Entropy production in oscillatory processes during photosynthesis

Abstract

The flow of matter and heat and the rate of enzymatic reactions are examined using two models of photosynthesis that exhibit sustained and damped oscillatory dynamics, with the objective of calculating the rate of entropy generation and studying the effects of temperature and kinetic constants on the thermodynamic efficiency of photosynthesis. The global coefficient of heat transfer and the direct and inverse constants of the formation reaction of the RuBisCO–CO₂ complex were used as control parameters. Results show that when the system moves from isothermal to non-isothermal conditions, the transition from a steady state to oscillations facilitates an increase in the energy efficiency of the process. The simulations were carried out for two photosynthetic models in a system on a chloroplast reactor scale.

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Introduction

One of the challenges faced by humanity in recent decades has been the development of renewable energy sources. Fuels such as bioethanol, biodiesel, biogas and biohydrogen are obtained from materials that come from photosynthetic organisms that convert solar energy into chemical energy.^1,2 In recent years, advances in our understanding of the mechanisms and basic principles that regulate photosynthetic processes have facilitated the development of artificial systems that show how it is possible to harness solar energy to produce fuels such as hydrogen.^3,4 In order to design efficient artificial systems, it is necessary to further investigate the basic regulatory mechanisms involved in transforming light energy into chemical energy in different organisms, as well as the effects of various external factors, such as light intensity, the concentration of atmospheric gases and temperature. The phenomenon of photoinhibition, discovered by Ewart in 1896,⁵ shows that by increasing the light intensity it is possible to inhibit the flow of electrons towards photosystem II.^5–7 Regarding temperature, it has been determined that in higher plants there are temperature gradients of up to 0.8 K in the surface of plant leaves. This is related to the processes of gas exchange and leaf transpiration, which cause a latent heat flux that results in the cooling of a leaf with respect to its surroundings.⁸ It has also been experimentally observed that the temporal dynamics of the photosynthetic process in certain organisms are not altered by small changes in the ambient temperature.⁹ Our modern understanding of photosynthesis allows us to partially explain why this is the case. For this reason, it is useful to have simplified mathematical models that explain the most relevant characteristics of this natural process.^10–12

The surge in studies and production of biofuels as renewable energy sources has also been criticized by some for its possible future impact, primarily because of the current efficiency of the processes.^13–15 The thermodynamic efficiency of the photosynthetic process has been analysed by several authors, with calculated magnitude varying depending on the available information and the way that information is interpreted.^16–20 Based on thermochemical data such as the standard Gibbs free energies of formation, efficiency is estimated to be between 20% and 99%, depending on whether the entire process is analysed or just one of the steps, such as light absorption. If the energy efficiency of a process is evaluated by taking the maximum efficiency of a reversible process as a criterion of the second law of thermodynamics, it follows that the magnitude of the generated entropy is directly related to the real efficiency of irreversible processes.^21–25 By calculating the thermodynamic efficiency of non-equilibrium states, we can consider the different dynamical regimes that the system can exhibit during the process.^26,27 In this study, we use the second law of thermodynamics to analyse entropy production during photosynthesis, using simplified mathematical models that describe the steady and oscillatory transport in chloroplasts.^28–30 The effect of temperature on this process is also examined in the analysis.

Models for photosynthetic oscillations

The phenomenon of oscillations during photosynthesis was discovered over 60 years ago as a result of experimental studies performed on plants, research on the carbon cycle, and kinetic modelling.^18,31–35 Several experimental methods, such as isotopic gas exchange and electrochemical and fluorescence techniques,^36–39 have been developed to study the interaction between energy and matter at a molecular level with the highest precision possible, by determining the parameters involved in the regulation and dynamics of the photosynthetic process. Using these methods, scientists have produced a series of hypotheses regarding the destabilizing processes that induce oscillations, such as a decompensation between the ATP/NADPH flow in the Calvin cycle,⁴⁰ a competitive kinetic inhibition between kinases for the ATP in the Calvin cycle⁴¹ and the autocatalytic activity of fructose 2,6-biphosphate during the synthesis of sugars,^42,43 among others.⁴⁴ Oscillations that occur in the CO₂/O₂ flow during photosynthesis can also be coupled with asynchronous dynamic patterns of opening and closing of the stomata in the leaf during the Calvin cycle, as was observed in the surface of Glechoma hederacea leaves when exposed to high concentrations of atmospheric CO₂.⁴⁵ A similar phenomenon has been observed in the extracellular space of tobacco leaves,⁴⁶ and is caused by alternation between the steps of CO₂ assimilation and photorespiration. Therefore, oscillations in the flow of CO₂/O₂ can be explained through cellular metabolism as follows: the assimilation of CO₂ in the Calvin cycle is mediated by the enzyme RuBisCO, which performs carboxylase and oxygenase catalytic activities. During the carboxylation of the ribulose-1,5-biphosphate substrate, CO₂ is consumed and the oxygenation leads to the liberation of CO₂ in the mitochondria through photorespiration, as a result of decarboxylation of glycine. Furthermore, for some values of CO₂ concentration, the oxygenase function is inhibited in RuBisCO. This introduces a negative feedback into the dynamics of the process of CO₂ liberation through photorespiration in the mitochondria, and in the CO₂ assimilation process in the chloroplasts. The existence of this feedback has been studied theoretically in order to analyse the cause of the oscillatory phenomenon.^47–49 In this work, we examine two models that describe the oscillatory dynamics of photosynthesis using complementary approaches; one is a two substrate model with kinetic competition of substrates to bind an enzyme and with feedback control for one substrate inflow,²⁹ and the other is a one substrate model for a specific enzyme activity and with an enzymatic description of the reaction product.

The Dubinsky–Ivlev–Igamberdiev model²⁸

The work of Alexander Ivlev and collaborators on the isotopic fractionation of carbon in plants^50–53 demonstrated that two opposite isotopic effects occur, one during the process of CO₂ assimilation and the other during photorespiration, when the enzyme RuBisCO can act as a switch that induces oscillations. This hypothesis was supported by the work of Roussel, Ivlev and Igamberdiev,⁴⁶ which proved that oscillations occur in Nicotiana tabacum leaves exposed to low concentrations of CO₂. In later works on the oscillatory dynamics of photosynthesis, Dubinsky, Ivlev and Igamberdiev²⁸ used concepts from Ivlev et al. to propose a simplified mathematical model of photosynthesis, which only used the CO₂ assimilation stage for the enzyme RuBisCO and the efflux of sugars or carbohydrates to find the range of parameter values for which the model exhibited a sustained oscillatory regime of a few seconds. The Dubinsky–Ivlev–Igamberdiev model²⁸ (D–I–I model) is represented by the system of eqn (1) and (2):

dX/dt = (1/5)k_cXC − V_out(X/(X + K_out)) (1)

dC/dt = −k_cXC + k_d(C_i − C) (2)

where X and C represent the molar concentration of carbohydrates and CO₂ respectively, V_out is for the maximum rate of carbohydrate consumption, K_out is the Michaelis constant of the pseudo enzyme that models the total consumption of carbohydrates (X), k_d is the diffusion coefficient of CO₂ from the external medium to the chloroplast, C_i is the CO₂ concentration in the external medium, and k_c is the kinetic constant of carboxylation of ribulose-1,5-biphosphate (RuBP).

The Ngo and Roussel model²⁹

Ngo and Roussel²⁹ proposed an enzymatic model (N–R model) of competitive inhibition, with a flow of substrates that exhibit damped and sustained oscillations when one of the flows is controlled by an allosteric feedback and for different association/dissociation rate values of each substrate with the enzyme. Based on an analysis of different reaction mechanisms proposed to explain oscillations during photosynthesis, Roussel and Igamberdiev⁴⁹ conclude that oscillations are linked to the feedback that occurs between CO₂ assimilation and photorespiration, while Dubinsky and Ivlev⁴⁷ show through the study of a photosynthesis model that CO₂ and O₂ concentrations oscillate in opposite phases because the enzyme RuBisCO continually shifts between serving carboxylase and oxygenase functions. The N–R model²⁹ is described by the system of eqn (3)–(5):

dC/dt = k_f − k₁(E)(C) + k₋₁(EC) (3)

dO/dt = −k₃(E)(O) + k₋₃(EO) (4)

dE/dt = −k₁(E)(C) + k₋₁(EC) − k₂(EC) − k₃(E)(O) + k₋₃(EO) (5)

where k_f is the constant inflow of CO₂ to the chloroplast, k₁ and k₋₁ are the direct and inverse constants of the formation of the RuBisCO–CO₂ complex, and k₃ and k₋₃ are the inverse and direct constants of formation of the RuBisCO–O₂ complex. In this model, CO₂ is the main substrate of the enzyme, whereas O₂ acts as an inhibitor, modelling the oxygenase function of the enzyme; the total amounts of the enzyme and the inhibitor remain constant, as seen in eqn (6) and (7):

E_t = E + (EC) + (EO) (6)

O_t = O + (EO) (7)

Energy balance of an open chloroplast reactor

The possibility that climate change will have future effects on the Earth's biomass, particularly in regard to food,⁵⁴ has led to a great deal of research about the effect of temperature on photosynthesis. Studies examining photosynthesis in plant leaves usually measure variations in the rate of CO₂ assimilation in relation to different parameters, such as the external concentration of CO₂ and O₂, light intensity and temperature. There are no organelles or physical, chemical or metabolic processes in plants that act as a thermometer; rather, they respond to thermic stimuli through a complex network of interconnected devices (membranes, cytoskeletons, proteins, enzymes, etc.), with the particularity that every device has a selective response to temperature changes.⁵⁵ Therefore, it is reasonable to believe that photosynthesis can take place under non-isothermal conditions, as suggested by the temperature gradients experimentally measured in plant leaves. With higher concentrations of atmospheric CO₂, and at higher room temperatures, the transport capacity of electrons and the stability of RuBisCO activase become the limiting factors of photosynthesis. In this regard, it has been suggested that plants can protect the thylakoid membrane from irreversible thermal damage by activating a redox cycle between the photosystems, as follows: PSI(ox)/PSII(red) ⇌ PSI(red)/PSII(ox).⁵⁶

At the optimal temperature interval, in which the plant does not suffer irreversible damages, the rate of CO₂ assimilation for different intracellular concentrations of CO₂ is regulated by the extent to which any of the three following processes occurs: the consumption of RuBP by RuBisCO, the regeneration of RuBP through the Calvin cycle and the reactions in the thylakoid, and the regeneration of inorganic phosphorus through the consumption of triose-phosphate during the synthesis of starch and sugars.⁵⁷ Recent studies on the effect of temperature on the global net assimilation of CO₂ or on the previously mentioned specific metabolic processes show that the dependence on temperature, in a wide interval of temperatures and concentrations, can be adequately described through an Arrhenius type law, as indicated by the results with the factor Q₁₀.^58,59

In order to include the effect of temperature in the D–I–I and N–R mathematical models, we take the chloroplast as an ideal open reactor at constant volume, in which the processes of photosynthesis take place as a homogeneous reaction mixture in liquid phase, with constant stirring, exchanging matter and energy through the walls, as illustrated in Fig. 1. In order to determine the energy balance in the chloroplast reactor, we examined the changes in enthalpy due to the flow of reactants, the enthalpies of reaction of the processes mediated by RuBisCO, and the enzyme-substrate binding and protonation enthalpies. To determine the transfer of energy through the walls, we used the Newton law of cooling. The energy balance that involves all the terms previously mentioned is shown in eqn (8),

Fig. 1 Schematic representation of the chloroplast reactor. (a) The setup configuration for the Dubinsky–Ivlev–Igamberdiev model;²⁸ (b) the setup configuration for the Ngo and Roussel model.²⁹

In the previous equation, the first term on the right side corresponds to changes in enthalpy due to the flow of reactants and products, the second term is for the Newton cooling law, , and the third term is the reaction term. In regard to the variables, ρ represents the density of the reaction mixture, c_p is heat capacity, F^α is the volumetric flow into or out of the chloroplast reactor, c_pi is the CO₂ heat capacity, C_i is the inflow concentration of CO₂, U is the global coefficient of heat transfer, A is the surface area for the heat transfer of the reactor chloroplast, ΔH_Ri is the enthalpy of the enzymatic reaction, ΔH_bi is the bond enthalpy of the enzyme-substrate, ΔH_pi is the enthalpy of protonation, r_i is the rate of the reaction, T^α is the external temperature, and V is the reactor volume.

Kinetic parameters of the models are considered temperature dependent according to the Arrhenius equation. For this reason, for the D–I–I model, eqn (1), constants k_c and V_out are written as indicated in eqn (9) and (10):

k_c(T) = k_c,0exp((E_a/R)(1/T⁰ − 1/T)) (9)

V_out(T) = V_out,0exp((E_a/R)(1/T⁰ − 1/T)) (10)

where k_c,0 and V_out,0 are constants and their value corresponds to that of k_c and V_out at room temperature T⁰, E_a is the activation energy of the reaction when the enzyme RuBisCO is involved, and R is the constant for ideal gases. For the N–R model, eqn (2)–(5), the constant k₂ is also temperature dependent, as seen in eqn (11):

k₂(T) = k_2,0exp((E_a/R)(1/T⁰ − 1/T)) (11)

k_2,0 is a constant and T⁰ is the magnitude of k₂ at room temperature.

Thermodynamic considerations

Non-equilibrium states of open, oscillatory or steady state systems continually degrade the available energy into less useful forms, such as heat. Energy stored in the gradients of chemical potential, electrochemical field or pressure, and temperature is used by natural systems to carry out all kinds of coupled or interconnected processes. These include processes that take place in photosynthesis during the biosynthesis of macromolecules, the active transport of ions or the flow of electrons. ATP-catalysed hydrolysis is the main source of energy used by the cells in most metabolic processes. In this work, we do not consider thermodynamic coupling between processes, but rather that which occur spontaneously in the direction indicated by the gradient, such as the flow of heat, the transport of matter and the chemical reaction. All natural processes are irreversible and produce entropy. The second law of thermodynamics allows us to establish a direct relationship between the entropy generation rate and the energy efficiency of the processes. One of the most important reasons to study the energy efficiency of a process is to understand how a system manages cost–benefit relationships depending on the restrictions imposed. An understanding of energy efficiency in natural systems serves as a guideline for the implementation of more efficient industrial processes.²⁴ Efficiency is calculated based on the second law of thermodynamics, using the following eqn (12):

η_II = 1 − W_lost/W_ideal (12)

where W_lost = T^α(d_iS/dt), the expression in parentheses is for the entropy generation rate in the non-equilibrium state and W_ideal represents the work associated with the processes at the ideal limit of reversibility. The rate of entropy generation contains the conjugate pairs of forces, X, and flows, J, of all processes that take place in the system, as shown in the following equation:

σ = (JX)_mass + (JX)_heat + (JX)_chem (13)

Each factor that contributes to the generation of entropy in the previous equation, such as flow of heat, transport of matter and biochemical reactions, is calculated using the expressions shown in eqn (14)–(17). These are obtained by following the thermodynamic procedures for irreversible processes.^22,60 The entropy generation rate resulting from thermal gradients between the chloroplast reactor and the environment is calculated using the balance of entropy and the Newton law of cooling, see eqn (14):

dS_heat/dt = UA((T − T^α)²)/TT^α (14)

where U is the global heat transfer coefficient and A is the transfer area, T^α is the external temperature and T is the temperature of the chloroplast reactor, which is calculated using the energy balance, eqn (8). The rate of entropy generation due to the transport of matter, driven by the chemical potential gradient between the chloroplast and the environment, is calculated by eqn (15):²⁶

dS_mass/dt = −((μ₂ − μ₁)/T)(dξ/dt) = −R(r_D − r_I)ln(C) (15)

where ξ represents the extent of the reaction, and r_D and r_I are the direct and inverse rates of the reaction, for example where CO₂ (C) is involved. Finally, the rate of entropy generation due to chemical reactions is calculated using the gradient of chemical potential and the total reaction rate, as shown in eqn (16) and (17):

dS_i/dt = (−Δ_rG/T)(dξ/dt) > 0 (16)

dS_r/dt = R(r_D − r_I)ln(r_I/r_D) (17)

When the non-equilibrium state corresponds to oscillatory dynamics, the rate of entropy generation per cycle, which we call thermodynamic dissipation, <σ> is calculated using the following expression, eqn (18):²⁶

where τ is the oscillation period.

Results and discussion

The D–I–I model,²⁸ shown in eqn (1) and (2), was simultaneously solved with the energy balance represented in eqn (8), using the data from Tables 1 and 2 as parameters. The numeric simulation was carried out using the subroutine DLSODE,⁶¹ see Fig. 2.

Fig. 2 The limit cycle attractor of the D–I–I model. The phase space trajectory is obtained by solving eqn (1), (2) and (8) for the parameter values given in Tables 1 and 2.

Table 1 Set of parameter values used in models of photosynthesis. The * means that the value was adjusted for the temperature effect

N–R model^29		D–I–I model^28
Parameter	Value	Parameter	Value
k f	5 × 10^−5 s^−1	C i	10 μM
k 1	1 × 10^8 M^−1 s^−1	k d	2.5 s^−1
k −1	1 s^−1	V out_0	5 μM s^−1*
k 2	9 s^−1*	K out	0.25 μM
k 3	1 × 10^5 M^−1 s^−1	k c,0	15 μM^−1 s^−1*
k −3	1.0 × 10^−2 s^−1	—	—
E t	1.5 × 10^−5 M	—	—
Ot	2.5 × 10^−5 M	—	—

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Table 2 Set of parameter values used in the energy balance equation of the N–R and D–I–I models of photosynthesis. The ** means that the value was estimated for a chloroplast with a membrane thickness of 60 Å and radii of 10 μm

Parameter	Value	Reference
E a	79.43 kJ mol^−1	62
T ^α	298.15 K	Estimated
T Si	302 K	Estimated for the D–I–I model
	302.15 K	Estimated for the N–R model
U	0.1 W m^−2 K^−1	Estimated for the N–R model
	30 W m^−2 K^−1	Estimated for the D–I–I model
A	1.88 × 10^−13 m^2**	Estimated
V	4.18 × 10^−15 m^3**	Estimated
C p	7.5 × 10^−2 kJ mol^−1 K^−1	63
C pi	0.037 kJ mol^−1 K^−1	63
	6.5 × 10^4 mol m^−3	63
ΔHr, O	−319.1 kJ mol^−1	64
ΔHr, C	−21.3 kJ mol^−1	64
ΔHp	−30.0 kJ mol^−1	65
ΔHb	−56.5 kJ mol^−1	66
D CO2	5 × 10^−12 m^2 s^−1	67

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Fig. 2 shows the trajectory in phase space of the non-isothermal model for the variables of temperature, the concentration of CO₂, and the concentration of carbohydrates. The limit cycle attractor shown in this figure indicates dynamics of sustained oscillations for the model, with a temperature variation of approximately 0.5 K in the chloroplast reactor. This result is typical of the D–I–I model and is in keeping with experimental observations of sustained oscillations in CO₂ concentration and temperature gradients in the leaves of higher plants.^8,46

The N–R model, represented by eqn (3)–(5), was solved numerically in the same way as the D–I–I model, using the values in Tables 1 and 2 as parameters, see Fig. 3. In Fig. 3, we observe in the phase space that the system evolves by spiralling towards a fixed-point attractor or steady state. Transient and damped oscillations are typical of this model, as observed in experiments performed by Siebke and Weis.^42,45 The figure shows that the temperature changes are less than 1 K until reaching the steady state with a temperature slightly above room temperature.

Fig. 3 The fixed-point attractor—damped oscillations—of the Ngo and Roussel (1997) model. The phase space trajectory is obtained by solving eqn (3)–(5) and (8) for the parameter values given in Tables 1 and 2.

The global coefficient of heat transfer, U, was used as a control parameter for studying the energy efficiency of the process, as a function of the dynamics of the process and of the energetic interaction between the chloroplast reactor system and its environment. The coefficient U acts as a thermal resistance ranging between two extremes: when values are close to 0 the reactor is adiabatic and when values are high enough, the reactor is isothermal. A numerical analysis of the dynamics of the D–I–I non-isothermal model as a function of the global coefficient of heat transfer shows that the system exhibits sustained oscillations for values less than 38 W m⁻² K⁻¹, with frequency and amplitude decreasing as the value rises. From 38 to 50 W m⁻² K⁻¹, U shows the dynamic of damped transient oscillations. Finally, for values greater than 50 W m⁻² K⁻¹ the system remains in an isothermal steady state, with the temperature approaching that of the environment as the value rises. The entropy generation rate, eqn (18), is calculated by determining the entropy contributions of each chemical reaction and the transfer of matter and heat. In the D–I–I model, the reaction state is kinetically irreversible, so it is not possible to quantify the generation of entropy according to eqn (17).⁶⁸ Instead, for this model we evaluate the rate of entropy generation due to transfer of heat <σ_heat>, eqn (14), and transfer of matter <σ_mass>, eqn (15). Results are shown in Fig. 4.

Fig. 4 Entropy production phase diagram for the D–I–I model. (a) The behavior of the average entropy production for <σ_heat> (dashed line, left Y-axis) and <σ_mass> (solid line, right Y-axis) as a function of the global heat transfer coefficient (U); (b) time series of the heat entropy production for three different values of parameter U, see (a); (c) time series of the mass flow entropy production for three different values of parameter U, see (a).

As shown in Fig. 4(a), the magnitude of the entropy generated in the chloroplast reactor as a result of heat transfer is very small compared to that generated by the transfer of matter. This is a clear indication of the energy loss associated with changes in composition during photosynthesis. It is also important to note the particularity observed in Fig. 4(a) for the D–I–I model, where the maxima of entropy generation for transfer of matter and transfer of heat are offset in relation to the parameter U, in 20 W m⁻² K⁻¹ for <σ_heat> and in 38 W m⁻² K⁻¹ for <σ_mass>. In Fig. 4(b) and (c) the behaviour of entropy generation for different values of U is shown in detail, demonstrating that <σ_heat> allows us to quantify changes in the amplitude and frequency of the oscillations, whereas <σ_mass> is affected by the dynamic behaviour of the system. In conclusion, Fig. 4 shows that as the system moves from isothermal conditions in the chloroplast reactor (high U values) to non-isothermal conditions (low U values), the rate of entropy generation increases in steady states and decreases in oscillatory states.

For the N–R model the average generation of entropy was calculated for transfer of matter, transfer of heat and chemical reaction. <σ_mass> and <σ_heat> do not significantly change in relation to U, because the system exhibits damped transient oscillations that quickly fall into a stationary state. Therefore, in Fig. 5 we only show the results for entropy generation due to chemical reaction <σ_r>, as a function of the kinetic constants k₁ and k₋₁ involved in the formation of the enzyme-substrate complex.

Fig. 5 The average rate of entropy production for the Ngo and Roussel (1997) model as a function of kinetic constants: k₁ (solid line, right Y-axis) and k₋₁ (dashed line, left Y-axis).

Fig. 5 shows the change in the rate of entropy generation when the chemical equilibrium is shifted. When k₁ increases, the reaction shifts to the formation of products and the process is more irreversible. When k₋₁ increases, on the other hand, the reaction shifts to the reactants and the process is more reversible. These kinetic constants are related to the Michaelis constant (k_M = (k₋₁ + k₂)/k₁), which at low values shows a greater affinity for the formation of the RuBisCO–CO₂ complex. The cross point of the two curves in Fig. 5 can be understood as a balance between the cost–benefit relationship, entropy generation and synthesis of products.

For a better understanding of the results shown in Fig. 4 for the D–I–I model, we calculated the energy efficiency of the second law, η_II, using eqn (12). Results are shown in Fig. 6.

Fig. 6 Second law efficiency (η_II) for the D–I–I model as a function of the heat transfer parameter, U. The upper limit of the graph is determined by calculating W_ideal as the standard free energy of glucose formation in photosynthesis (dotted line, ΔG⁰_r = 480 kJ mol⁻¹) and the lower edge of W_ideal is computed as the free energy standard of glucose formation from RuBP in the Calvin cycle (dashed line, ΔG⁰_r = 236.95 kJ mol⁻¹ taken from the MetaCyc Database).

Fig. 6 shows that as the global heat transfer coefficient decreases to U = 38 W m⁻² K⁻¹, the thermodynamic efficiency of the process in the chloroplast reactor decreases (branch of stationary states), and that if U decreases below this value, the efficiency increases (branch of oscillations). Fig. 6 is read right to left as it is in this direction that the system makes the transition from isothermal to non-isothermal conditions, as was shown in Fig. 4. Values obtained for the efficiency of the sustained oscillations range between 86% and 96%, which are close to those reported for the net efficiency of the Calvin cycle reactions.¹⁸ The energy efficiency for the N–R model, evaluated only for chemical reactions, increases as k₋₁ increases. (The figure is not shown.)

The results of this study show how oscillations improve the energy efficiency of the photosynthetic process, and are therefore a logical response for plants facing adverse climatic conditions, as has been discussed by different authors in several contexts.^9,30,69–72

Conclusions

Faced with concerns over the effects of climate change on the Earth's available biomass, it is necessary to understand the responses of plants to adverse external conditions. In the context of this work, two earlier studies attracted our attention: the phenomenon of thermal compensation studied by Sorek and Levy⁹ regarding the importance of oscillations when faced with temperature changes, and efforts to genetically improve the affinity of the RuBisCO enzyme to CO₂ in order to facilitate the assimilation phase of photosynthesis.^73,74

To better understand these phenomena, we studied the rate of entropy generation as a function of temperature for the simple mathematical models proposed by Dubinsky et al.²⁸ and by Ngo and Roussel,²⁹ which describe the main features of the oscillatory dynamics experimentally observed during photosynthesis, based on assimilation cycles and photorespiration.

Results show that the oscillatory state in photosynthesis helps improve the thermodynamic efficiency of the process when faced with the presence of thermal gradients between the plant and its environment. These dynamics are part of a highly complex process with a regulatory role in the control of energy consumption in the global process.

This study also shows that oscillations, initially measured in CO₂/O₂ flow, are also present in other flows that form part of the process, such as heat transport, matter transfer and the cascade of biochemical reactions. Therefore, in order to clearly understand how CO₂ is assimilated during photosynthesis, it is necessary to consider the thermodynamic coupling that takes place between other flows and forces of the process with the catalytic activity of the RuBisCO enzyme in the Calvin cycle.

Acknowledgements

The research for this project has been supported by the Faculty of Sciences and DIME of the Universidad Nacional de Colombia in Medellín.

Notes and references

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