Culture experiments

Culture experiments were undertaken in the Department of Earth Sciences at the University of Oxford. Duplicate monoclonal batch cultures of four strains of coccolithophore belonging to the family Noëlaerhabdaceae were grown in sterile filtered (0.2 μm) artificial seawater prepared according to ESAW68 adapted for a range of DIC concentrations ((DIC)=1.380 mM, 2.147 mM, 3.067 mM and 6.135 mM) at constant pH total scale (8.2) by varying sodium bicarbonate addition and titration with HCl and with nitrate (442 μM), phosphate (5.00 μM), vitamins, trace metals and ethylenediaminetetraacetic acid according to K/2 (ref. 69). Cultures were maintained at 15 °C with an incident photon flux of 250 μE and a 12/12 light/dark cycle. Cells were acclimated for >20 generations in dilute batch culture for each experimental condition before inoculation. Cells were inoculated in 2.4l polycarbonate flasks, with no headspace and sealed off to the air with teflon lined caps. Removal of medium during the experiment was unavoidable due to the need to count and measure cells, and resulted in a maximum headspace of 20 cm3 at harvest. To minimize the drift in culture conditions throughout the course of the experiment, cells were harvested at ∼1–2% (and never >4%) of maximum cell density, which was determined for each experimental condition and strain combination via prelimenary experimentation. Strains were AC478 (RCC1211 G. oceanica from Portuguese coast in Atlantic Ocean), AC472 (RCC1216 E. huxleyi, from Tasman Sea in Pacific Ocean), AC448 (RCC1256 E. huxleyi, Icelandic coast in Atlantic Ocean) and AC279 (RCC1314 G. oceanica, French coast in Atlantic Ocean) from the Roscoff culture collection (RCC). Particulate material was harvested by dry filtration onto pre-weighed membranes with 0.2 μm pore-size, and rinsed of salt with a minimal amount of neutralized deionised water. Coccolithophore size and concentration were obtained using a Beckman Z2 Coulter Counter. Coccosphere and cell size were measured three times each respectively pre- and post-decalcification both morning and evening on the harvest day and the preceding day. Cells were decalcified by reducing the pH of the suspension with HCl addition to 5.0 with for around 20 min. The Coulter counter was calibrated to use ESAW+K/2 medium as an electrolyte, and for use with the acidified electrolyte, to accommodate for the difference in ionic strength. Cell division is synchronized under the light/dark cycle and cell size was assumed to increase linearly throughout the day70. By measuring cell and coccosphere size morning and evening, the bias introduced due to the time of day of measurement can be removed by interpolation to the same time of day. This also removes the daily variation of cell size70. Culture health was monitored by cell counts and microscope inspection on alternate days. Molar PIC and POC were measured with a Rock Eval analyser in the Earth Sciences department at Oxford University.

Isotope measurements

Carbon and oxygen isotopic compositions of calcite were measured at the University of Oxford using a VG Isogas Prism II mass spectrometer with an on-line VG Isocarb common acid bath preparation system. Samples were first rinsed with neutralized deionised water to remove any salt. Samples were then dosed with acetone and dried at 60 °C for at least 30 min. In the instrument they were reacted with purified phosphoric acid at 90 °C. Calibration to PDB standard was via the international standard NBS-19 using the Oxford in-house (NOCZ) Carrara marble standard. Reproducibility of replicated standards was better than 0.1‰ (1σ) for δ13C and δ18O expressed relative to the V-PDB standard. Carbon isotopic composition of organic material was measured on an automated carbon and nitrogen elemental analyzer (Carlo Erba EA1108) at the Research Laboratory for Archaeology and the History of Art at the University of Oxford. Samples were decalcified with HCl, and rinsed with MilliQ water at least three times before being weighed into tin capsules. The internal alanine standard reproducibility is ∼0.16‰ expressed relative to the V-PDB standard.

Assumptions of the model

The assumptions on which this model is based, are:

Cells are in steady-state growth. The concentration of every carbon pool (that is, each carbon species in each intracellular compartment) is constant, which implies that all fluxes associated with that pool sum to zero.

The cell and compartments are assumed to be isomorphic across all strains and conditions.

CO 2 and ions move across membranes via passive diffusion and facilitated diffusion respectively. Both chemical species are assumed to move across membranes with a flux proportional to surface area of the membrane and the concentration of the carbon species on the proximal side of the membrane. There is no concentration-independent active uptake of bicarbonate or carbon dioxide. Membranes are impermeable to carbonate ions. There is no other source of carbon to the cell.

Loss of carbon from the cell is via carbon fixation, calcification, passive diffusion of CO 2 , and co-transport of .

The membrane permeability to CO 2 is constant.

The effective membrane permeability to , which incorporates the concentrations of coported ions, is a linear function of background carbon utilization ( U 0 ; defined in equation (2)), representing the ability of cells to upregulate transport proteins. The special cases whereby permeability is non-adaptive, or is zero are explicitly allowed.

The constants describing the permeability of membranes to CO 2 and to are identical across all species, and for all membranes except the cell membrane where the strong gradient of Na + from the extra-cellular to intra-cellular environment is assumed to dominate transport via Na + coport, thus introducing an asymmetric permeability. As this is an electroneutral process, the membrane potential does does affect this flux, which is assumed to be a product of mass-action.

pH in each compartment is constant, and is prescribed a priori , according to the values measured by Anning et al . 33 .

Organic matter is not remobilised from the organic pool to the cytosol via mitochondrial respiration.

Movement of carbon across membranes does not impart a kinetic isotopic fractionation.

All cells are assumed to exhibit active external CA (see Supplementary Note 1).

The chloroplast is assumed to consist of a single compartment with no pyrenoid. The large isotopic fractionation of carbon that occurs during fixation catalysed by the enzyme RuBisCO , is here factored into the model as an effective pyrenoid/RuBisCO black box fractionation ( fe ).

Membrane permeabilities

A putative bicarbonate transport protein has been observed to be upregulated of at low [CO 2 ]42. Here these results are interpreted as consistent with an approximately linear increase in transcript abundance with the degree of carbon utilization (Supplementary Fig. 2). This physiological result is assumed to reflect an increase in synthesis of transport proteins, and thus the density of transport proteins in the membrane, manifest as an increase in the effective permeability of membranes to . The flux of through membranes facilitated by SCL4-like exchange proteins is proportional the the product of on the proximal side of the membrane and Na+ on the proximal side of the membrane for coport, and that of Cl− on the distal side of the membrane for antiport. For simplicity, we assume transport to be driven by Na+ coport, and the gradients of the coported ion to be negligible across all membranes, except the cellular membrane where it is substantial. This assumption is factored into the model via an additional universal constant, which describes the extra-cellular to intra-cellular concentration gradient of the coported ion. This assumption is more coherent with the current biological literature than equivalent assumptions of previous models, including concentration-independent implicitly ATP-driven active uptake of , assumed by Keller et al.24 to scale with growth rate, or by Bolton and Stoll2 to be independent of all other parameters.

In this model, the permeability of membranes to CO 2 (P Ccell ) is assumed to be constant. The permeability of membranes to (P Bcell ) is assumed to increase linearly with utilization of background (that is, before upregulation of anion exchange proteins) carbon supply (U 0 ), which is defined as:

where C in0 is the net carbon supply when membrane permeabilities to CO 2 and to are at their background values, and F FIX +F CAL is the rate of fixation of carbon into organic and inorganic matter. The membrane permeability to bicarbonate is therefore:

where T 0 is the base-level membrane permeability to and T U is the gradient of the increase in membrane permeability to with increasing utilization; F FIX and F CAL are the rates of carbon fixation and calcification respectively, C e and e are the concentrations of CO 2 and in the ambient medium respectively, and SA cell is the surface area of the cell. The large Na+ gradient at the cell membrane is included as an additional constant, G. This line of reasoning is a set of assumptions factored into the model via four universal constants describing the membrane permeability to CO 2 and to (P Ccell , T 0 , T U and G). These four constants will be constrained by the data, explicitly leaving open the option of non-upregulated transport (through T U =0), complete impermeability of membranes to (through T 0 =T U =0), and through up- and down-gradient movement of dependent on the inferred Na+ ion gradient.

Compartment shapes and sizes

Each intracellular compartment is assumed to be an oblate spheroid. The equatorial axis of the spheroid, a, is assumed to have a constant ratio with the cell radius. This is referred to as the scaling factor (s f c × r=a c ). The polar axis, c, is assumed to have a constant ratio with the equatorial axis. This ratio is referred to as the aspect ratio factor (a f c × a c =c c ). Isometry is assumed across species, and in this way, two constants are used to describe the relative size and shape of each compartment, and therefore their volumes and surface areas. The surface area of an oblate spheroid is given by:

The volume of a spheroid is given by:

Intracellular carbonic anhydrase

The CAs are a family of zinc-containing metalloenzymes responsible for catalysing the hydration and dehydration of CO 2 and respectively. Their behaviour is well described by Michaelis–Menten kinetics45. Given the general form of an enzymatically catalysed reaction:

where E, S, ES and P denote enzyme, substrate, enzyme–substrate complex and product concentrations, respectively. k f , k r and k cat are rate constants respectively for the binding and unbinding of substrate to the enzyme, and for the maximum catalytic throughput of the reaction.

According to the Michaelis–Menten equations, the velocity of a reaction such as equation (7) is given by:

where K m is the Michaelis constant, or half-saturation constant , and describes the substrate concentration when v= . When [S]<<K m , equation (8) becomes:

and is the specific activity of the enzyme in units of M−1 s−1. This is analogous to a rate constant (k p ), or activity, when multiplied by the concentration of the enzyme. The rate constant for CA-catalysed hydration of CO 2 is then:

As enzymes catalyse reactions but do not alter the position of equilibrium, the rate constant for the reverse reaction, CA-catalysed dehydration of , is given by the hydration rate constant divided by the equilibrium constant:

A value of 2.7 × 107 M−1 s−1 was determined by Uchikawa et al.45 for the of the hydration reaction catalysed by bovine erythrocyte CA.

Model derivation

Fluxes of CO 2 through membranes are the product of membrane surface area (SA), CO 2 concentration on the source side of the membrane, and membrane permeability to the specific carbon species. From Fig. 1 expressions for transmembrane CO 2 fluxes are:

and likewise for fluxes:

where all fluxes are defined in Fig. 1. G is the constant that describes the effective asymmetry of the cell membrane to due to the transmembrane Na+ gradient.

In each compartment, carbon dioxide and bicarbonate ions are interconverted by the reversible hydration and hydroxylation reactions, and by hydration in the presence of CA:

where arrow annotations denote rate constants (see section for CA rate constants). Combining the rate constants of equation (14) at constant pH and [CA] in a generic compartment, a, gives:

where k CBa denotes the reaction rate constant for the conversion of CO 2 to , and k BCa denotes the rate constant for the reverse reaction. H a , OH a and CA a refer to the concentrations of H+ ions, OH− ions and CA in compartment, a. As compartment-specific pH and [CA] are defined apriori, k CBa and k BCa can be treated as pH and [CA] dependent compound rate constants. The rates of the reactions described by equation (14) in each compartment are therefore:

Carbon fluxes throughout the cell can be fully described with equations (12), (13) and (17), and with two additional output fluxes; the rate of calcification (F CAL ) and the rate of photosynthetic carbon fixation (F FIX ; Fig. 1). Assuming steady state, the rate of change of the amount of each carbon species in each compartment is zero, giving:

At steady state these differential equations become linear functions of the concentration of the different inorganic carbon species. Substituting in equations (12), (13) and (17):

This set of interdependent equations can be written as a linear system of the form:

where A is the coefficient matrix of the linear system, N is the nonhomogeneous term and Φ is the unknown vector containing the carbon species concentrations in each compartment. Equation (20) can be solved for Φ, where A and N, defined by the dynamic carbon species concentration equations, are as follows:

[13C] is very low compared to [12C], carbon fluxes are assumed to represent 12C. 13C dynamics are therefore determined by prefixing each C flux with corresponding R, where:

By definition,

so the assumption of balanced growth allows both offset R standard and scale factor R standard to be eliminated while preserving the same system of equations. Effectively, we can thus prefix each flux directly with the associated δ13C value. For fractionation fluxes, this leads to a prefix with the sum of the δ13C of the source pool and the process-specific fractionation factor ( ). In the following, δ Θa refers to the δ13C of carbon species, Θ, in compartment, a.

The following fractionation factors (in ‰)44, are assumed constant:

where the subscripts are consistent with those from equation (14). Analogous to the compound rate constants of equations (15) and (16), the interconversion of carbon dioxide and bicarbonate has a pH and [CA] dependent, compartment-specific, compound isotopic fractionation factor.

When the linear system described for carbon fluxes is solved, all carbon species concentrations in all compartments are known. Dynamic equations for the isotopic composition of each compartment, assuming balanced growth, and expressed in terms of carbon dioxide, and bicarbonate fluxes and fractionation factors are thus:

At steady state the differential equations become linear functions of δ13C of the different carbon species in the different compartments. Substituting in equations (12), (13) and (17):

This set of interdependent equations can also be written as a linear system of the form:

where A is the coefficient matrix of the linear system, N is the nonhomogeneous term and Φ is the unknown vector containing the carbon isotopic compositions of each carbon species in each compartment. Equation (36) can be solved for Φ, where A and N, defined by the dynamic carbon species concentration equations, are as follows:

Equilibrium limit special case

At the limit where CA activity is infinitely high in all compartments, complete chemical and isotopic equilibrium with respect to carbon can be assumed. In this scenario, the total [DIC] (D) in each compartment and its isotopic composition, along with (prescribed) pH, completely specifies the concentrations and isotopic compositions of the different inorganic carbon species in each compartment.

The dynamic equations for total DIC (D) in the three different compartments are:

In steady state all three equations must equal 0, which implies:

Inserting these into equation (37), the expression for cytoplasm total DIC, gives:

The four fluxes across the cell membrane equal:

where the permeability for relates to the permeability for CO 2 through background utilization:

As a result, all cross-membrane fluxes are proportional to P C SA and equation (42) can be divided through by this factor. With chemical equilibrium and prescribed pH c in the cytoplasm, B c is proportional to C c for any compartment c∈{e, i, x, v}:

where is the temperature and salinity-dependent chemical equilibrium constant, and H c the H+ concentration (H c = ). Thus equation (42) can be rewritten as:

Isolating C i and divide through by C e gives:

Thus, concentrations and fluxes of carbon species in all compartments are known.

Following the same process, two additional fluxes across the chloroplast membrane equal:

where f x is the ratio of the product of cellular membrane permeability and surface area to that of the chloroplast. Equation (40) can therefore be rewritten:

and inserting equation (50) becomes:

Similarly, two additional fluxes across the coccolith vesicle membrane equal:

where f v is the ratio of the product of cellular membrane permeability and surface area to that of the coccolith vesicle. Equation (41) can therefore be rewritten:

and inserting equation (50) likewise becomes:

The values f x and f v are constants when all membranes are assumed to have the same permeability, and isometry is assumed.

Membrane permeability to bicarbonate is reliant on the background utilization of carbon, U 0 . The value of U 0 is given by:

At equilibrium, has a constant isotopic offset from CO 2 , denoted here as E. As both and CO 2 in each compartment are known if one is known, the whole system can be described by CO 2 alone. The dynamic equations for isotopic fluxes are:

As with chemical fluxes, substituting in expressions for concentration driven diffusion, membrane permeability to in terms of background utilization, concentration in terms of CO 2 and H+, and rearranging gives:

The chemical carbon flux equations (50), (54) and (58) can be solved simultaneously. Further, the carbon isotope flux equations (63)–(65), , can also be solved simultaneously, using the chemical carbon flux equations. In these expressions, including U 0 , the rates of fixation and calcification always occur divided by P C SAC e , and this is also the only appearance of these parameters. Thus,

are compound varibles that describe the entire system when constant and intracellular pH is assumed and known a priori. In this model, the fixation rate is related to cell size via the equation,

where ρ is the cellular carbon density and V and μ are the volume and division rate of a cell respectively. As PIC:POC is equal to , the compound parameters become:

and:

Therefore, at the limit, where intracellular CA is infinitely high such that all compartments are in chemical and isotopic equilibrium, the orthogonal variables that describe the entire system are:

τ is a parameter that is similar to the ratios that evolved through early work describing carbon fluxes in phytoplankton16,17,18,21,23,24,25,32. The more complicated disequilibrium model collapses to a state completely consistent with this earlier work when equilibrium is assumed and PIC:POC=0. When the cell is close to chemical and isotopic equilibrium with respect to carbon therefore, changes in cell size, growth rate and extracellular [CO 2 ] cannot be decoupled.

Fitted parameters

The results from culture were used to constrain P Ccell (the permeability of membranes to CO 2 ), T 0 , T U (the constants required to define the linear dependence of membrane permeability to on utilization (equation (4)), G (the asymmetry of the effective cell membrane permeability to , which may reflect the extra-cellular to intra-cellular concentration ratio of the coported ion), and fe (the effective enzymatic isotopic fractionation associated with carbon fixation by RuBisCO). Universal constants were fitted to the data by minimization of a misfit function, which quantified the sum of squared deviations of the modelled results from the measured data. Minimization of the misfit function was achieved using an R implementation71 of the constrained DIRECT_L algorithm72 within a defined parameter space. The model was prescribed to be undefined when concentrations are negative, or when the saturation state of calcite in the coccolith vesicle is below 0.1. Final parameter values were checked with an alternative implementation of the model in Python, using a combination of a broad search with the Differential Evolution algorithm73, followed by refinement of its result with the Nelder-Mead Simplex algorithm74. Only complete data sets were used to computationally constrain the universal parameters of the model. The required parameters are: cell size, PIC:POC, growth rate, medium pH and [DIC], δ13C DIC , δ13C calcite and δ13C org . The model was fitted to these data multiple times with pseudo-replicate datasets generated by resampling each value from an assumed gaussian distribution defined by the expected value and uncertainty associated with each measurement. All input parameters were resampled in this way. The fitted value of P Ccell is ∼9.3 × 10−4 ms−1, which is at the high end of the range suggested by Hopkinson et al.46 for diatoms. T 0 is ∼7.8 × 10−8 ms−1; also consistent with values given by Hopkinson et al.46 Based on these results, at low utilization, the membrane is 3–4 orders of magnitude less permeable to as to CO 2 . The constant T U describing the increase in membrane permeability to with increased utilization is ∼5.1 × 10−6 ms−1 per unit utilization. The parameter G, which describes the asymmetry of the cell membrane permeability to , is constrained to be 2.7. This value partially reflects the Na+ gradient, but is somewhat too low, so probably also represents Cl−-coupled anion exchange, which would drive bicarbonate out of the cell. When utilization is low, either at high DIC or in small cells, ∼10% of carbon enters the cell in the form of . When cells experience high utilization, at low DIC or in larger cells, the contribution of to total DIC supply can be as high as 50%. CA in all compartments is between 0.1 and 1 mM in all intracellular compartments, based on a specific activity of 2.7 × 107 M−1 s−1 (ref. 45). Values below 0.1 mM lead to extremely unstable model output and increasing this value above 0.1 has little effect on model output suggesting that compartments are approximately in equilibrium with respect to carbon. As it takes an order of magnitude longer to reach equilibrium with respect to oxygen, and we know that the cell cannot be in equilibrium with respect to oxygen due to the existence of oxygen isotopic vital effects in coccolith calcite, the CA concentration must exist within this small range. fe is ∼−14.3‰, which is far smaller than that used in the literature to date2,21,24,25,32 but is closer to the in vitro measured value of given for E. huxleyi (−11‰ (ref. 39)) and for the diatom Skeletonema costatum (−19‰ (ref. 53)). Parameters associated with the model input and output are summarized in Table 2.

Table 2 Parameters from the model, their definitions, derivation and units. Full size table

Data availability

The culture data that support the findings of this study are contained in the Supplementary Material, and in the literature cited throughout the manuscript.