Ecological Modelling 113 (1998) 41 – 54

Light-limited microalgal growth: a comparison of modelling approaches Cor Zonneveld * Department of Theoretical Biology, Vrije Uni6ersiteit Amsterdam, De Boelelaan 1087, 1081 HV, Amsterdam, The Netherlands

Abstract Microalgal growth models aim to predict growth rate as well as cellular composition, given the environmental conditions (such as temperature, nutrient availability, light, etc.). This generic purpose poses strong requirements to the structure of any model attempting to comply with it. First, cellular characteristics should be modelled, rather than taken as model input. Second, if one interprets ‘cellular composition’ as nutrient cell quota, rather than as nutrient ratios, models should be formulated at the level of the individual cell. Third, modelling should proceed by making assumptions about processes, rather than about states. This paper evaluates current modelling methodology in the light of these desiderata. The models examined mostly (but not exclusively) monospecific cultures in laboratory environments. Whereas the older models are inadequate in view of the generic purpose, recent models increasingly focus on cellular composition and on the processes that lead to changes in cellular composition. Models to be developed in the next few years thus have the potential to fulfil the generic purpose. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Cell composition; Growth; Models; Phytoplankton; Review

1. Microalgal growth models share a generic purpose ‘‘Would you tell me, please, which way I ought to go from here?’’ ‘‘That depends a good deal on where you want to get to,’’ said the Cat. (L. Carroll, Alice’s Adventures in Wonderland).

* Tel.: + 31 20 4447128; fax: + 31 20 4447123; e-mail: [email protected]

In the past two decades, sophisticated models dealing with the growth of phytoplankton have been published. These models were devised for different purposes, and start from different premises, hence a comparison is not a simple exercise. Some authors have examined a modest number of models side by side (Laws et al., 1983, 1985; Bannister, 1990; Cullen, 1990). The aim of

0304-3800/98/$ - see front matter © 1998 Elsevier Science B.V. All rights reserved. PII S0304-3800(98)00133-1

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this paper is to compare a broader range of modelling approaches. To compare widely different models, one needs a common yardstick. This may be provided by the generic purpose that underlies most, if not all, microalgal growth models. In essence, the purpose of microalgal growth modelling is to predict growth rate and biomass composition, given the environmental conditions. The verb ‘to predict’ should be interpreted in quite a strong sense: a model ‘predicts’ if and only if it is based on assumptions about underlying mechanisms. This conception of the generic purpose is an abstraction of various formulations concerning the purposes of modelling microalgal growth. The following citations may serve as examples. “ Shuter (1979): ‘‘In this paper, a simple model of the basic growth processes... is used to show that [patterns of response to changes in light intensity, inorganic nutrient availability, and temperature] can be explained in terms of a control system which responds to a change in environmental conditions by adjusting cellular composition so as to maximize growth rate.’’ “ Sakshaug et al. (1989): ‘‘... the ability to estimate primary production from maps of temperature, chlorophyll concentration, and, incident light... will depend on an adequate description of the relationship between growth rate and light absorption.’’ “ Kiefer and Cullen (1991): ‘‘These same environmental parameters [viz. light intensity, photoperiod, temperature, and nutrient supply]... affect the growth rate... of phytoplankton crop. Unfortunately, there has not been a complete mathematical description of the interaction of all four environmental parameters. This study presents an approach to describing these interactions.’’ “ Cullen et al. (1993): ‘‘The growth of phytoplankton is fundamentally important to biogeochemical cycling in the sea.... These models [of phytoplankton] must specify... the cellular chemical composition of phytoplankton... and photosynthesis/unit chlorophyll a as a function of irradiance. Models of global biogeochemical processes in the sea require information on the effects of day length, irradiance, temperature,

and nutrients on the biochemical and physiological properties of phytoplankton.’’ “ Arrigo and Sullivan (1994): ‘‘Unfortunately, few models account for the effects of more than two environmental factors in their formulation of phytoplankton growth... Here, we present in detail the biological component of that model, focusing on the physiological responses of ice microalgae to changes in environmental forcing, including temperature, diel variation in spectral irradiance, concentrations of inorganic macro-nutrients... and salinity.’’ The generic purpose imposes important constraints on the structure of any model attempting to serve that purpose. Therefore, in comparing microalgal growth models I focus on their structure. To characterise this structure, I first introduce some terminology borrowed from systems theory. This terminology enables a comparison between a number of prevalent microalgal growth models in relation to the generic purpose.

2. Various roles for input, state, and output variables The input variables of a model are those quantities which are considered to be given. Examples are the nutrient concentrations in the medium and the level of irradiance. A model does not predict the time-course of the input variables. Hence input variables are also known as independent variables. Alternatively, input variables are called forcing or driving variables. Input variables that do not vary in time might also be called parameters. The distinction between parameters and time-constant input variables is nevertheless useful, as these quantities play different roles. By designating some variable as an input variable, it becomes clear that one is interested in the behaviour of the system in response to changes in the value of the input variable. To call some characteristic a parameter signifies that either it is beyond experimental control (some cellular characteristic, for example), or that one keeps it deliberately constant (temperature, for instance).

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Table 1 Identification of model structure of some recent microalgal growth models Reference

Input variablea

State variable

Output variableb

Approachc

Bannister (1979) Shuter (1979)

E0, u E0, N, T

m(E0, u), r(m) m(I), m(N), u(E0), u(m(E0), T), u(m[N])I, RNA/C(E0), stored C/C(E0)

S O

Kiefer and Mitchell (1983) Sakshaug et al. (1989) Laws and Chalup (1990) Lancelot et al. (1991) Kiefer (1993) Arrigo and Sullivan (1994) Geider et al. (1996)

E0, u

— Stored nutrient, stored carbon, photosynthetic apparatus, synthetic apparatus —

(m+r)(E0, u)

S

E0, D, u

–

m(E0, D, u)

S

E0, m(N)

Stored carbon, dark reactions carbon, light reactions carbon Functional macromolecules, small metabolites, reserve products – –

m(E0), nitrogen/carbon [m(N)], u[m(N)] P(E0), Chl a (t) in mg l−1

O

u(E0, D, N, T), m(E0, D, N, T, u) Chl a(t) in mg m−2

S S

u(E0), m(E0, u); u(t), Chl a (t), C(t) after shift-down or shift-up of E0

D

u(E0), m(E0, u) u(E0, T, N), m(E0, u, T)

D D

Baumert (1996) Geider et al. (1997)

E0 E0, D, N, T E0, D, N, T, salinity E0

E0 E0, N, T, D

Light-harvesting apparatus, energy storage reserves, biosynthetic apparatus u Organic carbon concentration, chlorophyll a concentration

D

E0, scalar irradiance; N, nutrient concentration; D, day length; T, temperature; u, Chl:C; m(N), nutrient limited specific growth rate. b m, Specific growth rate; r, specific respiration; P, rate of photosynthesis. c S, static; O, optimality; D, dynamic. a

State variables describe the state of the cell. Knowledge of the state of the cell is sufficient to predict future states, given the values of the input variables. The values of state variables may change in response to changes in the values of the input variables, and/or to their own values. For example, a nutrient store (a state variable) may change through uptake from external nutrient (an input variable) as well as through utilization, which depends on the size of the store itself. A model may have any number of state variables, including none. Finally, output variables may depend on state variables, they may also depend on the input variables, but they affect neither. For example, the specific growth rate or the cell quota of a nutrient may be output variables. Output variables may be numerically identical to state variables. Output variables are also known as dependent variables.

Models without state variables lack the dynamic aspect, hence they are called static models. They relate input to output variables directly, without intervention of some ‘hidden’ variable. For instance, a simple relationship between light intensity and specific growth rate can be viewed as a model lacking state variables. On the one hand, this approach seems to be safe, as, in view of the small number of assumptions involved, few assumptions can be wrong. On the other hand, little can be learned from static models, since it is by studying the consequences of assumptions that something can be learned from models. Models with state variables are often written in the form of differential (or difference) equations that describe how state variables vary in time, in response to their current values and to the values of the input variables. However, optimality models may also contain state variables, but no assumptions are made on how these variables vary

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with time. Rather, such models help us find which value of the state variables optimizes some output variable, for instance the specific growth rate.

3. Overview of some recent models Table 1 presents the structure of eleven different microalgal growth models. I restrict the discussion to models dealing especially with light-limited growth, as light availability is most frequently thought to limit microalgal growth in nature.

3.1. Input 6ariables All models listed in Table 1 have irradiance as an input variable. The chlorophyll a to carbon ratio (Chl:C) and day length feature in three and four models, respectively. These three variables all deal with light availability, or the potential to absorb light. This does not necessarily mean that the role of nutrients is neglected, though. Nutrients may be implicitly accounted for via their effects on Chl:C, as low nutrient cell quota tend to decrease this ratio. ‘Irradiance’ is generally interpreted as photosynthetically active radiation (PAR), i.e. the scalar irradiance integrated over the wavelength range of 400 – 700 nm: PAR=

&

700 nm

E0(l) dl 400 nm

with E0 the wavelength-specific irradiance. Light absorption by chlorophyll is wavelength dependent. To account for this dependence, Morel (1978) introduced the concept of photosynthetically usable radiation (PUR), defined as: PUR =

&

&

700 nm

a¯ =

a(l)E0(l) dl

400 nm

PAR

with a(l) the absorption coefficient for photons of wavelength l. Incorporation of wavelength dependence adds to the realism of a model. However, it is not an essential ingredient of the model structure. Models that simply use E0 to characterize light intensity can always be refined to incorporate wavelength dependence of certain parameters. Day length is introduced to account for the effects of the dark period on growth of phytoplankton kept in a light–dark cycle. In the dark the cells cannot photosynthesize, so growth will depend on day length. To calculate the growth rate in a light–dark cycle, the growth rate under continuous illumination is multiplied by the fractional day length (light period/24 h). This amounts to taking an average of the growth rates during the light and the dark period and assuming a constant growth rate during the light period and no growth during the dark period. If irradiance varies, the average growth rate cannot be calculated simply by multiplying the growth rate under continuous illumination by the fractional day length. Day length as a input variable then loses its appeal. An alternative to the use of day length and irradiance as separate input variables is to specify irradiance as a function of time. This more general approach obliterates the need for day length as an input variable. However, in the description of field data day length as an input variable may offer an advantage in the use of available meteorological data. Hence, whether or nor to use day length as an input variable depends on the purpose of modelling.

3.2. State 6ariables

700 nm

E0(l)A(l) dl 400 nm

where A(l) is the probability that a photon of a given wavelength will be absorbed by an algal cell. The function A(l) derives from the absorption spectrum of the cells. A related approach is to calculate the spectrally averaged absorption coefficient, a¯, according to:

Half of the models in Table 1 do not contain state variables, the rest have at least one state variable. Models with state variables have between one and four of them. The models without state variables are mostly energy budgets based on carbon. Such models have the following overall structure:

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Table 2 Interrelations between state variables of models in Table 1 Geider et al. (1997)

Shuter (1979)

Chlorophyll Á Ã Ã Ã Í Ã Ã Ã Ä

Organic C

Photosynthetic apparatus Synthetic apparatus

Laws and Chalup (1990) Á ÃLight reactions C Í ÃDark reactions C Ä

Geider et al. (1996) Light harvesting apparatus Â Ã Â Ã Ì Ã Å

Biosynthetic apparatus

(Structural components)a (Structural C)

Stored C

Storage C

Lancelot et al. (1991)

Reserve products

Ã Ã Ì Ã Ã Ã Å

Functional macro molecules

Á ÃEnergy storage Í ÃSmall metabolites Ä

Stored nutrient State variables of each model are presented in columns; relations between variables become apparent by comparison over rows. The curly braces indicate that a particular variable in one model is split up into several variables in another model; or, vice versa, several variables in one model are lumped into one variable in another model. Notice that in two models the structural carbon pool is a parameter, not a variable. C, carbon. a These parameters are included for ease of interpretation.

m + r=F



Chl , I, ... C



with m the specific growth rate, r the specific rate of dark respiration, and F some function of Chl:C, irradiance I, and potentially some other variables. Chl:C is generally treated as an input variable, but Baumert (1996) treats it as a state variable. In this respect, the Baumert (1996) model is a generalization of the static energy budget models. The nomenclature of the state variables shows that modellers deem it desirable to represent certain functionally coherent classes of macromolecules by state variables (e.g. light-harvesting apparatus, dark reactions carbon). An explicit rationale for the choice of state variables is rarely given, however. As it is generally unknown to what extent model predictions depend on the choice of state variables, this lack in underpinning one’s choice is somewhat curious. Table 2 summarizes the relations between the state variables in the various models, based on the authors’ descriptions of the variables. As the authors’ use different terminology’s, some interpre-

tation on my part is involved. Baumert’s (1996) model is not included, as the relations between his single state variable, Chl:C, and those of the other models, are not easily indicated in this table. Chlorophyll a is part of the light-harvesting apparatus, while total cell carbon is the sum of the carbon in all model variables. Hence Chl:C corresponds to the ratio between the light-harvesting apparatus and total cell carbon. The various models in Table 2 diverge especially as regards the choice of state variables representing the photosynthetic machinery. This again highlights the focus on light-limited growth in current modelling attempts. Most models in Table 2 feature reserve pools. Lancelot et al. (1991) split up the reserves pool into an energy storage proper and a pool of small metabolites, which act as an intermediate pool mediating all energy fluxes. The background of this choice may be the desire to represent in the model the actual flows of carbon within the cell. This section ends with a remark on the visualization of the model structure as presented by most authors. The models are invariably portrayed as a box, suggestive of the cell wall or

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membrane. Inside the box various pools are indicated, which often represent the models’ state variables. Outside the box are the environmental variables affecting the cell’s behaviour (the input variables). Arrows indicate flows of materials or feedback mechanisms, or both. Such figures are certainly helpful, but they might convey more to the casual reader than the model actually offers. I experienced this myself with the figure representing Shuter’s model. In this figure, four different carbon pools are indicated, connected with arrows indicating carbon flows. As the model concerns the optimal allocation of carbon to the various pools, I expected all pools to be represented by state variables. However, Shuter (1979) assumes a fixed size for the structural carbon pool. So contrary to the suggestion of the figure representing the model, this pool size is a parameter, not a state variable.

3.3. The choice of the chlorophyll a to carbon ratio as a model 6ariable Although Chl:C is often considered a key factor in phytoplankton growth, it is rarely pointed out why this should be so. Geider (1987) observes that cell carbon is a measure of cell energy content, while chlorophyll a limits the energy supply rate. Langdon (1987) remarks that the Chl:C ratio reflects the amount of biomass which is supported by a unit amount of light-harvesting pigment. These assertions can be criticized on good grounds, however. It is true that energy can be stored in carbon-rich compounds, but not all carbon-rich compounds can be used to fuel energy demanding processes. The fraction of cellular carbon that can be mobilized for this purpose may differ from 10% in Prorocentrum micans to 60% in Isochrysis galbana (Zonneveld et al., 1997). So the carbon content of a cell is not necessarily an accurate indicator of the cell content of available energy. Furthermore, Chl:C may not be relevant in light-saturating conditions, since chlorophyll a probably does not limit the light-saturated rate of photosynthesis (Sukenik et al., 1987). Finally, the remark of Langdon (1987) implies that biomass has to be

supported by energy captured via chlorophyll. Although during harsh periods maintenance costs may contribute substantially to the energy budget, in more favorable conditions these costs are often quantitatively insignificant (e.g. Zonneveld et al., 1997). Hence Langdon’s remark does not suffice to motivate the general use of Chl:C. The importance attached to Chl:C might have more mundane reasons, too. Chlorophyll a is the most readily measured indicator of phytoplankton abundance. As a result, many physiological characteristics, including the rate of photosynthesis, are expressed on a per unit chlorophyll basis. The rate of photosynthesis (expressed as carbon, C, fixed per unit time) is then given as: dC = P Chl · Chl, dt with P Chl the rate of photosynthesis per unit chlorophyll, and Chl the amount of chlorophyll. However, phytoplankton ecologists are often more interested in the specific rather than the absolute rate of increase. Division of the left and the right hand sides of this equation by the carbon content of the standing crop, C, yields the desired expression: m=

Chl 1 dC = P Chl · C C dt

The ability to measure chlorophyll and the desire to model the specific growth rate thus give rise to a model structure in which Chl:C figures. Obviously, this does not suffice to establish the physiological relevance of the ratio. But the arguments from physiology are not altogether convincing either. This raises the question whether Chl:C should be a key variable in any microalgal growth model. Geider et al. (1998) develop models that have separate expressions for chlorophyll and organic carbon, but still relate the rate of photosynthesis to Chl:C. In a companion paper to this review, I show that it is well possible to model carbon cell quota and chlorophyll cell quota without reference to their ratio (Zonneveld, 1998). (Of course, as a corollary their ratio is described, too).

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Table 3 General form of PI-curves in the models of Table 1 Type of PI-curve

Photosynthesis proportional toa

References

Proportional Hyperbolic Generalized hyperbolic Inverted exponential Inverted exponential with photoinhibition term

E0 E0/EK+E0 1/m E0/(K m+E m 0) 1−exp(−aE0) [1−exp(−aE0)]× exp(−bE0)

Shuter, 1979 Kiefer and Mitchell, 1983; Laws and Chalup, 1990; Kiefer, 1993 Bannister, 1979 Sakshaug et al., 1989; Arrigo and Sullivan, 1994; Geider et al., 1996, 1997 Lancelot et al., 1991; Baumert, 1996

a

E0, scalar irradiance; EK, irradiance at which photosynthesis is half-maximal; K, saturation parameter; m, shape parameter of PI-curve; a, photosynthetic efficiency; b, parameter characterising strength of photoinhibition.

3.4. Light and photosynthesis: the PI-cur6e Photosynthesis depends on light intensity, an input variable of all models. The way the rate of photosynthesis depends on light is often called the photosynthesis –irradiance curve (PI-curve) — a convention followed here. As PI-curves are of much interest to phytoplankton ecology, they are dealt with here in a separate section. Table 3 catalogues the types of PI-curves used in the models examined. Notice that they are probably intended to be steady state curves; short-term variations are not accounted for. Most models feature a saturating PI-curve, the (generalized) hyperbolic function, or an inverted exponential function of light intensity. Photoinhibition is accounted for by two models. In these models a saturating function is multiplied by a term representing photoinhibition. The only model not assuming some form of saturation is Shuter (1979). In his model, growth becomes nutrient-limited at high light intensities, and the remainder of the fixed carbon is stored. Compared to the hyperbolic function, the inverted exponential often yields better descriptions of experimental data regarding the PI-curve (e.g. Lederman and Tett, 1981). However, the inverted exponential is insufficiently grounded in mechanistic assumptions (Zonneveld, 1997). The inverted exponential PI-curve is based on a model for the oxygen yield per light flash as a function of flash intensity (Ley and Mauzerall, 1982). According to this model, the probability that a photosynthetic unit is hit by a certain number of

photons follows a Poisson distribution. A photosynthetic unit is activated by a flash if it is hit by at least one photon. Based on the Poisson distribution, the probability of ] one hit= P (at least one hit)= 1−exp(− s E), with E the flash intensity and s the absorption cross section. Dubinsky et al. (1986) modified this Poisson model to render it also applicable to continuous light. They concluded that the modified model is valid for the statistical analysis of the PI-curve in continuous light; they made no claims regarding the mechanistic underpinning of the model. A model for the PI-curve in continuous light based on the Poisson model should account for photon absorption by a photosynthetic unit, the handling of the resulting exciton, and absorption of a subsequent photon. After an exciton has been handled by a photosynthetic unit, the unit awaits a new hit, so it is idle for some time. Current derivations of the inverted exponential PI-curve inadvertently neglect this idle time (e.g. Baumert, 1996). Once this idle time is accounted for, the resulting PI-curve will be a three- rather then a two-parameter model. Although rarely treated as such, the static energy budget models have much in common with models for the PI-curve, as noted by Cullen (1990). However, various factors tend to obscure the basic identity of models for PI-curves and static energy budget models. First, PI-curves proper describe the instantaneous rate of photosynthesis, whereas the growth models describe steady-state rates of photosynthesis. Second, photosynthesis is often expressed as oxygen produc-

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tion or carbon fixation per unit carbon per unit time. This is the so-called specific rate of photosynthesis. Growth models deal with the specific growth rate, i.e. the amount of biomass fixed per unit time per amount of biomass. Apparently these are different phenomena, but with a fixed carbon:biomass ratio, the specific growth rate for biomass is proportional to the specific rate of photosynthesis (assuming zero maintenance costs). Third, mathematical descriptions of the PI-curve are often simple expressions with as few parameters as possible. These parameters are not derived from knowledge of the underlying physiological processes. By contrast, the more elaborate growth models provide detailed descriptions with many parameters that represent biophysical properties. Finally, respiration is explicitly accounted for in growth models, whereas PI-curves proper often deal only with carbon fixation.

3.5. The role of nutrients Although current modelling focuses on lightlimited growth, many models deal, in one way or another, with the possibility of nutrient limitation. There is no consensus in the way the various models deal with nutrients. This section briefly describes the various ways nutrients are accounted for. Also, it pays attention as to how the models distinguish between light-limited and nutrient-limited growth. The models of Lancelot et al. (1991), Baumert (1996), and Geider et al. (1996) do not include the role of nutrients. These models are thus restricted to light-limited growth. Note, however, that they do not specify the conditions under which light limitation applies. Nutrients may limit the growth rate by restricting the synthesis of chlorophyll, thus affecting Chl:C. For this reason, the role of nutrients can be thought of as being implicitly accounted for in the static energy budget models of Bannister (1979), Kiefer and Mitchell (1983), and Sakshaug et al. (1989), which all feature Chl:C as an input variable. The static energy budgets then apply to both light and nutrient-limited growth. This is a remarkable achievement, but at high cost: the approach only works if Chl:C is treated as an

input variable. If Chl:C were treated as a state variable, one should have to specify how the ratio changes in response to nutrient availability. This approach is followed by Geider et al. (1997). In their model, the maximum carbon-specific rate of photosynthesis varies according to the Monod equation with the nutrient concentration in the environment. Arrigo and Sullivan (1994) relate the nutrientlimited growth rate to the external nutrient concentration according to the Monod relationship. They compare this potential nutrient-limited growth rate with the potential growth rate under light limitation. They then take the lower of the two values, and in this way determine whether growth is either nutrient-limited or light-limited. The energy budget model of Kiefer (1993) approaches nutrient limitation more explicitly. He first derives an equation for the daily carbon-specific rate of photosynthesis. Two of the quantities in this equation, the maximum instantaneous carbon specific rate of photosynthesis and the carbon to chlorophyll ratio, are subsequently considered to depend on the nutrient concentration of the environment (and also on temperature, see below). In this approach neither nutrient nor light is exclusively limiting; both nutrient and light simultaneously affect the rate of photosynthesis. Shuter’s model features a pool of stored nutrient, but somehow the size of this pool plays no role in the derivation of the model (Shuter, 1979). He derives an expression for the optimal nutrient uptake rate, based on the assumption that the growth rate is maximized. Growth is nutrient-limited if the optimal nutrient demand exceeds the potential uptake rate. The potential uptake rate depends in a Michaelis–Menten fashion on the external nutrient concentration. According to the graphs shown, nutrient cell quota increase with the growth rate under conditions of nutrient limitation. How this comes about is not clear, as no equations are presented to describe the size of the nutrient pool. Laws and Chalup (1990) state that Shuter’s model does not account for the fact that under nitrogen limitation the nitrogen to carbon ratio is uniquely related to the relative growth rate. For this and a number of other reasons, they devel-

C. Zonne6eld / Ecological Modelling 113 (1998) 41–54

oped a new model, following Shuter’s approach. They imposed the observed dependence of the nitrogen to carbon ratio on the nutrient-limited specific growth rate. Notice, however, that their model does not predict this growth rate. In contrast, their model does predict the (nutrient-saturated) light-limited growth rate, but not the nitrogen to carbon ratio under these conditions. The model does not predict when either nutrient or light is limiting.

3.6. Modulating factors: temperature and salinity Phytoplankton cells take up nutrients and absorb light. Nutrients and light are, therefore, exhaustible resources. This does not apply to temperature and salinity, although these certainly affect microalgal growth. The latter properties may therefore be called modulating factors. Most models bypass modulating factors by assuming constant values for them. Temperature is explicitly accounted for by four models, salinity by one. Here we briefly discuss how these models deal with modulating factors. In Shuter’s model, temperature affects some parameter values. Four parameters representing rates of various processes vary with temperature with a Q10 of 2.0, while the half-saturation constant of nutrient uptake kinetics has a Q10 in the range of 1.0–2.0. The remaining parameters (some of which are rates) are independent of temperature. A similar approach is followed by Geider et al. (1997). In their model, only the maximum carbonspecific photosynthesis rate is affected by temperature, according to the Arrhenius relation. As this parameter occurs in the expressions for Chl:C and for the specific growth rate, both quantities are temperature dependent. Arrigo and Sullivan (1994) let the maximum (nonlimited) specific growth rate vary exponentially with temperature. The temperature dependence of respiration is modelled similarly. Notice that none of the underlying processes leading to growth vary with temperature; only the final expression for the growth rate is temperature adjusted. The resulting value for the growth rate is also multiplied by a salinity-dependent growth

49

coefficient which is calculated as a fifth-degree polynomial in Sb, the brine-salinity. Kiefer (1993) assumes that two model parameters, the maximum instantaneous carbon specific rate of photosynthesis and the carbon to chlorophyll ratio, depend on either the nutrient concentration of the environment, or on the temperature. The function g(N, T)=min{gn (N), gt (T)} describes this dependence, with gn = g mN/(KN + N) and gt = aT exp(bTT}. This choice is motivated by an appeal to Liebig’s Law of the minimum. However, temperature will affect growth even if nutrients are growth limiting (see for instance Rhee and Gotham, 1981). In general, Liebig’s law is better not applied to modulating factors.

3.7. Output 6ariables: the link between models and data If model variables do not coincide with measured variables, auxiliary hypotheses are necessary to relate them. For example, Shuter’s model is formulated in terms of carbon pools, whereas he aims to describe, among other things, the cellular RNA:carbon ratio as dependent on light intensity. To arrive at a model prediction for this ratio, auxiliary hypotheses relating RNA content to the sizes of the various carbon pools had to be made. Such hypotheses do not belong to the core of the model, hence the term ‘auxiliary hypotheses’. The resulting variable describing the measurements is called the ‘output variable’. The concept might seem superfluous when the state variables themselves are directly observable quantities, like Chl:C or the specific growth rate. Yet, for consistency of terminology, I use the term ‘output variable’ whenever I refer to the description of data by a model. Table 1 lists the output variables of the models. As an output variable is allowed to be given by any function of state and input variables, the number of potential output variables is essentially unlimited. My presentation restricts to those output variables that were actually used in the cited references to describe experimental data. I use the notation y(x) to indicate that the output variable y is considered a function of the variable x. Most often, x is one of the input

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variables. But x itself may be an output variable, for example when the specific respiration is described as dependent on the specific growth rate, which in turn depends on irradiance. A correct representation of these relationships is r[m(E0)]. In Table 1 I abbreviated this to r(m) if the dependence m(E0) was already given, and no ambiguity resulted. For many phytoplankton ecologists and physiologists, a model is useful only if it has something to say about the real world. Probably for this reason, most modellers pay considerable attention to the ability of their models to describe experimental data. However, it is far from easy to arrive at firm conclusions about such descriptions. I briefly mention a few of the problems that play a role in evaluating the correspondence between model prediction and data. On logical grounds, satisfactory model descriptions of data do not count as conclusive evidence in favor of the model, as such descriptions may result from wrong assumptions. Though this argument is correct, its strength depends much on the scope of a model. The scope of a model may be provisionally defined as its number of output variables. A model with only one output variable, for instance the dependence of the specific growth rate on irradiance, has a very limited scope. Many different models might lead to comparable relationships. So even if data were in line with a particular model, the support would be meagre; the data would be in line with many other models, too. But suppose the model were also to predict rates of photosynthesis and respiration, and the relevant data turned out to be well described by the predictions. Then the support would be much better, as it is far more difficult to develop a model that correctly describes three output variables. The potential of data to vote against a model thus depends very much on the model’s scope. The wider the scope of a model, the more likely it is that the data will discredit the model, so the more useful a comparison with data is. But how is such a comparison to be made? Deviations between the model prediction and observations derive from two sources. First, biological variability and measurement errors together manifest themselves as observational

variation. Second, there may be a lack of fit even in the absence of observational variation; only this latter lack of fit can be attributed to the model. To disentangle the two sources of error, replicate data are necessary, which allow the estimation of the observational variation. Once the magnitude of the observational variation is known, it becomes possible to gauge the magnitude of the lack of fit due to imperfections of the model. So the goodness of fit can only be tested with replicated data. However, replicated data are often not available. None of the models discussed in this paper has been tested against replicated data. But even if one neglects this problem, it is not clear what kind of support a good model description of data (i.e. no significant lack of fit) lends to the model. A typical example where statistics is of scant use is provided by the Shuter (1979) model. In general, the descriptions generated by his model are quite acceptable; however, the most salient features, for instance kinks in the curves coinciding with the transition from light to nutrient limitation, are weakly if at all supported. The support derived from a good model fit also depends on some other aspects that are hard to quantify. Some models can describe almost any data set (polynomials are an example), whereas other models are less flexible. The more flexible the model, the less support a good fit provides. If parameter values are chosen on the basis of prior knowledge (e.g. Shuter, 1979), minor deviations of the model from the data may be acceptable. The model fit certainly has to comply to more strict requirements if parameter values are estimated from the data to be described. Some further complication may arise due to the data themselves. These can be ‘noisy’, showing much variation that is not explained by the independent variables under study. Even worse, the data may be unreliable. In re-analysing the data in Droop (1974), I stumbled upon such a problem (Zonneveld, 1996). The data analysed concerned cell quota of a non-limiting nutrient, of cells grown in a chemostat in steady state. In two sets of experiments, the uptake mechanism for this nutrient was always fully saturated, according to the measured concentrations of the nutrient in the chemostat. If so, the cell quota of this nutrient

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must be similar at identical growth rates, but that was certainly not the case. Thus the data on medium concentration and cell quota were inconsistent. In such cases, the failure to accurately describe the data should probably not be attributed to the model, but to the data. These remarks make clear that evaluating the correspondence between model prediction and data poses some sticky methodological problems. So how are we going to use data to evaluate a model’s usefulness? I have no firm answer to this important question, but the following suggestions might be considered. First, the data to be used should be examined carefully against the background of the model’s assumptions. A good fit can only provide evidence in favor of a model if the experimental set-up is consistent with the models assumptions. For instance, a model may assume the absence of lightdark periodicity. Data used to test such a model should then derive from experiments with algae grown in continuous light. Second, the data selected to test a model should not be noisy, as they should be able to expose deviations of model predictions from reality. Third, in the absence of rigorous statistical tools, one might judge a fit by eye. Though subjective, the eye is a sensitive instrument, well able to detect major deviations between predictions and data.

4. That model structure would serve the generic purpose? As the generic purpose inferred earlier in this paper is a generalization of different purposes of various models, it is formulated rather imprecisely. For this reason, the generic purpose cannot fully and unambiguously determine the structure of a model. Nonetheless, it does impose certain constraints on the structure of models aiming to comply with it. In this section I explore these constraints. Mathematical modelling is often seen as an art, rather than a technique that follows some set of prescribed simple rules. Hence, in specifying a possible model structure, there is ample room for difference of opinion. The follow-

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ing remarks thus necessarily mirror my ideas about modelling, to some extent.

4.1. Input 6ariables Physico-chemical variables characterising the environment, for example, nutrient concentrations, light availability, and temperature, should play the role of input variables. The reason for this is that environmental variables are outside the scope of the model. By contrast, cell characteristics should be predicted by the model, so they should not figure as input variables. Some of the earlier models take Chl:C as an input variable (see Table 1). These models were meant in the first place as an aid in the estimation of primary production. Nevertheless, even within this framework it is possible to come closer to the generic purpose. Kiefer (1993) presents an empirical description of the dependence of Chl:C on environmental conditions. Though an improvement, the empirical character compares unfavorably with the ideal. Baumert (1996) treats the Chl:C ratio as a state variable. By doing so, he comes across questions regarding the regulation of Chl:C. This in itself is already rewarding, as such questions draw the attention towards the underlying mechanisms. Geider et al. (1997) use Chl:C in their model, but they derived separate equations for the dynamics of chlorophyll and carbon. Temperature interferes with metabolism by changing the rates at which biochemical processes proceed. Differences in biochemical composition of the cell might arise as various biochemical processes are differently affected by temperature. Temperature effects can thus be accounted for by assuming that all rate parameters (i.e. all parameters which have ‘per time’ in their physical dimensions) are temperature dependent. The temperature dependence may for instance be described by the Arrhenius relation, or on the basis of Q10-values. Non-rate parameters might also be temperature dependent, for instance when they represent ratios of two rates that differ in their temperature sensitivity. In the absence of any knowledge on temperature dependence, one might assume that non-rate parameters do not depend on temperature. Few models explicitly eke this

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approach. Only Shuter (1979) models temperature dependence in this way. Notice, however, that temperature dependence can be incorporated in any model as just described.

4.2. State 6ariables Variations in cell composition in response to variation in the environment should be accounted for. This demand hints at the use of state variables. Cell composition might also be accounted for by simple empirical relationships, thus bypassing the need for state variables. However, in this way one cannot predict on basis of the underlying mechanisms. Hence a static approach, not involving state variables, does not compare favorably with the demands of the generic purpose. Cullen et al. (1993) proposed to develop a so-called DENT-model, where the acronym ‘DENT’ stands for day length, irradiance (often denoted by E), nutrient, and temperature. This DENT-model should be an energy budget, based on carbon, explicitly addressing the estimation of primary production in the sea. Moreover, it would serve as a guide for a numerical model that is appropriate for dynamic situations. To me it seems unlikely that a steady-state model can guide the development of a dynamic model. Important decisions to be made in the development of a dynamic model concern the choice of state variables, and how these change in response to varying inputs. A steady-state model without state variables seems to be of little help in this. The generic purpose implies that at least one of the state variables should characterise the cell size, if one identifies ‘biomass composition’ with cell quota of various nutrients, not with ratios of cell quota. (Note that I interpret cell quota as amount per cell). In his review, Droop (1983) is curiously silent about what he exactly means by ‘cell quota’. The data in his 1974 paper are expressed as mol/cell, so my interpretation is in accordance with the way he uses the concept (Droop, 1974). Given nutrient cell quotas, nutrient ratios can always be determined; but given ratios, cell quotas cannot be determined. So if cell quota are to be modelled, cell size should somehow be accounted

for, since the cell quota is the amount of nutrient divided by cell volume. Most dynamic models (e.g. Lancelot et al., 1991; Geider et al., 1996, 1997) bypass the level of the cell. The model variables are expressed as the chlorophyll a or the carbon content of the water. By contrast, the optimality models of Shuter (1979) and Laws and Chalup (1990) feature a variable characterising cell size. But Shuter’s model assumes that total cell carbon is given; only the partitioning within this total is subject of his model. One might question whether cell size can be bypassed as a model variable in a dynamic model. Individual phytoplankton cells are organized to function as integrated units. Some phenomena may therefore be best understood from models formulated at the level of the individual cell. This might for instance apply to the the so-called package effect, which plays a role if cells are compared that are acclimated to different irradiances. Due to the package effect, the chlorophyll-specific absorption coefficient is not constant. This is most easily modelled if cell size is accounted for (Zonneveld, this issue). State variables do not have to coincide with measured variables (i.e. they may be ‘hidden variables’). There is a quite strong tendency to choose model variables that are easily determined experimentally. For example, chlorophyll a is often used as a measure for biomass in biological oceanography (and with good reason); it also features in many microalgal growth models. However, chlorophyll a is but one of the many molecules necessary to harvest light energy. Instead of chlorophyll a, one might also use the more neutral term ‘photosynthetic pigments’. This is more than a semantic trick. Using chlorophyll a as a model variable implies some tacit assumptions, for instance that the ratio of chlorophyll a to other pigments is constant. This assumption may be irrelevant to the core of a model. Moreover, one may unnecessarily restrict the applicability of the model to certain taxonomic groups. By using ‘photosynthetic pigments’ one can focus on the model itself, not hindered by connotations sticking to chlorophyll a. Only when the model is to be compared with data, one has to make some auxiliary assumptions relating the model variable to the measured variables (Section 3.7).

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In my view, it is important to frame assumptions in terms of processes, or mechanisms, rather than in terms of states. Assumptions about uptake and utilization processes define the budget, which is subject to the mass conservation law. Also, cells regulate states by adjusting rates at which processes proceed. For example, in response to nutrient shortage, cells may change characteristics of the nutrient uptake system. This, in turn, may affect the nutrient status of the cell. Assumptions about processes thus lie ‘closer’ to the underlying mechanisms. (Nowadays, there is a strong tendency to equate ‘mechanism’ with ‘biochemical mechanism’. However, a model that does not account for all biochemical details is not, for that reason, an empirical model. Mechanisms can be studied at levels other than that of biochemistry, too.) The optimality approach features state variables, however rules of how these change in time are absent. Rather, this approach aims to find the values of the state variables that optimize the specific growth rate. The axiom that cells optimize their growth rate is relatively straightforward to implement in models that assume a constant environment; whether this also holds for changing environments is not clear. One probably has to define the changes that occur, before the optimality approach can provide predictions on optimal growth rates and cellular composition. This severely restricts the use of the optimality approach. The dynamic approach seems naturally suited to meet the requirements of the generic purpose. The use of state variables allows one to describe changes in cellular composition in response to changes in the environment. The need to describe changes leads to a focus on the processes that underlie the behavior of the cell in relation to the environment. Recently, the dynamic approach is more often taken in microalgal growth modelling.

5. Conclusions Microalgal growth models share a generic purpose: to predict growth rate and cellular composition in response to environmental conditions. This

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generic purpose imposes quite stringent demands on the structure of any model aiming to fulfil it. The environmental conditions should be represented by input variables, whereas state variables should describe the cell. The model should describe changes of the state variables in response to environmental conditions. Current modelling methodology does not fully comply with these requirements. Models that chiefly aim to predict primary production are mostly formulated as steady-state carbon budgets. These models lack state variables that are necessary to describe variations in cellular composition. Some models aiming to understand balanced growth follow an optimality approach. The basic point of departure of these models is that the cell partitions its resources in such a way that the growth rate is maximized. Application of this principle is restricted to steady state growth. Whether this point of departure can also be applied to cells in a fluctuating environment remains unclear, yet a fluctuating environment will be the rule rather than the exception. This might severely restrict the applicability of this approach. Dynamic models are potentially best suited to study the regulation of growth in response to variations in the environment. Recently, microalgal growth models are increasingly formulated as dynamic models. However, few dynamic models take the cell as the basic point of departure. As an alternative to models formulated at the population level, models at the individual level are also worthwhile to pursue.

Acknowledgements I gratefully acknowledge, in alphabetical order, the constructive criticisms of John Beardall, Hugo van den Berg, Ad van Dommelen, Ger Ernsting, Winfried Gieskes, Jef Huisman, Bas Kooijman, Tarzan Legovic´, Astrid Schoenmaker, Peter West broek and an anonymous referee. Jolanda de Jong corrected the English language. 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

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References Arrigo, K.R., Sullivan, C.W., 1994. A high resolution big-optical model of microalgal growth: tests using sea-ice algal community time-series data. Limnol. Oceanogr. 39, 609– 631. Bannister, T.T., 1979. Quantitative description of steady state, nutrient-saturated algal growth, including adaptation. Limnol. Oceanogr. 24, 76–96. Bannister, T.T., 1990. Comparison of Kiefer–Mitchell and Bannister – Laws models. Limnol. Oceanogr. 35, 972–979. 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. Cullen, J.J., 1990. On models of growth and photosynthesis in phytoplankton. Deep-Sea Res. 37, 667–683. Cullen, J.J., Geider, R.J., Ishizaka, J., et al., 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, Berlin, pp. 153–176. Droop, M.R., 1974. The nutrient status of algal cells in continuous culture. J. Mar. Biol. Ass. UK 54, 825–855. Droop, M.R., 1983. 25 years of algal growth kinetics. Bot. Mar. 26, 99 – 112. Dubinsky, Z., Falkowski, P.G., Wyman, K., 1986. Light harvesting and utilization by phytoplankton. Plant Cell Physiol. 27, 1335 – 1349. 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 dymanic regulatory model of phytoplankton acclimation to light, nutrients and temperature. Limnol. Oceanogr. 43, 679–694. 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, Berlin, pp. 93–120. Kiefer, D.A., Cullen, J.J., 1991. Phytoplankton growth and light absorption as regulated by light, temperature, and nutrients. In: Sakshaug, E., Hopkins, C., Øritsland, N. (Eds.), Proceedings of the Pro Mare Symposium on Polar Marine Ecology 12 – 16 May 1990, Trondhiem, Polar Res., 10.

Kiefer, D.A., Mitchell, B.G., 1983. A simple, steady state description of phytoplankton growth based on absorption cross sections and quantum efficiency. Limnol. Oceanogr. 28, 770 – 776. Lancelot, C., Veth, C., Mathot, S., 1991. Modelling ice-edge phytoplankton bloom in the Scotia-Weddell sea sector of the southern ocean during spring 1988. J. Mar. Syst. 2, 333 – 346. Langdon, C., 1987. On the causes of interspecific differences in the growth-irradiance relationship for phytoplankton: part I: a comparative study of the growth-irradiance relationship of three marine phytoplankton species: Skeletonema costatum, Olisthodiscus luteus and Gonyoulax tamarensis. J. Plankton Res. 9, 459 – 482. Laws, E.A., Chalup, M.S., 1990. A microalgal growth model. Limnol. Oceanogr. 35, 597 – 608. Laws, E.A., Jones, D.R., Terry, K.L., Hirata, J.A., 1985. Modifications in recent models of phytoplankton growth: theoretical developments and experimental examination of predictions. J. Theor. Biol. 114, 323 – 341. Laws, E.A., Redalje, D.G., Karl, D.M., Chalup, M.S., 1983. A theoretical and experimental examination of the predictions of two recent models of phytoplankton growth. J. Theor. Biol. 105, 469 – 491. Lederman, T.C., Tett, P., 1981. Problems in modelling the photosynthesis-light relationship for phytoplankton. Bot. Mar. 24, 125 – 134. Ley, A.C., Mauzerall, D.C., 1982. Absolute absorption crosssections for photosystem II and the minimum quantum requirement for photosynthesis in Chlorella 6ulgaris. Biochim. Biophys. Acta 680, 95 – 106. Morel, A., 1978. Available, usable, and stored radiant energy in relation to marine photosynthesis. Deep-Sea Res. 25, 673 – 688. Rhee, G.Y., Gotham, I.J., 1981. The effect of environmental factors on phytoplankton growth: light and the interaction of light with nitrate limitation. Limnol. Oceanogr. 26, 649 – 659. 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. Shuter, B., 1979. A model of physiological adaptation in unicellular algae. J. Theor. Biol. 78, 519 – 552. Sukenik, A., Bennett, J., Falkowski, P.G., 1987. Light-saturated photosynthesis-limitation by electron transport or carbon fixation? Biochim. Biophys. Acta 891, 205 – 215. Zonneveld, C., 1996. Modelling the kinetics of non-limiting nutrients in microalgae. J. Mar. Syst. 9, 121 – 136. Zonneveld, C., 1997. Modelling 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. Modelling carbon cell quota in light-limited phytoplankton. J. Theor. Biol. 188, 215 – 226. Zonneveld, C., 1998. A cell-based model for the chlorophyll a to carbon ratio in phytoplankton, Ecol. Modell., 113, 55 – 70.