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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
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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TOP
Abstract
INTRODUCTIONDeveloped 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),
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, 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)].
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], 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.
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[Cf. equation (2)
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; 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).
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], 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)].
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. NC declines with Cu.
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) 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.
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] 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.
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[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).
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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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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).
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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)].
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] 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.
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] 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).
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] 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)].
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[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.
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], 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.]
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] 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.
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, giving PCu [equation (28)]. Here the constants given in John and Flynn (John and Flynn, 2001) have been used.
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). 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).
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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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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. |
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.
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] 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].
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, 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.
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; 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. |
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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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.
Estimation of constants A1 to A4 for ammonium transport, and N1 to N4 for nitrate, together with NC-quota constants NCm, NCo and NCk are considered in Flynn et al. (Flynn et al., 1999). It should be pointed out that transport capacities for silicate, phosphate and iron doubtless also develop in response to changes in their respective nutrient stresses. If such information were available then transport of these nutrients could be described using equations similar to (2) and (4) rather than simply being indexed to Um as at present. Determination of half saturation constants for transport (Akt, Nkt, Fkt, Pkt and Skt) is discussed by Flynn (Flynn, 1998) and the parameterization of iron-dependent processes by Flynn and Hipkin (Flynn and Hipkin, 1999). Determination of the kinetics of iron transport and of the concentration of bioavailable iron in the water is difficult; optimization to data sets may be best attained by modification of the value for Fkt. Similarly, in the absence of data for parameterization of the transport of other nutrients, modification of the transport half saturation constants may prove the easiest route for optimization (provided these remain plausible).
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RESULTS AND DISCUSSION |
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The model, or its predecessors, have been run to simulate experimental data for ammonium–nitrate interactions (Page et al., 1999
), nitrogen–silicon interactions (Flynn and Martin-Jézéquel, 2000), temperature–growth–cell size (this work), for nitrogen–phosphorus interactions in a toxic dinoflagellate (John and Flynn, 2001), and for nitrogen–photoacclimative interactions (Flynn, Marshall and Geider, in preparation). In the simulations presented here, the constants given in Table III have been employed (unless indicated). Growth kinetics and the N-source interaction
Steady-state relationships between growth and the amount of a single nutrient or light are shown in Figure 3. Those simulations where nitrogen was not limiting used 20 µM nitrate (N) with 1 µm ammonium (A). Theresultant half saturation constants for growth can be judged from Figure 3. Note that the plot for iron is when PFD is saturating; when light is limiting the requirement for iron increases rapidly (Flynn and Hipkin, 1999) and hence so does the half saturation constant for growth. The particular constants used here in equations (2) to (5), defining ammonium and nitrate transport kinetics, give a higher growth rate using ammonium [see (Flynn et al., 1999; Flynn and Hipkin, 1999)]. The resultant ammonium–nitrate interaction, with shifts in the f-ratio (where f = Nt/(At+Nt)) are shown in Figure 4a.
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The true nature, and the environmental relevance, of the ammonium–nitrate interaction with a dual kinetic nitrate transporter [see (Flynn, 1999
)] require further study. Within the model the kinetics of different nitrate transporters can be altered readily by modification of the constants in the second section of equation (5). More traditionally the ammonium–nitrate interaction has been considered using either an inhibition equation (Fasham, 1993) or a ‘top-up’ equation (Armstrong, 1999). While neither of these approaches is biologically valid, an inhibition equation can simulate the decrease in the effect of ammonium with increasing concentrations of nitrate as would be appropriate with a dual kinetic nitrate transporter.
Flynn et al. give comparisons of the behaviour of the inhibition model versus ammonium–nitrate interactions simulated using the type of model described here but with a single nitrate porter type (Flynn et al., 1999). The equation for the f ratio using the inhibition model is given by :
where A and N are ammonium and nitrate concentrations, KA and KN are half saturation constants for assimilation (transport + incorporation), and Ki is the inhibition constant. Figure 4b shows a comparison between the performance at 5 and 20 µM nitrate of the dual-kinetic model against the inhibition model tuned to the dual-kinetic output for nitrate concentrations of 5, 10 and 20 µM (as in Figure 4a). This tuning returned KN = 5.5 µM, KA = 0.77 µM and Ki = –0.6 µM–1. The problem is that KN is very high which means that the ability to use nitrate at low concentrations is impaired, while the mechanistic model with its true dual-kinetic porter still gives a low half saturation constant of ca. 0.5 µM, (Figure 3).
Silicon deposition
Scello and Scellm define, respectively, the minimum Si-cell quota and the maximum content under optimal growth conditions. Any event other than silicon-stress that depresses the growth rate enhances silicon deposition because the cell cycle interphase period G2, when most silicon transport and deposition occurs, is prolonged [reviewed by (Martin-Jézéquel et al., 2000)]. Under these conditions Scell may exceed Scellm. The two constants, celluP and StP, are important for modulating the inter-actions between silicon deposition, C and cell-specific growth rates, controlling the extent to which Scell can exceed Scellm. Maximum enhancement of deposition is attained with a high celluP and low StP [equations (9) and (24)]. Figure 5 shows how the silicon content can vary over the best part of an order of magnitude at different non-silicon-limiting growth rates by altering the ratio of these two constants. The model was run to steady state under iron-limited growth conditions (nitrate, silicon and light in excess) giving the growth rates (µ) indicated (see Figure 3 for the relationship between iron concentrations and growth rate). Values of celluP ranged from 2.2 to 4, and StP from 3.8 to 2, giving the indicated ratios of celluP:StP. The values of celluP and StP given in Table III, and used for all other simulations, are both 3 (i.e. ratio of 1 in Figure 5). The ability to be able to make Si deposition a function of growth rate in this manner may be used to develop a generic diatom model that has different Si:N depending on whether it is growing in Fe-sufficient coastal waters rather than Fe-depleted polar waters, where phytoplankton Si:N is elevated [see discussion in (Pondaven et al., 2000)].
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Variability of state variables with growth limited by different factors
Figure 6 shows model behaviour in steady state when growth is limited by different factors. The relationships between µ and NC when either A or N is controlling growth is similar to that expected from the NC quota curve despite the fact that this relationship no longer has a direct control of growth [see also (Flynn and Fasham, 1997)]. The relationship between P:C and µ underP-stress is also similar to the respective quota curve. That the performance under steady-state conditions follows quota model predictions is important. It means that the model can be configured using N:C and P:C quota relationships obtained from traditional experimental methods and employed in standard quota models. Only light limitation results in an increase in ChlC, other limitations leading to a decline. Iron-stress results in a decline in FC, while other limitations result in an increase. The effect of different nutrients on the f-ratio operates via changes in NC [these affect Atq and Ntq in equations (2) and (4)] and in GC due to changes in the supply of C (hence light and Fe limitation have a pronounced effect).
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With the model constructed as described, any nutrient other than silicon limiting growth will have the same effect on silicon deposition (Figure 6
). If a differential effect is required then celluP and/or StP could be made auxiliaries varying with NCu, PCu, Fcon, or PS/Pqm (Flynn and Martin-Jézéquel, 2000). Using the diatom equation for cellu [equation (24)], limitations other than by silicon result in a decline in C content per cell (Figure 7a, see also Figure 5b); silicon limitation achieves the opposite. Using the non-diatom equation [equation (30)], P-stress results in an increase in Ccell, while other limitations (iron-stress is shown in Figure 7b) lead to a decline, as expected (Raven, 1990; Raven, 1998; Falkowski and Raven, 1997). A hypothetical example of multi-nutrient algal growth dynamics
Figures 8 and 9 show the results of a simulated hypothetical interaction between a diatom and a flagellate, demonstrating how different nutrients, together with light, can interact to regulate growth and how the model can be manipulated. The diatom was set with Um twice that of the flagellate and, unlike the latter, it could also sink out of the system; sinking rate was related to Ccell (Bienfang et al., 1982). Initially growth uses nutrients supplied at the start (akin to a batch culture) but from day 15 an additional nutrient input is given that is low in phosphate and silicate.
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For the first 5 days growth is relatively unrestricted. However, as the biomass increases (Figure 8
), self-shading leads to a decrease in available PFD, matched by an enhanced ChlC (Figure 9). Note that the increase in biomass as estimated by chlorophyll (as would typically be measured in environmental studies) is twice that of C-biomass indicating the dangers inherent in using pigment as a biomass marker. The increase in ChlC places an additional burden on iron supply, so iron now becomes co-limiting (Fcon falls; Figure 9). Shortly after 10 days the macronutrients are all at very low concentrations (Figure 8). Growth becomes co-limited by nitrogen and phos-phorus for both species (Figure 9) and hence ChlC declines. Silicon limitation of cell division in diatoms at 13 days results in an elevated Ccell for these organisms, hence sinking rates increase, causing a rapid decline in diatom numbers (Figure 8).
An inflow of nutrients starts at 15 days that is relatively rich in ammonium but poor in phosphate and silicate. Because the diatoms are still silicon-stressed they recover most rapidly from nitrogen and phosphorus-stress (indicated by NCu and PCu; Figure 9). Low growth rates, low ChlC and higher PFD (because the attenuation of light falls following the loss of diatoms with sinking) all relieve the iron-stress (Fcon increases; Figure 9). Towards the end, growth of both diatoms and flagellates is increasingly limited by phosphate. Nitrate and even ammonium increase; the inflow is dominated by ammonium but this is the ‘preferred’ nitrogen source and hence nitrateaccumulates in the water more rapidly. The diatoms outgrow the flagellates throughout, but the latter finally dominate because the declining diatom growth rate barely matches their sinking rate.
Temperature
As a demonstration of the potential of the model to simulate temperature effects, model output is compared with data of Rhee and Gotham (Rhee and Gotham, 1981). Other than selecting for non-diatom and introducing equations (31) and (32), the remainder of the model was not changed. Tref was set at 20°C, Uref at 1.3 d–1, with Q10t and Q10g at 1.75 and 2 respectively; no attempt was made to optimize the model further. Fitting equation (32) is not easy because Rhee and Gotham (Rhee and Gotham, 1981) do not report extremes of cell size with N-limited growth at different temperatures. The relationships used here are shown in Figure 10a, together with the results of steady-state simulations co-plotted with original data for N-limited growth of a chlorophyte at three different temperatures (Figure 10b). The fit to the data is good.
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As a concluding comment, this demonstration for temperature control also shows that it is not necessary to rigorously optimize all components to achieve satisfactory fits to real data. Although mechanistic models are inevitably more complex than empirically formulated models, they are also more robust. They may thus be more likely to give realistic outputs when run in test scenarios using environmental conditions beyond that in data sets used for their parameterization, such as in global warming and eutrophication predictions.
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Acknowledgments |
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This work was supported by the Natural Environment Research Council of the UK. Comments by Mike Fasham and Veronique Martin-Jézéquel are greatly appreciated, as are the recommendations made by referees.
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Received on July 12, 2000
; accepted on April 10, 2001
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