Skip Navigation

This Article
Right arrow Abstract Freely available
Right arrow FREE Full Text (PDF) Freely available
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Similar articles in ISI Web of Science
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrow Search for citing articles in:
ISI Web of Science (41)
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Flynn, K. J.
Right arrow Search for Related Content
PubMed
Right arrow Articles by Flynn, K. J.
Social Bookmarking
 Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

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


    Abstract
 TOP
 Abstract
 INTRODUCTION
 DESCRIPTION OF THE MODEL
 NITROGEN
 SILICON ASSIMILATION IN DIATOMS
 IRON
 PHOTOSYNTHESIS, CARBON AND...
 CELL DIVISION IN DIATOMS
 INTRODUCING PHOSPHORUS
 CELL SIZE IN NON-DIATOMS
 TEMPERATURE
 SIMPLIFICATIONS
 PARAMETERIZATION
 RESULTS AND DISCUSSION
 REFERENCES
 
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.


    INTRODUCTION
 TOP
 Abstract
 INTRODUCTION
 DESCRIPTION OF THE MODEL
 NITROGEN
 SILICON ASSIMILATION IN DIATOMS
 IRON
 PHOTOSYNTHESIS, CARBON AND...
 CELL DIVISION IN DIATOMS
 INTRODUCING PHOSPHORUS
 CELL SIZE IN NON-DIATOMS
 TEMPERATURE
 SIMPLIFICATIONS
 PARAMETERIZATION
 RESULTS AND DISCUSSION
 REFERENCES
 
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, 1997Go)]. 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, 2000Go)]. 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, 1996Go; Sommer, 1998Go) and mesozooplankton (e.g. Miralto et al., 1999). Models of these transfers into predators [e.g. (Anderson, 1992Go; Anderson, 1994Go; Davidson, 1996Go)] 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, 1998Go) as functions of nutrient status and algal group [e.g. (Penna et al., 1999Go)]. Modelling the operation of microbial planktonic systems, such as that described by Priddle et al. (Priddle et al., 1995Go), 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., 2000Go) and Lenton and Watson (Lenton and Watson, 2000Go) may be criticized (Hoppema and Goeyens, 1999Go; Thomas et al., 1999Go; Pahlow and Riebesell, 2000Go) for their assumption of fixed Redfield ratios (Redfield, 1934Go) 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, 1999Go) 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, 1992Go; Dunne et al., 1999Go). Iron availability also interacts with other nutrient assimilations and with silicification (Hutchins and Bruland, 1998Go; Takeda, 1998Go). Discussions of the NP-limitation of marine primary production (Downing, 1997Go; Falkowski, 1997Go; Benitez-Nelson and Buesseler, 1999Go; Tyrrell, 1999Go) 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, 1999Go) and also in some marine systems; e.g. Emiliania huxleyi blooms with high irradiance, high nitrate and low phosphate availability (Tyrrell and Taylor, 1996Go).

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, 1942Go; Caperon, 1968Go; Droop, 1968Go). 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, 1996Go; Andersen, 1997Go); 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., 1997Go; Flynn and Fasham, 1997Go) were followed by models describing light acclimation and nitrite release (Flynn and Flynn, 1998Go), iron stress (Flynn and Hipkin, 1999Go), silicate (Flynn and Martin-Jézéquel, 2000Go) and phosphate nutrition (John and Flynn, 2000Go). 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., 1982Go)] can also be included within the ecosystem model.


    DESCRIPTION OF THE MODEL
 TOP
 Abstract
 INTRODUCTION
 DESCRIPTION OF THE MODEL
 NITROGEN
 SILICON ASSIMILATION IN DIATOMS
 IRON
 PHOTOSYNTHESIS, CARBON AND...
 CELL DIVISION IN DIATOMS
 INTRODUCING PHOSPHORUS
 CELL SIZE IN NON-DIATOMS
 TEMPERATURE
 SIMPLIFICATIONS
 PARAMETERIZATION
 RESULTS AND DISCUSSION
 REFERENCES
 
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., 1997Go)], and the indexing of rate processes to the maximum growth rate. A schematic of the model is given in Figure 1Go; 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 IIIGoGoGo, with the external parameters (i.e. nutrients) given in Table IVGo. 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.



View larger version (25K):
[in this window]
[in a new window]
 
Fig. 1. Schematic of model structure showing the major flows in and out of state variables (solid arrows and boxes) from the external parameters (nutrients and light, PFD), and the major feedback processes (dashed arrows). State variables and external parameters are explained in Tables I and IVGoGo respectively.

 

View this table:
[in this window]
[in a new window]
 
Table I: State variables. Equations (Eq) for derivation are given
 

View this table:
[in this window]
[in a new window]
 
Table II: Auxiliary variables
 

View this table:
[in this window]
[in a new window]
 
Table III: Constants; those marked * are the most obvious candidates for parameterization
 

View this table:
[in this window]
[in a new window]
 
Table IV: External parameters. Note the time unit for PFD must be the same as for Um in Table IIIGo
 

    NITROGEN
 TOP
 Abstract
 INTRODUCTION
 DESCRIPTION OF THE MODEL
 NITROGEN
 SILICON ASSIMILATION IN DIATOMS
 IRON
 PHOTOSYNTHESIS, CARBON AND...
 CELL DIVISION IN DIATOMS
 INTRODUCING PHOSPHORUS
 CELL SIZE IN NON-DIATOMS
 TEMPERATURE
 SIMPLIFICATIONS
 PARAMETERIZATION
 RESULTS AND DISCUSSION
 REFERENCES
 
The maximum value of ammonium transport varies with the N-status of the cells (Flynn et al., 1997Go, 1999Go), indexed to the maximum growth rate, Um [equation (2)Go].


Ammonium transport [equation (3)Go], 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)Go [Cf. equation (2)Go].


There is evidence that nitrate transport in phytoplankton may be biphasic (Collos et al., 1997Go; Lomas and Glibert, 1999Go; Flynn, 1999Go), as it is in higher plants (Nissen, 1991Go). 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)Go 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 1Go in (Flynn, 1999Go)]. 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 1Go). 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 IIIGo). 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)Go] (Flynn et al., 1997Go; Flynn and Fasham, 1997Go).


Nitrogen from ammonium and/or nitrate leads to changes in the concentration of glutamine, GC [equation (6)Go], in turn providing a feedback regulation of transport (Flynn et al., 1997Go). 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)Go], thus nitrogen-starved cells exhibit a lag in being able to process new nitrogen (Flynn et al., 1997Go). The availability of C for this process is indexed by CAAs [equation (18)Go]. 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)Go].


Changes in NC are given by equation (7)Go. NC declines with Cu.


A normalized quota equation (Flynn et al., 1999Go) is used to describe the relative growth rate at a given NC, giving the quotient NCu [equation (8)Go]. 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.



    SILICON ASSIMILATION IN DIATOMS
 TOP
 Abstract
 INTRODUCTION
 DESCRIPTION OF THE MODEL
 NITROGEN
 SILICON ASSIMILATION IN DIATOMS
 IRON
 PHOTOSYNTHESIS, CARBON AND...
 CELL DIVISION IN DIATOMS
 INTRODUCING PHOSPHORUS
 CELL SIZE IN NON-DIATOMS
 TEMPERATURE
 SIMPLIFICATIONS
 PARAMETERIZATION
 RESULTS AND DISCUSSION
 REFERENCES
 
The simplified silicon model of Flynn and Martin-Jézéquel (Flynn and Martin-Jézéquel, 2000Go) has been used. Silicate transport [equation (9)Go], 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)Go] 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)Go] 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)Go]. Note the silicon content is a cell, not C, quota.



    IRON
 TOP
 Abstract
 INTRODUCTION
 DESCRIPTION OF THE MODEL
 NITROGEN
 SILICON ASSIMILATION IN DIATOMS
 IRON
 PHOTOSYNTHESIS, CARBON AND...
 CELL DIVISION IN DIATOMS
 INTRODUCING PHOSPHORUS
 CELL SIZE IN NON-DIATOMS
 TEMPERATURE
 SIMPLIFICATIONS
 PARAMETERIZATION
 RESULTS AND DISCUSSION
 REFERENCES
 
Details for the basis of these equations are given in Flynn and Hipkin (Flynn and Hipkin, 1999Go). 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)Go]. 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)Go [see (Flynn and Hipkin, 1999Go) 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, 1999Go)]. The photosynthesis subunit (PSU) may contain 500–1300 chlorophyll a (Chl a) molecules (Falkowski et al., 1981Go) and each PSU contains 23 Fe atoms (Raven, 1990Go). At 500 Chl a PSU–1, this gives 0.003 g Fe g–1 Chl a, as used in equation (12)Go. 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, 1999Go).

As a simplification, only the synthesis of the most iron-resource-expensive component, the photosystems [see Figure 7Go in (Flynn and Hipkin, 1999Go)], 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, 1999Go). Fcon [equation (13)Go] 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).




View larger version (18K):
[in this window]
[in a new window]
 
Fig. 7. Changes in cell size for a diatom (a) when growth is limited by the indicated nutrient or by light (see Figure 6Go 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.

 

    PHOTOSYNTHESIS, CARBON AND RESPIRATION
 TOP
 Abstract
 INTRODUCTION
 DESCRIPTION OF THE MODEL
 NITROGEN
 SILICON ASSIMILATION IN DIATOMS
 IRON
 PHOTOSYNTHESIS, CARBON AND...
 CELL DIVISION IN DIATOMS
 INTRODUCING PHOSPHORUS
 CELL SIZE IN NON-DIATOMS
 TEMPERATURE
 SIMPLIFICATIONS
 PARAMETERIZATION
 RESULTS AND DISCUSSION
 REFERENCES
 
Changes in chlorophyll content, ChlC, are described by equation (14)Go. This has been modified from equations given for photoacclimation (Flynn and Flynn, 1998Go), for cell size control (Flynn and Martin-Jézéquel, 2000Go), and for iron control (Flynn and Hipkin, 1999Go). Synthesis is a function of the N-status [NCu, equation (8)Go], 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)Go]. Fine-tuning of the rate of photoacclimation may be achieved using scalar M. This value is typically between 1 (Flynn and Flynn, 1998Go) and 3 (Flynn and Hipkin, 1999Go), 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)Go] 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)Go] 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, 1998Go) and by Geider et al. (Geider et al., 1998Go).


Note in both equations (14) and (15)GoGo NCu (as indicated by *) is replaced by NPCu [see equation (29)Go] 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, 1998Go). The current maximum rate of photosynthesis [Pqm, equation (15)Go] 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, 1999Go)].

Basal respiration [basres, equation (16)Go] is indexed to Um assuming a rate of 5%; the value varies greatly, being much higher in dinoflagellates for example (Harris, 1978Go). 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)Go] describes the level of reserve C available for metabolic processes in darkness or at very low light when photosynthesis is inadequate (Flynn et al., 1997Go).


The value of Cresv is then used in the calculation of the quotients CAAs [equation (18)Go] and Nred [equation (19)Go] describing, respectively, the availability of C for amino acid and ChlC synthesis and the regulation of nitrate reduction to ammonium [see (Flynn et al., 1997Go)].




Equation (20)Go [modified after (Jassby and Platt, 1976Go)] 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)Go], 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, 1998Go)], synthesis of NC and basal respiration (Flynn and Hipkin, 1999Go). A hyperbolic function prevents cell size exceeding a maximum value as described by RS [equation (23)Go]; implications for the release of DOC are mentioned by Flynn and Martin-Jézéquel (Flynn and Martin-Jézéquel, 2000Go). The increase inC-biomass is then given by equation (22)Go.





    CELL DIVISION IN DIATOMS
 TOP
 Abstract
 INTRODUCTION
 DESCRIPTION OF THE MODEL
 NITROGEN
 SILICON ASSIMILATION IN DIATOMS
 IRON
 PHOTOSYNTHESIS, CARBON AND...
 CELL DIVISION IN DIATOMS
 INTRODUCING PHOSPHORUS
 CELL SIZE IN NON-DIATOMS
 TEMPERATURE
 SIMPLIFICATIONS
 PARAMETERIZATION
 RESULTS AND DISCUSSION
 REFERENCES
 
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)Go] and silicate transport [equation (9)Go], is given by equation (23)Go. 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, 2000Go) where RS=1 indicated the minimum rather than the maximum size.]


Changes in cell number [equation (25)Go] occur with cell division [equation (24)Go], which in turn is affected by the current silicate transport rate [equation (9)Go]. Cell division becomes increasingly more likely as cell size becomes larger, controlled by the sigmoidal function to RS in equation (24)Go [the derivation of this equation is explained in (Flynn and Martin-Jézéquel, 2000Go)]. 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.





    INTRODUCING PHOSPHORUS
 TOP
 Abstract
 INTRODUCTION
 DESCRIPTION OF THE MODEL
 NITROGEN
 SILICON ASSIMILATION IN DIATOMS
 IRON
 PHOTOSYNTHESIS, CARBON AND...
 CELL DIVISION IN DIATOMS
 INTRODUCING PHOSPHORUS
 CELL SIZE IN NON-DIATOMS
 TEMPERATURE
 SIMPLIFICATIONS
 PARAMETERIZATION
 RESULTS AND DISCUSSION
 REFERENCES
 
For the simulations presented here the multiple pool phosphorus model, iPIM (John and Flynn, 2000Go), 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, 2001Go).

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)GoGo] 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)Go, giving PCu [equation (28)Go]. Here the constants given in John and Flynn (John and Flynn, 2001Go) 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, 1997Go). 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)Go.

Figure 2Go 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)Go 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.



View larger version (16K):
[in this window]
[in a new window]
 
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.

 
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)GoGo, 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)GoGo] resulted in occasional oscillatory model behaviour as a consequence of feedback processes within the model.


    CELL SIZE IN NON-DIATOMS
 TOP
 Abstract
 INTRODUCTION
 DESCRIPTION OF THE MODEL
 NITROGEN
 SILICON ASSIMILATION IN DIATOMS
 IRON
 PHOTOSYNTHESIS, CARBON AND...
 CELL DIVISION IN DIATOMS
 INTRODUCING PHOSPHORUS
 CELL SIZE IN NON-DIATOMS
 TEMPERATURE
 SIMPLIFICATIONS
 PARAMETERIZATION
 RESULTS AND DISCUSSION
 REFERENCES
 
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, 1990Go; Falkowski and Raven, 1997Go; Raven, 1998Go). The easiest way to incorporate P-stress with cell division is via PCu [equation (28)Go]. When PCu is 0, there is no growth and cell size will be maximal (i.e. RS, from equation (23)Go, tends to 1). When PCu is 1, with other nutrients in excess, the cell can divide rapidly. Equation (30)Go describes cellu as a sigmoidal function of RS.


The power in the equation [analogous to the constant celluP in equation (24)Go] 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)Go now replaces the final (non-diatom) section in equation (24)Go.

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, 2000Go).


    TEMPERATURE
 TOP
 Abstract
 INTRODUCTION
 DESCRIPTION OF THE MODEL
 NITROGEN
 SILICON ASSIMILATION IN DIATOMS
 IRON
 PHOTOSYNTHESIS, CARBON AND...
 CELL DIVISION IN DIATOMS
 INTRODUCING PHOSPHORUS
 CELL SIZE IN NON-DIATOMS
 TEMPERATURE
 SIMPLIFICATIONS
 PARAMETERIZATION
 RESULTS AND DISCUSSION
 REFERENCES
 
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., 1998Go) using the Arrhenius equation (Goldman, 1979Go; Raven and Geider, 1988Go). Haefner (Haefner, 1996Go) 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, 1981Go); 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, 1981Go) also reflect primarily post-nutrient transport processes. Flynn (Flynn, 1998Go) 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., 1998Go), as in Flynn's (1999) explorations of the results of Lomas and Glibert (Flynn, 1999Go; Lomas and Glibert, 1999Go).

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, 1988Go)]; 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)Go, respectively].

T is the operational temperature. Umt is then used to replace Um in equations (2), (4), (9), (11)GoGoGoGo, and the first part of (26)Go. 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, 1981Go) 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)Go 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, 1977Go; Furnas, 1978Go). An increase in silicon per cell with growth at low temperature (Durbin, 1977Go) will be handled by the model in the same way as any other non-silicon limitation of growth (Flynn and Martin-Jézéquel, 2000Go) provided the silicate transport rate is less susceptible to changes in temperature than intracellular processes.


    SIMPLIFICATIONS
 TOP
 Abstract
 INTRODUCTION
 DESCRIPTION OF THE MODEL
 NITROGEN
 SILICON ASSIMILATION IN DIATOMS
 IRON
 PHOTOSYNTHESIS, CARBON AND...
 CELL DIVISION IN DIATOMS
 INTRODUCING PHOSPHORUS
 CELL SIZE IN NON-DIATOMS
 TEMPERATURE
 SIMPLIFICATIONS
 PARAMETERIZATION
 RESULTS AND DISCUSSION
 REFERENCES
 
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)GoGoGo. CAAs should then be removed from equations (6), (7) and (14)GoGoGo, and Nred from equation (5)Go. If a dual kinetic nitrate transporter is not required then the second half of equation (5)Go is deleted. The form of equations (2) and (4)GoGo may also be simplified, though transport must become disabled at NC<NCm [see (Flynn et al., 1999Go)]. If only non-diatoms are to be considered then all reference to silicon [equations (9) and (10)GoGo] can be removed and all but the very last part of equation (24)Go becomes redundant, unless cell size information is required, in which instance equation (24)Go is replaced by (30)Go. Reference to cell size [RS in equations (14), (21), (23)GoGoGo] and to cells [equations (24), (25), (30), (32)GoGoGoGo] 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)GoGoGo are deleted, together with reference to Fcon in equation (14)Go. 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, 2000Go). Equations (31) and (32)GoGo are deleted if temperature is not required as an input.


    PARAMETERIZATION
 TOP
 Abstract
 INTRODUCTION
 DESCRIPTION OF THE MODEL
 NITROGEN
 SILICON ASSIMILATION IN DIATOMS
 IRON
 PHOTOSYNTHESIS, CARBON AND...
 CELL DIVISION IN DIATOMS
 INTRODUCING PHOSPHORUS
 CELL SIZE IN NON-DIATOMS
 TEMPERATURE
 SIMPLIFICATIONS
 PARAMETERIZATION
 RESULTS AND DISCUSSION
 REFERENCES
 
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 VGo. 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.


View this table:
[in this window]
[in a new window]
 
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 inTableIIIGo. 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, 1996Go); 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
 
It is thus important to appreciate that it is not necessary to parameterize rigorously all the constants given in Table IIIGo 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 IIIGo. 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, 1996Go)]. That assumes that the model behaves in a sensible fashion [accords to face and event validity; (Rykiel, 1996Go)], 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., 1999Go). 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)GoGo 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, 1998Go) and the parameterization of iron-dependent processes by Flynn and Hipkin (Flynn and Hipkin, 1999Go). 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).


    RESULTS AND DISCUSSION
 TOP
 Abstract
 INTRODUCTION
 DESCRIPTION OF THE MODEL
 NITROGEN
 SILICON ASSIMILATION IN DIATOMS
 IRON
 PHOTOSYNTHESIS, CARBON AND...
 CELL DIVISION IN DIATOMS
 INTRODUCING PHOSPHORUS
 CELL SIZE IN NON-DIATOMS
 TEMPERATURE
 SIMPLIFICATIONS
 PARAMETERIZATION
 RESULTS AND DISCUSSION
 REFERENCES
 
The model, or its predecessors, have been run to simulate experimental data for ammonium–nitrate interactions (Page et al., 1999Go), nitrogen–silicon interactions (Flynn and Martin-Jézéquel, 2000Go), temperature–growth–cell size (this work), for nitrogen–phosphorus interactions in a toxic dinoflagellate (John and Flynn, 2001Go), and for nitrogen–photoacclimative interactions (Flynn, Marshall and Geider, in preparation). In the simulations presented here, the constants given in Table IIIGo 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 3Go. 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 3Go. Note that the plot for iron is when PFD is saturating; when light is limiting the requirement for iron increases rapidly (Flynn and Hipkin, 1999Go) and hence so does the half saturation constant for growth. The particular constants used here in equations (2) to (5)GoGoGoGo, defining ammonium and nitrate transport kinetics, give a higher growth rate using ammonium [see (Flynn et al., 1999Go; Flynn and Hipkin, 1999Go)]. The resultant ammonium–nitrate interaction, with shifts in the f-ratio (where f = Nt/(At+Nt)) are shown in Figure 4aGo.



View larger version (21K):
[in this window]
[in a new window]
 
Fig. 3. Steady-state relationships between growth rate and the amount of the limiting nutrient (others being in excess).

 


View larger version (20K):
[in this window]
[in a new window]
 
Fig. 4. Implications of running a dual nitrate transporter on the interaction between ammonium (A) and nitrate (N). Plot (a) shows performance of the model as described. The depression of the f-ratio with increasing amounts of A is decreased with higher concentrations of N. For comparison, a plot is also given for a simulation where only the single high-affinity N transporter is present. Plot (b) compares model output for the dual nitrate transporter (solid lines) with behaviour of the inhibition model of Fasham (Fasham, 1993Go) (dashed lines) at two nitrate concentrations.

 
The true nature, and the environmental relevance, of the ammonium–nitrate interaction with a dual kinetic nitrate transporter [see (Flynn, 1999Go)] 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)Go. More traditionally the ammonium–nitrate interaction has been considered using either an inhibition equation (Fasham, 1993Go) or a ‘top-up’ equation (Armstrong, 1999Go). 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., 1999Go). 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 4bGo 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 4aGo). 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 3Go).

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., 2000Go)]. 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)GoGo]. Figure 5Go 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 3Go 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 IIIGo, and used for all other simulations, are both 3 (i.e. ratio of 1 in Figure 5Go). 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., 2000Go)].



View larger version (18K):
[in this window]
[in a new window]
 
Fig. 5. Effects of altering constants celluP and StP on the amount of silicon deposition (a) and cell size (b). For brevity these constants are shown here as varying with a ratio around 1 (as given when both constants have a value of 3 as used in the other simulations shown in this work). The constants can be varied independently.

 
Variability of state variables with growth limited by different factors

Figure 6Go 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, 1997Go)]. 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)GoGo] and in GC due to changes in the supply of C (hence light and Fe limitation have a pronounced effect).



View larger version (28K):
[in this window]
[in a new window]
 
Fig. 6. Steady-state values of state variables for a simulated diatom when growth is limited by the indicated nutrient or by light. A, ammonium; F, iron; N, nitrate; S, silicate; P, phosphorus; PFD, photon flux density. Default nutrient values were A = 1 µM, N = 20 µM, S = 20 µM, F = 0.1 µM, P = 20 µM, PFD = 200 µmol photons m–2 s–1. Um was 1 d–1.

 
With the model constructed as described, any nutrient other than silicon limiting growth will have the same effect on silicon deposition (Figure 6Go). 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, 2000Go). Using the diatom equation for cellu [equation (24)Go], limitations other than by silicon result in a decline in C content per cell (Figure 7aGo, see also Figure 5bGo); silicon limitation achieves the opposite. Using the non-diatom equation [equation (30)Go], P-stress results in an increase in Ccell, while other limitations (iron-stress is shown in Figure 7bGo) lead to a decline, as expected (Raven, 1990Go; Raven, 1998Go; Falkowski and Raven, 1997Go).

A hypothetical example of multi-nutrient algal growth dynamics

Figures 8 and 9GoGo 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., 1982Go). 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.



View larger version (22K):
[in this window]
[in a new window]
 
Fig. 8. Simulation of a hypothetical interaction between a diatom and flagellate in a water body with a mixed depth of 5 m. At day 15, an input of nutrients started introducing 3 µM A, 0.5 µM N, 10 nM F, 0.04 µM P and 0.2 µM S. Surface PFD was 100 µmol m–2 s–1, attenuated by algal biomass within the water column. The growth parameters for the two algal species were the same except the diatom had a Um of 1 d–1 and used S while the flagellate (which did not use S) had a Um of 0.5 d–1. Diatoms sink out of the system with a rate related to their Ccell as given by RS, according to (0.25 + RS • 2) m d–1. Flagellates do not sink out.

 


View larger version (22K):
[in this window]
[in a new window]
 
Fig. 9. As Figure 8Go, but showing changes in cell size, pigment content (ChlC) and nutrient status. NCu and PCu show the relative nitrogen and phosphorus status of the organisms, NPCu the combined NP status, and Fcon the iron status; in all instances 1 is optimal.

 
For the first 5 days growth is relatively unrestricted. However, as the biomass increases (Figure 8Go), self-shading leads to a decrease in available PFD, matched by an enhanced ChlC (Figure 9Go). 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 9Go). Shortly after 10 days the macronutrients are all at very low concentrations (Figure 8Go). Growth becomes co-limited by nitrogen and phos-phorus for both species (Figure 9Go) 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 8Go).

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 9Go). 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 9Go). 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, 1981Go). Other than selecting for non-diatom and introducing equations (31) and (32)GoGo, 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)Go is not easy because Rhee and Gotham (Rhee and Gotham, 1981Go) do not report extremes of cell size with N-limited growth at different temperatures. The relationships used here are shown in Figure 10aGo, 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 10bGo). The fit to the data is good.



View larger version (17K):
[in this window]
[in a new window]
 
Fig. 10. Operation of the model using a temperature function. Plot (a) shows the data (solid symbols) of Rhee and Gotham (Rhee and Gotham, 1981Go) recalculated to show changes in Ccell for growth of Scenedesmus at different temperatures and model output (open symbols) over a wide range of N-limited growth at three temperatures. The temperature-Ccell relationships for Ccello and Ccellm are indicated together with the regression line (dashed) through the original data. Plot (b) shows the data (symbols) of Rhee and Gotham (Rhee and Gotham, 1981Go) for variations of the N-cell quota for the same organism under N-limited growth at three temperatures together with model output (lines).

 
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.


    Acknowledgments
 
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.


    REFERENCES
 TOP
 Abstract
 INTRODUCTION
 DESCRIPTION OF THE MODEL
 NITROGEN
 SILICON ASSIMILATION IN DIATOMS
 IRON
 PHOTOSYNTHESIS, CARBON AND...
 CELL DIVISION IN DIATOMS
 INTRODUCING PHOSPHORUS
 CELL SIZE IN NON-DIATOMS
 TEMPERATURE
 SIMPLIFICATIONS
 PARAMETERIZATION
 RESULTS AND DISCUSSION
 REFERENCES
 
Andersen, T. (1997) Pelagic Nutrient Cycles. Ecological Studies 129. Springer-Verlag, Berlin.

Anderson, T. R. (1992) Modelling the influence of food C:N ratio, and respiration on growth and nitrogen excretion in marine zooplankton and bacteria. J. Plankton Res., 14, 1645–1671.[Abstract/Free Full Text]

Anderson, T. R. (1994) Relating C:N ratios in zooplankton food and faecal pellets using a biochemical model. J. Exp. Mar. Biol. Ecol., 184, 183–199.

Anderson, T. R. and Williams, P. J. le B. (1998) Modelling the seasonal cycle of dissolved organic carbon at Station E1 in the English Channel. Est. Coastal Shelf Sci., 46, 93–109.

Armstrong, R. A. (1999) An optimization-based model of iron-light-ammonium co-limitation of nitrate uptake and phytoplankton growth. Limnol. Oceanogr., 44, 1436–1446.

Benitez-Nelson, C. R. and Buesseler, K. O. (1999) Variability of inorganic and organic phosphorus turnover rates in the coastal ocean. Nature, 398, 502–505.

Bienfang, P. K., Harrison, P. J. and Quarmby, L. M. (1982) Sinking rates response to depletion of nitrate, phosphate and silicate in four marine diatoms. Mar. Biol., 67, 295–302.

Caperon, J. (1968) Population growth in micro-organisms limited by food supply. Ecology, 49, 715–721.

Collos, Y., Vaquer, A., Bibent, B., Slawyk, G., Garcia, N. and Souchu, P. (1997) Variability in nitrate uptake kinetics of phytoplankton communities in a Mediterranean coastal lagoon. Est. Coast. Shelf Sci., 44, 369–375.

Davidson, K. (1996) Modelling microbial food webs. Mar. Ecol. Prog. Ser., 145, 279–296.

Davidson, K. and Cunningham, A. (1996) Accounting for nutrient processing time in mathematical models of phytoplankton growth. Limnol. Oceanogr., 41, 779–783.

Downing, J. A. (1997) Marine nitrogen:phosphorus stoichiometry and the global N:P cycle. Biogeochemistry, 37, 237–252.

Droop, M. R. (1968) Vitamin B12 and marine ecology IV. Kinetics of uptake, growth and inhibition in Monochrysis lutheri. J. Mar. Biol. Assoc. UK., 48, 689–733.

Dunne, J. P., Murray, J. W., Aufdenkampe, A. K., Blain, S. and Rodier, M. (1999) Silicon-nitrogen coupling in the equatorial Pacific upwelling zone. Glob. Biogeochem. Cycle, 13, 715–726.

Durbin, E. G. (1977) Studies on the autoecology of the marine diatom Thalassosira nordenskioeldii. II. The influence of cell size on growth rate and carbon, nitrogen, chlorophyll a and silica content. J. Phycol., 27, 8–14.

Evans, G. T. and Garçon, V. C. (1997) (eds) One Dimensional Models of Water Column Chemistry. JGOFS report 23/97. JGOFS Bergen, Norway.

Falkowski, P. G. (1997) Evolution of the nitrogen cycle and its influence on the biological sequestration of CO2 in the ocean. Nature, 387, 272–275.

Falkowski, P. G., Owens, T. G., Ley, A. C. and Mauzerall, D. C. (1981) Effect of growth irradiance levels on the ratio of reaction centers in 2 species of marine phytoplankton. Plant Physiol., 68, 969–973.[Abstract/Free Full Text]

Falkowski, P. G. and Raven, J. A. (1997) Aquatic Photosynthesis. Blackwell Science, London.

Fasham, M. J. R. (1993) Modelling the marine biota. In Heimann, M. (ed.), The Global Carbon Cycle, NATO ASI ser. I 15. Springer-Verlag, Berlin, pp. 458–504.

Fasham, M. J. R. and Evans, G. T. (2000) Advances in ecosystem modelling within JGOFS. In Ducklow, H. W, Field, J. G. and Hanson, R. (eds) The Changing Ocean Carbon Cycle: a Mid-term Synthesis of the Joint Global Ocean Flux Study. Cambridge University Press, Cambridge, pp. 417–446.

Flynn, K. J. (1998) Estimation of kinetic parameters for the transport of nitrate and ammonium into marine phytoplankton. Mar. Ecol. Prog. Ser., 169, 13–28.

Flynn, K. J. (1999) Nitrate transport and ammonium–nitrate interactions at high nitrate concentration and low temperature. Mar. Ecol. Prog. Ser., 189, 283–287.

Flynn, K. J. and Fasham, M. J. R. (1997) A short version of the ammonium–nitrate interaction model. J. Plankton Res., 19, 1881–1897.[Abstract/Free Full Text]

Flynn, K. J. and Flynn, K. (1998) The release of nitrite by marine dinoflagellates—development of a mathematical simulation. Mar. Biol., 130, 455–470.

Flynn, K. J. and Hipkin, C. R. (1999) Interactions between iron, light, ammonium and nitrate; insights from the construction of a dynamic model of algal physiology. J. Phycol., 35, 1171–1190.[Web of Science]

Flynn, K. J. and Martin-Jézéquel, V. (2000) Modelling Si–N limited growth of diatoms. J. Plankton Res., 22, 447–472.[Abstract/Free Full Text]

Flynn, K. J., Fasham, M. J. R. and Hipkin, C. R. (1997) Modelling the interaction between ammonium and nitrate uptake in marine phytoplankton. Philos. Trans. R.. Soc. Lond. B., 352, 1625–1645.

Flynn, K. J., Page, S., Wood, G. and Hipkin, C. R. (1999) Variations in the maximum transport rates for ammonium and nitrate in the prymnesiophyte Emiliania huxleyi and the raphidophyte Heterosigma carterae. J. Plankton Res., 21, 355–371.[Abstract/Free Full Text]

Furnas, M. J. (1978) Influence of temperature and cell size on the division rate and chemical content of the diatom Chaetoceros curvisetum Cleve. J. Exp. Mar. Biol. Ecol., 34, 97–109.

Geider, R. J., MacIntyre, H. L. and Kana, T. M. (1998) A dynamic regulatory model of phytoplankton acclimation to light, nutrients and temperature. Limnol. Oceanogr., 43, 79–694.

Goldman, J. C. (1979) Temperature effects on steady-state growth, phosphate uptake, and the chemical composition of a marine phytoplankter. Microbial Ecol., 5, 153–166.

Haefner, J. W. (1996) Modeling Biological Systems. Chapman and Hall, New York.

Harris, G. P. (1978) Photosynthesis, productivity and growth: the physiological ecology of phytoplankton. Arch. Hydrobiol. Ergeb. Limnol., 10, 1–171.

Hoppema, M. and Goeyens, L. (1999) Redfield behavior of carbon, nitrogen, and phosphorus depletions in Antarctic surface water. Limnol. Oceanogr., 44, 220–224.

Hotinski, R. M., Kump, L. R. and Najjar, R. G. (2000) Opening Pandora's Box: The impact of open system modeling on interpretations of anoxia. Paleoceanography, 15, 267–279.

Hutchins, D. A. and Bruland, K. W. (1998) Iron-limited diatom growth and Si:N uptake ratios in a coastal upwelling regime. Nature, 393, 561–564.

Jassby, A. D. and Platt, T. (1976) Mathematical formulations of the relationship between photosynthesis and light for phytoplankton. Limnol. Oceanogr., 21, 540–547.

John, E. H. and Flynn, K. J. (2000) Modelling phosphate transport and assimilation in microalgae; how much complexity is warranted?. Ecol. Modelling, 125, 145–157.

John, E. H. and Flynn, K. J. (2001) Modelling changes in paralytic shellfish toxin content of dinoflagellates in response to nitrogen and phosphorous supply. Mar. Ecol. Prog. Ser., in press.

Lenton, T. M. and Watson, A. J. (2000) Redfield revisited 1. Regulation of nitrate, phosphate, and oxygen in the ocean. Glob. Biogeochem. Cycle, 14, 225–248.[Web of Science]

Lomas, M. W. and Glibert, P. M. (1999) Temperature regulation of nitrate uptake: a novel hypothesis about nitrate uptake and reduction in cool water diatoms. Limnol. Oceanogr., 44, 556–572.

Martin-Jézéquel, V., Hildebrand, M. and Brzezinski, M. (2000) Silicon metabolism in diatoms: implications for growth. J. Phycol., 36, 821–840.[Web of Science]

Miralto, A., Barone, G., Romano, G., Poulet, S. A., Ianora, A., Russo, G. L., Buttino, I., Mazzarella, G., Laabir, M. et al., (1999) The insidious effect of diatoms on copepod reproduction. Nature, 402, 173–176.

Monod, J. (1942) Recherches sur la croissance des cultures bactériennes. Actualités scientifiques et industrielles. Annu. Rev. Microbiol., 3, 3–71.

Nissen, P. (1991) Multiphasic uptake mechanisms in plants. Int. Rev. Cytol., 126, 89–134.

Page, S., Hipkin, C. R. and Flynn, K. J. (1999) Interactions between nitrate and ammonium in Emiliania huxleyi. J. Exp. Mar. Biol. Ecol., 236, 307–319.

Pahlow, M. and Riebesell, U. (2000) Temporal trends in deep ocean Redfield ratios. Science, 287, 831–833.[Abstract/Free Full Text]

Penna, A., Berluti, S., Penna, N. and Magnani, M. (1999) Influence of nutrient ratios on the in vitro extracellular polysaccharide production by marine diatoms from the Adriatic Sea. J. Plankton Res., 21, 1681–1690.[Abstract/Free Full Text]

Pondaven, P., Ruiz-Pino, D. Fravalo, C., Tréguer, P., and Jeandel, C. (2000) Interannual variability of Si and N cycles at the time-series station KERFIX between 1990 and 1995—a 1-D modelling study. Deep-Sea Res., 47, 223–257.

Priddle, J., Leakey, R., Symon, C., Whitehouse, M., Robins, D., Cripps, G., Murphy, E. and Owens, N. (1995) Nutrient cycling by Antarctic marine microbial plankton. Mar. Ecol. Prog. Ser., 116, 181–198.

Raven, J. A. (1990) Predictions of Mn and Fe use efficiencies ofphototrophic growth as a function of light availability for growth and of C assimilation pathways. New Phytol., 116, 1–18.[Web of Science]

Raven, J. A. (1998) The twelfth Tansley Lecture: Small is beautiful: the picophytoplankton. Functional Ecol., 12, 503–513.

Raven, J. A. and Geider, R. J. (1988) Temperature and algal growth. New Phytol., 110, 441–461.

Redfield, A. C. (1934) On the proportions of organic derivatives in sea water and their relation to the composition of plankton. In Daniel,R. (ed.) James Johnstone Memorial Volume. University Press of Liverpool, Liverpool, pp.177–192.

Reynolds, C. S. (1999) Non-determinism to probability, or N:P in the community ecology of phytoplankton. Arch. Hydrobiol., 146, 23–35.

Rhee, G-Y. and Gotham, I. J. (1981) The effect of environmental factors on phytoplankton growth: Temperature and the interactions of temperature with nutrient limitation. Limnol. Oceanogr., 26, 635–648.

Rykiel, E. J. Jr (1996) Testing ecological models: the means of validation. Ecol. Model., 90, 229–244.

Sommer, U. (1998) From algal competition to animal production: Enhanced ecological efficiency of Brachionus plicatilis with a mixed diet. Limnol. Oceanogr., 43, 1393–1396.

Takeda, S. (1998) Influence of iron availability on nutrient consumption ratio of diatoms in oceanic waters. Nature, 393, 774–777.

Thomas, H., Ittekkot, V., Osterroht, C. and Schneider, B. (1999) Preferential recycling of nutrients – the ocean's way to increase new production and to pass nutrient limitation? Limnol. Oceanogr., 44, 1999–2004.

Treguer, P. and Jacques, G. (1992) Dynamics of nutrients and phytoplankton, and fluxes of carbon, nitrogen and silicon in the Antarctic ocean. Polar Biol., 12, 149–162.

Tyrrell, T. (1999) The relative influences of nitrogen and phosphorus on oceanic primary production. Nature, 400, 525–531.

Tyrrell, T. and Taylor, A. H. (1996) A modelling study of Emiliania huxleyi in the NE Atlantic. J. Mar. Sys., 9, 83–112.

Wong, C. S. and Matear, R. J. (1999) Sporadic silicate limitation of phytoplankton productivity in the subarctic NE Pacific. Deep-Sea Res. Part II-Top. Stud. Oceanogr., 46, 2539–2555.

Received on July 12, 2000 ; accepted on April 10, 2001
Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?


This article has been cited by other articles:


Home page
J PLANKTON RESHome page
K. J. Flynn and A. Mitra
Building the "perfect beast": modelling mixotrophic plankton
J. Plankton Res., June 28, 2009; (2009) fbp044v1.
[Abstract] [Full Text] [PDF]


Home page
Proc R Soc BHome page
K. J Flynn
Food-density-dependent inefficiency in animals with a gut as a stabilizing mechanism in trophic dynamics
Proc R Soc B, March 22, 2009; 276(1659): 1147 - 1152.
[Abstract] [Full Text] [PDF]


Home page
Phil Trans R Soc AHome page
R.S Lampitt, E.P Achterberg, T.R Anderson, J.A Hughes, M.D Iglesias-Rodriguez, B.A Kelly-Gerreyn, M Lucas, E.E Popova, R Sanders, J.G Shepherd, et al.
Ocean fertilization: a potential means of geoengineering?
Phil Trans R Soc A, November 13, 2008; 366(1882): 3919 - 3945.
[Abstract] [Full Text] [PDF]


Home page
Phil Trans R Soc BHome page
M. J Behrenfeld, K. H Halsey, and A. J Milligan
Evolved physiological responses of phytoplankton to their integrated growth environment
Phil Trans R Soc B, August 27, 2008; 363(1504): 2687 - 2703.
[Abstract] [Full Text] [PDF]


Home page
J PLANKTON RESHome page
K. J. Flynn
The importance of the form of the quota curve and control of non-limiting nutrient transport in phytoplankton models
J. Plankton Res., April 1, 2008; 30(4): 423 - 438.
[Abstract] [Full Text] [PDF]


Home page
J PLANKTON RESHome page
A. Nicklisch, T. Shatwell, and J. Kohler
Analysis and modelling of the interactive effects of temperature and light on phytoplankton growth and relevance for the spring bloom
J. Plankton Res., January 1, 2008; 30(1): 75 - 91.
[Abstract] [Full Text] [PDF]


Home page
J PLANKTON RESHome page
M. Ragni and M. R. d'Alcala
Circadian variability in the photobiology of Phaeodactylum tricornutum: pigment content
J. Plankton Res., February 1, 2007; 29(2): 141 - 156.
[Abstract] [Full Text] [PDF]


Home page
J PLANKTON RESHome page
A. Mitra
A multi-nutrient model for the description of stoichiometric modulation of predation in micro- and mesozooplankton
J. Plankton Res., June 1, 2006; 28(6): 597 - 611.
[Abstract] [Full Text] [PDF]


Home page
J PLANKTON RESHome page
K. J. Flynn
Castles built on sand: dysfunctionality in plankton models and the inadequacy of dialogue between biologists and modellers
J. Plankton Res., December 1, 2005; 27(12): 1205 - 1210.
[Abstract] [Full Text] [PDF]


Home page
J PLANKTON RESHome page
K. J. Flynn and M. J. R. Fasham
Operation of light-dark cycles within simple ecosystem models of primary production and the consequences of using phytoplankton models with different abilities to assimilate N in darkness
J. Plankton Res., January 1, 2003; 25(1): 83 - 92.
[Abstract] [Full Text] [PDF]


This Article
Right arrow Abstract Freely available
Right arrow FREE Full Text (PDF) Freely available
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Similar articles in ISI Web of Science
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrow Search for citing articles in:
ISI Web of Science (41)
Right arrowRequest Permissions
Google Scholar
Right arrow Articles by Flynn, K. J.
Right arrow Search for Related Content
PubMed
Right arrow Articles by Flynn, K. J.
Social Bookmarking
 Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?