- Email: [email protected]
A cell-based model for the chlorophyll a to carbon ratio in phytoplankton
Ecological Modelling 113 (1998) 55 – 70
A cell-based model for the chlorophyll a to carbon ratio in phytoplankton C. Zonneveld * Department of Theoretical Biology, Vrije Uni6ersiteit Amsterdam, De Boelelaan 1087, 1081 HV, Amsterdam, The Netherlands
Abstract The ratio of chlorophyll a per cell and carbon per cell is considered to be a key quantity in phytoplankton growth. The quantity varies with nutrient and light availability. The aim of this paper is to predict how the chlorophyll a to carbon ratio varies in relation to environmental conditions. A cell-based model is presented, that allows the description of carbon and chlorophyll a cell quota, and thus the chlorophyll a to carbon ratio, in nutrient-limited as well as light-limited growth. The model predictions are in line with experimental data. Variations in Chl:C result, among others, from photoacclimation. The analysis in this paper shows that the impact photoacclimation on phytoplankton growth is rather small, despite large variations in Chl:C. This is mainly due to the package effect, a phenomenon that can be easily accounted for especially in cell-based models. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Chl:C; Light-limited growth; Model; Nutrient-limited growth; Photoacclimation; Phytoplankton
1. Introduction The ratio of the amount of chlorophyll a per cell and the amount of carbon per cell, Chl:C, features as a key variable in many microalgal growth models (Bannister, 1979; Sakshaug et al., 1989; Cullen et al., 1993; Kiefer, 1993; Baumert, 1996; Geider et al., 1996). The widespread use of Chl:C as a model variable is basically rooted in the ease with which chlorophyll a concentrations * Tel.: + 31 20 4447128; fax: + 31 20 4447123; e-mail: [email protected]
are experimentally determined. Chlorophyll a is therefore often used as a proxy for phytoplankton biomass. However, it is a poor measure, and one needs knowledge of Chl:C to translate chlorophyll a measurements to the more useful phytoplankton carbon measure (Geider et al., 1997). The argument outlined above favors the use of Chl:C as a model variable. Yet this use can have its drawbacks too, especially from the modelling perspective. The possibility of direct measurements on a model variable is appealing, but other aspects are also important. For example, the characteristic should play a vital role in the physiology
0304-3800/98/$ - see front matter © 1998 Elsevier Science B.V. All rights reserved. PII S0304-3800(98)00134-3
56
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
of the organism, and this role should preferably be easy to model. Although the physiological importance of chlorophyll is self-evident, taking chlorophyll as a basic model variable not necessarily yields the most parsimonious model. For instance, light-saturated rates of photosynthesis per unit chlorophyll strongly depend on the state of photoacclimation. The primary reason for this dependence is that while photoacclimation affects the amounts of chlorophyll per cell, the amount of RUBISCO (the enzyme limiting the light-saturated rate of photosynthesis) per cell is often unaffected. So the relevance of chlorophyll (and consequently of Chl:C) for light-saturated rates of photosynthesis can be questioned. Another disadvantage of Chl:C is that the use of the ratio of two cellular quantities, both subject to regulation, hinders a better understanding of the regulation of both quantities separately. In this paper, I present a cell-based model of Chl:C. An important difference with current modeling approaches is that a cell-based model enables one to model carbon cell quota, nitrogen cell quota and chlorophyll cell quota separately. As a result, expressions for the Chl:C ratio can also be derived. To judge the relevance of the model, I use data from the literature. Finally, I present an analysis of photoacclimation on the basis of the model presented in this paper. This analysis shows that the impact of photoacclimation on phytoplankton growth is less severe than might be expected in view of the dramatic effect of photoacclimation on Chl:C.
To deal with light-limited as well as nutrientlimited growth, I reformulate a model for growth on two potentially growth-limiting essential nutrients (Zonneveld, 1996). In this paper, nitrogen and carbon play the role of the two potentially growth-limiting essential nutrients. The model characterises a phytoplankton cell in terms of three variables: permanent volume, with a fixed C:N ratio, and two transient nutrient pools, one for carbon and one for nitrogen (see Fig. 1). Assimilation acts to increase the transient nutrient pools. The light-dependence of the carbon assimilation is taken into account, but the lightdependence of the nitrogen assimilation (Curtis and Megard, 1987) is neglected here, as studies on nutrient-limited growth are generally restricted to one level of irradiance. Carbon and nitrogen are mobilized from the pools to form permanent volume. Variations in the size of the carbon and nitrogen pool relative to the permanent volume lead to variation in cellular composition. The general ideas of this model structure can be traced back to the models of Williams (1967) and, especially, Kooijman (1993). When carbon is in lower supply relative to need than nitrogen, growth is said to be light-limited, otherwise it is nitrogen-limited. Thus limitation is of the either/or type, and co-limitation of two nutrients is not considered here. The cellular physiology is geared to the limiting nutrient, which implies that the uptake rate of the non-limiting nutrient depends on the cell quota of the limiting nutrient. Similarly, the kinetics of chloro-
2. Basic model structure Phytoplankton ecologists often consider growth to be either nutrient-limited or light-limited, a convention I follow here. For nutrient-limited growth, the model must specify how the growth rate depends on the cellular nutrient availability. Furthermore, the model must specify the carbon and chlorophyll cell quota. For light-limited growth, the growth rate as dependent on irradiance must be specified, and, again, the carbon and chlorophyll cell quota.
Fig. 1. Basic model structure.
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
phyll is thought to depend on the limiting nutrient. I shall compare the model formulation with data from chemostat cultures. The comparison concerns predictions regarding the cell quotas for the limiting and the non-limiting nutrient, and for chlorophyll. The comparison is restricted to steady-state states. The model derived below deals with nutrientlimited as well as with light-limited growth, assimilation of nutrients, etc. It is unavoidable to introduce a considerable number of variables and parameters. Despite its cumbersome appearance, there is a certain rationale behind the notation, explained in Appendix A.
3. Nutrient-limited growth Let N denote the amount of nitrogen in the transient nitrogen pool. It is often more convenient to work with the nitrogen density [N]: the amount of nitrogen in the transient pool divided by permanent volume V. (Permanent volume thus excludes the volume occupied by the transient nutrient pools.) I assume that the nitrogen density of the cells eventually reaches a constant value, say [N m l ], when cells are grown at saturating concentrations of the nitrogen source in the environment for a sufficiently long period of time. This value cannot be exceeded as long as the nutrient is limiting indeed. Furthermore, I assume that nutrient uptake rate is proportional to cell volume, and depends hyperbolically on the environmental concentration (i.e. Michaelis – Menten uptake kinetics, see for instance McCarty, 1981). Throughout this paper, fN indicates the hyperbolic dependence of the uptake rate on the substrate concentration xN (i.e., fN =xN /(KN +xN )). The rate of change nutrient density is assumed to obey d [N ]= [A m N,l]fN −nN [Nl ], dt l
(1)
i.e. the nitrogen density follows first order kinetics. The quantity nN [Nl ] represents the usage of nitrogen from the transient pool; nN is a proportionality constant. If the external nitrogen-source
57
concentration is saturating ( fN = 1) for a sufficiently long period of time, the nitrogen density no longer changes, so (d/dt)[Nl ]= 0. From this m condition one obtains nN = [A m N,l ]/[N l ]. On the one hand, the utilization rate equals the assimilation rate AN,l = [A m N,l ]VfN minus the rate of change of the amount of nutrient, d/dt(Nl )= d/ dt([Nl ]V). On the other hand, if nitrogen is used only for growth (no extrusion of nitrogen from the cell) and nitrogen contents of a unit structural volume, [GN ], is constant, one knows that the utilization rate must equal [GN ]d/dt(V), so that AN,l −
d d d ([N]V)= [Nl ] nNV− V = [GN ] V. dt dt dt
(The first equality is obtained by application of the chain rule for differentiation and substitution of Eq. (1) and the expression for the assimilation rate. The second equality holds by assumption.) This equation can be solved for the growth rate (d/dt)V. It is convenient to focus on the specific growth rate, denoted by m; from the above equation, this is found to be mN
1 d [Nl ] V= nN . V dt [GN ]+ [Nl ]
(2)
At steady-state in a chemostat, the specific growth rate m equals the dilution rate, D. Combined with Eq. (2) this yields an expression for the density of the transient nutrient pool in steadystate, as dependent on the dilution rate: [N. l ]=
[GN ]D . nN − D
(3)
3.1. Carbon in nitrogen-limited phytoplankton In a previous paper (Zonneveld, 1996), I showed that the uptake capacity (i.e. the uptake rate given saturating conditions of the nutrient in the environment) for a non-limiting nutrient can be considered to depend on the density of the transient pool of the limiting nutrient. A similar approach is followed by Geider et al. (1998). The precise form of this dependence may differ between species and the nutrients considered. I here postulate three basic patterns, see Fig. 2. The uptake capacity may be independent of the nutri-
58
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
Fig. 2. Dependence of the assimilation capacity for a non-limiting nutrient on the density of the transient nutrient pool of the limiting nutrient. The left panel shows the basic patterns; the assimilation capacity may be (A) independent of, or (B) proportional to the density of the transient nutrient pool of the limiting nutrient, or (C) be proportional to the specific growth rate, so hyperbolic in the nutrient density (Eq. (2)). The middle and the right panel shows all possible combinations of the three basic patterns: D= A +B, E= A +C, F= B+C, G= A+ B+ C.
ent density, it may be proportional to the nutrient density, or it may be proportional to the specific growth rate, i.e. hyperbolic in the nutrient density (Eq. (2)). For a particular phytoplankton species, any linear combination of the three basic patterns is allowed. Experimental data may give clues to what extent each of the basic patterns contributes to the actual pattern of density dependence. The fixation of CO2 in the form of reduced carbon compounds can be thought of as the assimilation of a nutrient. Consequently, I assume that the maximum rate of photosynthesis depends on the density of the limiting nutrient. Such a relationship has been shown experimentally in the diatom, Phaeodaclylum tricornutum Bohlin (Osborne and Geider, 1986). Photosynthesis depends on the light intensity as well as on the concentration of dissolved inorganic carbon (DIC) in the water (for the dependence of photosynthesis on DIC, see Nielsen, 1995). Sufficient availability of DIC is usually taken for granted, hence rates of photosynthesis are mostly related only to the light intensity. This approach is valid as long as the CO2 concentration is either constant or saturating. In this paper, I assume that either of these conditions holds. Studies on nutrient-limited growth are generally performed at one irradiance only, so information on the effects of irradiance on carbon cell quota in nutrient-limited phytoplankton is mostly lacking. Consequently, this topic is not addressed in
this paper, and I simply indicate the relationship between the rate of photosynthesis and irradiance by fI [A m C,l ], with fI the rate of photosynthesis as a fraction of its maximum value [A m C,l ]. Carbon is used for maintenance, growth, and respiration related to growth. Maintenance requirements are taken proportional to permanent volume, and growth-related respiration is taken proportional to the growth rate. The rate of change of the carbon pool is thus given by d t [C ]= fI [A m C,n]− [MC ]− m([G C]+ [Cn ]) dt N
(4)
The aggregate assimilation capacity [A m C,n ] obeys [Nl ] [Nl ] 0 p [A m + [A hC,n] , C ]= [A C,n]+ [A C,n] ] [N m [G l N ]+ [Nl ] (5) where [A 0C,n ], [A pC,n ] and [A hC,n ] are weighing coefficients corresponding to patterns A, B and C, respectively, in Fig. 2. In steady-state d/dt[Cn ]= 0. With Eq. (3), the steady-sate carbon cell quota QC = [G nC ]+ [Cn ] can be expressed as QC =
fI [A 0C,n]− [Mc ] [A pC,n] [GN ] + fI [A hC,n]/nN + fI D nN − D [N m l ] − [G rC].
(6)
This expression might look unwieldy, but its qualitative behaviour is completely determined by only three dimensionless parameters. Introduction of non-dimensional variables and parameters allows the following simplification:
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
F=
a0 ap + +ah D 1− D
with the following dimensionless variables and parameters: F = Qc /[G rC] D = D/nN a0=
scaled carbon cell quota scaled dilution rate
0 C,n
fI [A ]− [Mc ] nN [G rC]
scaled weighing coefficient for pattern A in Fig. 2 [A l ][G ] a p = fI C,nr Nr nN [G C][N l ]
59
3 clearly shows this. At high growth rates, pattern B in Fig. 2 dominates the carbon uptake. Hence the behavior of the carbon cell quota at high growth rates is mainly determined by the value of a p, which characterises the contribution of pattern B. Studies on nutrient-limited phytoplankton sometimes show that the carbon cell quota are almost independent of the growth rate (Goldman and Peavey, 1979; Elrifi and Turpin, 1985). Fig. 3 shows that, with an appropriate choice of parameter values, such a pattern can be described. Complete independence of the growth rate is obtained for a p = a 0 = 0. Below I shall analyse the data of Goldman and Peavey (1979).
scaled weighing coefficient for pattern B in Fig. 2 ah=
fI [A hC,n]/nN − [G rC] [G rC]
scaled weighing coefficient for pattern C in Fig. 2. Basically, the introduction of non-dimensional variables and parameters amounts to a judicious choice of the units in which they are expressed. For example, the dilution rate is expressed in multiples of the parameter nN. This choice is particularly useful, as the maximum specific growth rate, and so the maximum steady-state dilution rate, is less then nN (from Eq. (2) one has lim[Nl] m= nN ), so the non-dimensionless dilution rate takes values between 0 and 1. The remaining quantities are likewise rendered dimensionless. Fig. 3 shows how the carbon cell quota change with growth rate. The parameter a h characterises the uptake proportional to the growth rate, hence changes in its value do not affect the carbon cell quota differently at different growth rates. The effect of an increase in a h is just to elevate the entire curve. The potential values of a h range from −1 (for [A hC,n ]= 0) to . From Eq. (2) it is clear that low growth rates correlate with small nitrogen densities. Hence the contributions to carbon uptake of patterns B and C in Fig. 2, characterised by a h and a p, become very small at low growth rates. The parameter a 0 thus controls the behaviour of the carbon cell quota carbon cell quota at low growth rates. Fig.
Fig. 3. Dependence of carbon cell quota on the dilution rate, for various parameter values, in nitrogen-limited phytoplankton.
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
60
3.2. Nutrient-limited Chl:C I assume that when cells are nitrogen-limited, the absolute rate at which chlorophyll is produced depends on the amount of nitrogen in the transient pool. Furthermore, chlorophyll cell quota decrease because of dilution through growth, and may also decrease through chlorophyll destruction (characterised by a rate constant r). By assumption, the chlorophyll production rate equals kNl, with k a proportionality constant. (The chlorophyll production rate may also depend on irradiance, in which case k would depend on the irradiance. As studies on nutrient-limited growth are generally restricted to one level of irradiance, information on this aspect is lacking. I therefore treat k as a constant.) This implies that the cellular chlorophyll concentration changes according to d [Chl]= k[Nl ]− (m +r) · [Chl] dt so that at steady-state (when m = D) [Chl] =
r=0 k[GN ]D k[GN ] = . (nN −D)(D + r) (nN −D)
For a negligible destruction rate (r = 0) this expression is similar to that for the nitrogen cell quota, Eq. (3). Based on the analysis of experimental data in section 4.3 I assume henceforth that r =0, which simplifies the equation for Chl:C, while it hardly affects its accuracy. The chlorophyll to carbon ratio, u, now follows from the expressions for the chlorophyll cell quota and the carbon cell quota: u=
[Chl] = Qc
3.3. Test against data The relevance of the above derived expressions will now be evaluated by comparing them to experimental data. To this end, I choose data on nitrogen-limited Dunaliella tertiolecta Butcher, presented by Goldman and Peavey (1979) and Goldman (1980). These articles contain data regarding nitrogen, carbon and chlorophyll cell quota of steady-state growing cells. The alga was limited by various nitrogen sources, but the nitrogen source (nitrate, nitrite, ammonium or urea) did not affect the growth response, nor the carbon or chlorophyll cell quota. Therefore I do not distinguish between the different nitrogen sources on which the cells were grown. (See the original publications for further details.) The following equations describe the data: nitrogen cell quota QN = [GN ]+ [N. l ]=
[GN ]nN nN − D
carbon cell quota QC =
ap a0 + +ah D n− D
with
k[GN ]/(n−D) . (fI [A 0C,n]−[Mc ])/D+(fI [A hC,n]/nN−[G rC])+fI [A pC,n][GN ]/[N m l ](nN−D)
Employing the dimensionless parameters introduced before, this equation simplifies to U−1 = a p + a 0
Fig. 4 compares the chlorophyll to carbon (left panels) with the carbon to chlorophyll ratio (right panels). Though they convey the same information, the impact of parameter values on Chl:C is more easily recognized. Chl:C is proportional to the dilution rate if a h = 0 and a p = a 0 " 0. Such a proportionality has been found in Thalassiosira fin6iatilis Hustedt, for example (Laws and Bannister, 1980). Depending on the parameter values, other patterns may be also be observed.
(1− D) + (1 −D)a h D
with U= ((nN [G ])/(k[GN ]))u. r C
(7)
a 0 = fI [A 0C,n]− [MC ] a p = fI [A pC,n][GN ]/[N m l ] a h = fI [A hC,n]/nN − [G rC] chlorophyll cell quota [Chl]=
k[GN ] nN − D
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
61
Fig. 4. Qualitative behaviour of the chlorophyll to carbon ratio (left) and its inverse, the carbon to chlorophyll ratio (right), as dependent on the dimensionless parameters.
Fig. 5 shows the data with their model description. The nitrogen cell quota curve closely follows the data. This is not surprising, as the model is similar to the Droop relationship, well-known to describe such data nicely.
The carbon cell quota are also well described by the model. One may express doubts on the validity of the ascending left branch of the curve, as it seems not strongly supported by the data. As outlined above, the carbon cell quota at low
62
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
growth rates depends especially on the parameter a 0, which characterises the rate of photosynthesis at zero growth rate. To see whether this parameter is really important, I also fitted the model with a 0 = 0. The dashed curve shows the resulting model fit. The systematic deviations from the data suggest that the assumption that a 0 =0 is not warranted. Biologically this means that gross photosynthesize at zero growth rate exceeds the maintenance requirements.
The amounts of chlorophyll per cell also obey a Droop-type relationship against the growth rate. Destruction of chlorophyll may become apparent only at very low dilution rates, where no data were available. As the model describes both carbon and chlorophyll cell quota well, one might expect that the model for the chlorophyll to carbon ratio also fits the data. Except for minor deviations, this expectation is indeed fulfilled.
Fig. 5. (A) Nitrogen cell quota; (B) carbon cell quota; (C) chlorophyll cell quota and (D) Chl:C in nitrogen-limited Dunaliella tertiolecta. Parameter values: [GN ] =1.25 pg N cell − 1, nN = 1.55 d − 1, a 0 =1.27, a p =9.30, a h =9.81 pg C/(cell · d − 1), k = 0.207 pg Chl/(pg N · d − 1). The parameter values were simultaneously estimated from the data on nitrogen, carbon and chlorophyll cell quota. The dashed curve describing the carbon cell quota is the best fit for a 0 =0 (with a p =6.56, a h =16.2). The solid curve describing the chlorophyll cell quota assumes a zero destruction rate; the dashed curve is the model with the destruction rate estimated from the data (r= 0.015 d − 1).
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
4. Light-limited growth In light-limited phytoplankton, the growth rate is determined by the transient carbon pool instead of the transient nitrogen pool. This means that to derive an expression for u one only has to model the carbon and chlorophyll cell quota, not the nitrogen cell quota. The transient carbon pool in light-limited phytoplankton plays the same role as the transient nitrogen pool in nitrogen-limited phytoplankton. Consequently, I use the same modelling approach. Light-limited growth and carbon cell quota have been dealt with more fully in Zonneveld et al. (1997). Moreover, the derivation of the expression for the light-limited growth rate is similar to that for the nutrient-limited growth rate. Hence I choose for a more succinct presentation in this section. As elaborated in Zonneveld et al. (1997), the rate of photosynthesis is taken to depend hyperbolically on light intensity, not on the concentration of inorganic carbon. Below I evaluate the consequences of photoacclimation, but for the moment I neglect this phenomenon. Proceeding in the same way as for nutrient-limited growth, the light-limited specific growth rate is obtained as follows. The density of the transient carbon pool [Cl ] follows first-order kinetics: d I [Cl ]= [A m −nc [Cl ], C,l] dt Is + I
(8)
Carbon serves both as a nutrient necessary for growth and as a storage of energy. With regard to energetic demands, carbon is used to cover maintenance requirements and growth-related respiration. Furthermore, it is used as a building material for new permanent volume. The utilization rate is thus given by [MC ]V +([G 6C]+ [G rC])
d V dt
On the other hand, the utilization rate equals [Cl ](nCV−d/dt(V)). By equating these two expressions one obtaines for the light-limited specific growth rate: m
1 d n [C ]−[MC ] V= c l . V dt [Cl ]+[GC ]
(9)
63
Continuous culture studies on nutrient-limited growth are performed in chemostats, in which the steady-state growth rate equals the dilution rate. Therefore, the dilution rate (often somewhat misleadingly called the specific growth rate) is treated as the independent variable. Continuous culture studies on light-limited growth are often performed in turbidostats, in which the dilution rate is adjusted until a steady-state is achieved. In such studies, the incident light intensity is taken as the independent variable. An expression for the steady-state specific growth rate as dependent on the light intensity is obtained as follows. From Eq. (8) one has [C. l ]= [C m l ]
I . Is + I
Substitution of this expression into Eq. (9) results, after some rearrangements, in m=
I([A m C,l]− [MC ])− [MC ]Is . t t I([C m l ]+ [G C])+ Is [G C]
(10)
The steady-state carbon cell quota are given by I . Is + I
QC = [G 6C]+ [C m l ]
4.1. Light-limited Chl:C Whereas during nutrient-limited growth the chlorophyll cell quota increases with the growth rate, the opposite is true for light-limited growth. But even if growth is light-saturated, the chlorophyll cell quota decrease with further increasing light intensity. This suggests that chlorophyll production rates in light-limited phytoplankton may be light-dependent. I am not aware of detailed quantitative studies on this topic, so I follow a somewhat provisional approach here. I assume that chlorophyll production is a function of light intensity, say p(I). I choose this function such that a close correspondence with experimental data results. I reanalyse data from Falkowski et al. (1985), on light-limited growth in Isochrysis galbana (Green) and Thalassiosira weisfeogii (Grunow) Fryxell and Hasle. I took p(I)= aI/(b + I+cI 2) to describe the chlorophyll production rate. Steady-state chlorophyll cell quota are then given by
64
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
Fig. 6. (A) Specific growth rate; (B) carbon cell quota; (C) chlorophyll per cell and (D) inferred chlorophyll production rates in light limited Isochrysis galbana. The chlorophyll cell quota are described by two curves, the solid one representing no destruction of chlorophyll (r =0), the dashed one representing r = 0.5 d − 1. Production rates are deduced from amount of chlorophyll per cell, the known specific growth rate, and a hypothetical destruction rate. Data from Falkowski et al. (1985). Parameter values: [G nC ]= 8.0 pg −1 −1 C cell − 1, [G rC ] =2.5 pg C cell − 1, [C m , Is = 150 mEinst m − 2 s − 1, [A m · d − 1, [MC ]= 0.84 pg C l ] = 14 pg C cell C ]= 35 pg C cell cell − 1 · d − 1, a = 0.35 pg Chl cell − 1 · d − 1, b= 75 mEinst m − 2s − 1, c = 2.2 m2 s mEinst − 1, r = 0.
Chl +
p(I) aI = , m(I)+ r (m(I) + r)(b +I +cI 2)
with m(I) given by Eq. (10). The chlorophyll cell quota decreases through dilution by growth and possibly by destruction of chlorophyll, characterised by a constant destruction rate r. Fig. 6 shows a nice description of the data, but as 3 (or even 4, including r) parameters are used to describe only five points, this at best weakly supports the model. The data are mute about the value of the specific rate of chlorophyll destruction: values between 0 and 0.5 d − 1 are all consistent with the data. Note that although the rate of chlorophyll production decreases with decreas-
ing irradiance (at low irradiances) the chlorophyll cell quota does increase. The reason for this increase is that dilution through growth becomes progressively less important with decreasing irradiance. An alternative explanation for the observed dependence of chlorophyll cell quota on irradiance is that the chlorophyll production rate increases monotonically with irradiance (or with the amount of carbon in the transient carbon pool), while the destruction rate increases with irradiance. These assumptions lead to similar fits as those in Fig. 6, but with unrealistically high chlorophyll destruction rates (up to 3 d − 1 at 600
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
mEinst m − 2 s − 1). Therefore I conclude that this alternative is not viable. Carbon cell quota are well described by the model, and so is u, as Fig. 7 shows. To compare the light-limited u with the nutrient-limited one, u
65
is also plotted as dependent on the specific growth rate. As the light-dependence of the specific growth rate is also liable to variation, the plot of u against specific growth rate shows slightly more scatter compared to the plot against light intensity. Nevertheless, the overall picture is satisfactory. Geider (1987) noted that the carbon to chlorophyll ratio (u − l) increases linearly with light level in all nutrient-sufficient, uni-algal cultures examined. Fig. 7 shows that this is approximately valid for Isochrysis galbana. The linear dependence of u − l on light intensity can also be derived analytically, by taking the limI for u − 1. For large I, the carbon cell quota converge towards [G nC ]+ [C m l ], and the growth rate converges towards the maximum growth rate, say mm. So for large irradiances, u − 1 obeys lim I
QC [G 6C]+ [C m l ] = Chl (aI/(mm + r)(b +I+ cI 2)) =
2 ([G 6C]+ [C m l ])(mm + r)(b +I+ cI ) aI
−1 = ([G 6C]+ [C m + cI/a) l ])(mm + r)(a
= k1 + k2I. increases linearly with light intensity Thus, u at high light levels. Fig. 7 suggests that this relationship approximately holds true even at quite low light levels. Geider et al. (1997) also arrived at a linear dependence of u − l on irradiance. Their description involves a maximum value for u at very low irradiance, and two parameters characterising the PI-curve. −1
4.2. Photoacclimation
Fig. 7. (A) Chlorophyll: Carbon ratio; (B) Carbon: Chlorophyll ratio, both as dependent on irradiance; and (C) Chlorophyll: Carbon ratio as dependent on the specific growth rate.
The present descriptions of carbon cell quota and the specific growth rate do not make use of u. This variable is one of the best documented characteristics of photoacclimation, so one is led to wonder why the model works while this aspect is neglected. The present model is formulated at the level of the individual cell. To see why this matters, let us focus on light-saturated rates of photosynthesis. Photoacclimation results in increased chlorophyll cell quota when cell are kept under low light
66
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
conditions. However, the maximum rate of photosynthesis per cell may be unaffected by photoacclimation (Harding et al., 1987; Sukenik et al., 1987). If so, the maximum rate of photosynthesis per milligram chlorophyll is strongly affected by photoacclimation, but this reflects changes in chlorophyll cell quota rather than changes in the photosynthetic capacity of the cell. A cell-based model circumvents this pitfall. Due to the so-called package effect the efficiency of light capture of chlorophyll decreases when chlorophyll cell quota increase. The increased amount of chlorophyll of cells grown in low light is packed in the same or (generally) a smaller volume, in comparison with cells grown in high light conditions. Self-shading of chlorophyll prohibits a proportional increase in light capture efficiency of the cell with the concentration of chlorophyll. As a result, light absorption is not proportional to u. The remainder of this section aims to show that photoacclimation can be neglected in a cell-based model without excessive loss of accuracy, due to the package effect. My approach to model the package effect is a convenient combination of mechanistic and descriptive elements, more fully explored in Zonneveld (1997). The focus is on the cellular absorption cross section, which equals the chlorophyll-specific absorption cross section times the chlorophyll concentration of the cell. I assume that cells without chlorophyll cannot harvest light, so their cellular absorption cross section is zero. Furthermore, at very high chlorophyll concentrations in the cell self-shading sets an upper bound to the cellular absorption cross section. Additional chlorophyll would not be able to capture more photons, as all photons entering the cell are already absorbed. I choose an arbitrary function satisfying these mechanistic elements, a rectangular hyperbola. Such a function can be characterised by the slope of the curve at zero chlorophyll and by its maximum. The slope is the ‘true chlorophyll-specific absorption cross section’, denoted by x(dm2 mg − 1 Chl). This is the absorption cross section unaffected by the package effect. The maximum cellular absorption cross section (mm2 · cell − l) is denoted by zm. The cellular absorption cross section is thus given by
z=
zm Chl . zm /x+Chl
(11)
The chlorophyll-specific absorption cross section s is then given by s = z/Chl=
zm . zm /x+Chl
(12)
Fig. 8 shows an application of Eqs. (11) and (12) to experimental data. The good fit supports the present approach to model the photoacclimational response of the absorption cross section. The cellular absorption cross section does increase with decreasing light intensity, but the increase is not pronounced. The cellular cross section increases by about 25%, whereas the chlorophyll cell quota increases almost 3-fold. This shows that the chlorophyll cell quota only weakly characterises the absorptive potential of a cell. The same applies to u. In Fig. 8, I also analyse the consequences of photoacclimation for the photosynthesis –irradiance (PI) curve. The saturation constant of the PI-curve is proportional to the cellular cross section (Zonneveld, 1997), which depends on the light intensity at which the cells grow. To fix the proportionality constant, I set the saturation constant of cells grown at a light intensity of 300 mmol quanta m2 s − 1 equal to 150 mmol quanta m2 s − 1. The saturation constants of cells grown at different light intensities can then be calculated. Fig. 8 shows inferred PI-curves for cells grown at each steady-state light intensity (dotted curves). Each curve is marked at the light intensity with which the curve corresponds. The solid curve represents a least squares fitted simple PI-curve to the five points, which does not account for photoacclimation. This simple PI-curve yields an excellent approximation to the PI-curves that account for photoacclimation. This foregoing analysis of photoacclimation suggests that it affects the PI-curve only mildly. It is important to realize that this conclusion applies only to rates of photosynthesis expressed on a per cell basis. Rates of photosynthesis expressed per unit chlorophyll are dramatically affected by photoacclimation, but this is so mainly because the chlorophyll cell quota are affected.
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
67
Fig. 8. Photoacclimation in light-limited Thalassiosira weisflogii. Data from Falkowski et al. (1985). (A) Chlorophyll per cell; (B) chlorophyll-specific absorption cross section; (C) cellular absorption cross section and (D) inferred photosynthesis– irradiance curves. See text for explanation. Parameter values: maximum cellular cross-section zm =63.4 mm2 cell − 1, true chlorophyll-specific absorption cross section x= 19.3 m2 g − 1 Chl.
5. Discussion The present model successfully describes the carbon and chlorophyll cell quota, and, as a corollary, the chlorophyll to carbon ratio u. This applies to nutrient-limited as well as to light-limited growth. The model descriptions find their basis in assumptions about uptake and utilization of nutrient. This contrasts with models that relate u to environmental or physiological variables via statistical models (Cloern et al., 1995). My approach also contrasts with other mechanistic microalgal growth models in two important aspects: I take the phytoplankton cell as the point of departure, and I deny a pivotal role to u. Both aspects deserve to be commented upon.
Many phytoplankton species occur as single cells. Most models on phytoplankton growth do not account for this level of organisation. Ignoring this level in a model may be advantageous, but has its drawbacks, too. For example, photoacclimation may be dealt with more easily if one accounts for individual cells. The increase in chlorophyll content of a phytoplankton cell serves to enhance the harvesting of light energy, when energy needed to fix carbon dioxide is in short supply. The light-saturated rate of photosynthesis of a phytoplankton cell may well be unaffected by an increase in chlorophyll per cell, as the energy supply is not limiting the fixation of carbon dioxide. However, the light-saturated rate of photosynthesis per unit chlorophyll is strongly affected
68
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
by photoacclimation, and one somehow has to account for this dependency. This can possibly be done via u, which is an important indicator of the state of photoacclimation. However, the problem had better be avoided in the first place, by recognizing the importance of the cell as a basic biological entity. One might argue that cell size (permanent volume) as a basic model variable is inappropriate in view of the variation in size encountered in algal communities, which might span several orders of magnitude. Admittedly, my approach is not suited for studies on algal communities with unknown species composition. But I doubt whether the use of any sophisticated model is justified if the system to be studied is ill characterised. With only a rough measure of biomass at our disposal, such as the chlorophyll content of a water body with unknown species composition, one should use simple models, or even rules of thumb. The predictive value of a model depends to a considerable
extent on the quality of its input. If the input is ill defined, the output cannot be improved by elaborating a model. My approach is most suited for algal growth in controlled environments. This does not imply, however, that it does not bear on more realistic situations. It may underpin simpler models, which can be applied to realistic situations.
Acknowledgements The work presented in this paper was supported by the Dutch Government, National Research Programme on global air pollution and climate change, Contract No. 013/1204.10. I greatfully acknowledge the constructive criticisms of Hugo van den Berg and Bas Kooijman. The comments of Tarzan Legovic´ and Claudio Rossi, which improved the presentation of the paper, are also greatfully acknowledged.
Appendix A. Nomenclature Square brackets […] express parameters representing amounts of nutrients per unit permanent volume. For instance, [GN ] represents the amount of nitrogen per unit permanent volume. Since estimates of the parameters are based on chemostat data in which only quota per cell are given, the parameter estimates also are given as amounts of nutrient per ‘average cell’. The usage of some symbols corresponds to the conventional one: Qx Cell quota of x, i.e. total amount of x in a cell (pa per cell) D Dilution rate of chemostat (per day) m Specific growth rate (per day) [Chl] Chlorophyll cell quota (pa per cell) u Chlorophyll to Carbon ratio (g g−1) The letters used to represent variables and parameters often (but not always) provide a clue to what they represent: Permanent volume (mm3) V [C−] Density of transient carbon pool (pg per mm3). A subscript is used to indicate whether growth is light-limited (1 ) or nitrogen-limited (n) [C m Maximum density of transient carbon pool (pg per mm3) for light-limited growth l ] [N] Density of transient nitrogen pool (pg per mm3). In contrast to carbon, no subscript is necessary to indicate the role of nitrogen, as equations concerning nitrogen always deal with nitrogen-limited growth [N m] Maximum density of transient nitrogen pool (pg per mm3) [Gx ] Growth-costs parameters: quantity of nutrient x required to for the growth of a unit permanent volume, excluding nutrient in the transient pool (pg per mm3)
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
69
For carbon, a superscript may be used to distinguish between carbon built into the structural volume (6) or carbon respired during growth (r). The superscript t indicates the total carbon costs of a unit permanent volume. [A m x, ] –
Assimilation capacity of nutrient x (pg per mm3 per day). A second subscript may be used to indicate whether nutrient x is limiting (l) or not (n). The aggregate assimilation capacity for carbon in nitrogen-limited growth is the sum of three different uptake capacities, each of which may be indicated by its own superscript. These three uptake capacities are either independent of the nitrogen density (superscript 0), proportional to the nitrogen density (superscript p), or proportional to the specific growth rate, which implies hyperbolically dependent on the nitrogen density (superscript h).
[MC ] XN fN I fI nx
Maintenance costs (pg carbon per mm3 per day) Nitrogen concentration in the environment (molar) Hyperbolically transformed nitrogen concentration: fN = xN /(KN+xN ) (dimensionless), with KN the half-saturation constant of the uptake kinetics Irradiance or light intensity (mEinst m−2 s−1) Hyperbolically transformed light intensity (dimensionless): fI = I/(Is+I) with Is is the irradiance at which the photosynthesis process is half saturated Nutrient rate of x (per day). This parameter is a measure for the rate at which the transient nutrient pool can be replenished
References Bannister, T.T., 1979. Quantitative description of steady state, nutrient-saturated algal growth, including adaptation. Limnol. Oceanogr. 24, 76–96. Baumert, H., 1996. On the theory of photosynthesis and growth in phytoplankton. Part I. Light limitation and constant temperature. Int. Rev. Gesamten Hydrobiol. 81, 109 – 139. Cloern, J.E., Grenz, C., Vidergar-Lucas, L., 1995. An empirical model of the phytoplankton chlorophyll: carbon ratiothe conversion factor between productivity and growth rate. Limnol. Oceanogr. 40, 1313–1321. Cullen, J.J., Geider, R.J., Ishizaka, J., Kiefer, D.A., Marra, J., Sakshaug, E., Raven, J.A., 1993. Towards a general description of phytoplankton growth for biogeochemical models. In: Evans, G.T., Fasham, M.J.R. (Eds.), Towards a Model of Ocean Biogeochemical Processes Volume 10 of NATO Special Programme on Global Environmental Change. Springer-Verlag, Germany, pp. 153–176. Curtis, P.J., Megard, R.O., 1987. Interactions among irradiance, oxygen evolution and nitrite uptake by Chlamydomonas (Chlorophyceae). New Phytol. 23, 608–613. Elrifi, I.R., Turpin, D.H., 1985. Steady-state luxury consumption and the concept of optimum nutrient ratios: a study with phosphate and nitrate limited Selenastrum minutum (Chlorophyta). J. Phycol. 21, 592–602.
Falkowski, P.G., Dubinsky, Z., Wyman, K., 1985. Growth-irradiance relationships in phytoplanton. Limnol. Oceanogr. 30, 311 – 321. Geider, R.J., 1987. Light and temperature dependence of the carbon to chlorophyll a ratio in microalgae and cyanobacteria: implications for physiology and growth of phytoplankton. New Phytol. 106, 1 – 34. Geider, R.J., MacIntyre, H.L., Kana, T.M., 1996. A dynamic model of photoadaptation in phytoplankton. Limnol. Oceanogr. 41, 1 – 15. Geider, R.J., MacIntyre, H.L., Kana, T.M., 1997. A dynamic model of phytoplankton growth and acclimation: responses of the balanced growth rate and the chlorophyll a: carbon ratio to light, nutrient-limitation and temperature. Mar. Ecol. Prog. Ser. 148, 187 – 200. Geider, R.J., MacIntyre, H.L., Kana, T.M., 1998. A dynamic regulatory model of phytoplankton acclimation to light, nutrients and temperature. Limnol. Oceanogr. 43, 679 – 694. Goldman, J.C., 1980. Physiological processes, nutrient availability, and the concept of relative growth rate in marine phytoplankton ecology. In: Falkowski, P.G. (Ed.), Primary Productivity in the Sea Volume 19 of Environmental Science Research. Plenum Press, NY, pp. 179 – 194. Goldman, J.C., Peavey, D.G., 1979. Steady-state growth and chemical composition of the marine Chlorophyte Dunaliella tertiolecta in nitrogen-limited continuous cultures. Appl. Environ. Microbiol. 38, 894 – 901.
70
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
Harding, L.W.J., Fisher, T.R.J., Tyler, M.A., 1987. Adaptive responses of photosynthesis in phytoplankton: specificity to time-scale of change in light. Biol. Oceanogr. 4, 403– 437. Kiefer, D.A., 1993. Modeling growth and light absorption in the marine diatom Skeletonema costatum. In: Evans, G.T., Fasham, M.J.R. (Eds.), Towards a Model of Ocean Biogeochemical Processes Volume 10 of NATO Special Programme on Global Environmental Change. SpringerVerlag, Germany, pp. 93–120. Kooijman, S.A.L.M., 1993. Dynamic Energy Budgets in Biological Systems-Theory and Applications in Ecotoxicology. Cambridge University Press, Cambridge. Laws, E.A., Bannister, T.T., 1980. Nutrient and light-limited growth of Thalassiosira flu6iatilis in continuous culture, with implications for phytoplankton growth in the ocean. Limnol. Oceanogr. 25, 457–473. McCarty, J.J., 1981. The kinetics of nutrient utilization. In: Platt, T. (Ed.), Physiological Bases of Phytoplankton Ecology Volume 210 of Can. Bull. Fish. Aquat. Sci., pp. 211 – 233. Nielsen, M.V., 1995. Photosynthetic characteristics of the coccolithophorid Emiliania huxleyi (Prymnesiophyceae) ex-
.
posed to elevated concentrations of dissolved inorganic carbon. J. Phycol. 31, 715 – 719. Osborne, B.A., Geider, R.J., 1986. Effect of nitrate-nitrogen limitation on photosynthesis of the diatom Phaeodaclylum tricornutum Bohlin (Bacillariophyceae). Plant Cell Environ. 9, 617 – 625. Sakshaug, E., Andresen, K., Kiefer, D.A., 1989. A steady state description of growth and light absorption in the marine planktonic diatom Skeletonema costatum. Limnol. Oceanogr. 34, 198 – 205. Sukenik, A., Bennett, J., Falkowski, P., 1987. Light-saturated photosynthesis-limitation by electron transport or carbon fixation? Biochim. Biophys. Acta 891, 205 – 215. Williams, F.M., 1967. A model of cell growth dynamics. J. Theor. Biol. 15, 190 – 207. Zonneveld, C., 1996. Modelling the kinetics of non-limiting nutrients in microalgae. J. Mar. Syst. 9, 121 – 136. Zonneveld, C., 1997. Modeling the effects of photoadaptation on the Photosynthesis Irradiance curve. J. Theor. Biol. 186, 381 – 388. Zonneveld, C., Van den Berg, H.A., Kooijman, S.A.L.M., 1997. Modeling carbon cell quota in light-limited phytoplankton. J. Theor. Biol. 188, 215 – 226.
A cell-based model for the chlorophyll a to carbon ratio in phytoplankton C. Zonneveld * Department of Theoretical Biology, Vrije Uni6ersiteit Amsterdam, De Boelelaan 1087, 1081 HV, Amsterdam, The Netherlands
Abstract The ratio of chlorophyll a per cell and carbon per cell is considered to be a key quantity in phytoplankton growth. The quantity varies with nutrient and light availability. The aim of this paper is to predict how the chlorophyll a to carbon ratio varies in relation to environmental conditions. A cell-based model is presented, that allows the description of carbon and chlorophyll a cell quota, and thus the chlorophyll a to carbon ratio, in nutrient-limited as well as light-limited growth. The model predictions are in line with experimental data. Variations in Chl:C result, among others, from photoacclimation. The analysis in this paper shows that the impact photoacclimation on phytoplankton growth is rather small, despite large variations in Chl:C. This is mainly due to the package effect, a phenomenon that can be easily accounted for especially in cell-based models. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Chl:C; Light-limited growth; Model; Nutrient-limited growth; Photoacclimation; Phytoplankton
1. Introduction The ratio of the amount of chlorophyll a per cell and the amount of carbon per cell, Chl:C, features as a key variable in many microalgal growth models (Bannister, 1979; Sakshaug et al., 1989; Cullen et al., 1993; Kiefer, 1993; Baumert, 1996; Geider et al., 1996). The widespread use of Chl:C as a model variable is basically rooted in the ease with which chlorophyll a concentrations * Tel.: + 31 20 4447128; fax: + 31 20 4447123; e-mail: [email protected]
are experimentally determined. Chlorophyll a is therefore often used as a proxy for phytoplankton biomass. However, it is a poor measure, and one needs knowledge of Chl:C to translate chlorophyll a measurements to the more useful phytoplankton carbon measure (Geider et al., 1997). The argument outlined above favors the use of Chl:C as a model variable. Yet this use can have its drawbacks too, especially from the modelling perspective. The possibility of direct measurements on a model variable is appealing, but other aspects are also important. For example, the characteristic should play a vital role in the physiology
0304-3800/98/$ - see front matter © 1998 Elsevier Science B.V. All rights reserved. PII S0304-3800(98)00134-3
56
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
of the organism, and this role should preferably be easy to model. Although the physiological importance of chlorophyll is self-evident, taking chlorophyll as a basic model variable not necessarily yields the most parsimonious model. For instance, light-saturated rates of photosynthesis per unit chlorophyll strongly depend on the state of photoacclimation. The primary reason for this dependence is that while photoacclimation affects the amounts of chlorophyll per cell, the amount of RUBISCO (the enzyme limiting the light-saturated rate of photosynthesis) per cell is often unaffected. So the relevance of chlorophyll (and consequently of Chl:C) for light-saturated rates of photosynthesis can be questioned. Another disadvantage of Chl:C is that the use of the ratio of two cellular quantities, both subject to regulation, hinders a better understanding of the regulation of both quantities separately. In this paper, I present a cell-based model of Chl:C. An important difference with current modeling approaches is that a cell-based model enables one to model carbon cell quota, nitrogen cell quota and chlorophyll cell quota separately. As a result, expressions for the Chl:C ratio can also be derived. To judge the relevance of the model, I use data from the literature. Finally, I present an analysis of photoacclimation on the basis of the model presented in this paper. This analysis shows that the impact of photoacclimation on phytoplankton growth is less severe than might be expected in view of the dramatic effect of photoacclimation on Chl:C.
To deal with light-limited as well as nutrientlimited growth, I reformulate a model for growth on two potentially growth-limiting essential nutrients (Zonneveld, 1996). In this paper, nitrogen and carbon play the role of the two potentially growth-limiting essential nutrients. The model characterises a phytoplankton cell in terms of three variables: permanent volume, with a fixed C:N ratio, and two transient nutrient pools, one for carbon and one for nitrogen (see Fig. 1). Assimilation acts to increase the transient nutrient pools. The light-dependence of the carbon assimilation is taken into account, but the lightdependence of the nitrogen assimilation (Curtis and Megard, 1987) is neglected here, as studies on nutrient-limited growth are generally restricted to one level of irradiance. Carbon and nitrogen are mobilized from the pools to form permanent volume. Variations in the size of the carbon and nitrogen pool relative to the permanent volume lead to variation in cellular composition. The general ideas of this model structure can be traced back to the models of Williams (1967) and, especially, Kooijman (1993). When carbon is in lower supply relative to need than nitrogen, growth is said to be light-limited, otherwise it is nitrogen-limited. Thus limitation is of the either/or type, and co-limitation of two nutrients is not considered here. The cellular physiology is geared to the limiting nutrient, which implies that the uptake rate of the non-limiting nutrient depends on the cell quota of the limiting nutrient. Similarly, the kinetics of chloro-
2. Basic model structure Phytoplankton ecologists often consider growth to be either nutrient-limited or light-limited, a convention I follow here. For nutrient-limited growth, the model must specify how the growth rate depends on the cellular nutrient availability. Furthermore, the model must specify the carbon and chlorophyll cell quota. For light-limited growth, the growth rate as dependent on irradiance must be specified, and, again, the carbon and chlorophyll cell quota.
Fig. 1. Basic model structure.
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
phyll is thought to depend on the limiting nutrient. I shall compare the model formulation with data from chemostat cultures. The comparison concerns predictions regarding the cell quotas for the limiting and the non-limiting nutrient, and for chlorophyll. The comparison is restricted to steady-state states. The model derived below deals with nutrientlimited as well as with light-limited growth, assimilation of nutrients, etc. It is unavoidable to introduce a considerable number of variables and parameters. Despite its cumbersome appearance, there is a certain rationale behind the notation, explained in Appendix A.
3. Nutrient-limited growth Let N denote the amount of nitrogen in the transient nitrogen pool. It is often more convenient to work with the nitrogen density [N]: the amount of nitrogen in the transient pool divided by permanent volume V. (Permanent volume thus excludes the volume occupied by the transient nutrient pools.) I assume that the nitrogen density of the cells eventually reaches a constant value, say [N m l ], when cells are grown at saturating concentrations of the nitrogen source in the environment for a sufficiently long period of time. This value cannot be exceeded as long as the nutrient is limiting indeed. Furthermore, I assume that nutrient uptake rate is proportional to cell volume, and depends hyperbolically on the environmental concentration (i.e. Michaelis – Menten uptake kinetics, see for instance McCarty, 1981). Throughout this paper, fN indicates the hyperbolic dependence of the uptake rate on the substrate concentration xN (i.e., fN =xN /(KN +xN )). The rate of change nutrient density is assumed to obey d [N ]= [A m N,l]fN −nN [Nl ], dt l
(1)
i.e. the nitrogen density follows first order kinetics. The quantity nN [Nl ] represents the usage of nitrogen from the transient pool; nN is a proportionality constant. If the external nitrogen-source
57
concentration is saturating ( fN = 1) for a sufficiently long period of time, the nitrogen density no longer changes, so (d/dt)[Nl ]= 0. From this m condition one obtains nN = [A m N,l ]/[N l ]. On the one hand, the utilization rate equals the assimilation rate AN,l = [A m N,l ]VfN minus the rate of change of the amount of nutrient, d/dt(Nl )= d/ dt([Nl ]V). On the other hand, if nitrogen is used only for growth (no extrusion of nitrogen from the cell) and nitrogen contents of a unit structural volume, [GN ], is constant, one knows that the utilization rate must equal [GN ]d/dt(V), so that AN,l −
d d d ([N]V)= [Nl ] nNV− V = [GN ] V. dt dt dt
(The first equality is obtained by application of the chain rule for differentiation and substitution of Eq. (1) and the expression for the assimilation rate. The second equality holds by assumption.) This equation can be solved for the growth rate (d/dt)V. It is convenient to focus on the specific growth rate, denoted by m; from the above equation, this is found to be mN
1 d [Nl ] V= nN . V dt [GN ]+ [Nl ]
(2)
At steady-state in a chemostat, the specific growth rate m equals the dilution rate, D. Combined with Eq. (2) this yields an expression for the density of the transient nutrient pool in steadystate, as dependent on the dilution rate: [N. l ]=
[GN ]D . nN − D
(3)
3.1. Carbon in nitrogen-limited phytoplankton In a previous paper (Zonneveld, 1996), I showed that the uptake capacity (i.e. the uptake rate given saturating conditions of the nutrient in the environment) for a non-limiting nutrient can be considered to depend on the density of the transient pool of the limiting nutrient. A similar approach is followed by Geider et al. (1998). The precise form of this dependence may differ between species and the nutrients considered. I here postulate three basic patterns, see Fig. 2. The uptake capacity may be independent of the nutri-
58
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
Fig. 2. Dependence of the assimilation capacity for a non-limiting nutrient on the density of the transient nutrient pool of the limiting nutrient. The left panel shows the basic patterns; the assimilation capacity may be (A) independent of, or (B) proportional to the density of the transient nutrient pool of the limiting nutrient, or (C) be proportional to the specific growth rate, so hyperbolic in the nutrient density (Eq. (2)). The middle and the right panel shows all possible combinations of the three basic patterns: D= A +B, E= A +C, F= B+C, G= A+ B+ C.
ent density, it may be proportional to the nutrient density, or it may be proportional to the specific growth rate, i.e. hyperbolic in the nutrient density (Eq. (2)). For a particular phytoplankton species, any linear combination of the three basic patterns is allowed. Experimental data may give clues to what extent each of the basic patterns contributes to the actual pattern of density dependence. The fixation of CO2 in the form of reduced carbon compounds can be thought of as the assimilation of a nutrient. Consequently, I assume that the maximum rate of photosynthesis depends on the density of the limiting nutrient. Such a relationship has been shown experimentally in the diatom, Phaeodaclylum tricornutum Bohlin (Osborne and Geider, 1986). Photosynthesis depends on the light intensity as well as on the concentration of dissolved inorganic carbon (DIC) in the water (for the dependence of photosynthesis on DIC, see Nielsen, 1995). Sufficient availability of DIC is usually taken for granted, hence rates of photosynthesis are mostly related only to the light intensity. This approach is valid as long as the CO2 concentration is either constant or saturating. In this paper, I assume that either of these conditions holds. Studies on nutrient-limited growth are generally performed at one irradiance only, so information on the effects of irradiance on carbon cell quota in nutrient-limited phytoplankton is mostly lacking. Consequently, this topic is not addressed in
this paper, and I simply indicate the relationship between the rate of photosynthesis and irradiance by fI [A m C,l ], with fI the rate of photosynthesis as a fraction of its maximum value [A m C,l ]. Carbon is used for maintenance, growth, and respiration related to growth. Maintenance requirements are taken proportional to permanent volume, and growth-related respiration is taken proportional to the growth rate. The rate of change of the carbon pool is thus given by d t [C ]= fI [A m C,n]− [MC ]− m([G C]+ [Cn ]) dt N
(4)
The aggregate assimilation capacity [A m C,n ] obeys [Nl ] [Nl ] 0 p [A m + [A hC,n] , C ]= [A C,n]+ [A C,n] ] [N m [G l N ]+ [Nl ] (5) where [A 0C,n ], [A pC,n ] and [A hC,n ] are weighing coefficients corresponding to patterns A, B and C, respectively, in Fig. 2. In steady-state d/dt[Cn ]= 0. With Eq. (3), the steady-sate carbon cell quota QC = [G nC ]+ [Cn ] can be expressed as QC =
fI [A 0C,n]− [Mc ] [A pC,n] [GN ] + fI [A hC,n]/nN + fI D nN − D [N m l ] − [G rC].
(6)
This expression might look unwieldy, but its qualitative behaviour is completely determined by only three dimensionless parameters. Introduction of non-dimensional variables and parameters allows the following simplification:
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
F=
a0 ap + +ah D 1− D
with the following dimensionless variables and parameters: F = Qc /[G rC] D = D/nN a0=
scaled carbon cell quota scaled dilution rate
0 C,n
fI [A ]− [Mc ] nN [G rC]
scaled weighing coefficient for pattern A in Fig. 2 [A l ][G ] a p = fI C,nr Nr nN [G C][N l ]
59
3 clearly shows this. At high growth rates, pattern B in Fig. 2 dominates the carbon uptake. Hence the behavior of the carbon cell quota at high growth rates is mainly determined by the value of a p, which characterises the contribution of pattern B. Studies on nutrient-limited phytoplankton sometimes show that the carbon cell quota are almost independent of the growth rate (Goldman and Peavey, 1979; Elrifi and Turpin, 1985). Fig. 3 shows that, with an appropriate choice of parameter values, such a pattern can be described. Complete independence of the growth rate is obtained for a p = a 0 = 0. Below I shall analyse the data of Goldman and Peavey (1979).
scaled weighing coefficient for pattern B in Fig. 2 ah=
fI [A hC,n]/nN − [G rC] [G rC]
scaled weighing coefficient for pattern C in Fig. 2. Basically, the introduction of non-dimensional variables and parameters amounts to a judicious choice of the units in which they are expressed. For example, the dilution rate is expressed in multiples of the parameter nN. This choice is particularly useful, as the maximum specific growth rate, and so the maximum steady-state dilution rate, is less then nN (from Eq. (2) one has lim[Nl] m= nN ), so the non-dimensionless dilution rate takes values between 0 and 1. The remaining quantities are likewise rendered dimensionless. Fig. 3 shows how the carbon cell quota change with growth rate. The parameter a h characterises the uptake proportional to the growth rate, hence changes in its value do not affect the carbon cell quota differently at different growth rates. The effect of an increase in a h is just to elevate the entire curve. The potential values of a h range from −1 (for [A hC,n ]= 0) to . From Eq. (2) it is clear that low growth rates correlate with small nitrogen densities. Hence the contributions to carbon uptake of patterns B and C in Fig. 2, characterised by a h and a p, become very small at low growth rates. The parameter a 0 thus controls the behaviour of the carbon cell quota carbon cell quota at low growth rates. Fig.
Fig. 3. Dependence of carbon cell quota on the dilution rate, for various parameter values, in nitrogen-limited phytoplankton.
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
60
3.2. Nutrient-limited Chl:C I assume that when cells are nitrogen-limited, the absolute rate at which chlorophyll is produced depends on the amount of nitrogen in the transient pool. Furthermore, chlorophyll cell quota decrease because of dilution through growth, and may also decrease through chlorophyll destruction (characterised by a rate constant r). By assumption, the chlorophyll production rate equals kNl, with k a proportionality constant. (The chlorophyll production rate may also depend on irradiance, in which case k would depend on the irradiance. As studies on nutrient-limited growth are generally restricted to one level of irradiance, information on this aspect is lacking. I therefore treat k as a constant.) This implies that the cellular chlorophyll concentration changes according to d [Chl]= k[Nl ]− (m +r) · [Chl] dt so that at steady-state (when m = D) [Chl] =
r=0 k[GN ]D k[GN ] = . (nN −D)(D + r) (nN −D)
For a negligible destruction rate (r = 0) this expression is similar to that for the nitrogen cell quota, Eq. (3). Based on the analysis of experimental data in section 4.3 I assume henceforth that r =0, which simplifies the equation for Chl:C, while it hardly affects its accuracy. The chlorophyll to carbon ratio, u, now follows from the expressions for the chlorophyll cell quota and the carbon cell quota: u=
[Chl] = Qc
3.3. Test against data The relevance of the above derived expressions will now be evaluated by comparing them to experimental data. To this end, I choose data on nitrogen-limited Dunaliella tertiolecta Butcher, presented by Goldman and Peavey (1979) and Goldman (1980). These articles contain data regarding nitrogen, carbon and chlorophyll cell quota of steady-state growing cells. The alga was limited by various nitrogen sources, but the nitrogen source (nitrate, nitrite, ammonium or urea) did not affect the growth response, nor the carbon or chlorophyll cell quota. Therefore I do not distinguish between the different nitrogen sources on which the cells were grown. (See the original publications for further details.) The following equations describe the data: nitrogen cell quota QN = [GN ]+ [N. l ]=
[GN ]nN nN − D
carbon cell quota QC =
ap a0 + +ah D n− D
with
k[GN ]/(n−D) . (fI [A 0C,n]−[Mc ])/D+(fI [A hC,n]/nN−[G rC])+fI [A pC,n][GN ]/[N m l ](nN−D)
Employing the dimensionless parameters introduced before, this equation simplifies to U−1 = a p + a 0
Fig. 4 compares the chlorophyll to carbon (left panels) with the carbon to chlorophyll ratio (right panels). Though they convey the same information, the impact of parameter values on Chl:C is more easily recognized. Chl:C is proportional to the dilution rate if a h = 0 and a p = a 0 " 0. Such a proportionality has been found in Thalassiosira fin6iatilis Hustedt, for example (Laws and Bannister, 1980). Depending on the parameter values, other patterns may be also be observed.
(1− D) + (1 −D)a h D
with U= ((nN [G ])/(k[GN ]))u. r C
(7)
a 0 = fI [A 0C,n]− [MC ] a p = fI [A pC,n][GN ]/[N m l ] a h = fI [A hC,n]/nN − [G rC] chlorophyll cell quota [Chl]=
k[GN ] nN − D
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
61
Fig. 4. Qualitative behaviour of the chlorophyll to carbon ratio (left) and its inverse, the carbon to chlorophyll ratio (right), as dependent on the dimensionless parameters.
Fig. 5 shows the data with their model description. The nitrogen cell quota curve closely follows the data. This is not surprising, as the model is similar to the Droop relationship, well-known to describe such data nicely.
The carbon cell quota are also well described by the model. One may express doubts on the validity of the ascending left branch of the curve, as it seems not strongly supported by the data. As outlined above, the carbon cell quota at low
62
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
growth rates depends especially on the parameter a 0, which characterises the rate of photosynthesis at zero growth rate. To see whether this parameter is really important, I also fitted the model with a 0 = 0. The dashed curve shows the resulting model fit. The systematic deviations from the data suggest that the assumption that a 0 =0 is not warranted. Biologically this means that gross photosynthesize at zero growth rate exceeds the maintenance requirements.
The amounts of chlorophyll per cell also obey a Droop-type relationship against the growth rate. Destruction of chlorophyll may become apparent only at very low dilution rates, where no data were available. As the model describes both carbon and chlorophyll cell quota well, one might expect that the model for the chlorophyll to carbon ratio also fits the data. Except for minor deviations, this expectation is indeed fulfilled.
Fig. 5. (A) Nitrogen cell quota; (B) carbon cell quota; (C) chlorophyll cell quota and (D) Chl:C in nitrogen-limited Dunaliella tertiolecta. Parameter values: [GN ] =1.25 pg N cell − 1, nN = 1.55 d − 1, a 0 =1.27, a p =9.30, a h =9.81 pg C/(cell · d − 1), k = 0.207 pg Chl/(pg N · d − 1). The parameter values were simultaneously estimated from the data on nitrogen, carbon and chlorophyll cell quota. The dashed curve describing the carbon cell quota is the best fit for a 0 =0 (with a p =6.56, a h =16.2). The solid curve describing the chlorophyll cell quota assumes a zero destruction rate; the dashed curve is the model with the destruction rate estimated from the data (r= 0.015 d − 1).
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
4. Light-limited growth In light-limited phytoplankton, the growth rate is determined by the transient carbon pool instead of the transient nitrogen pool. This means that to derive an expression for u one only has to model the carbon and chlorophyll cell quota, not the nitrogen cell quota. The transient carbon pool in light-limited phytoplankton plays the same role as the transient nitrogen pool in nitrogen-limited phytoplankton. Consequently, I use the same modelling approach. Light-limited growth and carbon cell quota have been dealt with more fully in Zonneveld et al. (1997). Moreover, the derivation of the expression for the light-limited growth rate is similar to that for the nutrient-limited growth rate. Hence I choose for a more succinct presentation in this section. As elaborated in Zonneveld et al. (1997), the rate of photosynthesis is taken to depend hyperbolically on light intensity, not on the concentration of inorganic carbon. Below I evaluate the consequences of photoacclimation, but for the moment I neglect this phenomenon. Proceeding in the same way as for nutrient-limited growth, the light-limited specific growth rate is obtained as follows. The density of the transient carbon pool [Cl ] follows first-order kinetics: d I [Cl ]= [A m −nc [Cl ], C,l] dt Is + I
(8)
Carbon serves both as a nutrient necessary for growth and as a storage of energy. With regard to energetic demands, carbon is used to cover maintenance requirements and growth-related respiration. Furthermore, it is used as a building material for new permanent volume. The utilization rate is thus given by [MC ]V +([G 6C]+ [G rC])
d V dt
On the other hand, the utilization rate equals [Cl ](nCV−d/dt(V)). By equating these two expressions one obtaines for the light-limited specific growth rate: m
1 d n [C ]−[MC ] V= c l . V dt [Cl ]+[GC ]
(9)
63
Continuous culture studies on nutrient-limited growth are performed in chemostats, in which the steady-state growth rate equals the dilution rate. Therefore, the dilution rate (often somewhat misleadingly called the specific growth rate) is treated as the independent variable. Continuous culture studies on light-limited growth are often performed in turbidostats, in which the dilution rate is adjusted until a steady-state is achieved. In such studies, the incident light intensity is taken as the independent variable. An expression for the steady-state specific growth rate as dependent on the light intensity is obtained as follows. From Eq. (8) one has [C. l ]= [C m l ]
I . Is + I
Substitution of this expression into Eq. (9) results, after some rearrangements, in m=
I([A m C,l]− [MC ])− [MC ]Is . t t I([C m l ]+ [G C])+ Is [G C]
(10)
The steady-state carbon cell quota are given by I . Is + I
QC = [G 6C]+ [C m l ]
4.1. Light-limited Chl:C Whereas during nutrient-limited growth the chlorophyll cell quota increases with the growth rate, the opposite is true for light-limited growth. But even if growth is light-saturated, the chlorophyll cell quota decrease with further increasing light intensity. This suggests that chlorophyll production rates in light-limited phytoplankton may be light-dependent. I am not aware of detailed quantitative studies on this topic, so I follow a somewhat provisional approach here. I assume that chlorophyll production is a function of light intensity, say p(I). I choose this function such that a close correspondence with experimental data results. I reanalyse data from Falkowski et al. (1985), on light-limited growth in Isochrysis galbana (Green) and Thalassiosira weisfeogii (Grunow) Fryxell and Hasle. I took p(I)= aI/(b + I+cI 2) to describe the chlorophyll production rate. Steady-state chlorophyll cell quota are then given by
64
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
Fig. 6. (A) Specific growth rate; (B) carbon cell quota; (C) chlorophyll per cell and (D) inferred chlorophyll production rates in light limited Isochrysis galbana. The chlorophyll cell quota are described by two curves, the solid one representing no destruction of chlorophyll (r =0), the dashed one representing r = 0.5 d − 1. Production rates are deduced from amount of chlorophyll per cell, the known specific growth rate, and a hypothetical destruction rate. Data from Falkowski et al. (1985). Parameter values: [G nC ]= 8.0 pg −1 −1 C cell − 1, [G rC ] =2.5 pg C cell − 1, [C m , Is = 150 mEinst m − 2 s − 1, [A m · d − 1, [MC ]= 0.84 pg C l ] = 14 pg C cell C ]= 35 pg C cell cell − 1 · d − 1, a = 0.35 pg Chl cell − 1 · d − 1, b= 75 mEinst m − 2s − 1, c = 2.2 m2 s mEinst − 1, r = 0.
Chl +
p(I) aI = , m(I)+ r (m(I) + r)(b +I +cI 2)
with m(I) given by Eq. (10). The chlorophyll cell quota decreases through dilution by growth and possibly by destruction of chlorophyll, characterised by a constant destruction rate r. Fig. 6 shows a nice description of the data, but as 3 (or even 4, including r) parameters are used to describe only five points, this at best weakly supports the model. The data are mute about the value of the specific rate of chlorophyll destruction: values between 0 and 0.5 d − 1 are all consistent with the data. Note that although the rate of chlorophyll production decreases with decreas-
ing irradiance (at low irradiances) the chlorophyll cell quota does increase. The reason for this increase is that dilution through growth becomes progressively less important with decreasing irradiance. An alternative explanation for the observed dependence of chlorophyll cell quota on irradiance is that the chlorophyll production rate increases monotonically with irradiance (or with the amount of carbon in the transient carbon pool), while the destruction rate increases with irradiance. These assumptions lead to similar fits as those in Fig. 6, but with unrealistically high chlorophyll destruction rates (up to 3 d − 1 at 600
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
mEinst m − 2 s − 1). Therefore I conclude that this alternative is not viable. Carbon cell quota are well described by the model, and so is u, as Fig. 7 shows. To compare the light-limited u with the nutrient-limited one, u
65
is also plotted as dependent on the specific growth rate. As the light-dependence of the specific growth rate is also liable to variation, the plot of u against specific growth rate shows slightly more scatter compared to the plot against light intensity. Nevertheless, the overall picture is satisfactory. Geider (1987) noted that the carbon to chlorophyll ratio (u − l) increases linearly with light level in all nutrient-sufficient, uni-algal cultures examined. Fig. 7 shows that this is approximately valid for Isochrysis galbana. The linear dependence of u − l on light intensity can also be derived analytically, by taking the limI for u − 1. For large I, the carbon cell quota converge towards [G nC ]+ [C m l ], and the growth rate converges towards the maximum growth rate, say mm. So for large irradiances, u − 1 obeys lim I
QC [G 6C]+ [C m l ] = Chl (aI/(mm + r)(b +I+ cI 2)) =
2 ([G 6C]+ [C m l ])(mm + r)(b +I+ cI ) aI
−1 = ([G 6C]+ [C m + cI/a) l ])(mm + r)(a
= k1 + k2I. increases linearly with light intensity Thus, u at high light levels. Fig. 7 suggests that this relationship approximately holds true even at quite low light levels. Geider et al. (1997) also arrived at a linear dependence of u − l on irradiance. Their description involves a maximum value for u at very low irradiance, and two parameters characterising the PI-curve. −1
4.2. Photoacclimation
Fig. 7. (A) Chlorophyll: Carbon ratio; (B) Carbon: Chlorophyll ratio, both as dependent on irradiance; and (C) Chlorophyll: Carbon ratio as dependent on the specific growth rate.
The present descriptions of carbon cell quota and the specific growth rate do not make use of u. This variable is one of the best documented characteristics of photoacclimation, so one is led to wonder why the model works while this aspect is neglected. The present model is formulated at the level of the individual cell. To see why this matters, let us focus on light-saturated rates of photosynthesis. Photoacclimation results in increased chlorophyll cell quota when cell are kept under low light
66
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
conditions. However, the maximum rate of photosynthesis per cell may be unaffected by photoacclimation (Harding et al., 1987; Sukenik et al., 1987). If so, the maximum rate of photosynthesis per milligram chlorophyll is strongly affected by photoacclimation, but this reflects changes in chlorophyll cell quota rather than changes in the photosynthetic capacity of the cell. A cell-based model circumvents this pitfall. Due to the so-called package effect the efficiency of light capture of chlorophyll decreases when chlorophyll cell quota increase. The increased amount of chlorophyll of cells grown in low light is packed in the same or (generally) a smaller volume, in comparison with cells grown in high light conditions. Self-shading of chlorophyll prohibits a proportional increase in light capture efficiency of the cell with the concentration of chlorophyll. As a result, light absorption is not proportional to u. The remainder of this section aims to show that photoacclimation can be neglected in a cell-based model without excessive loss of accuracy, due to the package effect. My approach to model the package effect is a convenient combination of mechanistic and descriptive elements, more fully explored in Zonneveld (1997). The focus is on the cellular absorption cross section, which equals the chlorophyll-specific absorption cross section times the chlorophyll concentration of the cell. I assume that cells without chlorophyll cannot harvest light, so their cellular absorption cross section is zero. Furthermore, at very high chlorophyll concentrations in the cell self-shading sets an upper bound to the cellular absorption cross section. Additional chlorophyll would not be able to capture more photons, as all photons entering the cell are already absorbed. I choose an arbitrary function satisfying these mechanistic elements, a rectangular hyperbola. Such a function can be characterised by the slope of the curve at zero chlorophyll and by its maximum. The slope is the ‘true chlorophyll-specific absorption cross section’, denoted by x(dm2 mg − 1 Chl). This is the absorption cross section unaffected by the package effect. The maximum cellular absorption cross section (mm2 · cell − l) is denoted by zm. The cellular absorption cross section is thus given by
z=
zm Chl . zm /x+Chl
(11)
The chlorophyll-specific absorption cross section s is then given by s = z/Chl=
zm . zm /x+Chl
(12)
Fig. 8 shows an application of Eqs. (11) and (12) to experimental data. The good fit supports the present approach to model the photoacclimational response of the absorption cross section. The cellular absorption cross section does increase with decreasing light intensity, but the increase is not pronounced. The cellular cross section increases by about 25%, whereas the chlorophyll cell quota increases almost 3-fold. This shows that the chlorophyll cell quota only weakly characterises the absorptive potential of a cell. The same applies to u. In Fig. 8, I also analyse the consequences of photoacclimation for the photosynthesis –irradiance (PI) curve. The saturation constant of the PI-curve is proportional to the cellular cross section (Zonneveld, 1997), which depends on the light intensity at which the cells grow. To fix the proportionality constant, I set the saturation constant of cells grown at a light intensity of 300 mmol quanta m2 s − 1 equal to 150 mmol quanta m2 s − 1. The saturation constants of cells grown at different light intensities can then be calculated. Fig. 8 shows inferred PI-curves for cells grown at each steady-state light intensity (dotted curves). Each curve is marked at the light intensity with which the curve corresponds. The solid curve represents a least squares fitted simple PI-curve to the five points, which does not account for photoacclimation. This simple PI-curve yields an excellent approximation to the PI-curves that account for photoacclimation. This foregoing analysis of photoacclimation suggests that it affects the PI-curve only mildly. It is important to realize that this conclusion applies only to rates of photosynthesis expressed on a per cell basis. Rates of photosynthesis expressed per unit chlorophyll are dramatically affected by photoacclimation, but this is so mainly because the chlorophyll cell quota are affected.
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
67
Fig. 8. Photoacclimation in light-limited Thalassiosira weisflogii. Data from Falkowski et al. (1985). (A) Chlorophyll per cell; (B) chlorophyll-specific absorption cross section; (C) cellular absorption cross section and (D) inferred photosynthesis– irradiance curves. See text for explanation. Parameter values: maximum cellular cross-section zm =63.4 mm2 cell − 1, true chlorophyll-specific absorption cross section x= 19.3 m2 g − 1 Chl.
5. Discussion The present model successfully describes the carbon and chlorophyll cell quota, and, as a corollary, the chlorophyll to carbon ratio u. This applies to nutrient-limited as well as to light-limited growth. The model descriptions find their basis in assumptions about uptake and utilization of nutrient. This contrasts with models that relate u to environmental or physiological variables via statistical models (Cloern et al., 1995). My approach also contrasts with other mechanistic microalgal growth models in two important aspects: I take the phytoplankton cell as the point of departure, and I deny a pivotal role to u. Both aspects deserve to be commented upon.
Many phytoplankton species occur as single cells. Most models on phytoplankton growth do not account for this level of organisation. Ignoring this level in a model may be advantageous, but has its drawbacks, too. For example, photoacclimation may be dealt with more easily if one accounts for individual cells. The increase in chlorophyll content of a phytoplankton cell serves to enhance the harvesting of light energy, when energy needed to fix carbon dioxide is in short supply. The light-saturated rate of photosynthesis of a phytoplankton cell may well be unaffected by an increase in chlorophyll per cell, as the energy supply is not limiting the fixation of carbon dioxide. However, the light-saturated rate of photosynthesis per unit chlorophyll is strongly affected
68
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
by photoacclimation, and one somehow has to account for this dependency. This can possibly be done via u, which is an important indicator of the state of photoacclimation. However, the problem had better be avoided in the first place, by recognizing the importance of the cell as a basic biological entity. One might argue that cell size (permanent volume) as a basic model variable is inappropriate in view of the variation in size encountered in algal communities, which might span several orders of magnitude. Admittedly, my approach is not suited for studies on algal communities with unknown species composition. But I doubt whether the use of any sophisticated model is justified if the system to be studied is ill characterised. With only a rough measure of biomass at our disposal, such as the chlorophyll content of a water body with unknown species composition, one should use simple models, or even rules of thumb. The predictive value of a model depends to a considerable
extent on the quality of its input. If the input is ill defined, the output cannot be improved by elaborating a model. My approach is most suited for algal growth in controlled environments. This does not imply, however, that it does not bear on more realistic situations. It may underpin simpler models, which can be applied to realistic situations.
Acknowledgements The work presented in this paper was supported by the Dutch Government, National Research Programme on global air pollution and climate change, Contract No. 013/1204.10. I greatfully acknowledge the constructive criticisms of Hugo van den Berg and Bas Kooijman. The comments of Tarzan Legovic´ and Claudio Rossi, which improved the presentation of the paper, are also greatfully acknowledged.
Appendix A. Nomenclature Square brackets […] express parameters representing amounts of nutrients per unit permanent volume. For instance, [GN ] represents the amount of nitrogen per unit permanent volume. Since estimates of the parameters are based on chemostat data in which only quota per cell are given, the parameter estimates also are given as amounts of nutrient per ‘average cell’. The usage of some symbols corresponds to the conventional one: Qx Cell quota of x, i.e. total amount of x in a cell (pa per cell) D Dilution rate of chemostat (per day) m Specific growth rate (per day) [Chl] Chlorophyll cell quota (pa per cell) u Chlorophyll to Carbon ratio (g g−1) The letters used to represent variables and parameters often (but not always) provide a clue to what they represent: Permanent volume (mm3) V [C−] Density of transient carbon pool (pg per mm3). A subscript is used to indicate whether growth is light-limited (1 ) or nitrogen-limited (n) [C m Maximum density of transient carbon pool (pg per mm3) for light-limited growth l ] [N] Density of transient nitrogen pool (pg per mm3). In contrast to carbon, no subscript is necessary to indicate the role of nitrogen, as equations concerning nitrogen always deal with nitrogen-limited growth [N m] Maximum density of transient nitrogen pool (pg per mm3) [Gx ] Growth-costs parameters: quantity of nutrient x required to for the growth of a unit permanent volume, excluding nutrient in the transient pool (pg per mm3)
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
69
For carbon, a superscript may be used to distinguish between carbon built into the structural volume (6) or carbon respired during growth (r). The superscript t indicates the total carbon costs of a unit permanent volume. [A m x, ] –
Assimilation capacity of nutrient x (pg per mm3 per day). A second subscript may be used to indicate whether nutrient x is limiting (l) or not (n). The aggregate assimilation capacity for carbon in nitrogen-limited growth is the sum of three different uptake capacities, each of which may be indicated by its own superscript. These three uptake capacities are either independent of the nitrogen density (superscript 0), proportional to the nitrogen density (superscript p), or proportional to the specific growth rate, which implies hyperbolically dependent on the nitrogen density (superscript h).
[MC ] XN fN I fI nx
Maintenance costs (pg carbon per mm3 per day) Nitrogen concentration in the environment (molar) Hyperbolically transformed nitrogen concentration: fN = xN /(KN+xN ) (dimensionless), with KN the half-saturation constant of the uptake kinetics Irradiance or light intensity (mEinst m−2 s−1) Hyperbolically transformed light intensity (dimensionless): fI = I/(Is+I) with Is is the irradiance at which the photosynthesis process is half saturated Nutrient rate of x (per day). This parameter is a measure for the rate at which the transient nutrient pool can be replenished
References Bannister, T.T., 1979. Quantitative description of steady state, nutrient-saturated algal growth, including adaptation. Limnol. Oceanogr. 24, 76–96. Baumert, H., 1996. On the theory of photosynthesis and growth in phytoplankton. Part I. Light limitation and constant temperature. Int. Rev. Gesamten Hydrobiol. 81, 109 – 139. Cloern, J.E., Grenz, C., Vidergar-Lucas, L., 1995. An empirical model of the phytoplankton chlorophyll: carbon ratiothe conversion factor between productivity and growth rate. Limnol. Oceanogr. 40, 1313–1321. Cullen, J.J., Geider, R.J., Ishizaka, J., Kiefer, D.A., Marra, J., Sakshaug, E., Raven, J.A., 1993. Towards a general description of phytoplankton growth for biogeochemical models. In: Evans, G.T., Fasham, M.J.R. (Eds.), Towards a Model of Ocean Biogeochemical Processes Volume 10 of NATO Special Programme on Global Environmental Change. Springer-Verlag, Germany, pp. 153–176. Curtis, P.J., Megard, R.O., 1987. Interactions among irradiance, oxygen evolution and nitrite uptake by Chlamydomonas (Chlorophyceae). New Phytol. 23, 608–613. Elrifi, I.R., Turpin, D.H., 1985. Steady-state luxury consumption and the concept of optimum nutrient ratios: a study with phosphate and nitrate limited Selenastrum minutum (Chlorophyta). J. Phycol. 21, 592–602.
Falkowski, P.G., Dubinsky, Z., Wyman, K., 1985. Growth-irradiance relationships in phytoplanton. Limnol. Oceanogr. 30, 311 – 321. Geider, R.J., 1987. Light and temperature dependence of the carbon to chlorophyll a ratio in microalgae and cyanobacteria: implications for physiology and growth of phytoplankton. New Phytol. 106, 1 – 34. Geider, R.J., MacIntyre, H.L., Kana, T.M., 1996. A dynamic model of photoadaptation in phytoplankton. Limnol. Oceanogr. 41, 1 – 15. Geider, R.J., MacIntyre, H.L., Kana, T.M., 1997. A dynamic model of phytoplankton growth and acclimation: responses of the balanced growth rate and the chlorophyll a: carbon ratio to light, nutrient-limitation and temperature. Mar. Ecol. Prog. Ser. 148, 187 – 200. Geider, R.J., MacIntyre, H.L., Kana, T.M., 1998. A dynamic regulatory model of phytoplankton acclimation to light, nutrients and temperature. Limnol. Oceanogr. 43, 679 – 694. Goldman, J.C., 1980. Physiological processes, nutrient availability, and the concept of relative growth rate in marine phytoplankton ecology. In: Falkowski, P.G. (Ed.), Primary Productivity in the Sea Volume 19 of Environmental Science Research. Plenum Press, NY, pp. 179 – 194. Goldman, J.C., Peavey, D.G., 1979. Steady-state growth and chemical composition of the marine Chlorophyte Dunaliella tertiolecta in nitrogen-limited continuous cultures. Appl. Environ. Microbiol. 38, 894 – 901.
70
C. Zonne6eld / Ecological Modelling 113 (1998) 55–70
Harding, L.W.J., Fisher, T.R.J., Tyler, M.A., 1987. Adaptive responses of photosynthesis in phytoplankton: specificity to time-scale of change in light. Biol. Oceanogr. 4, 403– 437. Kiefer, D.A., 1993. Modeling growth and light absorption in the marine diatom Skeletonema costatum. In: Evans, G.T., Fasham, M.J.R. (Eds.), Towards a Model of Ocean Biogeochemical Processes Volume 10 of NATO Special Programme on Global Environmental Change. SpringerVerlag, Germany, pp. 93–120. Kooijman, S.A.L.M., 1993. Dynamic Energy Budgets in Biological Systems-Theory and Applications in Ecotoxicology. Cambridge University Press, Cambridge. Laws, E.A., Bannister, T.T., 1980. Nutrient and light-limited growth of Thalassiosira flu6iatilis in continuous culture, with implications for phytoplankton growth in the ocean. Limnol. Oceanogr. 25, 457–473. McCarty, J.J., 1981. The kinetics of nutrient utilization. In: Platt, T. (Ed.), Physiological Bases of Phytoplankton Ecology Volume 210 of Can. Bull. Fish. Aquat. Sci., pp. 211 – 233. Nielsen, M.V., 1995. Photosynthetic characteristics of the coccolithophorid Emiliania huxleyi (Prymnesiophyceae) ex-
.
posed to elevated concentrations of dissolved inorganic carbon. J. Phycol. 31, 715 – 719. Osborne, B.A., Geider, R.J., 1986. Effect of nitrate-nitrogen limitation on photosynthesis of the diatom Phaeodaclylum tricornutum Bohlin (Bacillariophyceae). Plant Cell Environ. 9, 617 – 625. Sakshaug, E., Andresen, K., Kiefer, D.A., 1989. A steady state description of growth and light absorption in the marine planktonic diatom Skeletonema costatum. Limnol. Oceanogr. 34, 198 – 205. Sukenik, A., Bennett, J., Falkowski, P., 1987. Light-saturated photosynthesis-limitation by electron transport or carbon fixation? Biochim. Biophys. Acta 891, 205 – 215. Williams, F.M., 1967. A model of cell growth dynamics. J. Theor. Biol. 15, 190 – 207. Zonneveld, C., 1996. Modelling the kinetics of non-limiting nutrients in microalgae. J. Mar. Syst. 9, 121 – 136. Zonneveld, C., 1997. Modeling the effects of photoadaptation on the Photosynthesis Irradiance curve. J. Theor. Biol. 186, 381 – 388. Zonneveld, C., Van den Berg, H.A., Kooijman, S.A.L.M., 1997. Modeling carbon cell quota in light-limited phytoplankton. J. Theor. Biol. 188, 215 – 226.