Towards a quantitative assessment of inorganic carbon cycling in photosynthetic microorganisms

Abstract Photosynthetic organisms developed various strategies to mitigate high light stress. For instance, aquatic organisms are able to spend excessive energy by exchanging dissolved CO2 (dCO2) and bicarbonate (HCO3−) with the environment. Simultaneous uptake and excretion of the two carbon species is referred to as inorganic carbon cycling. Often, inorganic carbon cycling is indicated by displacements of the extracellular dCO2 signal from the equilibrium value after changing the light conditions. In this work, we additionally use (i) the extracellular pH signal, which requires non‐ or weakly‐buffered medium, and (ii) a dynamic model of carbonate chemistry in the aquatic environment to detect and quantitatively describe inorganic carbon cycling. Based on simulations and experiments in precisely controlled photobioreactors, we show that the magnitude of the observed dCO2 displacement crucially depends on extracellular pH level and buffer concentration. Moreover, we find that the dCO2 displacement can also be caused by simultaneous uptake of both dCO2 and HCO3− (no inorganic carbon cycling). In a next step, the dynamic model of carbonate chemistry allows for a quantitative assessment of cellular dCO2, HCO3−, and H+ exchange rates from the measured dCO2 and pH signals. Limitations of the method are discussed.

ii. We assume equilibrium for the fast reactions 1 and 2 + /2 − , to eliminate the species H 2 CO 3 and CO 2− 3 .
iii. We introduce a carbonate pool which is unchanged by the fast reactions 1 and 2 + /2 − : In the last step, we assume [H + ] < 10 −2 corresponds to pH > pK 1 + 2 ≈ 3.6 + 2 = 5.6. In our experiments, we consider pH values from 6.5 to 8.1, and the assumption is well justified. iv. We compute the net rate of the reaction between CO 2 and the carbonate pool CP , determined by the slow reactions 1 and 1 + /1 − : Thereby, we • assume equilibrium for the fast reactions W and 1 (cf. i and ii), • use the equality of the equilibrium constants for the three mechanisms 1 + 1 , 1 + , and 1 − + W, • define the hydration rate "constant" v. Finally, we introduce total alkalinity which is unchanged by carbonate chemistry, water self-dissociation, and buffering (due to charge balancing in chemical reactions), Total alkalinity is only changed by the culture through HCO − 3 and H + exchange.
To summarize, we assume equilibrium for the fast reactions 1 , 2 + /2 − , W and B (and eliminate 5 out of 8 species), we introduce the carbonate pool CP and total alkalinity TA (and end up with 5 = 8 − 5 + 2 species), and we define the net hydration rate, determined by the slow reactions 1 and 1 Given the exchange rates qCO 2 , qHCO − 3 , qH + , we obtain 5 equations (3 ordinary differential equations and 2 algebraic equations) for the 5 variables [ Note that the ordinary differential equations (3abc) take this very simple form because of the introduction of the carbonate pool CP (and the use of total alkalinity TA). In particular, the effect of hydration (v) and exchange (qCO 2 , qHCO − 3 , qH + ) is very transparent. Solving the DAE system (3) requires the following values:

Model analysis
In order to explain the (fast) dynamics following the change of light conditions (dark/light or light/dark), we further analyze hydration.
For given exchange rates qCO 2 , qHCO − 3 , qH + and hydration rate constant k 1 , the hydration rate v is determined by the following ODE: Thereby, we introduce dimensionless variables α and β, which depend on HCO − 3 , H + , and buffer concentrations in a complicated way: See also Figure S2. The variables α and β are obtained by time differentiation of Eqns. (3de) and by solving the resulting linear system for the time derivatives dt in terms of dCP dt , dTA dt . As a result, we obtain a nonhomogeneous linear ODE for v with the "apparent" hydration rate constant k 1 (1 + α). In situations where α 1, the dynamics following an abrupt change of light conditions (dark/light or light/dark) is much faster than expected from the hydration rate constant k 1 .
In the main text, we often assume qHCO − 3 = qH + resulting from charge balance and implying TA = const. For notational simplicity, we also make this assumption here, and Eqn. (4) becomes 1

Abrupt light change
We consider an abrupt change in light conditions at t = 0 (either dark/light or light/dark), in particular, a step change in the exchange rates qCO 2 , qHCO − 3 , qH + . We denote the phase t < 0 by and the phase t ≥ 0 by ⊕. Assuming α ≈ const, Eqn. (5) can be solved is the quasi steady-state value of v; in particular,v is the quasi steady-state value of v for t < 0, that is,v = (qCO 2 − α qHCO − 3 )/(1 + α); analogously,v ⊕ for t ≥ 0. The hydration rate v starts fromv at t = 0 and approachesv ⊕ within t ≈ 1 (1+α) k 1 , leading to a "displacement" of ∆v =v ⊕ −v .
Using the solution for v, the ODE for [CO 2 ], Eqn. (3a), can be solved for t ≥ 0 as leading to a "displacement" of ∆[CO 2 ] = ∆v (1 + α) k 1 , followed by a slow linear change with a slope of The simplifying assumption α ≈ const yields analytical solutions that are in good agreement with numerical solutions of the DAE system (3).

Identification of exchange rates
and qCO 2 and qHCO − 3 can be identified from [CO 2 ] and [H + ] via Eqns. (3ab): The identification of exchange rates involves the time differentiation of measured concentrations which may lead to noise amplification.

Figure S2
Dependence of the parameter α (defined after Equation (4)) on pH and buffer (HEPES) concentration B . Panel A shows the entire range of the parameter α (for given pH and B ), panel B zooms in to α ≤ 1.

Figure S3
Dependence of the dCO2 hydration rate v (see Equation (1) for further details) during Dark-Light-Dark experiments on pH (panels A, B) and buffer concentration (panels B, C). Initial dCO2 concentration set to 30 µM, and buffer (HEPES) concentration set to 1 µM (panels A, B) or 17 mM (panel C). Exchange rates in dark set to +10 nmol L -1 s -1 for both CO2 and HCO 3 − To simulate inorganic carbon cycling (ICC, full lines), exchange rates at light were set to -150 nmol L -1 s -1 for qCO2 (uptake) and +100 nmol L -1 s -1 for qHCO 3 − (excretion). To simulate carbon uptake in the absence of ICC (dashed line), exchange rates at light were set to -50 nmol L -1 s -1 for qCO2 (uptake) and 0 nmol L -1 s -1 for qHCO 3 − . Grey rectangles: dark phases; white rectangles: light phases.

Figure S7
Comparison of the measured (absolute) slopes of dO2 (blue triangles) and dCO2 (red squares) at light with the carbon uptake rate -(qCO2 + qHCO 3 − ) as identified from the experimental data (green diamonds). For the computation of the carbon uptake rate, the quasi steady-state values of qCO2 and qCO2 were used ( Figure 6 of the main text, right column panels). The full dO2 and dCO2 dynamics within Dark-Light-Dark experiment are shown in Figure S6.