Introduction

Predictions for the trajectory of our global environment are reliant on our ability to accurately model the climate now and into the future under changing temperature and CO₂ regimes. A central component of global modelling is the ability to predict the rate at which photosynthetic species are removing CO₂ from the atmosphere, especially extrapolations to changes in CO₂ fixing capacity in response to increasing temperature and atmospheric CO₂ concentration. For example, models need to capture the changes in CO₂ fixing capacity over the predicted future temperature increases of up to four degrees, accompanied by up to a tripling of CO₂ concentrations by the year 2100 (RCP 8.5) [1].

Rates of net CO₂ exchange display a curved response to temperature, with increasing rates of CO₂ fixation up to peak in activity (T_opt), above which rates decline [2–5]. This curvature has been accounted for in various ways. Farquhar and colleagues have fit a Gaussian (which is curved and symmetric about an optimum temperature) to the electron transport component of photosynthesis [6]. In accounting for the temperature response of net CO₂ assimilation in sweet potato, the limiting processes of carboxylation (V_cmax), electron transport (J_max) and phosphate regeneration are modelled as curves [7]. Medlyn developed a ‘peaked’ model, which incorporates the negative effect of enzyme damage at high temperatures, which is applied to the temperature dependence of V_cmax and J_max [4]. Further, the temperature response of net CO₂ assimilation rates has recently been described in a three parameter single quadratic [2]. Currently, incorporation of CO₂ fixation at the global scale typically models leaf physiology based on the equations developed by Farquhar and colleagues, parametrising the temperature response of V_cmax and J_max based on the fit of modified Arrhenius functions to a range of data [4,8] and incorporating smoothing functions to transition between limiting factors and allow for co-limitation [9–11].

Macromolecular rate theory (MMRT) has been developed to account for the curved temperature dependence of enzyme catalysed rates [12,13]. The basis of MMRT is the expansion of the Eyring-Polanyi equation [14–16] to account for the unusual properties the arise for enzyme catalysis due to the large size of enzymes – hence macromolecular rate theory. Specifically, enzymatic reactions undergo a narrowing of the conformational space as the reaction progresses from the enzyme-substrate complex to the tight binding enzyme-transition state complex [17]. As the enzyme-transition state complex has a reduced heat capacity, enzymatic reactions are associated with a negative activation heat capacity () [16,17]. The incorporation of a negative into the Eyring-Polanyi equation quantifies the degree of negative curvature of enzyme catalysed rates with temperature [14,16]. This accounts for the temperature optimum and decreases in rates at high temperature observed in enzyme catalysis, without the need to invoke enzyme denaturation [12,13].

Given that enzyme catalysed reactions are the underlaying driver of biological rates at increasing scales of complexity, MMRT has been applied to the temperature response of multiple biological systems. This has included the temperature response of rates for both in vitro and in vivo metabolic pathways [18], as well as various soil processes [15], leaf respiration rates [19] and net ecosystem photosynthesis and respiration [20]. At these increasing scales of biological organisation, the temperature dependence of rates for these various processes are well described by MMRT. The basis for this scaling from enzymes to metabolic pathways has been investigated, showing that the temperature dependent curvature of a metabolic pathway is dependent on the temperature response of the constituent enzymes [18]. This raises the possibility for the application of MMRT to describe the temperature dependence of the CO₂ fixation as an in vivo metabolic pathway. In terms of global scale models of CO₂ fixation, utilising a curved function like MMRT to describe the temperature response eliminates the need to incorporate smoothing functions between limiting factors [9–11]. Compared to the curved functions which are also used [2], this would concurrently extract information on the thermodynamics of the enzyme driving the process [13,18].

Here, we address this possibility by applying MMRT to describe the temperature response of portions from the CO₂ fixation pathway of increasing complexity. We find that the temperature response of isolated RuBisCO enzyme, as a large complex macromolecule, is fully accounted for by MMRT. We also find the temperature response of V_cmax and J_max, representing portions of the CO₂ fixation pathway measured in vivo, are well described by MMRT. We further extend this to apply MMRT to model the temperature response of net CO₂ fixation rates in the C₃ species sweet potato (Ipomoea batatas) leaves across concurrent changes in temperature and CO₂ concentration. We find that MMRT, in conjunction with limitations imposed by RuBisCO enzyme kinetics and gas solubility, describes the temperature dependence of net CO₂ exchange in sweet potato leaves with just three fitted parameters. By incorporating MMRT into this model, the curved temperature response of the enzymes catalysing the CO₂ fixation process is mechanistically accounted for. This describes net CO₂ assimilation rates over a 30-degree temperature and 360 ppm_(g) CO₂ range and models the curved response of rates to temperature and associated changes in curvature with altered CO₂ concentration. Overall, this accounts for the changes in CO₂ fixation rates due to these two critical environmental factors with a combination of biological and physical parameters in a minimal biophysical model, underpinned by enzyme thermodynamics. For global scale modelling, MMRT presents a promising tool for incorporating a mechanistic understanding to the curvature of the CO₂ fixation process, while also simplifying input parameters and maintaining model accuracy.

Model description

We firstly define an artificial state of saturating CO₂ (substrate), no O₂ (inhibitor) and ideal light and moisture. The rate of this reaction is defined as:

(1)

Where c is a constant and k_p is the rate constant for the CO₂ fixation reaction. At saturating substrate, the reaction rate is not dependent on substrate and in the context of laboratory-based experiments, we assume that the number of CO₂ fixing centres, c, is constant (over the course of the experiment). We seek to calculate the value for k_p across the temperature range, k_p(T). If k_p(T) is well defined, then the rates of net CO₂ exchange in the lab/field may be calculated directly using this function and can be incorporated into models.

We use an existing dataset to define k_p(T) as follows: the CO₂ assimilation rate under experimental conditions is a function of ck_p, the concentration of substrate available to RuBisCO (dissolved CO₂ and its observed binding constant (), the inhibitor concentration (dissolved O₂ ) and its observed binding constant (), and the degree of cooperativity associated with the enzyme catalysed reaction (the hill coefficient, n). This is a Michaelis Menten equation with a competitive inhibitor (as in the Farquhar–von Caemmerer–Berry V_cmax equation [11]) with the addition of a cooperativity term (n). These parameters are defined by Equations 2–4. Both and are calculated directly from the partial pressures of these gases at any temperature, simplifying the myriad of complex components related to the bioavailability of CO₂ into a simple physical constant applicable to C₃ plants which rely on passive CO₂ dissolution (Equation 6). Due to this, the model is not applicable to C₄/CAM plants, as the CO₂ concentrating mechanisms in these species circumvent the physical constraints of CO₂ dissolution.

(2)

(3)

Combining (2) and (3) gives Equation 4

(4)

For sweet potato, Equation 4 was parametrised by a fit of intercellular CO₂ partial pressure (C_i) vs rate data with high and low O₂ concentrations to define the response of RuBisCO to substrate and inhibitor (Fig 3) [7]. These data were fit over three temperatures to define the temperature dependence of the parameters. Final model fitting of Equation 1 was achieved with independent of temperature (0.094 ppm_(aq)), whereas was linearly dependence on temperature, as defined by Equation 5.

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Fig 1. The solubility of CO₂ and O₂ in an aqueous system.

(A) The change in solubility constants with temperature for CO₂ and O₂ (Equation 6). Both gases become less soluble with increasing temperature. (A insert) The ratio of CO₂ and O₂ solubility. With increasing temperature, CO₂ becomes proportionally less soluble compared to O₂. (B) The effect of temperature on dissolved gas concentrations of current atmospheric O₂ and three CO₂ concentrations.

https://doi.org/10.1371/journal.pone.0319324.g001

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Fig 2. The temperature dependence of rates for isolated RuBisCO (A), V_cmax (B) and J_max (C).

Rates are fit to MMRT (Equation 7), with the exception of RuBisCO from T. thyasirae, which is fit using MMRT with a temperature dependent (see S2 supplementary in S1 File). Fitted values are reported in S1 Table in S1 File. RMSE values for the fitting from A-C are given in the table to the top right of the figure.

https://doi.org/10.1371/journal.pone.0319324.g002

(5)

The data show positive cooperativity, setting an average hill slope of 2.24 over all temperature and O₂ treatments for the final model. For all data, aqueous CO₂ and O₂ concentrations were calculated using the known solubility constants for the specific gases across the temperature range ( and respectively) [21,22].

The temperature dependence of values are defined by Henry’s law and physical constants for CO₂ and O₂ (Equation 6 and Fig 1).

(6)

The maximum rate at a given temperature (k_p) is defined by MMRT (Equation 7), where is the Boltzmann constant, h is Planck’s constant, R is the ideal gas constant, T is temperature in Kelvin, and are the activation enthalpy and entropy at the reference temperature (T₀), and is the change in heat capacity associated with the reaction. For the analysis here, T₀ was set to 4 °C below the temperature at which the fastest rates were measured (303 K; 30 °C). While the exact value of T₀ does not influence the fitting, this approach of selecting T₀ is consistent with standard MMRT fitting practise.

(7)

Just three parameters are fitted in this equation: , and . Furthermore, and are linked and define the magnitude of rates. then defines the curvature of the temperature response.

The above model was fit using GraphPad Prism (GraphPad Software, La Jolla, CA, www.graphpad.com; S1 Supplementary in S1 File). Sensitivity analysis of this model was performed by assessing the model fit (R²) through stepwise alteration of the , slope of (Equation 5) and n parameters about the typical error range of these values (as given in S4–S9 Tables in S1 File).

Results

The temperature and CO₂ response of net CO₂ exchange

Given the parametrised temperature dependence of gas solubilities and RuBisCO binding constants, along with the curved response of enzymatic pathways [18], accounting for the full temperature and CO₂ dependence of net CO₂ fixation rates is possible. Temperature data under different C_i concentrations have previously been collected in sweet potato [7]. These data at three CO₂ concentrations are simultaneously fit with Equation 1.

Here we find these data can be simply modelled in terms of an intrinsic temperature dependent rate constant k_p(T) based on MMRT, the exponential decreases in gas solubility with increases in temperature, and the effects this has on CO₂ fixation versus photorespiration rates due to and values respectively (Fig 4). The complete dataset, representing an in vivo metabolic pathway over a 10–40 °C temperature span, three CO₂ concentrations and rates varying up to nine fold, can be simultaneously fit with just three parameters (Equation 7), these being the magnitude ( and ) and the curvature () of rates.

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Fig 3. The temperature dependence of RuBisCO binding constants in sweet potato.

(A-C) Rate versus aqueous CO₂ concentration curves for high (200 mbar O₂(g); red) and low (30 mbar O₂(g); green) O₂ concentrations at 10, 25 and 31 °C respectively. Aqueous O₂ concentrations are given as ppm_(aq) within individual graphs. Each temperature is fit with Equation 4, to fit both O₂ treatments simultaneously to gain binding constants for CO₂ () and O₂ (). All CO₂ and O₂ concentrations are corrected from C_i values to account for gas solubility at the given temperature. (D) The temperature dependence of binding constants for sweet potato, as determined in fits A-C.

https://doi.org/10.1371/journal.pone.0319324.g003

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Fig 4. The temperature and CO₂ dependence of net CO₂ exchange in sweet potato under optimal light saturation.

(A) Global fit of Equation 1 to temperature curves at varying CO₂ concentrations. The T_opt under each condition are labelled. R² for the global fit =  0.9724. Fitting parameters and further statistics are provided in S3 Table in S1 File. (B) Intrinsic curvature (Equation 7) extrapolated from the fitted data, representing maximum rates of CO₂ fixation in unlimited conditions (saturating CO₂, low O₂, optimal light and moisture). (C) Realised proportion of maximum rates (k_p) given the CO₂ and O₂ concentration in solution and binding constants to RuBisCO with temperature. Colouration is the same as panel A for the three CO₂ concentrations. The deviations away from 100% capacity quantify the physical and biophysical restrictions placed on the CO₂ fixation system.

https://doi.org/10.1371/journal.pone.0319324.g004

Sensitivity analysis indicates the model parameters are well defined by the data and are essential for the full fitting of the CO₂ and temperature response. The ability of the global fit to accurately model the 140 ppm_(g) CO₂ curve is highly sensitive to the input of the , slope of and n. For example, increasing by 10% alters the R² for fitting of the 140 ppm_(g) CO₂ curve from 0.76 to 0.41, reducing the predictive power of the model at these low CO₂ concentrations. Further data for the sensitivity analysis is given in S4–S9 Tables in S1 File.

Discussion

CO₂ fixation displays a curved response to changes in temperature, with roughly symmetric decreases in rates either side of an optimum [26]. This curvature is characteristic of a range of biological processes, from enzymes to metabolic pathways, organism growth rates and various ecosystem processes [12,15,18,20]. Across these scales, curvature is well described by MMRT (Equation 7). At the enzyme level, the temperature dependent curvature of rates is a consequence of the large decrease in heat capacity () upon progression from the enzyme-substrate complex to the enzyme-transition state. This decrease in heat capacity occurs for the enzyme bound reaction system as the enzyme tightly binds to the transition state species, resulting in a narrowing of the conformational space. A large negative defines the temperature dependence of the free energy barrier for a reaction (), expanding the Eyring-Polanyi equation to account for curvature in enzyme rates that is independent of enzyme denaturation [12,13]. Recently, MMRT has been extended to show that the temperature dependent curvature of an in vitro metabolic pathway is a function of the curvature of the enzymes contributing to the catalytic cascade [18]. The temperature dependence of leaf [19] and soil [15] respiration, representing in vivo metabolic pathways, has also been described by MMRT. Further, analysis of a the global FLUXNET dataset has shown that MMRT can simultaneously describe the temperature dependence of ecosystem photosynthesis and respiration [20]. Considering CO₂ fixation as an in vivo metabolic pathway raises the possibility the process may be described by MMRT, where the temperature response of the pathway is a function of the thermodynamics of the individual enzymes in the cascade. This would then define the inherent temperature response of CO₂ fixation based on the enzymes catalysing the process, upon which other temperature limitations would layer. Defining the curvature of the photosynthetic pathway with MMRT gives a theoretical basis to the curvature of the temperature response (based on the curvature of individual enzymes), allowing access to information about the thermodynamics of the enzymes underpinning the process. This fitting is mathematically equivalent to the three-parameter single quadratic fit that has previously been used for net CO₂ assimilation rates [19,27].

To test the applicability of the MMRT equation to the CO₂ fixation pathway, we fit the temperature response of isolated RuBisCO, as well as V_cmax and J_max from a range of sapling species (Fig 2, S1 Fig in S1 File). Across the range of these data representing isolated RuBisCO enzyme and portions of the in vivo pathway, MMRT fully accounts for the temperature response of the rates. Further, enzyme data from bacterial type II RuBisCO from Thiomicrospira thyasirae requires the extension to include the temperature dependence of , a parameter previously applied to high quality enzyme data over wide temperature ranges [16].

We further extended this analysis to develop a minimal model to fully account for net CO₂ exchange in sweet potato leaves. For this, the response of RuBisCO to changing concentrations of dissolved CO₂ and O₂ was defined from existing data (Fig 3). The binding constants of CO₂ and O₂ were parametrised with temperature based on an enzymatic competitive inhibition model. The binding constant for CO₂ (), is relatively independent of temperature at 0.094 ppm_(aq) across this temperature range (Fig 3D), although this binding constant could be expected to increase sharply at higher temperatures [28]. Given a typical CO₂ concentration and temperature (C_i of 250 ppm_(g) and 18 °C), CO₂ dissolves to a concentration of 0.19 ppm_(aq). Thus, CO₂ fixation rates are not substrate saturated under typical conditions and are highly sensitive to changes in CO₂ concentration about current atmospheric levels. The binding constant for O₂ is two orders of magnitude lower compared to CO₂, consistent with the biological function of the enzyme. However, the affinity for O₂ increases with increasing temperature. Due to this, the rate of photorespiration will increase with temperature as the relative binding affinities change.

To be available for fixation, CO₂ must dissolve from the gas phase in the intercellular space into the aqueous environment of the leaf mesophyll. The movement of CO₂ from the gas phase to the chloroplast is complex and involves both membrane and cytosolic conductances, many of which are also temperature dependent, variable between species and poorly understood [29–32]. Whilst cognisant of these complexities, for the purposes of this minimal model, these are simplified into a single solubility parameter which captures the broad effects of reduced aqueous CO₂ with temperature in a well defined physical constant. At all temperatures, the solubility of CO₂ provides a significant limitation on the substrate available to C₃ plants for fixation as the equilibrium strongly favours the gas phase (by a factor of 1400 at 20 °C). This equilibrium under a given set of conditions is a physical constant, defined by an equilibrium constant (K_eq). With increasing temperature, CO₂ solubility decreases exponentially, placing further limits on substrate availability (Fig 1; Equation 6). Incorporating the limitations on CO₂ supply with temperature via well-defined fundamental restrictions of CO₂ solubility greatly simplifies the temperature considerations for C₃ plants, removing reliance on the nuances of membrane and cytosolic conductances.

The rate of net CO₂ assimilation is complicated by the competing reaction of O₂ with RuBisCO (photorespiration). Photorespiration temporarily takes a portion of the RuBisCO pool out of CO₂ fixing capacity, and also requires the release (as CO₂) of a fixed carbon to recycle the products of two photorespiratory O₂ turn overs [33]. Net CO₂ assimilation rates are thus the rate of CO₂ fixation offset by the rate of CO₂ release by photorespiration and respiration. The proportion of carboxylation to oxygenation is partially dependent on the aqueous concentrations of CO₂ and O₂ (along with the relative binding constants of the two molecules). O₂ is less soluble than CO₂, however due to the larger proportion of O₂ in air, is more prominent in the aqueous phase. For example, under current atmospheric conditions of 210,000 ppm_(g) O₂ (21%) and 400 ppm_(g) CO₂, at 25 °C the gases dissolve to 4.7 and 0.25 ppm_(aq) respectively. O₂ solubility also decreases with increasing temperature, however due to a more gradual decrease in solubility, relative concentrations of dissolved O₂ compared to CO₂ increase with temperature (Fig 1A insert). Thus, the physical limitation of gas solubility places further constraints on CO₂ fixation at elevated temperatures due to the relatively greater solubility of O₂ compared to CO₂ and the effects this has on photorespiration rates [26,34–36].

By taking these effects into account, the full response of sweet potato across three different CO₂ concentrations and a 30°C temperature range is accounted for (Fig 4). Rate decreases from a maximum rate curve (‘intrinsic pathway curvature’, Fig 4B) and shifts in the T_opt of CO₂ fixing capacity are captured by the changes in gas solubility (physical constraints), and the effect these have of RuBisCO CO₂ saturation and photorespiration rates (biological constraints). This maximum rate curve represents the temperature profile of CO₂ fixation rates in an unconstrained system of saturating CO₂, vanishingly low O₂, as well as optimal light and moisture. As CO₂ concentration is lowered, rates over the whole temperature range drop as RuBisCO becomes less saturated, and photorespiration is higher as the CO₂:O₂ ratio is decreased. As CO₂ is reduced, CO₂ fixing capacity peaks at a lower temperature due to the lower temperature at which inhibitory ratios of CO₂:O₂ are reached when CO₂ is decreased and O₂ held constant (Fig 4C). This shifts the T_opt of CO₂ fixation from 38 °C to 25 °C between a CO₂ saturated system (Fig 4B) and 140 ppm_(g) CO₂. The expansion of this analysis to other species is currently limited as this full data set is only available in sweet potato, however the general scale and magnitude of these responses is evident in the literature. A T_opt shift of similar scale has been reported previously under both high CO₂ and low O₂ conditions in Agropyron smithii (Western wheat grass) [34], as well as drought stressed Triticum aestivum (winter wheat) operating under reduced stomatal conductance and C_i [37]. Such shifts in the temperature optimum of CO₂ fixation are critical to accurately capture in models of a climate likely to experience more regular and sever temperatures spikes and increased atmospheric CO₂ [1].

The capacity of this simple model to incorporate the recent advances in our understanding of the temperature responses of enzymes and metabolic pathways, as well as accounting for the curvature and changes in T_opt under different CO₂ regimes has value for global scale modelling. Given the intertwined effects of changing temperature and CO₂, encompassing the extensive finer details of CO₂ transport through the mesophyll into a simple solubility term offers great benefit for incorporation into global models, along with other major limiting factors (light, water, nutrient availability). Given the exponential decreases of gas solubility with temperature, this presents a critical component for capturing the subtleties of CO₂ transport in C₃ plants into a defined physical constant for understanding large scale CO₂ fixation over short time scales (in the absence of adaptation mechanisms). To fully realise the potential of this, further data is needed to define the range and variability of the biological parameters of the model (the binding and inhibition constants) across multiple C₃ species, collet data across a wider temperature range, and assess if biome specific average of these parameters are a possibility.

Currently, terrestrial biosphere models employ a range of approaches to model carboxylation, electron transport and phosphate regeneration rates in response to incident radiation, CO₂ and temperature (compositions of the major models are summarised in [38]). These models simulate rates based primarily off the equations by Farquhar [11] or Collatz [39]. In addition, these models incorporate a rate-limiting selection between the three processes, and often a smoothing function to even out transitions between the limiting processes and allow for co-limitation [38]. By comparison, the approach presented here fits these limiting processes in one function (Equation 4), simply capturing the net effect of CO₂ assimilation rates of these three rate-limiting processes and transitions, along with the effects of O₂. This effectively halves input parameters and removes the need for smoothing functions, reducing model complexity while maintaining output accuracy.

CO₂ fertilisation has been presented as a mitigating process to rising atmospheric CO₂ concentrations [40]. The CO₂ fertilisation effect proposes that elevated CO₂ concentrations stimulate higher rates of carbon fixation, increasing atmospheric carbon removal to reverse some of the impact of increased anthropogenic inputs. However, when coupled to rising temperatures, this effect will be reduced due to the decreased solubility and bioavailability of the CO₂ at higher temperatures. For example, from 1991–2015 atmospheric CO₂ has increased by 40 ppm_(g), however the concurrent global average temperature rise of 0.5 °C mitigates the effect of this rise in terms of dissolved CO₂ bioavailable to plants (per the temperature dependence of CO₂ solubility; Equation 6). Over this period, no evidence of CO₂ fertilisation is measurable at a global scale [20]. Future predictions up to the year 2100 indicate that CO₂ rises (to 700 ppm) are significantly steep compared to projected temperature rises of three degrees to increase dissolved CO₂ overall from 0.34 to 0.53 ppm_(aq) (based on the RCP 6.0 scenario) [41]. This suggests that, in the absence of other major limiting factors [42], increased CO₂ concentrations will stimulate increased CO₂ removal. A full understanding of the extent to which CO₂ fertilisation will mitigate atmospheric CO₂ increases involves consideration of CO₂ solubility with temperature, and species-specific data such as RuBisCO binding and inhibition constants to predict wide scale responses to these changes. The analysis presented here provides a simple framework for this modelling, allowing the characterisation of CO₂ fixation rates of C₃ plants to concurrent changes in temperature and CO₂ concentration.

Conclusions

Here, we show that MMRT can successfully account for the intrinsic curvature of the CO₂ fixation pathway with temperature. This curvature is well described at the enzyme (RuBisCO), process (V_cmax and J_max), and full biochemical pathway level. This presents MMRT as a valuable tool for the incorporation into global scale models to account for the temperature response of the enzymes driving the CO₂ fixation pathway by incorporating fundamental enzyme kinetics. Here, the ‘real world’ limitations of substrate solubility and photorespiration have been successfully incorporated to account for reductions in carbon fixation potential under limited CO₂ in the C₃ species sweet potato. Further work is required to assess the range and variability of the parameter set determined here in a wider variety of C₃ plant species, and how other factors such as light, water and nutrient limitations layer on top of the underlaying enzymatic responses defined here. The model presented here incorporating MMRT along with the limitations imposed by substrate availability and the effects this has on enzyme rates illustrates the potential of this approach in capturing the nuance of temperature-based curvature in a complex metabolic system with a simple model based on fundamental enzyme behaviour. This provides a framework, based on the thermodynamics of enzyme activity, for building other limiting processes of the photosynthetic process onto, as a basis for incorporation into global scale models.

Supporting information