Competition between cyanobacteria and green algae at low versus elevated CO2: who will win, and why?

Contrary to the current paradigm, competition experiments showed that green algae defeated cyanobacteria at low CO2 levels, whereas cyanobacteria with high-flux carbon uptake systems became stronger competitors at elevated CO2.


Phytoplankton dynamics
Consider a number of i=1,…,n phytoplankton species. Let X i denote the population density of phytoplankton species i, and let Q i denote its cellular carbon content. The population dynamics of the competing phytoplankton species can then be written as a series of ordinary differential equations: where  i (Q i ) is the specific growth rate of species i as an increasing function of its carbon content, and m i is its loss rate.
Similar to previous models (Verspagen et al., 2014a,b), we assume that carbon assimilated by phytoplankton is allocated to structural carbon biomass and a transient carbon pool. We define Q min,i as the minimum amount of cellular carbon that needs to be incorporated into the structural biomass of species i. Furthermore, we define Q max,i as the maximum amount of carbon that can be stored in its cell. The relative size of the transient carbon pool, T i , is then given by: Accordingly, the relative size of the transient carbon pool is constrained to 0  T i  1.
The transient carbon pool can be invested to make new structural biomass, which contributes to further phytoplankton growth. The specific growth rate of a species is therefore determined by the size of its transient carbon pool: where µ max,i is the maximum specific growth rate of species i.

Carbon uptake
The carbon contents of the phytoplankton species increase through uptake of carbon dioxide (u CO2,i ) and bicarbonate (u HCO3,i ), and decrease through respiration (r i ) and dilution of the cellular carbon content by growth: Our model assumes that short-term uptake rates of carbon dioxide and bicarbonate are increasing functions of the ambient CO 2 and bicarbonate availability according to Michaelis-Menten kinetics. Since carbon uptake and assimilation require energy from the light reactions of photosynthesis, we further assume that the carbon uptake rates depend on photosynthetic activity. Finally, we assume that the carbon uptake rates decrease with an increasing size of the transient carbon pool (as in Verspagen et al., 2014a,b). This provides a very simple negative feedback loop, such that the carbon uptake systems have the highest activity under carbon-limiting conditions and are downregulated when cells are satiated with carbon. The uptake rates of CO 2 and bicarbonate by a phytoplankton species i can then be described by: 3 where u max,CO2,i and u max,HCO3,i are the maximum uptake rates of CO 2 and bicarbonate, respectively, H CO2,i and H HCO3,i are the half-saturation constants, T i is the relative size of the transient carbon pool (with 0  T i  1), and P i is the relative photosynthetic activity of phytoplankton species i (with 0  P i  1).
Light availability determines the photosynthetic rate, and thereby the amount of energy available for carbon assimilation. We therefore calculate the relative photosynthetic activity of a phytoplankton species from its depth-averaged photosynthetic rate (Huisman and Weissing, 1994): where p i is the photosynthetic rate of species i, and z max is the total depth of the water column. The notation p i (I(z)) indicates that the photosynthetic rate is a function p i of the local light intensity I, which in turn is a function of depth z.
The photosynthetic rate of a phytoplankton species is described by a simple Monod function of light intensity: where p max,i is the maximum photosynthetic rate of species i, and H I,i is its half-saturation constant for light. The maximum carbon uptake rate is already accounted for in equations (A5) and (A6). Therefore, without loss of generality, we can set p max,i = 1 (which constrains the depth integral in equation (A7) to 0 < P i < 1).

With the help of the Monod function (equation A8) and Lambert-Beer's law (equation 3 in the main text), the depth integral in equation (A7) can be solved
analytically (Huisman and Weissing, 1994). This yields: where r max,i is the maximum respiration rate when its cells are fully satiated with carbon.

DIC, alkalinity and pH
Changes in the concentration of total dissolved inorganic carbon, [DIC], are described by (Verspagen et al., 2014a): The first term on the right-hand side of this equation describes changes through the influx ([DIC] in ) and efflux of water containing DIC. The second term describes CO 2 exchange with the atmosphere, where g CO2 is the CO 2 flux across the air-water interface and division by z max converts the flux per unit surface area into the corresponding change in DIC concentration. The third term describes uptake of dissolved CO 2 and bicarbonate by the photosynthetic activity of the phytoplankton community. Finally, the fourth term describes CO 2 release by respiration of the phytoplankton species.
The CO 2 flux across the air-water interface, g CO2 , depends on the difference in partial pressure. More specifically, g CO2 depends on the difference between the expected concentration of dissolved CO 2 in water if in equilibrium with the partial pressure in the atmosphere and the actual dissolved CO 2 concentration (Siegenthaler and Sarmiento, 1993;Cole et al., 2010): where v is the gas transfer velocity (also known as piston velocity), K 0 is the solubility of CO 2 gas in water (also known as Henry's constant), pCO 2 is the partial pressure of CO 2 5 in the atmosphere, and [CO 2 ] is the dissolved CO 2 concentration. In chemostats, gas transfer will depend on the gas flow rate (a). We therefore assume that v = b a, where b is a constant of proportionality.
Concentrations of dissolved CO 2 , bicarbonate and carbonate can be calculated from [DIC] and pH (Stumm and Morgan, 1996). Changes in pH depend, in turn, on the alkalinity of water. In our application, alkalinity largely depends on the inorganic carbon and phosphate concentrations (Wolf-Gladrow et al., 2007): Changes in dissolved CO 2 do not change alkalinity (see the above equation), and uptake of bicarbonate by phytoplankton requires the simultaneous uptake of a proton to maintain charge balance and hence does not change alkalinity either. However, assimilation of nutrients such as nitrate, phosphate and sulfate is accompanied by proton consumption to maintain charge balance and therefore increases alkalinity (Wolf-Gladrow et al., 2007).
Our model therefore treats alkalinity as a dynamic variable: where ALK in is the alkalinity of the influx, and u N,i , u P,i and u S,i are the nitrate, phosphate and sulfate assimilation rates of phytoplankton species i. Nitrate and phosphate assimilation increase alkalinity by one mole equivalent, whereas sulfate assimilation increases alkalinity by 2 mole equivalents (Wolf-Gladrow et al., 2007). Accordingly, although nutrients do not limit phytoplankton growth in our model, the model does keep track of the nitrate, phosphate and sulfate concentration (see below) to calculate changes in alkalinity, and hence in pH and carbon speciation.
We used an iterative algorithm adapted from Portielje and Lijklema (1995) that, at each time step, calculates pH and the dissolved CO 2 , bicarbonate and carbonate concentration from the [DIC] and alkalinity predicted by Eqs. (A11) and (A14). The algorithm is described in the Supporting Information of Verspagen et al. (2014a).