Modeling with uncertainty quantification reveals the essentials of a non-canonical algal carbon-concentrating mechanism

Abstract

The thermoacidophilic red alga Cyanidioschyzon merolae survives its challenging environment likely in part by operating a carbon-concentrating mechanism (CCM). Here, we demonstrated that C. merolae's cellular affinity for CO₂ is stronger than the affinity of its rubisco for CO₂. This finding provided additional evidence that C. merolae operates a CCM while lacking the structures and functions characteristic of CCMs in other organisms. To test how such a CCM could function, we created a mathematical compartmental model of a simple CCM, distinct from those we have seen previously described in detail. The results of our modeling supported the feasibility of this proposed minimal and non-canonical CCM in C. merolae. To facilitate the robust modeling of this process, we measured and incorporated physiological and enzymatic parameters into the model. Additionally, we trained a surrogate machine-learning model to emulate the mechanistic model and characterized the effects of model parameters on key outputs. This parameter exploration enabled us to identify model features that influenced whether the model met the experimentally derived criteria for functional carbon concentration and efficient energy usage. Such parameters included cytosolic pH, bicarbonate pumping cost and kinetics, cell radius, carboxylation velocity, number of thylakoid membranes, and CO₂ membrane permeability. Our exploration thus suggested that a non-canonical CCM could exist in C. merolae and illuminated the essential features generally necessary for CCMs to function.

The thermoacidophilic red alga Cyanidioschyzon merolae possesses a unique carbon-concentrating mechanism (CCM), and a model illuminates the essential features generally required for CCM function.

Introduction

Cyanidioschyzon merolae is a red microalga found in moist environments surrounding geothermal sulfur springs. This species is extremophilic, with optimal laboratory growth conditions including low pH (∼2) and high temperatures (∼42°C) (Miyagishima and Wei 2017; Miyagishima et al. 2017). C. merolae and other thermoacidophilic red algae draw interest for their unique biology and simple characteristics, which position them as useful model organisms and as candidates for biotechnology applications (Rahman et al. 2017; Miyagishima and Tanaka 2021; Seger et al. 2023; Villegas-Valencia et al. 2023). For example, C. merolae is of interest because it is one of few organisms which relies on photosynthesis in geothermal spring environments, where hot and acidic conditions restrict the availability of inorganic carbon and challenge biological carbon fixation (Gross 2000; Miyagishima et al. 2017). Notably, organisms of acid waters can only access approximately 10 μM inorganic carbon, as the inorganic carbon pool at acid pH is primarily the volatile species CO_2. In comparison, organisms of near-neutral and alkaline waters may have access to several millimolar of inorganic carbon, due to accumulation of the involatile bicarbonate (Oesterhelt et al. 2007).

C. merolae is thought to survive in its challenging environment in part by operating a carbon-concentrating mechanism (CCM) (Zenvirth et al. 1985; Rademacher et al. 2017; Steensma et al. 2023). CCMs boost carbon-fixation efficiency by concentrating CO₂ around rubisco, providing ample substrate for carbon-fixation and inhibiting a competing oxygen-fixation reaction of rubisco. Evidence supporting a CCM in C. merolae includes measured accumulation of radiolabeled carbon in the cell, d¹³C consistent with a CCM, transcriptional response of potential CCM genes to CO₂ fluctuations, and substantial CO₂ assimilation at low environmental CO₂ concentrations (Zenvirth et al. 1985; Rademacher et al. 2017; Steensma et al. 2023). However, many of these indications of the CCM are not definitive: in particular, it is not known how much of C. merolae's ability to assimilate CO₂ efficiently could be explained by the affinity of C. merolae rubisco for CO₂. Thus, we here provide further evidence for the CCM in C. merolae by demonstrating that the affinity of C. merolae cells for CO₂ is better than could be explained by the affinity of C. merolae rubisco for CO₂.

C. merolae's CCM may be described as a “non-canonical” CCM, since the C. merolae CCM must operate differently from the few CCM types which are well-characterized.

For example, unlike algae and cyanobacteria with well-characterized CCMs, C. merolae is not able to take up external bicarbonate, and C. merolae lacks anatomy associated with the pyrenoid CCM organelle (Zenvirth et al. 1985; Badger et al. 1998; Misumi et al. 2005; Steensma et al. 2023). The absence of these CCM features in C. merolae challenges our understanding of what components are required for a functional CCM, and presents the opportunity to define essential CCM components. While previous work has discussed CO₂ as a source of carbon for the CCM (Fridlyand et al. 1996; Price 2011), there has been little quantitative exploration of whether a CCM could function while lacking both facilitated carbon uptake and specialized compartments such as the pyrenoid or carboxysome. We thus used mathematical modeling, informed by our experimental measurements, to explore how the C. merolae CCM may function.

Research on CCMs has long employed mathematical models to understand the components of functional CCMs in model cyanobacteria and algae, with a particular area of interest in CCM modeling being the possibility of boosting crop productivity by engineering CCMs into crops which lack CCMs (Price et al. 2013; McGrath and Long 2014; Fei et al. 2022; Kaste et al. 2024). By developing modeling approaches to robustly describe CCMs in organisms where biochemical data is limited, such as extremophile algae, we can better understand how organisms survive environmental challenges. Here we add to these engineering efforts by modeling a heat-tolerant CCM with minimal components which offers unique possibilities for plant synthetic biology (Misumi et al. 2017). To draw robust conclusions about cellular characteristics which can support a CCM, we used state-of-the-art statistical methods to define the effects of model parameters on the predicted photosynthetic phenotype while limiting unwarranted a priori assumptions. We demonstrate an interdisciplinary modeling approach which efficiently sampled from large parameter spaces and identified features (e.g. compartment permeability, pH, enzyme characteristics) that determine the function and energy cost of a simple CCM. This adds a useful tool for compartmental photosynthetic modeling, and could facilitate effective use of models to inform experiments and rational engineering.

Some sets of model input parameters produced model outputs which met empirically based criteria for functional carbon concentration and efficient energy usage, and we identified input parameters which have substantial impacts on the model outputs. Overall, our model of a hypothetical biophysical CCM which requires minimal enzymes and anatomical features (Fig. 1) appears to represent a feasible CCM structure in C. merolae, which invites further research into the sources of environmental resilience in extremophile algae.

Figure 1.

Cross-section of model structure. This model describes fluxes (indicated by arrows and “V#” notation) and pools (indicated by molecular formulas) of a simplified dissolved inorganic carbon system (CO₂, ${\mathrm{HCO}}_3^{-}$) and of oxygen (O₂). Fluxes are as defined in Supplementary Methods section “Model fluxes.” Molecule pools can be present in several well-mixed compartments: the bulk external medium surrounding the cell, an unstirred boundary layer of medium around the cell, the cytosol, or a central stromal space of the chloroplast. Circles mark enzymatically-catalyzed fluxes. Compartments are not drawn to scale. PR = photorespiratory CO₂ release, R_L = respiration in the light. All fluxes are reversible and are assigned an arbitrary direction, except those fluxes which represent producing or consuming material.

Results and discussion

Rubisco kinetics demonstrated that C. merolae operates a CCM

In previous work, we determine that if C. merolae has rubisco kinetics similar to other red algae, then this alga must operate a CCM to maintain its measured photosynthetic efficiency. Alternatively, its measured photosynthetic efficiency could be explained by unprecedented rubisco kinetics, meaning enzyme properties favoring carbon-fixation over oxygen-fixation to an unprecedented degree (Steensma et al. 2023). Here we confirmed that C. merolae rubisco kinetics are similar to those of other red-type (Form 1D) rubiscos (Read and Tabita 1994; Uemura et al. 1997; Whitney et al. 2001). C. merolae rubisco had a strong affinity for CO₂ (low K_C), a poor affinity for O₂ (high K_O), and a slow carboxylation rate (low kcat_C) (Fig. 2). Consistent with other studies, kcat_C and K_C were higher when measured at increased temperature, while K_O was lower. Although K_O is in the denominator of rubisco specificity (S_c/o) and S_c/o decreases with increased temperature, in vitro K_O is observed to decrease with increased assay temperature in some species (Jordan and Ogren 1984; Uemura et al. 1997; Prins et al. 2016).

Figure 2.

Experimental data incorporated into the model. A, B). Response of net assimilation in C. merolae to A) CO₂ availability and B) light availability. Points are mean ± SE (n = 3), and parameters calculated from the data are indicated in the upper left corner of each plot as mean ± SE. Dashed lines indicate trend fits used to determine Michaelis–Menten constant of CO₂ fixation (K_C) and respiration in the light (R_L). The linear fit used to determine the CO₂ compensation point (Γ_CO2) is not pictured but is described in Methods. C) Kinetic properties of C. merolae rubisco. Rubisco turnover rate for CO₂ fixation (kcat_C), Michaelis–Menten constant of CO₂ fixation (K_C), and Michaelis–Menten constant of O₂ fixation (K_O) were measured at 25 and 45°C. Data are mean ± SE, n = 4.

These kinetics findings indicated C. merolae does operate a CCM, as C. merolae cells had higher affinity for CO₂ than C. merolae rubisco (8.71 ± 1.7 µM cell K_C vs. 24.9 ± 3.2 µM rubisco K_C at 45°C, P = 0.008 by two-sample t-test) (Fig. 2). This result adds to the evidence of a CCM in C. merolae (Zenvirth et al. 1985; Rademacher et al. 2017; Steensma et al. 2023).

Quantitative modeling showed that a hypothesized CCM can explain C. merolae's carbon-concentrating behavior

To explore how the C. merolae CCM may operate, we constructed a functional model of a CCM (Fig. 1). This model demonstrated that there were parameter sets consistent with the empirical literature that result in a functional CCM, despite the minimal model structure lacking structures like a pyrenoid or carboxysome (Fig. 3). Cyanobacterial CCM models have also supported reduction to a simple model with only two compartments from the cell membrane inwards (Mangan and Brenner 2014).

Figure 3.

Values of key model outputs. A) Parameter sets are organized into a 2-dimensional histogram according to their output values of Γ_CO2 and ATP per CO₂, with dashed lines indicating bounds for acceptable values of these outputs. The dataset for the figure was the 240,000 total parameter sets. However, 80 parameter sets (0.03% of the total) are not pictured in this panel, as they produced negative ATP per CO₂ values and could not be log-transformed. B) Percentages of parameter sets meeting various combinations of output criteria. Model output criteria and associated units are as defined in Methods: Definition of reasonable model output values.

Our results provided quantitative support for a CCM taking inorganic carbon from the environment solely through CO₂ diffusion into the cell without specialized compartments, which we term a “non-canonical” CCM due to its differences in structure and function from CCMs that have been characterized in detail. C. merolae has a different structure and environment than the “canonical” CCMs of Chlamydomonas reinhardtii and of model cyanobacteria, which allowed us to explore a biology and a parameter space which are different from those in previous CCM models.

Though there is speculation that extremophilic red algae may use a C₄-like CCM, it has been previously proposed that acidophile algae may accumulate carbon by a “bicarbonate-trap” or “acid-loading” mechanism similar to our modeled CCM (Gehl and Colman 1985; Fridlyand 1997; Gross 2000; Rademacher et al. 2016; Curien et al. 2021; Fei et al. 2022). Briefly, this mechanism would involve bicarbonate being concentrated for enzymatic action by bringing inorganic carbon speciation near equilibrium in near-neutral cellular compartments, since the predominant inorganic carbon species from pH ∼6 to ∼10 is the poorly-membrane-permeable bicarbonate.

Various facilitated CO₂ uptake mechanisms exist in CCM-containing organisms, such as the NAD(P)H dehydrogenase-I complexes in cyanobacteria and the periplasmic carbonic anhydrase (CA) system in algae (Fridlyand et al. 1996; Moroney et al. 2011; Price 2011). We here test a different model where inorganic carbon enters the cell solely by passive CO₂ diffusion into the cytosol, followed by the action of non-vectorial cytosolic carbonic anhydrase. In contrast to the well-studied cyanobacterial and algal systems, where growth under limiting CO₂ is supported by active bicarbonate uptake and the accumulation of cytosolic bicarbonate above equilibrium levels (Price and Badger 1989; Price et al. 2004; Duanmu et al. 2009), our model functions as a CCM without taking any bicarbonate from the environment.

Another unique feature of our model is the nature of the diffusion barrier surrounding rubisco. Cyanobacteria encapsulate rubisco in a proteinaceous shell called the carboxysome, which is thought to provide a diffusion barrier to CO₂ (Price et al. 2008). The model alga C. reinhardtii aggregates rubisco into an organelle called the pyrenoid, which in wild-type cells is surrounded by a starch sheath that may serve as a diffusion barrier. In contrast to the well-studied system of C. reinhardtii, there has been comparatively less investigation into algae which lack starch sheaths or lack pyrenoids entirely (Morita et al. 1999; Barrett et al. 2021). Thus, to broaden our knowledge of CCM anatomy, we modeled an arrangement where rubisco is diffuse within a series of concentric thylakoid membranes. This allowed us to further investigate whether membranes, which are thought to be highly permeable to CO₂ (Gutknecht et al. 1977; Missner et al. 2008), could impact carbon concentration, and how carbon concentration could function without a carboxysome or pyrenoid.

To investigate these and other features of interest, we used two strategies to deeply explore the model parameter space and ensure that our conclusions were robust. First, the model included our experimental data on gas-exchange and rubisco parameters central to photosynthetic efficiency (Fig. 2). Second, we developed a method for thoroughly assessing the model's sensitivity to the value of model parameters of interest. Specifically, we were interested in 19 of the 43 model parameters which were biologically interesting in relation to the function of a hypothetical C. merolae CCM and which were not well-characterized physical constants (Supplementary Table S1). We thus sampled input parameter sets with varying numbers for these parameters of interest. We sampled parameter sets through a Latin hypercube design (McKay et al. 1979) which enhanced analysis accuracy by mitigating sampling bias, as it produced parameter sets distributed throughout the 19-dimensional parameter space of interest. Then, each input parameter set was used to parameterize the model and to generate a set of outputs for analysis.

Some of the input parameter sets produced outputs consistent with a functional CCM with reasonable energy cost. Of particular interest were the parameter sets which met all the empirically based criteria for a realistic and functional CCM (criteria selection described in Methods Supplementary S1 Supporting Information). Of 240,000 parameter sets, 13,998 (6%) parameter sets fulfilled the two competing objectives of functional carbon concentration (corresponding to outputs of low Γ_CO2, high stromal CO₂, and low v_o/v_c) and efficient energy usage (corresponding to output of low ATP per CO₂) (Fig. 2).

The generated parameter sets allowed us to explore the tradeoffs associated with various features related to the CCM. For example, adding additional concentric thylakoids slightly improved carbon concentration by presenting barriers to CO₂ leakage out of the chloroplast, but incurred additional energy costs of carbon transport (Fig. 4, Supplementary Figs. S1 and S2). This is consistent with other modeling studies indicating that thylakoid membranes could affect inorganic carbon diffusion and with observations of pyrenoids surrounded by layers of thylakoids in hornworts (Thoms et al. 2001; Fei et al. 2022; Robison et al. 2024).

Figure 4.

Effect of select input parameters on key model outputs. A, B) Effect of model input parameter membranes (x-axis) on key model outputs. Distribution of parameter set outputs for each value of membranes is represented by a box plot overlaid on a violin plot. Shaded areas represent unacceptable values of outputs. A) Effect of membranes on model output Γ_CO2. B) Effect of membranes on model output ATP per CO₂ 80 parameter sets (0.03% of total) are not pictured in this panel, as they produced negative ATP per CO₂ values and could not be log-transformed. C, D) Effect on key model outputs where carbonic anhydrases or bicarbonate transport activity at the chloroplast boundary are removed from the model. Distribution of parameter set outputs for each scenario is represented by a box plot overlaid on a violin plot. Shaded areas represent out-of-bounds values of outputs. The same sampling of input parameter sets was run through models representing each scenario. C) Γ_CO2 in model scenarios where various model features removed, with an indication of how many parameter sets met output criteria in each scenario. D) ATP per CO₂ in model scenarios where carbonic anhydrases or bicarbonate transport activity at the chloroplast boundary are removed from the model. 6,991 parameter sets producing negative ATP per CO₂ values (0.6% of total) are not pictured in this panel. All panels: Model output criteria and associated units are as defined in Methods: Definition of reasonable model output values. 240,000 total parameter sets were graphed for each panel of the figure, except where otherwise noted. The plots pictured were created with geom_violin and geom_boxplot in the R ggplot2 library. geom_boxplot visualizes the median, two hinges marking the first and third quartiles of the data, and two whiskers extending to the largest and smallest values no farther than 1.5 interquartile ranges from the hinge.

Machine-learning-based surrogate models identified the parameters that most influence CCM efficiency

Like most mathematical models of photosynthetic systems, this model faced the challenge of drawing robust conclusions while using parameters which, although bounded by their relationship to physical processes, have substantial uncertainty (Supplementary Table S1). To model a system with limited biochemical data while not constraining input parameters to a greater degree than was supported by the literature, it was important to assess uncertainties which seemed likely to have substantial and interdependent effects on the model. For example, the input parameter describing the permeability of a lipid bilayer to CO₂ (Plip_CO2) has reported values ranging over several orders of magnitude (Supplementary Table S1). Furthermore, the effect of Plip_CO2 in the model depended on the value of other parameters, such as the number of lipid bilayers which pose a barrier to carbon moving between the stroma and cytosol (Membranes). Various sensitivity analyses are available for ordinary differential equation (ODE) models, but Plip_CO2 and similar parameters were unlikely to be satisfactorily explored by classical local sensitivity analyses, which involve tracking model outputs when individual parameters are varied by a set fraction of the parameter's original value. Therefore, to reveal which model conditions were necessary for the modeled CCM to function biologically, and to identify interesting directions for future investigation, we used statistical methods to identify impactful parameters and to identify which input spaces corresponded to target output ranges. These statistical methods involved training a surrogate machine-learning model on our CCM model inputs and outputs. Interpretations of this surrogate model identified which zones in the input parameter space contained the most combinations fulfilling output criteria (Fig. 5lower left), quantified how much each input parameter affected the prediction of outputs by the surrogate model (Fig. 5upper right), and visualized the response of model outputs to inputs (Supplementary Figs. S4 to S7).

Figure 5.

Statistical investigation of parameters affecting model output. Upper right bar plots: Mean absolute SHapley Additive exPlanations (SHAP) plots showing which inputs most affect each output criterion based on the 240,000 samples of the surrogate model. Lower left density plots: Density plots of parameter sets meeting all output criteria, based on the 240,000 samples of the surrogate model and organized by selected pairwise input parameters (input parameters pictured are those input parameters with high SHAP values for all output criteria). Scale bars by each density plot indicate relative densities, with darker areas indicating areas where more parameter sets meeting criteria occur. (Scales of color vary for each plot.) Parameters and associated units are as defined in Supplementary Table S1.

Some input parameters had little impact on model outputs with the tested input ranges. For these parameters, values from across the input range were evenly represented in the parameter sets meeting all output criteria. The parameters with relatively little impact on outputs included values related to carbonic anhydrase concentration and kinetics ([CA], CAkcat, Km_CO2, and Km_HCO3− for carbonic anhydrases), chloroplast pH, and values related to bicarbonate membrane permeability (Plip_HCO3−, Q10_PlipHCO3−, Fig. 5, Supplementary Figs. S4 to S8). While it is possible that these aspects of the CCM may become impactful if varied beyond the tested range (e.g. if engineering efforts produce carbonic anhydrase concentrations falling outside the range of literature values we used), these parameters did not emerge as particularly impactful in our exploration. Due to how fast the interconversion of inorganic carbon species by carbonic anhydrase is, the enzyme is likely capable of keeping inorganic carbon species close to their equilibrium concentrations across the range of values we explored for its kinetics. Given this, it is unsurprising that model outputs varied little with respect to carbonic-anhydrase-related parameters, even though the complete absence of these enzymes was deleterious (Fig. 4).

Other parameters were more constraining in the model, indicating their importance in producing a functional CCM. For example, six parameters appeared to impact all four of the target model outputs in the mean absolute SHapley Additive exPlanations (SHAP) plots: V_c, Vmax_pump, Km_pump, pH in the cytosol, PlipCO₂, and Membranes (Fig. 5). Sobolʹ analysis (Sobol′ 2001) of the surrogate model produced similar results (Supplementary Fig. S9). As might be expected in a model relying on a cytosolic bicarbonate trap followed by bicarbonate pumping, parameter sets that successfully and efficiently concentrated carbon tended to have cytosolic pH at or above the pH where bicarbonate predominates (cytosol pH above 6) and tended to have a lower ATP cost of pumping bicarbonate (low Pump_cost), as well as faster and higher-affinity bicarbonate pumps (high Vmax_pump, low Km_pump) (Fig. 5).

Other features enriched in parameter sets meeting output criteria were a cell radius in the middle of the input range (moderate Radius_cell) and a lower CO₂ membrane permeability (low Plip_CO2, Fig. 5, Supplementary Figs. S4 to S9). This suggested an important relationship between the volumes where metabolism occurs and the surface areas which present diffusion barriers between compartments. As the radius of the cell increases, CO₂ loss from R_L may overcome the ability of the cell to acquire carbon through passive diffusion into the cell. Conversely, as the radius of the cell decreases, less absolute bicarbonate pumping would be necessary to achieve high rubisco saturation, especially when rubisco is slow (low V_c). In low-radius scenarios, “over-pumping” bicarbonate could reduce energy efficiency.

In silico knockouts identified experimental targets for further characterization of the C. merolae CCM

The modeling also suggested interesting directions for investigating enzymatic components of the CCM. Alternative models with CCM enzymes removed (carbonic anhydrases or bicarbonate pumping not functional) were less likely to meet the criterion of a Γ_CO2 indicative of functional carbon concentration, but tended to have lower ATP per CO₂ cost than the model with all enzymes present (Fig. 4, Supplementary Figs. S1 and S2).

The modeled CCM functioned without fine details of cellular structure that support photosynthesis in other organisms, such as rubisco aggregation into an area smaller than the stroma, carbonic anhydrases with restricted distributions and directions (i.e. lumenal and vectorial carbonic anhydrases), recapture of mitochondrially respired CO₂, and perforations or interconnections in concentric thylakoids (Nevo et al. 2007; Rademacher et al. 2017; Barrett et al. 2021). Our work thus expands on previous models with detailed chloroplast geometry (Fei et al. 2022) by demonstrating that efficient carbon capture may occur in a simple case when rubisco and carbonic anhydrase are diffuse within a series of concentric thylakoid spheres. It may still be of interest to explore what chloroplast structures support photosynthesis in C. merolae and to investigate the biochemical and molecular basis for this non-canonical CCM.

Further applications of surrogate modeling and uncertainty quantification

More broadly, the statistical approach adopted in this paper represents an advance in metabolic and biochemical modeling. By training a surrogate model on the parameter space of mechanistic biological models, we can understand and account for high-dimensional uncertainty in model parameters. Metabolic modeling in general, especially complex metabolic modeling, has been highlighted as a particularly promising application of surrogate modeling, as metabolic modeling has biotechnological potential but is challenged by the complexity of metabolism and by the “trial and error” process which is often required to produce a working metabolic model (Gherman et al. 2023). Surrogate modeling has found uses in dynamic flux balance analysis and process modeling for bioprocesses (Mountraki et al. 2020; de Oliveira et al. 2021). Our work expands on these investigations by demonstrating what is to our knowledge the first application of surrogate modeling to ODE-based compartmental modeling of biological systems. Our methods may be particularly valuable for models that have poorly defined parameters or are extremely computationally expensive. For example, the implementation of surrogate modeling described here could alleviate current limitations in interpreting reaction-diffusion models and genome-scale metabolic models (Gherman et al. 2023). Even for our relatively-simple model, the run time for 240,000 simulations was several hours and required the use of a computing cluster. In contrast, surrogate modeling could be run locally on a laptop computer and was able to generate 240,000 predictions for all four outputs of interest in less than 10 s, easily creating a large dataset for analysis and allowing for precise sensitivity estimation. We compared this with Sobolʹ sensitivity analysis (Sobol′ 2001) performed with the original model with a sample size of n = 163,840, comparable to the number of parameter sets and outputs used to train the surrogate model. Despite the generation of these samples taking several hours of computation time, this approach yielded extremely imprecise and uninterpretable results, suggesting that substantially more computational investment would be necessary to achieve acceptably precise sensitivity estimates (Supplementary Fig. S10). With normalized root-mean-square error (NRMSE) below 1.5% in our validation (Supplementary Table S2), the computational gains associated with the surrogate modeling approach outweighed the near-negligible potential error introduced by an inexact surrogate.

Important considerations in any surrogate modeling application include the sample size required to train the model and limitations of surrogate models for out-of-sample predictions. Surrogates should be used cautiously for out-of-sample predictions, particularly in high-dimensional settings where training data are limited (Forrester et al. 2008). Regarding the sample size, early studies (Chapman et al. 1994; Jones et al. 1998; Loeppky et al. 2009) suggested using around 10d samples, where d is the input dimension, for building an accurate Gaussian Process (GP) surrogate model. GP surrogates are particularly effective for small datasets and provide uncertainty quantification, which is valuable for assessing the confidence of out-of-sample predictions (Gramacy 2020). If the desired accuracy is not achieved, one can improve the model by increasing the sample size through adaptive strategies such as active learning (MacKay 1992), which allows for more efficient use of additional data to further enhance accuracy. Recent studies have also provided guidance on determining the run size required for a GP surrogate to achieve a pre-specified level of out-of-sample prediction accuracy (Harari et al. 2018). In scenarios where high extrapolation performance is critical, one may consider using physics-informed surrogates, which tend to be more reliable in out-of-sample contexts. These surrogate models incorporate physical laws into their training process and offer improved performance for out-of-sample predictions, especially when physical dynamics are a key feature of a model. Examples of physics-informed surrogates include a manifold-constrained GP surrogate that adheres to an underlying ODE system (Yang et al. 2021) or Physics-Informed Neural Networks (PINNs) (Raissi et al. 2019).

Effective parameter exploration and analysis may generally be useful in confronting global challenges. Here, we used statistical sampling, surrogate modeling, and uncertainty quantification methods to investigate how a particular aquatic organism achieves the high photosynthetic efficiency that enables them collectively to be responsible for approximately half of global photosynthetic CO₂ consumption (Field et al. 1998). Similar modeling techniques may be applied effectively to any system: for example, as part of engineering efforts for bioproduction, crop resilience, and other goals, it may be useful to in silico determine which features of a system are essential or inflexible throughout ranges of interest before devoting resources to in vivo experimentation.

Conclusions

The extremophilic red microalga C. merolae operates a CCM, as evidenced by this alga having gas-exchange behavior which was not explained by its rubisco properties. Mathematical modeling suggested that this CCM could consist of a minimal mechanism. Robust parameter exploration and statistical analysis, aided by the use of a surrogate model, allowed us to quantify the sensitivity of our model to parameter uncertainties, identify important parameter interactions, and identify key determinants of CCM efficiency. Therefore, in addition to supporting the presence of a non-canonical CCM in C. merolae, our results shed light on what conditions must be met for this CCM to function and the essential elements of biophysical CCMs in general.

Materials and methods

Experimental data collection: gas-exchange measurements

Cyanidioschyzon merolae 10D was grown as cultures in Erlenmeyer flasks in 50 mL of medium containing 40 mm (NH₄)₂SO₄, 4 mm MgSO₄ · 7H₂O, 8 mm KH₂PO₄, 0.75 mm CaCl₂ · 2H₂O, 1 mL L⁻¹ Hutner's Trace Elements solution, and H₂SO₄ to pH 2.7 (recipe modified from MA2 medium recipe of (Fujiwara and Ohnuma 2017)). Cultures were maintained at 40°C under 100 µmol m⁻² s⁻¹ white light, with aeration by shaking at 100 rpm. For gas-exchange measurements, three replicate cultures of OD₇₅₀ 1.0 to 1.2 were resuspended in growth medium to OD₇₅₀ 0.6 (1.60 × 10⁷ to 3.68 × 10⁷cells/mL). Gas-exchange parameters were measured in a LI-6800-18 Aquatic Chamber (LI-COR Biosciences) at 45°C and with normalization to cell count data from a hemocytometer slide, following the procedures of Steensma et al. (2023) and with a protocol similar to the study by Davey and Lawson (2024).

Experimental data collection: rubisco kinetics measurements

We purified rubisco from C. merolae biomass with a protocol adapted from the studies Miyagishima and Wei (2017) and Orr and Carmo-Silva (2018). Approximately 60 g of biomass was lysed by freeze–thawing followed by mechanical homogenization. Crude rubisco was polyethylene–glycol-precipitated from clarified homogenate and purified by fast protein liquid chromatography (FPLC). FPLC fractions eluting under the major UV trace peak were assayed by SDS–PAGE and by spectrophotometric rubisco activity assay (procedures adapted from Kubien et al. (2010) and Carter et al. (2013) (Supplementary Fig. S3). Fractions containing active semi-pure rubisco were pooled, concentrated with a 100-kDa centrifugal concentration filter, and snap-frozen for use in rubisco assays.

Purified rubisco was used in four technical replicate experiments to determine catalytic properties as described previously in detail (Prins et al. 2016), with some alterations to protein desalting and activation: concentrated protein aliquots were first diluted with activation mix containing 100 mm Bicine-NaOH pH 8.0, 20 mm MgCl₂, 10 mm NaHCO₃, and 1% (v/v) Plant Protease Inhibitor cocktail (Sigma-Aldrich, UK). Rubisco was then activated at 45°C for 15 min before being used in ¹⁴CO₂ consumption assays at either 25°C or 45°C with CO₂ concentrations of 8, 16, 24, 36, 68, and 100 µM. To determine K_O, these CO₂ concentrations were combined with concentrations of 0, 21, 40, or 70% (v/v) O₂. kcat_C was determined using measurements with 0% O₂. An aliquot of the activated protein was used for determination of Rubisco active sites via ¹⁴C-CABP binding using the method of (Sharwood et al. 2016). For ¹⁴C-CABP binding, protein aliquots were incubated at 45°C for 15 mins with ¹⁴C-CABP to maximize binding, prior to application to Sephadex columns as previously described (Loganathan et al. 2016). Aliquots were also analyzed via SDS–PAGE alongside known concentrations of plant-type Rubisco to strengthen estimates of Rubisco content.

Model details

The hypothetical CCM described in this study (Fig. 1) was modeled as a set of well-mixed compartments and represented as a system of ordinary differential equations (ODEs). In this minimal biophysical CCM, carbon diffuses into the cell as CO₂, is trapped in the cytosol as bicarbonate by the action of carbonic anhydrase, and is pumped into the chloroplast, where a second carbonic anhydrase provides CO₂ around rubisco. No pyrenoid diffusion barrier is present, as neither a starch sheath nor a clear organized subcompartment for rubisco has been described in C. merolae. However, we accounted for potential effects of the concentric thylakoids which are present in C. merolae and many other aquatic photosynthetic organisms (Ichinose and Iwane 2017). CAs and bicarbonate transporters are essential components of known biophysical CCMs and thus essential components of a CCM model (Beardall and Raven 2020). These components (V4, V11, and V8) are discussed in more detail below.

The model geometry is based on the cellular structure of C. merolae as apparent in published micrographs of this alga (Kuroiwa 1998; Miyagishima et al. 1998; Toda et al. 1998; Itoh et al. 1999; Yagisawa et al. 2012, 2016; Ichinose and Iwane 2017; Reimer et al. 2017; Sato et al. 2017; Moriyama et al. 2018). The modeled cell and its boundary layer form a series of concentric spherical well-mixed compartments. The cell is enclosed by a lipid bilayer of radius $Radiu{s}_{cell}$. The cell contains a cytosol of radius $Radiu{s}_{cell}$ and a chloroplast stroma space of radius $0.25*$. The cell is surrounded by a medium boundary layer of radius $2*$, beyond which lies an infinite external medium. Though varying fluid dynamic conditions strongly impact the size of boundary layers such as gas surface films or phycospheres, these layers are reported to be on the order of magnitude of 1 cell radius (Guterman and Ben-Yaakov 1987; Seymour et al. 2017).

Molecules cross the boundary of the stroma space according to diffusion or transport equations. For flux calculations, the boundary consists of 1–7 lipid bilayers of negligible thickness that are evenly spaced from $0.5*$ to $0.25*$. This boundary structure represents the fact that the C. merolae chloroplast is surrounded by a chloroplast envelope and by approximately 4 to 6 thylakoids which appear as concentric circles or spirals in microscopy examinations (Ichinose and Iwane 2017). A range of possible transport scenarios (how many membranes molecules must cross when crossing between the cytosol and stroma, and how much energy this crossing costs) are captured by varying parameters Membranes and Pump_cost.

Diffusion through lipid membranes (V1, V6, V5, V7, V15) was described using estimates of conductivity of lipid membranes to the chemical species in question:

$$\begin{array}{c}{J}_{membranediffusion}=Conductivit{y}_X*({[X]}_A-{[X]}_B)\end{array}$$ (1)

Where Conductivity_x is the conductivity—in units of µm³/s—of chemical species X through a lipid bilayer, and [X]_A and [X]_B are the concentrations of that species on the two sides of that lipid bilayer. Diffusion between the medium boundary layer and bulk medium (V18, V19) was described as an analogous simple diffusion flux, with conductivity determined according to diffusion coefficients through water at the boundary layer thickness. Lipid permeability coefficients for CO₂ and ${\mathrm{HCO}}_3^{-}$ and the water diffusion coefficient for O₂ were sourced from the literature (Supplementary Table S1), and other necessary gas permeability and diffusion coefficients were determined from the literature values by Graham's law of diffusion:

$$\begin{array}{c}{\displaystyle \frac{r_1}{r_2}=\sqrt{{\displaystyle \frac{M_1}{M_2}}}}\end{array}$$ (2)

where the rates of diffusion r₁ and r₂ for two different ideal gases, here CO₂ and O₂, are related according to their two molar masses M₁ and M₂.

To describe diffusion of CO₂ (V5), ${\mathrm{HCO}}_3^{-}$ (V7), and O₂ (V15) through variable numbers of stacked thylakoid membranes, an overall conductivity through all of the layers was calculated as:

$$\begin{array}{c}OverallConductivity={\left({\displaystyle \sum _{i=1}^n}{(4\pi r_n^2*Conductivit{y}_X)}^{-1}\right)}^{-1}\end{array}$$ (3)

where r_n is the radius of the sphere formed by the nth thylakoid membrane. This overall conductivity value is then used in (E1) to describe the movement of a chemical species from the outer stroma into the inner stroma space, as shown in Fig. 1. We assume that small gas molecules diffuse easily around membrane proteins, so that the diffusion of CO₂ and O₂ through any modeled membrane is potentially impeded by increased path length, but is not impeded by CO₂ and O₂ passing through high-resistance protein material.

Spontaneous interconversion of CO₂ and ${\mathrm{HCO}}_3^{-}$, as in V2, V3, V9, and V10 (E4-5), was described using simple first-order kinetics, according to the rate constant of the dehydration (slower) step of the interconversion:

$$\begin{array}{c}{J}_{C{O}_{2}hydration}=k_2[C{O}_2]\end{array}$$ (4)

$$\begin{array}{c}{J}_{HC{O}_3^{-}dehydration}=k_{-2}[HC{O}_3^{-}][H^{+}]\end{array}$$ (5)

Note that CO₂ must first be hydrated to H₂CO₃, which is then deprotonated to yield the ${\mathrm{HCO}}_3^{-}$ ion. However, because the interconversion of ${\mathrm{HCO}}_3^{-}$ and H₂CO₃ is essentially instantaneous relative to the hydration–dehydration reaction, here we ignore the H₂CO₃ species and approximate the spontaneous interconversion as the hydration–dehydration reaction. It was observed in Mangan et al. (2016) that the substantially higher permeability of H₂CO₃ relative to ${\mathrm{HCO}}_3^{-}$, coupled with the rapid interconversion of these species, results in a greater permeability through lipid membranes of this joint H₂CO₃/${\mathrm{HCO}}_3^{-}$ pool than would be expected from ${\mathrm{HCO}}_3^{-}$ permeability alone. To account for this while accommodating the simplification of not including the H₂CO₃ species, we explored a range of possible lipid permeabilities to ${\mathrm{HCO}}_3^{-}$ and CO₂ that substantially overlaps with the range of inorganic carbon permeability values from Mangan et al. (2016).

The interconversion of CO₂ and ${\mathrm{HCO}}_3^{-}$ by carbonic anhydrase (V4, V11) was described as in the study by McGrath and Long (2014):

$$\begin{array}{c}{J}_{CA}={\displaystyle \frac{[CA]*C{A}_{kcat}*\left([C{O}_2]-{\displaystyle \frac{[HC{O}_3^{-}][H^{+}]}{K_a}}\right)}{K_m^{C{O}_2}+[HC{O}_3^{-}]\left({\displaystyle \frac{K_m^{C{O}_2}}{K_m^{HC{O}_3^{-}}}}\right)+[C{O}_2]}}\end{array}$$ (6)

where the K_a value is the overall K_a for the CO₂/${\mathrm{HCO}}_3^{-}$ system. This value is temperature-sensitive and was calculated using the R package seacarb package (Lavigne et al. 2019). Other potentially temperature-sensitive parameters receive temperature adjustments according to Q₁₀ or Q₁₅ factors as in von Caemmerer (2000). In C. merolae, CA inhibitors have not been shown to affect oxygen evolution, but it remains plausible that CAs are involved in photosynthesis, since genes homologous to CCM CAs show transcript increases in response to lowered CO₂ availability (Rademacher et al. 2017; Parys et al. 2021). Of the two putative CAs with the most dramatic transcriptional response to CO₂, one protein has a computationally-predicted chloroplast targeting sequence and has been fluorescence-localized between the mitochondrion and chloroplast, while the other protein has no predicted targeting sequence and has been fluorescence-localized in the cytosol (Rademacher et al. 2017; Steensma et al. 2023).

Carboxylation by rubisco (V12) was described with the assumption that CO₂ is limiting, as in Farquhar et al. (1980):

$$\begin{array}{c}{v}_c={\displaystyle \frac{Vma{x}_{carboxylation}[C{O}_2]}{\left([C{O}_2]+K_m^{C{O}_2}\left(1+{\displaystyle \frac{[O_2]}{K_m^{O_2}}}\right)\right)}}\end{array}$$ (7)

To estimate oxygenation (V13), we estimate v_c/v_o (carboxylation flux over oxygenation flux) from the CO₂/O₂ specificity (S_c/o) of rubisco and chloroplast CO₂ and O₂ concentrations (E8), and then use this to arrive at v_o.

$$\begin{array}{c}{\displaystyle \frac{v_c}{v_o}=S_{co}\left({\displaystyle \frac{[C{O}_2]}{[O_2]}}\right)}\end{array}$$ (8)

The pumping of ${\mathrm{HCO}}_3^{-}$ across the stack of thylakoid membranes by a bicarbonate pump (V8) was described by simple Michaelis–Menten kinetics:

$$\begin{array}{c}{J}_{HC{O}_3^{-}pump}=\left({\displaystyle \frac{V_{max}[HC{O}_3^{-}]}{K_m+[HC{O}_3^{-}]}}\right)(SurfaceArea)\end{array}$$ (9)

Concerning what is known about bicarbonate transport in C. merolae, it is difficult to identify bicarbonate transporters by homology (Price and Howitt 2011; Steensma et al. 2023). C. merolae would have minimal access to extracellular bicarbonate even if bicarbonate were substantially available in its acidic environment, as is evident from radiolabeling and from gas-exchange conducted at varying pH (Zenvirth et al. 1985; Steensma et al. 2023). Bicarbonate transport at the chloroplast or thylakoids is a key feature of biophysical CCMs (Price et al. 2008; Spalding 2008).

Photorespiratory CO₂ release (V14) and photosynthetic oxygen evolution (V16) were determined by the stoichiometry described in Supplementary S1 Supporting Information. Non-photorespiratory CO₂ release occurring during photosynthesis, known as respiration in the light (R_L) (Xu et al. 2021), was estimated from gas-exchange data according to a modified Kok method (V17). Assimilation was measured under sub-saturating light intensities and extrapolated to estimate CO₂ release in the absence of light (Fig. 2B). The resulting mean measured value of R_L was normalized to cell size for use in the model: we assume that the empirical measurement of R_L we obtained was, on a per-cell basis, characteristic of a C. merolae cell of a radius of 1 µm. Under the assumption that R_L should vary proportionally with cell volume, we normalized R_L as follows:

$$\begin{array}{c}{R}_{Lnormalized}=R_{Lmeasured}{\displaystyle \frac{Volume}{Volum{e}_{1um}}}\end{array}$$ (10)

ATP costs for the cell were estimated as:

$$\begin{array}{c}AT{P}_{total}=3v_c+3.5v_o+(J_{HC{O}_3^{-}pump}*Membranes*Pum{p}_{cost})\end{array}$$ (11)

where Membranes is the number of thylakoid stacks and Pump_cost is the assumed cost, in ATP, of pumping a single ${\mathrm{HCO}}_3^{-}$ ion across a lipid bilayer by the hypothesized pump.

A full list of all flux equations and the system of ODEs used to describe the system can be found in Supplementary S1 Supporting Information.

Definition of reasonable model output values

To ensure the model reproduced experimental results we measured, or obtained from the literature, experimental data to set acceptable bounds for the following model outputs: CO₂ compensation point (Γ_CO2), the ratio of ATP consumption flux to net CO₂ assimilation flux (ATP per CO₂), the steady-state CO₂ concentration in the chloroplast stroma (stromal CO₂), and the ratio of oxygen-fixation flux to carbon-fixation flux (v_o/v_c).

CO₂ compensation point (Γ_CO2)

We accepted Γ_CO2 values less than or equal to 2.70 µM, corresponding to no more than twice the mean measured value (Fig. 2).

Ratio of ATP consumption flux to net CO₂ assimilation flux (ATP per CO₂)

We accepted ATP per CO₂ values which were less than or equal to 25 and greater than 0. These bounds are supported by measured light response curves which indicated how much additional light absorption drives a certain amount of additional CO₂ assimilation (Fig. 2). We used these data to estimate how much additional ATP production drives an additional CO₂ assimilation, using the photon per ATP values for various light-reaction pathways (Walker et al. 2020), the cylindrical geometry of the gas-exchange sample chamber, and the measured density of cells in the sample. The resulting estimated values were as follows: 13.8 ± 2.19 ATP produced/CO₂ assimilated (mean ± SE, assuming cyclic and linear electron flow operating equally) or 17.4 ± 2.76 ATP produced/CO₂ assimilated (mean ± SE, assuming linear electron flow only operating). This suggests that ATP per CO₂ values of up to ∼25 are supported by photosynthetic electron flow. The lower bound of the acceptable range excludes a few parameter sets outputting negative ATP per CO₂, since these parameter sets represented particularly non-functional CCM scenarios with negative net assimilation values under ambient CO₂ conditions.

Steady-state CO₂ concentration in the chloroplast stroma (stromal CO₂)

We accepted chloroplast CO₂ concentration values of greater than or equal to the CO₂ concentration in the medium under 400 ppm CO₂ atmosphere, by the logic that a functional CCM should result in rubisco accessing a greater CO₂ concentration than is available from ambient medium.

Ratio of oxygen-fixation flux to carbon-fixation flux (v_o/v_c)

We accepted v_o/v_c values less than or equal to 0.3, based on data and models indicating that plants without CCMs are unlikely to achieve v_o/v_c less than approximately 0.3 (Bellasio et al. 2014).

Model optimization and estimation of simulated compensation point

Steady-state fluxes and metabolite concentrations were solved using odeint() from Python's SciPy library (Virtanen et al. 2020) with error control handled by maintaining the following inequality:

$$max\left({\displaystyle \frac{errors(y)}{erro{r}_{weights}(y)}}\right)\le 1$$

where errors is a vector of local errors against computed outputs y and error_weights is a vector of weights:

$$erro{r}_{weights}=toleranc{e}_{relative}*|y|+toleranc{e}_{absolute}$$

where tolerance_relative and tolerance_absolute are the relative and absolute tolerance values set in the odeint() solver. We use the default value for these tolerances from SciPy version 1.10.0. All simulations were verified to reach steady state (metabolite concentration solutions changing 0.01% or less from the previous value). An end time of sufficient length was chosen to ensure that simulations successfully reached steady state. The maximum number of step sizes allowed for each time point was manually set to 5,000 as this was found to allow our simulations to reach steady-state without optimization difficulties. Other optimization parameters, such as the maximum and minimum step sizes, were left at their default settings as well and controlled by the optimizer. Using these settings, 100% (240,000/240,000) of all simulations successfully reached a steady-state solution in all model architectures.

In order to characterize the response of key outputs and robustness of conclusions to a wide range of possible parameterizations of the model, we used Latin Hypercube Sampling (McKay et al. 1979) to explore 240,000 parameter combinations according to the bounds specified in (Supplementary Table S1). These simulations were run on Michigan State University's High Performance Computing Cluster. CO₂ compensation point estimates were generated for every parameter set by running the model at external CO₂ concentrations ranging from 0.0001 to 1000 µM, constructing a cubic spline from the resulting curve of net CO₂ assimilation vs. external CO₂ concentration, and identifying the root of this spline to find the compensation point.

Parameter exploration and surrogate model selection

In order to thoroughly explore the 19-dimensional parameter space in a computationally feasible way, we trained a surrogate machine-learning model on the mechanistic CCM model. By emulating the intricacies of the mechanistic model, surrogate modeling faithfully captures the dynamics of complex systems while alleviating the substantial computational costs associated with obtaining additional results from a mechanistic model. Surrogate modeling additionally gave us access to powerful statistical tools for machine-learning model analysis, including SHAP (Lundberg and Lee 2017) and partial dependence (PD) plots (Friedman 2001).

To identify the optimal surrogate model for parameter exploration, we compared four popular machine-learning models: eXtreme Gradient Boosting (XGBoost) (Chen and Guestrin 2016), Local approximate Gaussian Process (laGP) (Gramacy and Apley 2015), single-layer Neural Network (NN) (James et al. 2013), and Deep Neural Network (DNN) (Chen and Guestrin 2016). We collected a 240,000-sized dataset, where the outputs were simulated from the mechanistic CCM model at space-filling input locations. 90% of the data was used for training the surrogate, and the remaining 10% was used as the test dataset to validate the model performance. The dataset was divided into training and test sets using a random sampling approach. Specifically, we used the sample() function in R with a fixed seed. The evaluation of prediction performance was based on the RMSE:

$$RMSE=\sqrt{{\displaystyle \sum _{i=1}^{n_{test}}}{\displaystyle \frac{{(y_i-{\hat{y}}_i)}^2}{n_{test}}}},$$

where $y_i$ is the ith test output and ${\hat{y}}_i$ is the i -th predicted model output.

Model outputs had varying scales and degrees of skew, so to effectively compare prediction performance on different model outputs, an NRMSE was calculated. The NRMSE was calculated as the RMSE divided by $y_{max}-y_{min}$, where $y_{max}$ is the highest test output and $y_{min}$ is the lowest test output.

From the model evaluation (Supplementary Table S2), it appears that XGBoost outperformed other models for v_o/v_c and ATP per CO₂, and remained comparable for Γ_CO2 and stromal CO_2. As such, XGBoost was used as the surrogate model for further analyses.

The XGBoost model was trained using a max number of boosting iterations of 1,000 with the evaluation metric of the root-mean-square error. The laGP model used the nearest neighbor method for prediction. The NN model is a simple feedforward neural network with a logistic activation function $(1/(1+e^{-x}))$ for regression tasks. The error function used for the calculation of the error was the sum of squared errors. The threshold parameter for the partial derivatives of the error function as stopping criteria for the NN model was set to half the range of the target variable.

The DNN model consists of 2 hidden layers containing 64 and 32 units respectively, both using rectified linear unit (ReLU) activation functions max(x, 0). The DNN model was trained using the adaptive moment estimation (Adam) optimizer and mean squared error (MSE) as the loss function. The model was trained for 40 epochs, with the learning algorithm processing the entire training dataset 40 times. A batch size of 240 was used, indicating the number of samples processed before updating the model's internal parameters. Moreover, 20% of the training data was set aside for validation purposes during the training process.

Accession numbers

Sequence data relevant to this article can be found in the KEGG and Cyanidioschyzon merolae Genome Project v3 data libraries under accession numbers:

EC 4.1.1.39, gene IDs CMV013C and CMV014C (Cyaniodioschyzon merolae rubisco).

Author contributions (CRediT taxonomy definitions)

Conceptualization: A.K.S., J.A.M.K., C.L.S., Y.S.H., B.J.W. Data Curation: A.K.S., J.A.M.K. Formal Analysis: A.K.S., J.A.M.K., J.H., D.J.O. Funding acquisition: A.K.S., J.A.M.K., C.L.S., Y.S.H., and B.J.W. Investigation: A.K.S., J.A.M.K., and D.J.O. Methodology: all authors. Project Administration: A.K.S., J.A.M.K., C.L.S., Y.S.H., B.J.W. Resources: all authors. Software: A.K.S., J.A.M.K., J.H. Supervision: A.K.S., J.A.M.K., C.L.S., Y.S.H., B.JW. Validation: A.K.S. and J.A.M.K. Visualization: A.K.S. and J.H. Writing—Original Draft Preparation: A.K.S. Writing—Review & Editing: all authors.

Supplementary data

The following materials are available in the online version of this article.

Supplementary methods

Supplementary Table S1. Parameter values or ranges used in the model.

Supplementary Table S2. The test root-mean-square errors (RMSEs) and normalized RMSEs (NRMSEs) of four machine-learning surrogate models.

Supplementary Figure S1. Effect of model input parameter Membranes (x-axis) on CO₂ leakage from the chloroplast.

Supplementary Figure S2. Effect on key model outputs when bicarbonate transport or carbonic anhydrases (CAs) are removed from the model.

Supplementary Figure S3. SDS–PAGE analysis of rubisco preparation

Supplementary Figure S4. Partial dependence (PD) plots of first-order effects for Γ_CO2.

Supplementary Figure S5. Partial dependence (PD) plots of first-order effects for stromal CO₂.

Supplementary Figure S6. Partial dependence (PD) plots of first-order effects for v_o/v_c.

Supplementary Figure S7. Partial dependence (PD) plots of first-order effects for ATP per CO₂.

Supplementary Figure S8. Ranges of parameters in all parameter sets versus in parameter sets meeting all output criteria.

Supplementary Figure S9. Total sensitivity of outputs to each parameter as determined by Sobolʹ analysis of the surrogate model.

Supplementary Figure S10. Total Sobolʹ sensitivity indices calculated by analyzing a sample set of size 163,840.

Funding

Work in the laboratory of BJW is supported by Grant Number DE-FG02-91ER20021 from the Division of Chemical Sciences, Geosciences and Biosciences, Office of Basic Energy Sciences of the United States Department of Energy. Work in the laboratory of YSH was supported by Grant Number DE-SC0018269 from the United States Department of Energy (https://www.energy.gov/). A.K.S. and J.A.M.K. were additionally supported by a predoctoral training award from Grant Number T32-GM110523 from the National Institute of General Medical Sciences of the National Institutes of Health. J.A.M.K. received additional support from the National Science Foundation Research Traineeship Program, grant number DGE-1828149. J.H. and C.L.S. are supported by Grant Number DMS-2113407 from the National Science Foundation. The contents of this publication are solely the responsibility of the authors and do not necessarily represent the official views of the funding agencies.

Data availability

Data and model code used in this study can be accessed via GitHub: https://github.com/anne-steensma/Cmerolae_CCM_model.

Preprint servers

This manuscript was deposited as a preprint at BioRciv under CC-BY license (https://doi.org/10.1101/2024.04.12.589284).

Dive Curated Terms

The following phenotypic, genotypic, and functional terms are of significance to the work described in this paper: