Functional analysis of Microcystis vertical migration: A dynamic model as a prospecting tool

Ecological Modelling 188 (2005) 386–403

Functional analysis of Microcystis vertical migration: A dynamic model as a prospecting tool I—Processes analysis Sophie Rabouille a,∗ , Marie-Jos´e Salenc¸on b , Jean-Marc Th´ebault a a

Laboratoire d’Ecologie des Hydrosyst`emes, FRE CNRS-UPS 2630, Universit´e Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France b Electricit´ e de France, Recherche et D´eveloppement, LNHE/GHEO, 6 Quai Watier, 78401 Chatou C´edex, France Received 27 August 2003; received in revised form 21 December 2004; accepted 17 February 2005 Available online 11 July 2005

Abstract YOYO is a deterministic model intended to represent the growth and vertical movements of Microcystis colonies in the water column. The fluctuations of the carbohydrate content in cells are submitted to the influence of photosynthesis and biosynthesis. They modify the colony density, which responds through a vertical movement of migration. The model simulates, at a daily scale, the fluctuations of Microcystis density resulting from the dynamics of the carbohydrate reserve metabolism during photosynthesis. A sensitivity analysis is carried out, under constant conditions of light, temperature and colony diameter, in order to identify the effect of light and temperature on the metabolism and migratory behaviour of colonies with different diameters. State equilibria, reached under permanent regimes of light and temperature close to culture conditions, are consistent with experimental results from the literature. A periodical light forcing is then applied, for one temperature and a large range of diameters. Simulations point out that migrations are characterised by a pair amplitude/period of the movement, which depends on the diameter. The migratory behaviour highlights an ecological advantage linked to the colony diameter. © 2005 Published by Elsevier B.V. Keywords: Microcystis; Photosynthesis; Carbohydrates; Vertical migration; Hydrodynamics; 1D mathematical model

1. Introduction Grangent reservoir (Loire, France) is a eutrophic reservoir with a high phytoplanktonic production in ∗ Corresponding author. Tel.: +31 113 577 472; fax: +31 113 573 616. E-mail address: [email protected] (S. Rabouille).

0304-3800/$ – see front matter © 2005 Published by Elsevier B.V. doi:10.1016/j.ecolmodel.2005.02.015

spring and summer. Each year, Cyanobacteria recurrently appear in the lake at the same period (August); however, some years they can develop earlier (from June) and occupy the water column until the autumn, disturbing the various uses of the water resource. The mechanism of their appearance remaining poorly known, any attempt to limit their development is quite hazardous.

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Planktonic Cyanobacteria are remarkable for their ability to regulate their buoyancy, which is supposed to facilitate their development in stratified waters. In order to determine whether this process has a key role in their dominance, a mathematical model has been developed to describe the growth and the vertical migration of the Cyanobacterium Microcystis sp. This approach is an exploratory phase as the model isolates Microcystis from the complexity of its real environment; it permits to highlight the dominant processes relative to Microcystis ecology. This work splits into two parts. The first one, decribed hereafter, is based on the conceptual diagram of Microcystis development previously presented (Rabouille et al., 2003). The model construction, its mathematical formulation and an analysis of the model behaviour under constant forcing conditions are detailed. The second part is an exploitation of the model through different simulations, run under unsteady environment conditions (thermal stratification and/or mixing in the water column), that allow pointing out situations where the ability to regulate the buoyancy is an advantage for Microcystis, compared to the nonmobile phytoplankton (Rabouille and Salenc¸on, 2005). The model, thus tested, is also used to simulate the behaviour of colonies under a real seasonal forcing (these results are presented in a forthcoming paper).

2. Relevant features about Microcystis ecology: selected hypotheses Cyanobacteria are pigmented and photoautotroph organisms. They regulate their buoyancy depending upon the surrounding conditions. Inside the cells, the presence of gas vacuoles, whose abundance lightens the cells enough to make them lighter than the water, is an outstanding character of planktonic Cyanobacteria. Depending on the genus, Cyanobacteria have developed different mechanisms to make cells heavier: strong or weak resistance of vesicles (composing the gas vacuoles) to internal pressure variations and intracellular accumulation of carbohydrates during photosynthesis (ballast effect). The ability of vertical migration that results would widely explain the dominance of some species in periodically stable environments (Reynolds and Walsby, 1975; Ibelings et al., 1991b; Klemer et al., 1996).

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Microcystis is the dominating genus in Grangent reservoir. Its cells are spherical or egg-shaped, with 3–9 ␮m diameter (Reynolds et al., 1981), and they are distributed in mucilage, organised in spherical colonies whose size influences the velocity of the vertical movements. Walsby (1994) presented an extended description of the gas vacuoles and their role in the ecology of Cyanobacteria. He relates that the resistance of these structures to the rise of osmotic pressure is different from one species to another (see also Walsby, 1971; Konopka et al., 1978, 1993; Walsby et al., 1983, 1991; Utkilen et al., 1985). In Microcystis sp., vesicles are too strong to be collapsed under natural conditions (Walsby, 1994): their flotation effect acts permanently. Thus, in Microcystis, the regulation mechanisms of cell density involve several antagonist processes: – Production of gas vacuoles as cell material that lightens the cells. – Storage of carbohydrates during the photosynthesis, with a ballast effect, because of the high molecular weight of these components. – Use of these reserves in the cell metabolism (production of cell material as proteins, lipids, RNA, DNA, . . . and energetic consumption linked to these transformations), lightening cells. In most of the species that present a specific growth rate inferior or equal to 1 d−1 , the cellular cycle is closely synchronised with the daily cycle (Chisholm and Costello, 1980; Chisholm, 1981; Vaulot et al., 1995; Vaulot and Marie, 1999). It was shown, in culture (Kromkamp et al., 1988) as well as in natural environment (Ibelings et al., 1991a), that the daily dynamics of carbohydrates is directly linked to the photosynthetic activity. Photosynthesis insures the fixation of the inorganic carbon dissolved in water, to feed the biosynthesis way. When the energy gained by photosynthetic antenna exceeds the one used in biosynthesis, the intermediate carbon photosynthates are in excess and stored as carbohydrates (Gibson, 1978). These dense components secondarily play the role of ballast (Thomas and Walsby, 1985, 1986; Van Rijn and Shilo, 1985; Kromkamp et al., 1988; Deacon and Walsby, 1990; Konopka et al., 1993). The intracellular carbon dynamics is thus fast and adjusted by the night/day alternation; it reflects changes in the cell growth cycle (Jacquet et al., 2001). We may consider that, at an hourly scale, the amount of gas vac-

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uoles in cells contributes to a reference density around which the fast carbon ballast fluctuations allow to obtain a fine adjustment of the density (Reynolds, 1987).

3. Modelled processes The growth model we designed for Microcystis is similar to the one developed by Th´ebault and Salenc¸on, (1993). It is based on a mass balance, and is equivalent to most of phytoplankton models currently developed (e.g. Salenc¸on and Th´ebault, 1996; Tusseau et al., 1997; Hakanson and Boulion, 2003; Canu et al., 2004; Robson and Hamilton, 2004). The particular feature in our model is the way we detailed photosynthesis. Instead of formulating it as a classical coupled function of light intensity (I) and temperature (T) (Salenc¸on and Th´ebault, 1996), it was dissociated into two distinct processes simulated through the intracellular carbon dynamics. Carbon is considered as an internal reserve; in this way, carbon uptake is similar to the Droop (1968, 1973) model. It is therefore possible to represent the photosynthesis mechanism at a finer scale, with the originality to point out the influence of each of these two factors upon the algae metabolism (Th´ebault and Rabouille, 2003). Besides, as we aimed at explicitly describing the vertical migration of colonies, and facing the informations obtained from the literature, we proposed a simulation of Microcystis behaviour through the dynamics of its intracellular carbon. One of the first models of buoyancy was developped by Okada and Aiba (1983a,b). More recently, Kromkamp and Walsby (1990) conceived a model based on a relation between the irradiance and the fluctuations of colony density. Howard et al. (1996) and Visser et al. (1997) took up this model and added the intracellular carbon dynamics as a connection process between the irradiance and the density fluctuations. The approach is based on a particle followup that does not allow the simulation of the whole biomass in a water body. Wallace and Hamilton (1999) have taken the same bases (Kromkamp and Walsby, 1990) to elaborate a simple model of buoyancy regulation, adapted to turbulent environments. They do not simulate the carbon dynamics but they introduce a lag time, between the fluctuation of light and the induced density changes, that represents the physiological adjustment of cells faced with light fluctuations.

The aim of this paper is to simulate the vertical distribution of the whole Microcystis population in time. Therefore, the intracellular carbon dynamics, through both mechanisms of photosynthesis and biosynthesis, is detailed as inspired from different sources: – Howard et al. (1996) and Visser et al. (1997) give a representation of a cell as different forms of carbon. – Visser et al. (1997) expose the two direct relations existing (i) between the irradiance and the carbohydrates and (ii) between carbohydrates and density. Thus, carbon reserves in cells are the link between the light regime and the density fluctuations. 3.1. Hypotheses and representation choices The system studied is a population of Microcystis sp. colonies interacting with their environment in a eutrophic lake. Field works conducted in Grangent associated with a literature review orientated the choice of dominant processes to be represented (Rabouille, 2002). Let us have six basic assumptions: (1) The simulated environment represents a lake, not a reservoir; in a first step, throughflow is not simulated. (2) Microcystis is not intensively grazed by zooplankton (Gliwicz, 1990; De Bernardi and Giussani, 1990; Blomqvist et al., 1994; Gregory, 1996). Predation is not represented. (3) The model simulates a homogeneous population composed of colonies with a constant diameter in time. The population increase is described in terms of biomass and colonies number. (4) In the model, colonies are represented by a fixed number of cells, i.e. the proportion of cells and mucilage is constant in time. In fact, the cells number fluctuates according to cells divisions, mortality and phenomena of aggregation and unbinding of colonies. The mucilage secretion also depends on the cells physiological state. Nevertheless, it would be hazardous to represent their fluctuation, especially because the regulating mechanisms of these processes are poorly known or uneasily quantifiable. (5) In this version of the model, proteins of gas vacuoles are considered as a constant fraction of cell material. They are not a state variable and their

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dynamics follows the biomass fluctuations. This simplifying assumption implies that the synthesis of gas vacuoles occurs with the same kinetic as for the whole cell material. The eventuality that gas vacuoles synthesis can be dependant on other anabolism processes is neglected here. Considering the estimations given by Post et al. (1985), Reynolds (1987), Kromkamp et al. (1988), Walsby (1994) and Visser (1995), a massive proportion of gas vacuoles should be comprised between 1.5 and 8% of the biomass, and the model used the maximum value, i.e. 8%. (6) Nutrients (P-PO4 3− , N-NO3 − , N-NH4 + ) are supposed non-limiting for growth. 3.2. Spatial structure of the model The model has a one-dimensional structure, essential to represent the migration of Microcystis. The water column is discretised with finite volumes, schematically representing a stack of horizontal layers with a constant thickness (dz = 0.25 m) whose volume and surface vary according to the depth, to represent the bathymetry of the lake bottom. The depth of the water column is 40 m. The deepest layer corresponds to the liquid layer of the sediments subsurface. Colonies sedimenting on the bottom accumulate there and can go up again into the water column. Cells in the last layer have the same metabolism as the other cells. 3.3. Forcing variables and state variables Hydrodynamic conditions (thermal profile, thermocline depth, depth of the well-mixed layer, irradiance and extinction coefficient) are introduced as forcing variables. They come from field data measured at Grangent station or simulation results from the vertical and thermal 1D model EOLE (Salenc¸on, 1997) applied to the Grangent reservoir. Colony diameter is also a forcing variable. The various combinations of forcing conditions applied to the system constitute the different scenarios, and will permit to test the model behaviour. Three state variables are represented: – The carbohydrate ballast: C (␮g C·L−1 ) represents the carbon fixed by photosynthesis, from the inorganic carbon dissolved in the water, and stored as carbohydrates. Carbon accumulated in this C com-

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partment constitutes (i) a reserve of molecules essential to the cell material synthesis and (ii) the energy required for this transformation. – The cell material M (␮g C·L−1 ) is the cell biomass without carbohydrate reserves and expressed as carbon. It is produced from the accumulated C reserves. The mucilage, which insures cells cohesion inside colonies, is included in this variable and its proportion remains constant in time (75% in the model). Gas vesicle proteins (Vesic (␮g C·L−1 )) are considered as a constant fraction Vp (Vp = 0.08) of cell material: Vesic = Vp ·M. – The dissolved carbon Cdis (␮g C·L−1 ): this variable, allowing the closure of the system, represents organic and mineral forms of dissolved carbon without any discrimination. In the present simulations, this variable is never limiting. According to this representation, the total amount of carbon contained in cells is represented by C + M. 4. Mathematical formulation of the fluxes Processes simulated by the model describe intracellular carbon dynamics, essential to adequately describe vertical movements. 4.1. Carbon fixation fcya is the rate of carbon fixation through photosynthesis (h−1 ), as a function of light intensity. This process regroups the photochemical phase of photosynthesis and the reduction of inorganic carbon dissolved in the water (Cdis ) as carbohydrates in cells. When the amount of fixed carbon exceeds that used in biosynthesis, excess carbon is stored; this first stage insures the filling of the C compartment (responsible for the ballast) and depends on incident light energy. As in many other species, photosynthesis inhibition has been observed in Microcystis (Ibelings et al., 1991b, 1994; Visser et al., 1997); the chosen function also takes into account this process through a lowering of the fixation rate under high light intensities (from Peeters and Eilers, 1978). There, the filling of the carbon reserve is supposed independent of temperature (Fig. 1a). Moreover, Reynolds (1990) emphasizes that under a sustained exposure to light, carbon can be fixed in excess, beyond the potentiality of carbohydrates accu-

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mulation in cells. This surplus is eliminated as CO2 through photorespiration and by excretion of intermediate products of photosynthesis, e.g. glycollates. A simplified process is considered here, with a maximum ability of carbon storage in cells as a function of a maximal threshold in their density, calculated from Visser’s data (1995). 4.2. Cell material synthesis The biosynthesis phase is the production of cell material (M) from carbon reserves (C) and nutrients. It is described through a carbon transfer Vm (h−1 ) from C to M and represents the growth activity. It does not require light and thus can occur night and day, whenever C reserves are available. This synthesis is influenced by temperature. The shape of the Parker (1974) limitation function was adjusted on data from the literature (Reynolds and Rogers, 1976; Thomas and Walsby, 1986; Visser, 1995) (Fig. 1b). 4.3. Respiration This process is a carbohydrates degradation that enables the generation of energy required for the

metabolism; it is expressed like a carbon loss as CO2 , represented by two different processes: – The energy required for the maintenance metabolism is obtained from C reserves as a priority; this carbon consumption is the basic respiration rate (Rb ) (h−1 ). It is proportional to the biomass M to be maintained and rises exponentially with temperature. – The energy required for the new cell material synthesis is also provided by the degradation of C reserves. This active respiration rate (Ra ) (h−1 ) depends on Vm . Thus, Ra only occurs during cell material synthesis. On the contrary, Rb is a permanent process that takes priority for survival and can lead to a lysis of the cell material if the C reserve is depleted. 4.4. Carbon excretion In healthy cells growing exponentially, excretion is about 0–10% of photosynthesis. It is represented in the model as a loss of cell material M, differentiating again two kinds of processes: a basic excretion (Eb ) and an activity excretion (Ea ) (Dauta, 1983) expressed in h−1 . As any metabolic function, the excretion depends on the environmental temperature. Following the assumptions, the excreted carbon feeds the dissolved carbon Cdis . 4.5. Colony migration

Fig. 1. Limitation coefficients: (a) for carbon fixation as a function of the irradiance I (Peeters & Eilers, 1978): lI and (b) for cell biosynthesis as a function of temperature T (Parker, 1974): lT .

Under favourable environmental conditions, the synthesis rate Vm is proportional to the amount of stored carbon. Environmental conditions modulate the Vm flux and finally determine the rate of the C reserve decrease. The ballast dynamics results from the unbalance between the two fluxes of carbon input (fixation) and use (Vm , Ra and Rb ). In a calm environment, the difference of density between water and Microcystis colonies governs their vertical movement. It seems relatively simple to apply the physical laws governing the particle movements in a fluid to Microcystis’ case, because of the spherical shape of colonies. On the other hand, the estimation of colony density presents a major difficulty as it depends on the cells metabolic history. To overcome this, the procedure followed that proposed by Howard et al. (1996) and carefully describes each step of this process.

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Table 1 System of differential equations in each layer ∂C ∂t

= fcya · M · (1 − Vp ) ·


 C   C  M max − M C M

max

− C · V m − Ra · C − R b · M ∂M ∂t

= C · V m − E a · C − Eb · M

∂Cdis ∂t

Fig. 2. Density of cells without vesicles and of colonies, for different C/M ratios calculated with an exponential function calibrated on Visser’s data (1995). Dots correspond to experimental data from this author, relative to cells without vesicles.

= −fcya · M · (1 − Vp ) ·


 C   C  M max − M C M

max

+ Ra · C + R b · M + E a · C + E b · M Vesic = M·Vp

ify the Stokes’ law applicability: – Calculation of the density of cells without vesicles, ρc = f(C/M) from Visser’s experimental data (1995). Because environmental conditions differ from those in laboratory, mainly the depth of the water column, values of C/M calculated with the model often overstep the range of values measured in the laboratory. In order to extend the validity domain of the experimental function proposed by Visser (1995), an exponential function was adjusted on Visser’s data, allowing to take into account a larger range of C/M values (Fig. 2). – Calculation of the density of cells with vesicles, ρcel , assuming a proportion Vp of vesicles in each cell, whose density is ρves = 150 kg·m−3 in Microcystis (from Reynolds, 1987). ρcel = (1 − Vp )ρc + Vp ρves – Calculation of the density of Microcystis colonies, ρcol , assuming that the proportion of cells in colony volume is Xcel = 0.25 and mucilage density is ρmuc = ρw + 0.7 (Reynolds et al., 1981), where ρw is the water density. Hence: ρcol = (1 − Xcel )ρmuc + Xcel · ρcel The velocity of the vertical migration Vs (m·s−1 ) is calculated with Stokes’ law (Reynolds, 1987), that takes into account the colony size (Vcol , volume in m3 ; R, radius in m) and their density ρcol . This law is applicable to Microcystis colonies which can be considered as spheres. As a convention, positive velocities match with sedimentation and negative velocities with flotation. Reynolds’ number Re, the ratio between viscosity and inertia forces, is calculated for each particle to ver-

Re =

ρw Vs 2R η

where η is the dynamic viscosity of the water, which depends on the water temperature. McNown and Malaika (1950) showed very little departure between measured and calculated velocity for Re values <0.1; this error remains <10% when Re <0.5. Reynolds (1984) relates that Re values calculated for many planktonic algae indicate that Stokes’ law can be applied to calculate the migration velocity. All those conditions are met in the model. The maximum velocity calculated was 1.5 × 10−4 m·s−1 , maximum Re was then 0.045 with a simulated diameter of 300 ␮m. A system of differential equations is used to describe the variations of state variables in each layer of the water column (Table 1). The mathematical formulation of the processes is given in Table 2 and the model parameters in Table 3.

5. Numerical resolution The modelling technique applies the fractional steps method that consists in calculating separately the uncoupled processes (Salenc¸on and Th´ebault, 1997). Two fractional steps are considered: – The biological system is integrated by the fourth order Runge and Kutta method (Legras, 1971). The selected integration step (0.25 h) allows an accurate description at the phenomena scale (fluctuations of intracellular carbon), for a reasonable simulation time over one year. One could notice a strong cou-

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Table 2 Modelled processes Microcystis metabolism Growth fcya , carbon fixation rate by photosynthesis (h−1 ) Vm , carbon transfer rate from C to M (h−1 )

fcya = fmax ·lI Vm = Vmax ·lT

Respiration Rb , basic respiration rate (h−1 ) Ra , activity respiration rate (h−1 ): carbon respiration rate by unit of newly synthesised M

Ra = ra ·Vm

Excretion Eb , basic excretion rate (h−1 ) Ea , activity excretion rate (h−1 )

Eb = eb · rb exp(T) Ea = eb ·Vmax exp(T)

Coefficients lI , light limitation coefficient of C fixation (from Peeters and Eilers’ equation, 1978) lT , temperature limitation coefficient of the transfer from C to M (Vm ) (Parker equation, 1974) exp(T), exponential function of temperature for Ea , Eb and Rb calculation Vertical migration Water density at the temperature T (kg·m−3 ) Water dynamic viscosity as a function of temperature (Pa·s)

lI = 2(l + βI )xi/(xi2 + 2 xi βI + 1) with xi = I/Iopt lT = ((T/θ opt )·(xU ))A with x = (T − θ l )/(θ opt − θ l ) U = (θ l − θ opt )/θ opt exp(T) = αe exp(βe ·(T − θ opt + θ r ) −6 2 ␳w = ρ0 ·(l +
δ) with δ = −6.63·10  (T − 4)



η = 10

−3

·

10

262 −1.65+ T +139



Mucilage density in a layer i (kg·m−3 ) Cells density in a layer i (kg·m−3 ) Colonies density in a layer i (kg·m−3 )

ρc = ρmin + (ρmax − ρmin ) · (1 − e−k·CMi ) with CMi = C/M in the ith layer ρmuc = ρw + 0.7 ρcel = (1 − Vp )·ρc + Vp ·ρvac ρcol = (1 − Xcel ).ρmuc + Xcel · ρcel

Migration velocity (m·h−1 , Stokes’ law)

VS =

Density of cells without vesicles in a layer i

(kg·m−3 )

Rb = rb exp(T)

pling between the physical forcing and biological processes, characterising a non-autonomous system. The physical forcing was thus linearised on the calculation time step of the biological system in order not to introduce some discontinuities that a nonautonomous system could amplify. – Colony migration is solved in an explicit scheme with a controlled time step. The carbon budget is checked throughout the simulation to ensure there is no drift. 6. Simulations First of all, focus was given on the determinism of colony sedimentation and flotation towards the surface. A first model analysis under permanent conditions of light and temperature in a non-turbulent water column was presented in Rabouille et al. (2003). This step per-

2·g·R2 ·(ρcol −ρw ) 9·ϕ·η

· 3600

mitted to separately investigate the influence of light and temperature on the migration behaviour of colonies with two different diameters. Results obtained highlighted a more pronounced migratory response in small colonies (150 ␮m) than in big ones (300 ␮m) with low temperatures. Here was explored, under the same permanent conditions, the effect of some other diameters (600, 300, 200 and 150 ␮m) associated with a wider range of temperatures (T = 20, 12, 8, 7, 5 and 0 ◦ C). Aim was at (i) pointing out the state equilibrium reached by the model under low temperatures and (ii) more subtly analysing of the influence of the colony diameter. Although some of these conditions are unrealistic from a biological point of view (e.g. a water temperature set to 0), they are very important from a mathematical point of view since they are necessary to test the behaviour of the model. Afterwards, the colony migration was simulated under a constant 20 ◦ C temperature and light forcing

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Table 3 Model parameters Parameter Definition

Unit

Known range

Genus or taxa and references

Cyanobacteria parameters CHLC Chlorophyl a:carbon ratio βI Shape factor for the function f = f(I) fmax Maximum carbon fixation rate

␮g ch1a/␮g C 0.02 – −0.01 h−1 0.114

0.02

Reynolds (1990)

0.104

Microcystis—Reynolds (1990), from Robarts and Zohary (1987)

28

25–35

◦C – – h−1

Cyanobacteria—Reynolds Walsby (1975)

35 4 Variable 0.05

0.0198

Microcystis—Reynolds (1990)

–

0.2

0.2

Phytoplankton—Langdon (1988)

h−1

0.004

0.001

Oscillatoria, maintenance constant—Van Liere and Mur (1979), cited by Post et al. (1986)

Proportion of carbon lost by excretion Coefficient in the exponential function exp(T) Coefficient in the exponential function exp(T) Temperature reference in the exponential function exp(T) Proportion of cells in the colony volume (1 − Xcel = % mucilage)

– – h−1 ◦C

0.1 0.286 0.05 25

%

25

19

Microcystis—Howard et al. (1996)

Percentage of gas vacuoles in cells, expresed as a massic fraction of M

%

8

<26 2–10

Microcystis—Reynolds et al. (1987) Cyanobacteria, % volume of the total proteins—Reynolds (1987) Microcystis, % mass of the total proteins—Kromkamp et al. (1988)

Instantaneous optimal intensity for W·m−2 photosynthesis (expressed as total spectrum) Parker equation (biomass M synthesis and activity respiration) ◦C θ opt Optimal growth temperature Iopt

θl A U Vmax

Lethal maximum temperature of growth Shape factor in Parker equation Coefficient in Parker equation Maximum carbon transfer Vm from C to M (=biosynthesis) Respiration and excretion ra Respired carbon by unit of newly synthesised carbon rb Basic respiration rate at 0 ◦ C

eb αe βe θr Xcel

Vp

Used value

250

up to 14 Light extinction CKe Extinction coefficient due to the water and suspended material Sedimentation parameters ρvac Gas vacuoles density Water density at 4 ◦ C ρ0 ρmin Minimum density of cells without gas vacuoles Maximum density of cells without gas vacuoles ρmax k Coefficient in the density equation ρc

m−1

0.92*

kg·m−3 kg·m−3 kg·m−3 kg·m−3 –

150 999.973 1037 1150 0.7

Modified Stokes’ equation ϕ Friction resistance due to the colony shape g Gravitational acceleration

– m·s−2

1.0 9.81

*

If the light extinction is constant. CKe = 0.92 corresponds to a euphotic zone Zeu = 5 m.

and

150

Microcystis—Reynolds (1987)

1065

Microcystis—from Visser’s (1995) Reynolds et al. (1987)

From Reynolds et al. (1987)

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conditions representative of the daily cycle as a sine curve. This scenario is called S0 . Photoperiod is 16/8 with a maximum irradiance of 650 W·m−2 at 12 a.m. (expressed as total spectrum). These simulations are run on colonies with different diameters, from 80 to 1100 ␮m. In all cases, the thermal profile is supposed homogeneous. 7. Results and discussion 7.1. Constant forcing Results obtained under constant forcing conditions are synthesised on Fig. 3, which displays, on the vertical axis, the amplitude of colony migration observed after 300 days of simulation under a constant 650 W·m−2 irradiance. X-coordinates indicate the different conditions of temperature we applied. Individually, each factor shows the following effects: - Temperature impacts the biosynthesis since the use of carbon reserves depends on this factor, leading to a rise in the C/M ratio under low temperatures and thus to a colony sedimentation as deep as T is low.

Fig. 3. Synthetic representation of simulations run over 300 days under a constant irradiance (650 W·m−2 ), for different constant conditions of T and colony diameter: comparison between the final state equilibrium reached by big colonies (500, 300 and 200 ␮m) and the behaviour of small colonies (150, 75 ␮m).

- Available light impacts the amount of carbon accumulated in cells during photosynthesis. This factor strongly affects fluctuations of colony density: colonies become heavier and tend to sink when exposed to a sustained irradiance. Sedimentation depth and velocity are also linked to the intensity of the irradiance they receive. - The size of the diameter impacts their displacement velocity and acts on the time they will spend in the lighted zone. Under a permanent regime of light I and temperature T, the model reaches a different state equilibrium according to temperature ranges: colony stabilisation at a constant depth depending on T when temperature is comprised between 20 and 8 ◦ C, and permanent oscillation below the euphotic zone when the temperature stands below 8 ◦ C. This value appears as a threshold for the migratory behaviour only; however, it does not correspond to any discontinuity in the physiological behaviour. Indeed, above 8 ◦ C the stabilisation in depth is possible because the rates of carbon fixation and carbon incorporation in the biomass are of the same magnitude. Below 8 ◦ C, the biosynthesis slows down so that oscillations appear. The migratory response of colonies (characterised by an amplitude–period pair of oscillations) then depends on the colony diameter: the smaller they are, the deeper they sink. Nevertheless, colonies with a diameter smaller than 190 ␮m are still in a transitory regime after 300 days of simulation; and when T is below 5 ◦ C they settle on the bottom of the lake since the amplitude of their oscillations is, in this case, greater than 40 m. From a biological point of view, these extreme cases are not realistic anymore; however, they were simulated to test the model behaviour. The permanent regime of the model can be reached for small colonies, but it requires a longer simulation as well as a deeper water column; a biomass re-sowing also becomes necessary. As an example, the state equilibrium reached for 150 ␮m colonies at 5 ◦ C corresponds to oscillations down to 22.25 m deep after 641 days. As previously argued (Rabouille et al., 2003), these results are supported by observations related in the literature; they strengthen the idea that temperature lowering is partly responsible for colonies sedimentation and bloom dispersal in autumn.

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Fig. 4. (a) Temporal variations of the density and (b) migration velocities of 150 and 300 ␮m colonies submitted to a periodical light regime (S0 ).

The situation where colonies settle at the bottom results from an amplification of the migratory response under low temperatures, and with a long migration period (>300 days). It is similar to that observed during

some winter field situations when “benthic” colonies spend the season lying on the bottom before they can float back up into the water column during the following spring.

Fig. 5. Vertical migration of 160, 165 and 175 ␮m colonies submitted to a periodical light regime (S0 ). The carbon fixation rate observed at the peak maximum is drawn in parallel. Iopt = optimal light intensity for growth. Zeu = limit of the euphotic zone (5 m).

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7.2. Periodical forcing Afterwards, a periodical light forcing was applied, which is a first step towards the simulation of more realistic environmental conditions. Under the environmental conditions of scenario S0 , the migratory behaviour of colonies with different sizes, from 80 to 1100 ␮m was simulated. The irradiance rhythm generates a periodical migration since the colony velocity depends on this factor (light intensity and extinction coefficient). The colony diameter thus plays a leading role, as it participates in the calculation of the migration velocity (calculation with R2 ). Fig. 4 illustrates that under the same environmental conditions and compared to big colonies, small colonies present slower displacements with reduced amplitude. As the metabolism was not connected to the colony size, small and large colonies present the same biological functioning. However, their displacement velocity is different, leading to a feedback effect on the metabolism, through the different conditions they are faced with during the migration. The acceleration of colonies with increasing diameters is analysed; it leads to a series of migratory behaviours of great interest (Figs. 5–9). 80 ␮m colonies are too slow to migrate and stay at 3 m deep. When diameter increases (up to 160 ␮m), they begin to regularly migrate, while remaining in the euphotic zone. Since colonies do not leave the euphotic zone, they are continuously subject to daily light fluctuations and can fix carbon for the whole light period; they thus present a 24 h periodicity. The buoyancy phase starts at night and is slowed down when the biomass peak catches the light, up to the uppest migration position, reached around 12 a.m. (Fig. 5). The amplitude of the migration is getting wider with the diameter, as well as the synthesised biomass. From 165 ␮m, a bimodal mode appears that makes connection with a 48 h periodicity. As the sedimentation time is getting longer with the diameter, the upper most point of migration is not identical from one period to the next. One time out of two, colonies float near from the surface (−2.5 m), whereas they remain a bit farther to it (−3.25 m) during the following period. From the upper most point (−2.5 m), colonies are stuffed a lot with carbon. The longer descent phase thus initiated postpones the moment when colonies get buoyant till the first hours of the following day (3 h). Colonies in

Fig. 6. Vertical migration of colonies with 200, 220 and 250 ␮m diameters submitted to a periodical light regime (S0 ). Zeu = limit of the euphotic zone (5 m).

the buoyant phase still remain in the euphotic zone; they begin to fix carbon as soon as the sun rises, what stops buoyancy at a deeper point (−3.25 m). A timid sedimentation is initiated since colonies remained deep in Zeu and thus stored a few carbon; for so they float up earlier in the evening (10 p.m.) and thus can reach the uppest point in the surface layers (−2.5 m) (Fig. 5). From 165 to 170 ␮m, the “small ascent” of the bimodal mode reduces and disappears, while the upper most point of migration comes closer to the surface

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Fig. 7. Under S0 conditions: (a) Time (h) that colonies spend at the minimal depth of the migration at night, and in the euphotic zone (Zeu ) during the light period. Each duration is calculated over 48 h. (b) Time (h) spent in the euphotic zone (Zeu ) according to the colony size. The two coloured bands represent the length of the light period. (c) Total biomass synthesised at the end of the simulation and amplitude of the migration (materialised by the minimum and maximum displacement depths), displayed as a function of the diameter. Zeu = limit of the euphotic zone (5 m).

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and is shifted towards the early morning, generating a 48 h periodicity (Fig. 5). The 200 ␮m diameter is particular since the vertical position where the buoyant phase sets is situated at the bottom of the euphotic zone and the position where sedimentation begins is located under the surface. Above this diameter, colonies move down below the euphotic zone, where fall in obscurity. Then another bimodal behaviour appears (220 ␮m); colonies reach the surface one period out of two and stay in the optimal zone to fix carbon (Fig. 6). They sink below the euphotic zone in 13 h and stay there the following day, until 6 p.m. The meaning of obscure episode for the colony is highlighted here: night or deep descent. They float up towards the surface at night. The more the diameter increases, the faster they move, the deeper they sink and come back earlier to the surface. Such behaviour stabilises the migrations amplitude and allows synchronization between the migratory and light cycles with a 48-h period, since algae located under the euphotic zone only receive one light phase out of two, given the process slowness. With increasing diameters, colonies spend less and less time in Zeu during the descent phase (14 h in 250 ␮m colonies, 11 h in 300 ␮m colonies and 10 h in 400 ␮m colonies): they all begin to sink at the sunrise, while their velocity increases. They come back in Zeu at night and have to wait for the fol-

lowing sunrise to fix carbon (Fig. 7b). They globally fix much less carbon and produce much less biomass. The maximum biomass, with the 48-h period, is obtained for 250 ␮m colonies, without standing at the surface at night, a situation that appears optimal under these environmental conditions (Fig. 7c). From 250 to 350 ␮m, the time spent at the surface at night increases (6 h in 350 ␮m colonies) and the biomass steeply drops (Fig. 7a and b). From 350 ␮m, colonies float back in Zeu at the end of the afternoon; then, the few carbon fixed leads to a small biomass increase. The 405 ␮m diameter marks the return to a 24 h period of migration because colonies move fast enough to come back up in the lighted zone within 24 h. Obviously, this transition is characterised by a discontinuity in the amplitude that changes from (0; −10.5 m) for 400 ␮m colonies, into (2; −6.25 m) for 405 ␮m: colonies do not float up to the surface (Fig. 8). From this diameter, the 24-h period is remaining and the oscillations amplify ((0; −11.75 m) for the 1000 ␮m colonies) (Fig. 9). According to the same process as previously described, the biomass increases until the uppest migration point reaches the surface (optimum situation for 650 ␮m colonies), then lowers when the biomass peak stands at the surface at night (Fig. 7a and b).

Fig. 8. Vertical migration of 400 and 405 ␮m colonies submitted to a periodical light regime (S0 ). The carbon fixation rate observed at the peak maximum is drawn in parallel. Iopt = optimal light intensity for growth. Zeu = limit of the euphotic zone (5 m).

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the biomass production is low. Contrary to what could be expected, the transition to a 48 h periodicity is not a disadvantage in terms of biomass production; the dark episode below Zeu is balanced by a better exploitation of the surface layers. However, the maximum biomass is observed when the periodicity comes back to 24 h, for the 650 ␮m colonies. Here, the model points out the importance of reserves functioning, from an ecological point of view. In algae capable to store nutrients, as here carbon, the growth will be maintained during periods when environmental conditions are unfavourable, as here an obscure period (Fig. 10). The model also emphasises the importance of the migration for the access to surface layers where light is not limiting. 7.3. Does the colony size have a meaning at the ecological level?

Fig. 9. Vertical migration of 500, 650 and 1000 ␮m colonies submitted to a periodical light regime (S0 ). The carbon fixation rate observed at the peak maximum is drawn in parallel. Iopt = optimal light intensity for growth. Zeu = limit of the euphotic zone.

The model displays a range of various migratory behaviours resulting from the coupling between different conditions of light, extinction coefficient and colony size (Fig. 7b and c). The combination incident light/euphotic zone/temperature constitutes a habitat that colonies exploit differently according to their size. In colonies with a small diameter, whose 24 h migrations are enclosed in Zeu without reaching the surface,

The simulation results incline us to favourably respond. A large diameter makes colonies more reactive to the light conditions, which is also simulated by Visser (1995); ability to “escape” in depth and to fastly float back to the surface. On the other hand, a very large diameter is not necessarily advantageous in terms of growth, because of the strongest reactivity of the colony; the least increase in density, consequent to carbon fixation, is promptly turned into a fast sedimentation that penalises carbon fixation. This is a self-regulating mechanism. The increase in velocity with the diameter induces, in the model, two migration periodicities, 24 and 48 h. Changes in the produced biomass result from the same process in both cases. This process can be described as three successive phases: (i) migration with a narrow amplitude, that do not reach the surface; the biomass is increasing, (ii) migrations that reach the surface, with the highest biomass and (iii) colonies pass faster and faster through Zeu , stay at the surface at night, and thus show a lowering of the synthesised biomass. For the 48 h periodicity, the maximum biomass is obtained for 250 ␮m colonies. It is inferior to the biomass produced for the 24 h periodicity with 650 ␮m colonies. Indeed, the time spent in Zeu during the day in 48 h is shorter for 250 ␮m colonies than for 650 ␮m colonies (respectively, 15 and 16 h), and the 650 ␮m colonies exploit more efficiently the surface light intensity as they begin their descent at the sunrise Fig. 9.

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Fig. 10. Scenario S0 : carbon flux in cells located in the maximum biomass peak and migration depth of 300 ␮m colonies. Zeu = limit of the euphotic zone (5 m). The coloured band materialises the light period.

Beyond 650 ␮m, the biomass slowly decreases with increasing diameters and all the big-sized colonies produce a quite similar biomass, showing the ability to maintain a minimal amount of biomass under quite unfavourable conditions. The high displacement velocities of the largest colonies do not appear as an advantage in the access to light, but can prove to be one in the access to the nutrient-rich hypolimnion. Our simulations show that in a stable environment, the migration is an advantage in terms of biomass production as soon as it can occur. The almost nonmotility of small colonies (<165 ␮m) does not allow them to take advantage of the migration to optimise the exploitation of favourable zones. This regime is not disadvantageous in the early spring, when turbulent motions carry away the phytoplankton in the light gradient. The advantage here lies in the ability of algae to physiologically adapt, at best, to the rapid changing environment. The 48 h periodicity obtained for mean diameters (up to 250 ␮m) allows to considerably increase the produced biomass with an amplitude of migrations inferior to 7.5 m. These migration scales correspond to spring conditions of stratification (intermittent thermal stability) when nutrients are still abundant in surface. From 650 ␮m, the maintenance of a biomass with migrations down to −11.75 m is better adapted to sum-

mer conditions (stratified water column); the access to hypolimnetic rich layers is made easier by large displacements (Fig. 9). A great size also seems useful in summer, to avoid the alteration of photosynthetic pigments (photo-inhibition) that occurs in cells close to the surface. The colony diameter thus could be an adaptive feature of algae faced with environmental conditions such as the amount of available light and/or nutrients. It is worth noting that ecological advantages linked to the diameter and underlined by the model correspond to the seasonal changes in the colony diameter of Microcystis, observed in situ in Grangent reservoir. This diameter ranges from ≈50 ␮m (or less) in winter to a size about the millimeter in summer (Duris-Latour, 2002).

8. Conclusion A biological model was elaborated, one-dimensional and vertical. Its implementation is motivated by the need to conduct a prospective approach of the ecology of the colonial Cyanobacterium Microcystis sp., which emphasises the separate influence of physical factors on the colony dynamics. In the model, the representation of each cell by means of two different state variables (the “energy reserve” carbon and

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the “metabolised” carbon) allows us to simulate the dynamics of the carbon reserves and also to uncouple the effect of light from the influence of temperature on the metabolism. Results show that, through the metabolism, light and temperature have opposed effects on the migration dynamics of Microcystis colonies. Light enhances the ballast effect of carbon reserves whereas temperature regulates the biosynthesis and makes colony lighter. According to the combination of these two factors, the migratory response of colonies shows a strong repercussion on their growth, as it permits or not the occupation of surface layers and a station in the lighted zone. Under a permanent regime of light I and temperature T, a different state equilibrium is reached according to temperature ranges; colonies stabilisation at a constant depth depending on T when 20 ◦ C ≥ T ≥ 8 ◦ C, regular oscillation under the euphotic zone when T <8 ◦ C. Above 8 ◦ C, rates of carbon fixation and incorporation into the biomass are of the same order of magnitude, whereas below 8 ◦ C, the synthesis slows down to trigger the colonies oscillations. The migratory response of colonies (characterised by an “amplitude–period” pair of oscillations) then depends on the colony diameter: the smaller they are, the deeper they sink and can deposit on the bottom of the lake. The application of a periodical light regime initiates a daily dynamics in Microcystis metabolism. The model then calculates an oscillatory migration dynamics where colonies periodically float back to the surface. Amplitude and period depend on the colony diameter. The combination incident light/euphotic zone/temperature constitutes a habitat that colonies exploit differently according to their size. Under the conditions of our simulations and beyond 200 ␮m, the vertical amplitude of this migration carries out the colonies below the euphotic zone, highlighting the concept of obscure episode for the alga (night or deep descent). In these scenarios, the model points out the synchronicity between migratory and light cycles. The buoyancy episode at the end of the day and the stay in surface at night trigger the descent the following morning, what constitutes an adjustment according to the solar rhythm and accounts for the reproducibility of this migratory dynamics with a period equal to a multiple of the daily cycle (24 or 48 h). A maximum biomass is reached for each periodicity; with the 250 ␮m colonies for the 48 h and 650 ␮m colonies for the 24 h. The tran-

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sition to a 24-h period for diameters bigger than 400 ␮m allows, because of an increased velocity, to maintain an important biomass with deeper migrations, in spite of a longer dark episode. This migration thus appears to result from the crossing between growth process and the light forcing. The migratory phenomenon simulated (via the variations of colony density in the water column) in Microcystis colonies is consistent with laboratory and in situ observations. Last of all, the colony size, which physically modifies the migration velocity, is ought to constitute an adaptive ecological factor at the seasonal scale. Acknowledgements This program was carried out in the context of a research program between Electricite De France, CESAC (Centre d’Ecologie des Systemes Aquatiques Continentaux, Universite Paul Sabatier, Toulouse) and the Laboratoire de Biologie Animale et Appliquee (Universite Jean Monnet, Saint-Etienne), and supported by Agence de l’eau Loire-Bretagne. References Blomqvist, P., Pettersson, A., Hyenstrand, P., 1994. Ammoniumnitrogen: a key regulatory factor causing dominance of nonnitrogen-fixing Cyanobacteria in aquatic systems. Arch. Hydrobiol. 132 (2), 141–164. Canu, D.M., Solidoro, C., Umgiesser, G., 2004. Erratum to “Modelling the responses of the Lagoon of Venice ecosystem to variations in physical forcings (vol. 170, p. 265, 2003)”. Ecol. Model. 175 (2), 197–216. Chisholm, S.W., 1981. Temporal patterns of cell division in unicellular algae. In: Platt, T. (Ed.), Physiological Bases of Phytoplankton Ecology, vol. 210. Can. Bull. Fish Aquat. Sci., pp. 150–181. Chisholm, S.W., Costello, J.C., 1980. Influence of environmental factors and population composition on the timing of cell division in Thalassiosira Fluviatilis (Bacillariophyceae) grown on light/dark cycles. J. Phycol. 16, 375–383. Dauta, A., 1983. Conditions de d´eveloppement du phytoplankton. Etude comparative du comportement de huit esp`eces en culture. Cin´etiques d’assimilation et de croissance: e´ tude exp´erimentale. Mod´elisation appliqu´ee a` un milieu naturel: le Lot. Ph.D. Thesis, Universit´e Paul Sabatier, Toulouse, France, 166 pp. Deacon, C., Walsby, A.E., 1990. Gas vesicle formation in the dark, and in light of different irradiances, by the cyanobacterium Microcystis sp. Br. Phycol. J. 25, 133–139. De Bernardi, R., Giussani, G., 1990. Are blue–green algae a suitable food for zooplankton ? An overview. Hydrobiologia 200/201, 21–41.

402

S. Rabouille et al. / Ecological Modelling 188 (2005) 386–403

Droop, M.R., 1968. Vitamin B12 and marine ecology. IV. The kinetics of uptake, growth, and inhibition in Monochrysis lutheri. J. Mar. Biol. Assoc. U.K. 48, 689–733. Droop, M.R., 1973. Some thoughts on nutrient limitation in algae. J. Phycol. 9, 264–272. Duris-Latour D., 2002. Vie planctonique et vie benthique de la Cyanobact´erie Microcystis aeruginosa sur la retenue de Grangent. Ph.D. Thesis, Universit´e Jean Monnet, St. Etienne, France, 193 pp. Gibson, C.E., 1978. Field and laboratory observations on the temporal and spatial variation of the carbohydrate content in planktonic blue–green algae in Lough Neagh, Northern Ireland. J. Ecol. 66, 97–115. Gliwicz, Z.M., 1990. Why do Cladocerans fail to control alga blooms? Hydrobiologia 200/201, 83–97. Gregory, T., 1996. Grazing on filamentous Cyanobacteria by Daphnia pulicaria. Limnol. Oceanogr. 3, 560–567. Hakanson, L., Boulion, V.V., 2003. A general dynamic model to predict biomass and) production of phytoplankton in lakes. Ecol. Model. 165 (2–3), 285–301. Howard, A., Irish, A.E., Reynolds, C.S., 1996. A new simulation of cyanobacterial underwater movement (SCUM’96). J. Plankton Res. 18 (8), 1375–1385. Ibelings, B.W., Mur, L.R., Walsby, A.E., 1991a. Diurnal changes in buoyancy and vertical distribution in populations of Microcystis in two shallow lakes. J. Plankton Res. 13 (2), 419–436. Ibelings, B.W., Mur, L.R., Kinsman, R., Walsby, A.E., 1991b. Microcystis changes its buoyancy in response to the average irradiance in the surface mixed layer. Arch. Hydrobiol. 120 (4), 385– 401. Ibelings, B.W., Kroon, B.M.A., Mur, L.R., 1994. Acclimation of photosystem II in a Cyanobacterium and a eukaryotic green alga to high and fluctuating photosynthetic photon flux densities, simulating light regimes induced by mixing in lakes. New Phytol. 128, 407–424. Jacquet, S., Partensky, F., Lennon, J.-F., Vaulot, D., 2001. Diel patterns of growth and division in marine picoplankton in culture. J. Phycol. 37, 357–369. Klemer, A.R., Cullen, J.J., Mageau, M.T., Hanson, K.M., Sundell, R.A., 1996. Cyanobacterial buoyancy regulation: the paradoxical roles of carbon. J. Phycol. 32, 47–53. Konopka, A., Brock, T.D., Walsby, A.E., 1978. Buoyancy regulation by planktonic blue–green algae in lake Mendota, Wisconsin. Arch. Hydrobiol. 83 (4), 524–537. Konopka, A.E., Klemer, A.R., Walsby, A.E., Ibelings, B.W., 1993. Effects of macronutrients upon buoyancy regulation by metalimnetic Oscillatoria agardhii in Deming lake, Minnesota. J. Plankton Res. 15 (9), 1019–1034. Kromkamp, J., Walsby, A.E., 1990. A computer model of buoyancy and vertical migration in Cyanobacteria. J. Plankton Res. 12 (1), 161–183. Kromkamp, J.C., Konopka, A., Mur, L.R., 1988. Buoyancy regulation in light-limited continuous cultures of Microcystis aeruginosa. J. Plankton Res. 10 (2), 171–183. Langdon, C., 1988. On the causes of interspecific differences in the growth–irradiance relationship for phytoplankton. II. A general review. J. Plankton Res. 10 (6), 1291–1312.

Legras, J., 1971. M´ethodes et Techniques de Fanalyse Num´erique. Dunod, Paris, 323 pp. McNown, J.S., Malaika, J., 1950. Effect of particle shape on settling velocity at low Reynolds numbers. Trans. Am. Geophys. Union 31, 74–82. Okada, M., Aiba, S., 1983a. Simulation of waterbloom in a eutrophic lake. II. Reassesment of buoyancy, gas vacuole and tugor pressure of Microcystis aeruginosa. Water Res. 17, 877–882. Okada, M., Aiba, S., 1983b. Simulation of waterbloom in a eutrophic lake. III. Modelling the vertical migration and growth of Microcystis aeruginosa. Water Res. 17, 883–893. Parker, A., 1974. Empirical functions relating metabolic processes in aquatic systems to environmental variables. J. Fish Res. Bd. Can. 31, 1550–1552. Peeters, J.C.H., Eilers, P.H.C., 1978. The relationship between light intensity and photosynthesis. A simple mathematical model. Hydrobiol. Bull. 12, 134–136. Post, A.F., Loogman, J.G., Mur, L.R., 1985. Regulation of growth and photosynthesis by Oscillatoria agardhii grown with a light/dark cycle. FEMS Microbiol. Ecol. 31, 97–102. Post, A.F., Loogman, J.G., Mur, L.R., 1986. Photosynthesis, carbon flows and growth of Oscillatoria agardhii Gomont in environments with a periodic supply of light. J. Gen. Microbiol. 132, 2129–2136. Rabouille, S., 2002. Mod´elisation de la dynamique des r´eserves carbon´ees chez Microcystis et de son influence sur la migration verticale—simulation d’une population sur un cycle annuel. Ph.D. Thesis, Universit´e Paul Sabatier Toulouse III, France, 157 pp. Rabouille, S., Th´ebault, J.-M., Salenc¸on, M.-J., 2003. Simulation of carbon reserve dynamics in Microcystis and its influence on vertical migration with YOYO model. C.R. Biol. 326 (4), 349–361. Rabouille, S., Salenc¸on, M.-J., 2005. Vertical migration of the Cyanobacterium Microcystis in stratified lakes analysed with YOYO model. II—Influence of mixing, thermal stratification and colony diameter on the biomass production. Aquat. Microb. Ecol. 39, 281–292. Reynolds, C.S., 1984. The Ecology of Freshwater Phytoplankton. Cambridge University Press, Cambridge. Reynolds, C.S., 1987. Cyanobacterial water blooms. Adv. Bot. Res. 13, 67–143. Reynolds, C.S., 1990. Temporal scales of variability in pelagic environments and the response of phytoplankton. Freshwater Biol. 23, 25–53. Reynolds, C.S., Walsby, A.E., 1975. Water-blooms. Biol. Rev. 50, 437–481. Reynolds, C.S., Rogers, D.A., 1976. Seasonal variations in the vertical distribution and buoyancy of Microcystis aeruginosa K¨utz, Emend. Elenkin in Rostherne Mere, England. Hydrobiologia 48, 17–23. Reynolds, C.S., Jaworski, G.H.M., Cmiech, H.A., Leedale, G.F., 1981. On the annual cycle of the blue-green alga Microcystis aeruginosa Kutz, Emend, Elenkin. Philos. Trans. R. Soc. Ser. B 293, 419–477. Reynolds, C.S., Oliver, R.L., Walsby, A.E., 1987. Cyanobacterial dominance: the role of buoyancy regulation in dynamic lake environments. N.Z. J. Mar. Freshwater Res. 21 (3), 379–390.

S. Rabouille et al. / Ecological Modelling 188 (2005) 386–403 Robarts, R.D., Zohary, T., 1987. Temperature effects on photosynthetic capacity, respiration, and growth rates of bloom-forming cyanobacteria. N.Z. J. Mar. Freshwater Res. 21, 391–399. Robson, B.J., Hamilton, D.P., 2004. Three-dimensional modelling of a Microcystis bloom event in the Swan River estuary, Western Australia. Ecol. Model. 174 (1–2), 203–222. Salenc¸on, M.-J., 1997. Study of the thermal dynamics of two dammed lakes (Pareloup and Rochebut, France), using the EOLE model. Ecol. Model. 104, 15–38. Salenc¸on, M.-J., Th´ebault, J.-M., 1996. Simulation model of a mesotrophic reservoir (Lac de Pareloup): MELODIA, an ecosystem reservoir management model. Ecol. Model. 84, 163–187. Salenc¸on, M.-J., Th´ebault, J.-M., 1997. Modelisation d’ecosysteme lacustre. Application a la retenue de Pareloup (Aveyron). Masson, Paris, 200 pp. Th´ebault, J.-M., Salenc¸on, M.-J., 1993. Simulation model of a mesotrophic reservoir (Lac de Pareloup): biological model. Ecol. Model. 65, 1–30. Th´ebault, J.-M., Rabouille, S., 2003. Comparison between two mathematical formulations of the phytoplankton specific growth rate as a function of light ant temperature, in two simulation model (ASTER and YOYO). Ecol. Model. 163, 145–151. Thomas, R.H., Walsby, A.E., 1985. Buoyancy regulation in a strain of Microcystis. J. Gen. Microbiol. 131, 799–809. Thomas, R.H., Walsby, A.E., 1986. The effect of temperature on recovery of buoyancy by Microcystis. J. Gen. Microbiol. 132, 1665–1672. Tusseau, M.H., Lancelot, C., Martin, J.M., Tassin, B., 1997. 1-D coupled physical–biological model of the northwestern Mediterranean Sea. Deep Sea Res. P. II 44 (3–4), 851– 880.

403

Utkilen, H.C., Oliver, R.L., Walsby, A.E., 1985. Buoyancy regulation in a red Oscillatoria unable to collapse gas vacuoles by tugor pressure. Arch. Hydrobiol. 102 (3), 319–329. Van Rijn, J., Shilo, M., 1985. Carbohydrate fluctuations, gas vacuolation, and vertical migration of a scum-forming cyanobacteria in fishponds. Limnol. Oceanogr. 30, 1219–1228. Vaulot, D., Marie, D., 1999. Diel variability of photosynthetic picoplankton in the equatorial Pacific. J. Geophys. Res. (C Oceans) 104 (2), 3297–3310. Vaulot, D., Marie, D., Olson, R.J., Chisholm, S.W., 1995. Growth of Prochlorococcus, a photosynthetic prokaryote, in the equatorial Pacific Ocean. Science (Wash.) 268 (5216), 1480–1482. Visser, P.M., 1995. Growth and vertical movement of the Cyanobacterium Microcystis in stable and artificially mixed water colums. Ph.D. Thesis, Amsterdam, the Netherlands, 147 pp. Visser, P.M., Passarge, J., Mur, L.R., 1997. Modelling vertical migration of the Cyanobacterium Microcystis. Hydrobiologia 349, 99–109. Wallace, B.B., Hamilton, D.P., 1999. The effect of variations in irradiance on buoyancy regulation in Microcystis aeruginosa. Limnol. Oceanogr. 44 (2), 273–281. Walsby, A.E., 1971. The pressure relationships of gas vacuoles. Proc. R. Soc. B 178, 301–326. Walsby, A.E., 1994. Gas vesicles. Microbiol. Rev. 58 (1), 94–144. Walsby, A.E., Utkilen, H.C., Johnsen, I.J., 1983. Buoyancy changes of a red coloured Oscillatoria agardhii in lake Gjersjeen, Norway. Arch. Hydrobiol. 97 (1), 18–38. Walsby, A.E., Kinsman, R., Ibelings, B.W., Reynolds, C.S., 1991. Highly buoyant colonies of the Cyanobacterium Anabaena lemmermannii form persistent surface waterblooms. Arch. Hydrobiol. 121 (3), 261–280.