Theoretical analysis of a temperature-dependent model of respiratory O2 consumption using the kinetics of the cytochrome and alternative pathways.

Temperature dependence of plant respiratory O2 -consumption has been empirically described by the Arrhenius equation. The slope of the Arrhenius plot (which is proportional to activation energy) sometimes deviates from a constant value. We conducted kinetic model simulations of mitochondrial electron flow dynamics to clarify factors affecting the shape of the Arrhenius plot. We constructed a kinetic model of respiration in which competitive O2 -consumption by the cytochrome pathway (CP) and the alternative pathway (AP) were considered, and we used this model to describe the temperature dependence of respiratory O2 -consumption of Arabidopsis. The model indicated that the electron partitioning and activation energy differences between CP and AP were reflected in the slope and magnitude of the dependent variables of the Arrhenius plot. When the electron partitioning and activation energies of CP and AP were constant with temperature change, our model suggested that the Arrhenius plot would be almost linear. When the electron partitioning or activation energy of CP, or both, rapidly changed with temperature, the Arrhenius plot deviated from linearity, as reported in previous experimental studies. Our simulation analysis quantitatively linked the kinetic model parameters with physiological mechanisms underlying the instantaneous temperature dependence of plant respiration rate.

Introduction

In the respiratory chain of plant mitochondria, complex IV (cytochrome c oxidase; COX) and the alternative oxidase (AOX) pass electrons to oxygen (O2) (Fig. 1). The cytochrome pathway (CP) includes complex III, cytochrome c and COX; the alternative pathway (AP) includes AOX. These two pathways share a common component – the branching point between them, identified as ubiquinone (UQ) (Del‐Saz et al., 2018). Reported observations support the ‘electron competition model’ that CP and AP competitively consume electrons (Ribas‐Carbo et al., 1995; Del‐Saz et al., 2018). Bypassing CP by partitioning electrons to AP does not contribute to the proton electrochemical gradient, and it decreases respiratory ATP production by approximately two‐thirds. Although AP seems to be energetically wasteful, it contributes to the thermogenic metabolism of some highly specialized flowers (Seymour, 2001), and in many nonthermogenic plants, rapid upregulation of AOX decreases the concentrations of reactive oxygen species under various environmental stresses (Vanlerberghe & McIntosh, 1997; Moore et al., 2002; Millenaar & Lambers, 2003). Alternative oxidase most likely represents an adaptive feature of sessile plants. The responses of these two O2‐consuming pathways have intrigued many researchers studying the physiological responses of plants to their living environments.

Fig. 1

Electron flow in the respiratory chain. Blue arrows indicate electron flow, and red arrows indicate oxygen reduction to water in AP and CP. AOX, alternative oxidase; COX, cytochrome c oxidase; e−, electron; FADH2, flavin adenine dinucleotide; NADH, nicotinamide adenine dinucleotide.

The O2‐consumption rate is a convenient index for assessing electron flow in the respiratory system. The rate of consumption of O2 by the mitochondrial respiratory chain can be explained by a chemical reaction in which O2 is reduced to water. The rate of a chemical reaction is controlled by the energy barrier (i.e. activation energy E) between the reactants (O2) and the products (H2O). The activation energy can be calculated from the instantaneous temperature dependence of the reaction rate, which is the slope of the Arrhenius plot given by the following equation:where k is the rate constant of the reaction (s−1), A is the pre‐exponential factor (s−1), which is often called frequency factor, E is the activation energy (J mol−1), R is the gas constant (8.314 J mol−1 K−1) and T is absolute temperature (K). In previous studies, the following Q 10‐model has often been used for the instantaneous temperature dependence of plant respiration rates (Tjoelker et al., 2001; Atkin & Tjoelker, 2003):where r is the rate of respiration at a certain temperature T and r is the rate at a temperature 10ºC lower than T (T). Lloyd & Taylor (1994) have suggested a modified Arrhenius equation for plant respiration (Eqn 3), in which the instantaneous temperature dependence of the plant respiration rate is quantified by a parameter E, which is assumed to be constant.

However, in some cases, Arrhenius plots of experimentally measured respiratory O2‐consumption and CO2 efflux rates deviate from linearity. This deviation indicates that the E for plant respiration is not necessarily constant and could be temperature dependent. Kruse & Adams (2008) have used a second‐order polynomial approximation to describe the instantaneous temperature dependence of E:where E (T) is the temperature dependence of the plant respiration rate at the reference temperature and δ(T) is a factor that accounts for the dynamic response of E to temperature. The integrated value of the right side of Eqn 4 over a distinct temperature interval is called Cap(r) and is considered to be a measure of respiratory capacity (Kruse et al., 2011). This model fits the experimental data in nonlinear Arrhenius plots well (Kruse et al., 2018), and calculating the values of E (T), δ(T) and Cap(r) enables the exploration of respiratory properties (Noguchi et al., 2015). Heskel et al. (2016) have analysed a large number of measurements of leaf CO2 efflux rates for 231 species from 7 biomes by using a second‐order log‐polynomial (LP) model, and they have estimated the coefficients needed to predict leaf CO2 respiration–temperature relationships. In the LP model, the quadratic coefficient, c, represents the temperature dependence of the slope of the loge(r) vs T relationship (O’Sullivan et al., 2013). However, few details are known about the mechanism that determines the instantaneous temperature dependence of E or how the factors δ(T) in the modified Arrhenius model and c in the LP model should be interpreted.

Hobbs et al. (2013) have advocated macromolecular rate theory (MMRT) to explain the instantaneous temperature dependence of Arrhenius plots in enzyme‐catalysed biochemical reactions. The theory is that the heat capacity (C p) for the enzyme–transition state complex is generally lower than that of the enzyme–substrate complex, which is not as rigid, and the difference in those heat capacities, ΔC p ‡, is not negligible in biochemical reactions because of the high molecular weight of enzymes. On the basis of MMRT, the natural logarithm of the rate constant of the reaction (k) is given by the following equation:where k B and h are Boltzmann’s and Planck’s constants, respectively, and and are the differences in enthalpy and entropy between the ground state and the transition state, respectively, at reference temperature T.

This model clearly accounts for the fact that the slope of the Arrhenius plot of enzyme‐catalysed biochemical reactions can deviate from linearity (Schipper et al., 2014; Arcus et al., 2016). Also, MMRT can explain the short‐term temperature dependence of leaf CO2 efflux rates as well as the LP model (Liang et al., 2018). The MMRT can therefore be applied to respiratory O2‐consumption in plants.

In plant respiratory systems, respiratory CO2 is produced mainly by the pyruvate dehydrogenase complex and the TCA cycle, and its efflux is regulated by various cellular processes. By contrast, plant O2‐consumption is mediated by two pathways, CP and AP, and the temperature dependence of the O2‐consumption rate should therefore be determined by the kinetics of CP and AP. Kruse et al. (2011) have surveyed reports on the instantaneous temperature dependence of plant O2‐consumption rates and have concluded that, in most cases, E depends on which pathway – CP or AP – dominates the electron flux. The electron partitioning between CP and AP is regulated by the collision frequency at each enzyme and the activation energy for each pathway. A change in electron partitioning is therefore a possible explanation for the deviation from linearity of some Arrhenius plots of plant O2‐consumption rates. Kerbler et al. (2019) have found that the instantaneous temperature dependence of ATP synthase hydrolysis differs from that of NADH oxidation in isolated mitochondria of Arabidopsis. This result suggests that the rate‐determining step within the CP, which consists of multiple components, can vary, and a change of the rate‐determining step with temperature change may cause a change in the E of the CP and lead to nonlinearity of the Arrhenius plot of O2‐consumption rates. However, it is still unclear how two factors – the temperature dependence of the E of CP and electron partitioning between CP and AP – quantitatively affect the nonlinearity of the Arrhenius plot of O2‐consumption rates.

In this study, we constructed a kinetic model of plant O2‐consumption rates that took into consideration competitive O2‐consumption by CP and AP. We used the kinetic model to describe the instantaneous temperature dependence of respiratory O2‐consumption. We conducted several simulations to identify how competitive O2‐consumption could affect the shape of the Arrhenius plot of O2‐consumption rate. Next, we fitted the kinetic model to the experimental leaf O2‐consumption‐rate data, and we examined what parameters in the kinetic model were most responsible for the nonlinearity of the Arrhenius plot of O2‐consumption rates. Finally, we discuss the physiological mechanisms underlying the instantaneous temperature dependence of respiratory O2‐consumption on the basis of the kinetic model.

Materials and Methods

Temperature dependence of the rate of oxygen consumption by plant mitochondria

O2 reduction by an enzyme can be expressed by the following equation, based on Michaelis–Menten kinetics:where [E] is the concentration of the active enzyme, [EO2] is the concentration of the O2–enzyme complex in the transition state, and the k parameters are the rate constants corresponding to the reactions indicated by the arrows.

When electrons are continuously supplied in plant cells from the upstream citric acid cycle and the CP and AP simultaneously share electrons, the [O2] dynamics in the measurement system can be expressed as follows. (Supporting Information Notes S1 contains a detailed description.)where [E]0cox is the concentration of the active components of CP (complex III, cytochrome c and COX) in the measurement system, [E]0aox is the concentration of active AOX, and α is the ratio of [E]0aox to [E]0cox. The parameter K is the observed rate constant for the rate of O2‐consumption in the measurement system and is defined in Eqn 8. The Michaelis constants for CP (km cox) and AP (km aox) are defined by Eqns 9 and 10, respectively.

The relationship between K–1 and [O2]–1 is expressed as follows:

We can therefore calculate (αk2aox + k2cox) from the reciprocal of the y‐intercept of the plot of K–1 vs [O2]–1 – that is, by substituting zero for [O2]–1 in Eqn 11, or by measuring K when [O2] is much greater than km cox and km aox (i.e. the reaction rate is a maximum). More details are provided in Notes S1.

On the basis of the Arrhenius equation (Eqn 1), the instantaneous temperature dependence of (αk2aox + k2cox), which can be measured experimentally, is expressed as follows:where A cox and A aox are pre‐exponential factors related to CP and AP, respectively, ΔE is the difference between the activation energies associated with O2 reduction by CP (E cox) and AP (E aox), and ɛ is the AP‐contribution term expressed by Eqn 13.

The apparent y‐intercept of the plot of loge(αk2aox + k2cox) vs T –1, loge A overall, can be obtained from Eqn 14 (derived by substituting zero for T –1 in Eqn 12):

The apparent slope (i.e apparent activation energy) can be obtained by differentiation of the right side of Eqn 12 with respect to T –1, as follows. (Details are provided in Notes S1).where γ is the AP‐contribution term expressed by Eqn 16.

In this study, the value of α reflected the relative amounts of active enzymes of CP and AP. Two other metrics, ɛ and γ, reflected the additional effects of the AP on the O2‐consumption rate and the slope of the Arrhenius plot (i.e. activation energy).

Simulations

With these models, we examined whether two factors, α, the ratio [E]0aox : [E]0cox, and ΔE, the difference between E cox and E aox, affected the nonlinearity of the Arrhenius plot. We then simulated the instantaneous temperature dependence of plant O2‐consumption rates under various combinations of α and ΔE in the temperature range 10–40°C. In accordance with the results of previous studies (Kruse et al., 2011), we set E aox to be lower than E cox (ΔE > 0). We first examined the effects of α and ΔE, not only on O2‐consumption rates but also on the AP‐contribution terms, ɛ and γ. We performed the first three of the following simulations by using various combinations of ΔE (5, 10, 20 and 40 kJ mol–1) and α (0.1, 0.3 and 0.5) and assuming ΔE and α to be independent of temperature. In simulations (4) and (5), α and ΔE, respectively, were functions of temperature.

Instantaneous temperature dependence of plant O2‐consumption rate (Eqn 12)

Instantaneous temperature dependence of the AP‐contribution term ɛ (Eqn 13)

Instantaneous temperature dependence of the AP‐contribution term γ (Eqn 16)

Instantaneous temperature dependence of plant O2‐consumption rate on the assumption that α gradually changed from 0.0 to 0.5 in response to temperature changes.

Instantaneous temperature dependence of plant O2‐consumption rate on the assumption that the activation energy of CP, E cox, gradually changed from 25 to 30 kJ mol–1 in response to temperature changes.

In simulations (1) and (3), A aox/A cox was equated to α under the assumption that the collision frequency (quantified by the pre‐exponential factor A) corresponded to the concentration of active enzyme, either [E]0cox or [E]0aox. In simulation (1), the intercept loge A was set to 28, and the activation energy of AP (E aox) was set as 20 kJ mol–1.

Next, we validated the effect on the curvature of the Arrhenius plot of an abrupt change of α or ΔE due to a change in temperature. When ΔE changed with the measurement temperature, we assumed that the temperature dependence of CP could vary for the following two reasons. First, CP consists of multiple components, including complex III, cytochrome c and COX; we therefore assumed that the activation energy would vary as a function of the status of each component. In addition, the CP components interact with each other and form an active super‐complex that functions as an oxidative phosphorylation system (Dudkina et al., 2010; Milenkovic et al., 2017). The activation energy of CP could thus vary because of the large heat capacity of the components in the context of MMRT. By contrast, we fixed the activation energy of AP because electron transport through AP is performed by a single small enzyme, AOX. Changes are therefore less likely in E aox than in E cox. However, if the AOX structure is drastically changed by its activation state or the surrounding conditions, or both, E aox may be changed. In this study, we fixed E aox to simplify the simulations. We performed the following two simulations on the assumption that either α or ΔE could change in response to instantaneous temperature changes. In simulations (4) and (5), the intercept loge A was set to 28, and the activation energy of AP (E aox) was set as 20 kJ mol–1. In simulation (4), ΔE was set as 10 kJ mol−1.

The simulations were performed using R v.3.6.2 (R Core Team, 2019).

Measurement of leaf O2‐consumption rate and fitting the kinetic model to experimental data

Experimental O2‐consumption rates were measured with leaves of Arabidopsis thaliana (Fei‐0) for the Arrhenius plot. Arabidopsis seeds were sown in plastic pots filled with a water‐saturated, 1 : 1 mixture of soil (Pro‐Mix HP, Premier Tech Horticulture, Pointe‐Lebel, Quebec, Canada) and vermiculite (Fukushima Vermi Ltd, Fukushima, Japan). After being incubated at 4ºC in the dark for 5 d, plants were grown for 2 wk at 15ºC or 25ºC, 60% relative humidity, and 100 µmol quanta m–2 s–1 of visible light for 10 h d–1 in growth chambers. From that time onward, the plants were fertilised twice a day until the end of the study with a Hyponex nutrient solution (N : P : K = 6 : 10 : 5) (HYPONeX Japan, Osaka, Japan) diluted 1 : 2000 with water. After 2 wk of cultivation, individual leaf O2 respiration rates were measured for five randomly selected seedlings at each growth temperature. For each seedling, leaves were detached into a 50‐ml glass vial, and the O2‐consumption rate was measured with a fluorescence O2 sensor (FDO925; Xylem Analytics, Freistaat Bayern, Germany) in the dark at 15, 20, 25, 30 and 35ºC. After the O2‐consumption rate had been recorded at each temperature, the sample in the vial was retrieved and dried at 80ºC until the weight was constant. The final weight was recorded. The O2‐consumption rate was expressed on this dry‐weight basis.

The modified Arrhenius model (Eqn 4) was fitted to the experimental data to obtain the curvatures of the Arrhenius plots, and then the kinetic model (Eqn 12) was fitted to the curves to examine the effects of α and E cox on the shape of the Arrhenius plots. Although Eqn 12 has five variables, three variables (E aox, A aox, and A cox) were assumed to be constant, and thus the model included two independent variables, α and ΔE. The fitting exercises were performed by letting α vary in the range 0.1–2.0 as a function of the measurement temperature on the assumption that E aox and E cox were independent of temperature, and letting E cox vary as a function of the measurement temperature on the assumption that α and E aox were both independent of temperature. E aox was set to 15 kJ mol−1 or 20 kJ mol−1, and E cox was chosen subject to the constraint that E cox > E aox. The intercept, loge A, was set to 18 because the Arrhenius plot obtained when only CP was engaged (i.e. α = 0) was lower than the observed curves. The simulations were performed using R v.3.6.2 (R Core Team, 2019).

Results

Simulation results when α and ΔE were constant, independent of measurement temperature

The instantaneous temperature dependence of plant O2‐consumption rates are potentially influenced by both CP and AP. According to Eqn 12, the temperature dependence of the O2‐consumption rate can be obtained by adding the AP‐contribution term ɛ (Eqn 13) to the Arrhenius equation obtained when only CP is engaged (i.e. α = 0).

The AP‐contribution term ɛ is always positive (i.e. the logarithm of ). Thus, as long as electrons flowed to AP, the Arrhenius plot of plant O2‐consumption rate was always above the plot obtained when only CP was engaged (i.e, α = 0; black solid line, Fig. 2a–d). Here, we assumed that E aox < E cox on the basis of previous reports that the activation energy of CP is often, but not always, larger than that of AP (Kruse et al., 2011). If, instead, the activation energy of AP was larger than that of CP (E aox > E cox), then as long as electrons flowed to AP, the Arrhenius plot would be lower than the plot when only CP was present (not shown here).

Fig. 2

Simulated Arrhenius plots, based on the assumption that ΔE and α are constant across the temperature range (a–d), and there is instantaneous temperature dependence of the AP‐contribution term ɛ (e–h). The simulation was conducted with various combinations of ΔE (5, 10, 20 and 40 kJ mol−1) and α (0.1, 0.3 and 0.5). E aox was set to 20 kJ mol−1. In (a–d), the black line indicates the Arrhenius plot when α = 0 (no electron flow to AP). The temperature dependence of ɛ is zero when α = 0. A cox, pre‐exponential factors related to CP; α, the ratio [E]0aox : [E]0cox; [E]0aox, the concentration of active AOX in the measurement system; [E]0cox, the concentration of active CP in the measurement system; ΔE, the difference in activation energy between O2 reduction by CP (E cox) and by AP (E aox); ɛ, the AP‐contribution term in Eqn 13.

On the assumption that α and ΔE were constant while the temperatures were being varied, the AP‐contribution term ɛ was negatively correlated with temperature (i.e. positively correlated with T –1 in Eqn 13, Fig. 2e–h). Furthermore, ɛ became more sensitive to temperature as ΔE increased (Fig. 2e–h). Fig. 3 shows the slope deviation, γ, for several combinations of α and ΔE. According to Eqn 16, γ is defined as the product of ΔE and the temperature‐dependent term . Because the temperature‐dependent term ranges from 0 to 1, γ always falls between 0 and ΔE. The value of γ approaches ΔE as the relative weight ratio of the active AOX enzyme and electron flow to AP increase – that is, as A aox and α increase (Eqn 16). If E aox < E cox, the fact that γ > 0 indicates that electron flow to AP has the effect of decreasing the slope of the Arrhenius plot (Eqn 15). Because γ depends on temperature, γ can cause the apparent overall activation energy to depend on the instantaneous temperature. However, the effects of temperature on γ in the simulations, which were based on realistic situations, were extremely small (Fig. 3), and Arrhenius plots of O2‐consumption rate vs temperature were almost linear (Fig. 2a–d). As shown in Eqn 14, the apparent y‐intercept of the Arrhenius plot, which can be interpreted as a collision frequency, can be expressed in terms of α and the pre‐exponential factors A for both CP and AP. The implication is that the apparent y‐intercept can be used as a metric of the contents of the CP and AOX or O2 accessibility to these terminal enzymes, or both.

Fig. 3

Instantaneous temperature dependence of the AP‐contribution term γ under various combinations of α (0.1, 0.3 and 0.5) (a–c) and ΔE (5, 10, 20 and 40 kJ mol−1) (d–g). E aox was set to 20 kJ mol−1. α, the ratio [E]0aox : [E]0cox; [E]0aox, the concentration of active AOX in the measurement system; [E]0cox, the concentration of active CP in the measurement system; ΔE, the difference in the activation energy between O2 reduction by CP (E cox) and by AP (E aox); γ, the AP‐contribution term in Eqn 16.

Simulation results when α or ΔE varies in response to measurement temperature

Simulations (1)–(3), described in ‘Simulations’ in the Materials and Methods section, above, suggested that the Arrhenius plot is almost linear when E and α are independent of temperature, and thus the effect of a rapid change of E cox or α caused by a temperature change on the curvature of the Arrhenius plot was validated here. In simulation (4), when α changed (0.1–0.5) monotonically with increasing temperature (10–40°C), the simulated Arrhenius plots deviated from linearity (Fig. 4, Fig. S1). The shapes of the plots resembled the shapes of the corresponding plots of the instantaneous temperature dependence of α (compare Fig. 4a and e, b and f, c and g, and d and h). The shape of the Arrhenius plot was convex (Fig. 4) or concave (Fig. S1), depending on the behaviour of α. In some cases, the slope of the Arrhenius plot was positive when α decreased with increasing temperature (Fig. 4f, g, Fig. S1f, g).

Fig. 4

Profiles of α vs temperature (a–d) that were assumed for the simulations, and the Arrhenius plots produced based on these assumptions (e–h, respectively). In (e–h), the black line indicates the Arrhenius plot when α = 0 (no electrons flow to AP). The various convex profiles of α vs temperature were obtained for α in the range 0.0–0.5. A cox, pre‐exponential factor related to CP; α, the ratio [E]0aox : [E]0cox; [E]0aox, the concentration of active AOX in the measurement system; [E]0cox, the concentration of active CP in the measurement system; E cox, the activation energy of O2 reduction by CP; ɛ, the AP‐contribution term in Eqn 13. E aox and E cox were set to 20 kJ mol−1and 30 kJ mol−1, respectively.

In simulation (5), ΔE changed (5–10 kJ mol–1) monotonically with increasing temperature (10–40°C), and the slope of the Arrhenius plot increased in accordance with the low α (Fig. 5). The shape of the Arrhenius plot was convex when the shape of ΔE changes became concave, and it was concave when the shape of the ΔE changes became convex. Furthermore, in some cases, the sign of the slope of the Arrhenius plot was positive when ΔE increased with increasing temperature. In those cases, the activation energy of the CP increased with temperature. The activation energy of AP was assumed to be constant at 20 kJ mol–1 for all cases in this simulation.

Fig. 5

Profiles of E cox vs temperature (a, b) that were assumed for the simulations, and the Arrhenius plots generated using these assumptions (c and d, respectively). The values of E cox were in the range 25–30 kJ mol−1 in both convex (a) and concave (b) plots. E aox was set to 20 kJ mol−1. A cox, pre‐exponential factors related to CP; E cox, the activation energy of the O2 reduction by CP; ɛ, the AP‐contribution term in Eqn 13.

Fitting the kinetic model to experimental data

According to simulations (1)–(5), a curved Arrhenius plot was satisfactorily explained by allowing α or E cox to vary with the measurement temperature. Fitting the kinetic model to experimental leaf O2‐consumption rates identified possible values of α and E cox, as well as the contribution of the temperature dependence of each parameter on the slope of Arrhenius plot.

Leaf O2‐consumption rates were higher for plants grown at 15ºC than at 25ºC throughout the range of measurement temperatures (Fig. 6a). The Arrhenius plot obtained deviated from linearity in both growth–temperature treatments, and the shape of the curve was convex for the plants grown at 25ºC (i.e. there was less temperature dependence as the temperature increased) and concave for the plants grown at 15ºC (i.e. there was more temperature dependence as the temperature increased) (Fig. 6a).

Fig. 6

An Arrhenius plot that deviated from linearity for O2‐consumption rates in leaves of Arabidopsis thaliana (Fei‐0) (a). Plants were grown for 2 wk at 15°C (blue circles) or 25°C (red circles). The solid lines in (a) indicate the fitted curve of the model described by Eqn 4 (R 2 = 0.831 in 15°C‐grown plants and R 2 = 0.829 in 25ºC‐grown plants). The profiles of α in the fitting simulation assumed that E cox was constant (i.e. independent of measurement temperature) (b), and the profiles of E cox in the fitting simulation assumed that α was constant (i.e. independent of measurement temperature) (c). See the Materials and Methods section for details.

The kinetic model given by Eqn 12 was fitted to the obtained curve at each growth temperature. In the fitting procedure, we changed α or E cox with measurement temperature, subject to the constraint that the other parameter was constant. The kinetic model described the curvature of the observed Arrhenius plots well (solid lines in Fig. 6a).

When E cox and E aox were independent of measurement temperature, the value of α corresponding to the observed Arrhenius curvature was plotted against the measurement temperature (Fig. 6b). The calculations were done by using two values of E aox, namely 15 and 20 kJ mol−1. For both E aox values, α was higher in the plants grown at 15ºC than at 25ºC throughout the range of measurement temperatures (Fig. 6b). The value of α increased as the leaf O2‐consumption rate increased at higher measurement temperatures, and the shapes of the α‐curves were similar to the observed Arrhenius plots: convex in plants grown at 25ºC and concave in plants grown at 15ºC. Fig. 6b shows the behaviour of the temperature dependence of α when E cox was increased from 21 kJ mol−1 in increments of 2 kJ mol−1. The value of α increased as the E cox increased, and the α‐curves tended to converge. Whereas the shape of the convergent α curves depended on the value of E aox, α was higher in plants grown at 15ºC than at 25ºC throughout the range of measurement temperatures. As a result, the kinetic model that incorporated the ‘temperature dependence of α’ explained the experimental data well.

Next, the E cox corresponding to the observed Arrhenius curvature was plotted against the measurement temperature subject to the condition that α was independent of temperature (Fig. 6c). As in Fig. 6b, we performed the calculations at two values of E aox, namely 15 and 20 kJ mol−1. The candidate values of α satisfying the observed Arrhenius curvature were bounded. The possible range of α depended on the combination of E aox and growth temperature: 0.1 < α < 0.3 when E aox = 15 kJ mol−1 for plants grown at 15ºC; α = 1.3 when E aox = 20 kJ mol−1 for plants grown at 15ºC; 0.1 < α < 0.2 when E aox = 15 kJ mol−1 for plants grown at 25ºC; and 0.2 < α < 0.7 when E aox = 20 kJ mol−1 for plants grown at 25ºC (Fig. 6c). The range of E cox values was higher for plants grown at 25°C (21.5–29.0 kJ mol−1 when E aox equalled 15 kJ mol−1 and 21.2–32.0 kJ mol−1 when E aox equalled 20 kJ mol−1) than for plants grown at 15°C (19.7–22.5 kJ mol−1 when E aox equalled 15 kJ mol−1 and 20.2–24.0 kJ mol−1 when E aox equalled 20 kJ mol−1) (Fig. 6c). At both growth temperatures, E cox decreased as the leaf O2‐consumption rate increased at higher measurement temperatures. As a result, the kinetic model incorporating ‘temperature dependence of E cox’ also explained the experimental data well, even though the possible values of α were bounded.

Discussion

We constructed a kinetic model in which CP and AP competitively consumed O2, and the model enabled us to analyse the temperature dependence of plant O2‐consumption rates and the activation energies of the two pathways. The kinetic model incorporating a temperature dependence of α or E cox satisfactorily reproduced the observed deviations from linearity of the Arrhenius plot of plant O2‐consumption rates, as reported in some previous experimental studies. Indeed, in the fitted simulation of the experimental Arabidopsis leaf O2‐consumption rates in this study, the fact that the curvature of the Arrhenius plot of plant O2‐consumption was fitted well, subject to the condition that α or ΔE changed with the measurement temperature, suggested that rapid changes of α or ΔE, or both, occurred. The fitted simulation suggested that the number of conditions satisfying the observed curvature was higher in the model with variable α than in the model with variable E cox. The pre‐exponential factor in the Arrhenius plot has been related to temperature in other models based on transition‐state theory or collision theory, whereas we assumed no such relationship for the values of A cox and A aox, the pre‐exponential factors related to the CP and AP, respectively, in our kinetic model. Therefore, our kinetic model was more flexible than other models and could include factors other than temperature that might affect the pre‐exponential factors in the CP and AP models.

Interpretation of changes of α

Because α in the model is defined as the ratio of [E]0aox to [E]0cox, changes in the amount of active AOX protein and electron partitioning through AP (τa) should affect the α value. Many previous studies have shown that the amounts of protein in AOX are influenced by environmental factors such as growth temperature (González‐Meler et al., 1999; Kurimoto et al., 2004; Fiorani et al., 2005; Watanabe et al., 2008; Umbach et al., 2009), irradiance (Noguchi et al., 2005), plant age (Millar et al., 1998), season (Searle & Turnbull, 2011) and nitrogen availability (Noguchi & Terashima, 2006). Some other studies using the 18O discrimination technique have reported that τa is changed by various factors, as follows. During long‐term acclimation, τa was changed by growth temperature in the leaves of Populus canadensis (Searle & Turnbull, 2011) and Quercus rubra (Searle et al., 2011a), by the amounts of photosynthetically active radiation in leaves of Chionochloa rubra and Chionochloa pallens (Searle et al., 2011b), by atmospheric CO2 concentration in cladodes of Opuntia ficus‐indica (Gomez‐Casanovas et al., 2007), by age in root seedlings of Glycine max (Millar et al., 1998) and by the degree of water stress in leaves of G. max (Ribas‐Carbo et al., 2005). According to our kinetic model, such long‐term acclimation is accompanied by changes of α that will be reflected by changes in the magnitude and shape of the Arrhenius plot of plant O2‐consumption rates (Fig. 2).

To our knowledge, only a few studies have described the response of α to rapid temperature changes. Macfarlane et al. (2009) monitored electron partitioning through AP (τa) and CP (τc) in leaf tissues of three plant species (Cucurbita pepo, Nicotiana sativa and Vicia faba) by measuring 18O/16O discrimination and O2‐consumption rates at temperatures of 17–36ºC. They reported that τa values in leaf tissues of the three plant species were invariant with temperature. By contrast, the τa in the leaves varied with temperature in the case of Arabidopsis (Armstrong et al., 2008) and P. canadensis (Searle & Turnbull, 2011). A slight change of τa was also observed in the leaves of Nicotina tabacum (Guy & Vanlerberghe, 2005) and in the leaves and hypocotyl of Vigna radiata (González‐Meler et al., 1999). Still, little is known about the pattern of τa response to temperature. Considering that plant mitochondria frequently undergo fusion and fission within minutes (Arimura et al., 2004; Wakamatsu et al., 2010), rapid changes of α and thus τa in response to a temperature change may occur. We can now examine the rapid changes of τa in response to temperature by fitting the kinetic model to the curvature of the Arrhenius plot of plant O2‐consumption rates (Fig. 6a,b). The accumulation of data on the instantaneous temperature dependency of τa and the fitting of our model to that data may reveal how variations of τa affect deviations of Arrhenius plots.

Interpretation of changes of ∆E

Our kinetic model will enable us to examine how the activation energies of the CP and AP respond to temperature changes. Some studies have shown that the instantaneous temperature dependence of O2‐consumption rates differs between CP and AP (McNulty & Cummins, 1987; Collier & Cummins, 1990; Weger & Guy, 1991; Armstrong et al., 2008; Kruse & Adams, 2008; Macfarlane et al., 2009; Kruse et al., 2011). Furthermore, some studies have reported that the instantaneous temperature dependence of the O2‐consumption rates of the two pathways varies when the plants are grown at different temperatures. McNulty & Cummins (1987) reported that Q 10 – a measure of the instantaneous temperature dependence of leaf O2‐consumption rates – was lower for Saxifraga cernua grown at 10°C than at 20°C. The instantaneous temperature dependence of the hypocotyl O2‐consumption rate was enhanced when Vigna radiana was grown at 19°C vs 28°C (González‐Meler et al., 1999). Kurimoto et al. (2004) have studied several wheat and rice cultivars; they found that the responses of the instantaneous temperature dependence of root O2‐consumption rate on growth temperature depended on the cultivar. Furthermore, the instantaneous temperature dependence of the O2‐consumption rate (including CP and AP) was influenced by the measurement temperature (Kruse & Adams, 2008; Kruse et al., 2011, 2018). These results suggest that E cox, E aox and ΔE can be changed in response to both long‐term and short‐term temperature changes. The fact that we also observed differences in the curvatures of Arrhenius plots for Arabidopsis leaf O2‐consumption between plants grown at different temperatures suggested that long‐term acclimation to growth temperature occurred, and that this long‐term acclimation led to differences in the temperature sensitivity to measurement temperature. Our simulation results suggest that electron flow to the AP results in a decrement of the slope of the Arrhenius plot of plant O2‐consumption (Figs 2, 3), and a rapid change of ΔE results in curvature of the Arrhenius plot (Figs 5, 6a,c). Further research addressing the underlying mechanisms of long‐term and short‐term changes in the ambient temperature will be needed.

According to MMRT, the fact that ΔC p ‡ < 0 explains the phenomenon that the temperature sensitivity of enzyme‐catalysed biochemical reactions decreases with an increase in temperature, and the reaction rate reaches a maximum at a certain temperature T opt (Hobbs et al., 2013; Arcus et al., 2016). In plant O2‐consumption, however, concave curvature of an Arrhenius plot – that is, increasing temperature sensitivity of the O2‐consumption rate with increasing measurement temperature – has sometimes been observed. Examples include the work of Kruse et al. (2018) and the plants grown at 15°C in our study. In our fitting simulation, in which E aox, α, A cox, and A aox were assumed to be independent of the measurement temperature, we were able to fit a concave Arrhenius curve if E cox decreased as the temperature rose (Fig. 6c). This behaviour is consistent with MMRT. The existence of another parallel pathway (e.g. AP) may be an explanation for the concave curvature of Arrhenius plots.

The assumption that ΔC p ‡ < 0 in MMRT also explains the negative slope in the Arrhenius plot (i.e. the decrease in the reaction rate with a rise in temperature) at temperatures higher than T opt (Arcus et al., 2016). Umekawa et al. (2016) reported a negative activation energy of the O2‐consumption rate in the heat‐producing flowers of Symplocarpus renifolius. They suggested a model in which a negative activation energy could result from biochemical pre‐equilibrium reactions comprised of reversible reactions catalysed by cellular dehydrogenases and a rate‐determining reaction catalysed by AOX and COX. In their model, the apparent overall activation energy will be negative when the activation energy for the reverse path of the pre‐equilibrium reaction (endothermic reaction) becomes larger than that of the forward reactions (exothermic reactions). Their model is persuasive for understanding plants with heat‐producing flowers, and it raises the possibility that the activation energy of some pre‐equilibrium reactions may be negative in the respiration of other plants. In addition, our simulation results suggest that negative activation energy may be associated with the reactions of mitochondrial terminal oxidases (AOX and COX) per se.

In some cases, the overall activation energy of CO2 release by plant leaves is also changed by instantaneous temperature changes (Kruse et al., 2011; Noguchi et al., 2015; Kruse et al., 2018). Furthermore, the fact that the overall activation energy of CO2 release at the growth temperature, E (growth), shows homeostasis against the growth temperature (Noguchi et al., 2015; Kruse et al., 2018) suggests that the NADH/NAD+ ratio may be maintained independent of growth temperature in the mitochondrial matrix by adjustment of the rate‐determining step of CO2 release in leaves. Analysis of the instantaneous temperature dependence of plant CO2 release rates on the basis of a kinetic model may facilitate identification of the rate‐determining step of CO2 release in plants.

Conclusions

We conducted a sensitivity analysis of a theoretical model of plant respiratory O2‐consumption by using the kinetics of CP and AP. Our simulations satisfactorily explained the nonlinearity of Arrhenius plots obtained from experimental rates of plant O2‐consumption. The simulation results indicated the possibility that the AP contribution or activation energy of CP, or both, rapidly changed in response to ambient temperature and that these changes could be the causes of the slope deviations. The fitting simulation suggested that the number of conditions satisfying the observed curvature was higher in the model with variable α than in the model with variable E cox. Simultaneous measurements of the temperature dependence of the O2‐consumption rate and τa will reveal in detail the contributions of α and E cox. Furthermore, the effect of the change in the heat capacity, ΔC p ‡, between the reactant and transition state complex in each pathway will be obtained by measuring the O2‐consumption rates of intact tissue or isolated mitochondria (or both) by using inhibitors of CP and AP, respectively. In vivo measurements of electron flows to CP and AP and further analyses of the relationships between activation energies and ambient environmental conditions will help us to understand the plant respiration system.

Author contributions

Both authors discussed and interpreted the data and contributed to drafting and finalising the manuscript.

Fig. S1 Concave profiles of α vs temperature that were assumed for the simulations, and the Arrhenius plots produced under these assumptions.

Notes S1 Additional details on derivation of equations.

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