jézéquel, 2000
) provided the
silicate transport rate is less susceptible to changes in temperature
than intracellular processes.
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SIMPLIFICATIONS
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As stated at the beginning of this description, an important
feature of the construction of this model is flexibility. Components
may be modified to customize the structure, deleting sections
that are not applicable for example.
If the model is to be run in a scenario where irradiance is supplied as a daily dose, with no periods of darkness, the structure can be simplified by deleting equations (17) to (19). CAAs should then be removed from equations (6), (7) and (14), and Nred from equation (5). If a dual kinetic nitrate transporter is not required then the second half of equation (5) is deleted. The form of equations (2) and (4) may also be simplified, though transport must become disabled at NC<NCm [see (Flynn et al., 1999)]. If only non-diatoms are to be considered then all reference to silicon [equations (9) and (10)] can be removed and all but the very last part of equation (24) becomes redundant, unless cell size information is required, in which instance equation (24) is replaced by (30). Reference to cell size [RS in equations (14), (21), (23)] and to cells [equations (24), (25), (30), (32)] may all be omitted if only C-biomass is to be simulated with no silicon input. If iron is not required as an input, equations (11) to (13) are deleted, together with reference to Fcon in equation (14). The inclusion, or otherwise, of phosphorus is described in the section on this nutrient, above. If it is not necessary to simulate both inorganic (polyphosphate) and organic pools of phosphorus, then the phosphorus submodel can be simplified. This topic is discussed by John and Flynn (John and Flynn, 2000). Equations (31) and (32) are deleted if temperature is not required as an input.
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PARAMETERIZATION
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Parameterization of complex models is inevitably problematic.
It is important that the model behaves in a robust fashion,
with no undue sensitivity to variation in the values of constants.
Sensitivity analyses have been conducted on previously published
subcomponents of the model (referenced above). These have shown
the construction to be robust; individual constants can be altered,
or nutrient submodels deleted, and the model will continue to
behave reasonably from a physiological standpoint.
The robust behaviour of the model to changes in Um and cell-size-related parameters is demonstrated clearly in Table V. The model was run to steady state using different values of Um, and minimum/maximum values of Ccell and Scell. Altering the constants Ccello and Ccellm only had a major impact on cell size (Ccell), and likewise Scello and Scellm only affected Scell. Cell size and silicon content (for diatoms) may thus be safely altered as required to match values for particular species or groups of organisms. Changing Um only affected the growth rate, Cu, ChlC and FC (when Fe–light co-limited), responses that are to be expected. Um can thus be altered safely to match growth rates of different algae, or linked with temperature to generate a temperature dependent growth rate.
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Table V: Sensitivity analysis. Simulations were run to steady-state with either all nutrients and light in excess (optimal conditions), with just one nutrient or light limiting, nitrogen and phosphorus co-limiting, nitrogen and silicon co-limiting, or iron and light co-limiting. For the Ccell and Scell series, Um was set at 1 d–1and the pairs of (Ccello and Ccellm) or (Scello and Scellm) halved or doubled relative to the values given inTableIII. For the Um series, Um was set at 1 (control), 0.5 or 0.25 d–1. A single point sensitivity analysis was performed (Haefner, 1996); 0 indicates no change (zero sensitivity), 1 indicates a pro rata change in the same direction (i.e. double constant doubles the response) and –1 a pro rata change in the opposite direction (double constant, halves the response). The effects are given for non-limited growth (optimal conditions), and as a mean and range from all treatments
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It is thus important to appreciate that it is
not necessary
to parameterize rigorously all the constants given in Table
III
for individual species or groups of organisms before the
model will give, what will for most purposes be, a quite satisfactory
output. Tuning these components will, of course, improve simulations
of multi-species multi-nutrient interactions; the most obvious
candidates for tuning are indicated in Table III
. However, assuming
that simulating a response to a given nutrient is important,
inclusion of partially parameterized processes for that nutrient
in mechanistic models is still preferable to their exclusion
[‘it is better to be almost right than completely wrong’;
after (Haefner, 1996
)]. That assumes that the model behaves
in a sensible fashion [accords to face and event validity; (Rykiel,
1996
)], but mechanistic models are more likely to do so than
empirically formulated structures. Methods of parameterizing
parts of the model to fine-tune the output are described below.
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Abstract
INTRODUCTION
