Thermodynamic and kinetic characterisation of individual haems in multicentre cytochromes c3

The characterisation of individual centres in multihaem proteins is difficult due to the similarities in the redox and spectroscopic properties of the centres. NMR has been used successfully to distinguish redox centres and allow the determination of the microscopic thermodynamic parameters in several multihaem cytochromes c₃ isolated from different sulphate-reducing bacteria. In this article we show that it is also possible to discriminate the kinetic properties of individual centres in multihaem proteins, if the complete microscopic thermodynamic characterisation is available and the system displays fast intramolecular equilibration in the time scale of the kinetic experiment. The deconvolution of the kinetic traces using a model of thermodynamic control provides a reference rate constant for each haem that does not depend on driving force and can be related to structural factors. The thermodynamic characterisation of three tetrahaem cytochromes and their kinetics of reduction by sodium dithionite are reported in this paper. Thermodynamic and kinetic data were fitted simultaneously to a model to obtain microscopic reduction potentials, haem-haem and haem-proton interacting potentials, and reference rate constants for the haems. The kinetic information obtained for these cytochromes and recently published data for other multihaem cytochromes is discussed with respect to the structural factors that determine the reference rates. The accessibility for the reducing agent seems to play an important role in controlling the kinetic rates, although is clearly not the only factor.     

Thermodynamic and kinetic characterisation of individual haems in multicentre cytochromes c3

The characterisation of individual centres in multihaem proteins is difficult due to the similarities in the redox and spectroscopic properties of the centres. NMR has been used successfully to distinguish redox centres and allow the determination of the microscopic thermodynamic parameters in several multihaem cytochromes c(3) isolated from different sulphate-reducing bacteria. In this article we show that it is also possible to discriminate the kinetic properties of individual centres in multihaem proteins, if the complete microscopic thermodynamic characterisation is available and the system displays fast intramolecular equilibration in the time scale of the kinetic experiment. The deconvolution of the kinetic traces using a model of thermodynamic control provides a reference rate constant for each haem that does not depend on driving force and can be related to structural factors. The thermodynamic characterisation of three tetrahaem cytochromes and their kinetics of reduction by sodium dithionite are reported in this paper. Thermodynamic and kinetic data were fitted simultaneously to a model to obtain microscopic reduction potentials, haem-haem and haem-proton interacting potentials, and reference rate constants for the haems. The kinetic information obtained for these cytochromes and recently published data for other multihaem cytochromes is discussed with respect to the structural factors that determine the reference rates. The accessibility for the reducing agent seems to play an important role in controlling the kinetic rates, although is clearly not the only factor.     

Reaction of myeloperoxidase with its product HOCl.

The reaction of human myeloperoxidase with its product, hypochlorous acid was investigated using both rapid-scan spectrophotometry and the stopped-flow technique. In the reaction of myeloperoxidase with hypochlorous acid a primary compound is found with properties similar to that of compound I and which is converted into compound II. The primary reaction is strongly pH-dependent. At pH 7.2 the reaction is too fast to be measured but at higher pH values it is possible to determine the apparent second-order rate constant. Its value decreases to about 2 x 10(7) M-1.s-1 at pH 8.3 and to 2.3 (+/- 0.4) x 10(6) M-1.s-1 at pH 9.2, respectively. The dissociation constant for the formation of the primary compound is 25.7 (+/- 15.3) microM at pH 9.2 and about 2.5 microM at pH 8.3. The apparent second-order rate constant for the formation of compound II is hardly affected by pH and varies between 2 to 5 x 10(4) M-1.s-1 at pH 10.2 and pH 8.3, respectively. Reaction of myeloperoxidase with hypochlorous acid also resulted in irreversible partial bleaching of the chromophore. Chloride, which is a substrate of the enzyme not only protects myeloperoxidase against bleaching by hypochlorous acid but also competitively inhibits the binding of hypochlorous acid to myeloperoxidase, a process which also has been observed in the reaction with hydrogen peroxide. It is concluded that hypochlorous acid binds at the heme iron to form compound I.     

A miniaturized technique for assessing protein thermodynamics and function using fast determination of quantitative cysteine reactivity

Protein thermodynamic stability is a fundamental physical characteristic that determines biological function. Furthermore, alteration of thermodynamic stability by macromolecular interactions or biochemical modifications is a powerful tool for assessing the relationship between protein structure, stability, and biological function. High-throughput approaches for quantifying protein stability are beginning to emerge that enable thermodynamic measurements on small amounts of material, in short periods of time, and using readily accessible instrumentation. Here we present such a method, fast quantitative cysteine reactivity, which exploits the linkage between protein stability, sidechain protection by protein structure, and structural dynamics to characterize the thermodynamic and kinetic properties of proteins. In this approach, the reaction of a protected cysteine and thiol-reactive fluorogenic indicator is monitored over a gradient of temperatures after a short incubation time. These labeling data can be used to determine the midpoint of thermal unfolding, measure the temperature dependence of protein stability, quantify ligand-binding affinity, and, under certain conditions, estimate folding rate constants. Here, we demonstrate the fQCR method by characterizing these thermodynamic and kinetic properties for variants of Staphylococcal nuclease and E. coli ribose-binding protein engineered to contain single, protected cysteines. These straightforward, information-rich experiments are likely to find applications in protein engineering and functional genomics.     

Reactions of soybean peroxidase and hydrogen peroxide pH 2.4-12.0, and veratryl alcohol at pH 2.4

Peroxidase from soybean seed coat (SBP) has properties that makes it particularly suited for practical applications. Therefore, it is essential to know its fundamental enzymatic properties. Stopped-flow techniques were used to investigate the pH dependence of the reaction of SBP and hydrogen peroxide. The reaction is linearly dependent on hydrogen peroxide concentration at acidic and neutral pH with the second order rate constant k(1)=2.0x10(7) M(-1) s(-1), pH 4-8. From pH 9.3 to 10.2 the reaction is biphasic, a novel observation for a peroxidase at alkaline pH. A fast reaction has the characteristics of the reaction at neutral pH, and a slow reaction shows hyperbolic dependence on hydrogen peroxide concentration. At pH >10.5 only the slow reaction is seen. The shift in mechanism is coincident with the change in haem iron co-ordination to a six-coordinate low spin hydroxy ligated alkaline form. The pK(a) value for the alkaline transition was observed at 9.7+/-0.1, 9.6+/-0.1 and 9.9+/-0.2 by spectrophotometric titration, the fast phase amplitude, and decrease in the apparent second order rate constant, respectively. An acidic pK(a) at 3.2+/-0.3 was also determined from the apparent second order rate constant. The reactions of soybean peroxidase compounds I and II with veratryl alcohol at pH 2.44 give very similar second order rate constants, k(2)=(2.5+/-0.1)x10(4) M(-1) s(-1) and k(3)=(2.2+/-0.1)x10(4) M(-1) s(-1), respectively, which is unusual. The electronic absorption spectra of compounds I, II and III at pH 7.07 show characteristic bands at 400 and 651 nm (compound I), 416, 527 and 555 nm (compound II), and 414, 541 and 576 nm (compound III). No additional intermediates were observed.     

A network of hydrogen bonds in the reaction centers of Rhodobacter sphaeroides serves as a regulatory factor of the temperature dependence of the recombination rate constant of photooxidized bacteriochlorophyll and primary quinone acceptors

The dark recombination rate constant for the photooxidized bacteriochlorophyll (P) and reduced primary quinone acceptor (QA) in the photosynthetic reaction centers (RC) from purple bacterium Rhodobacter sphaeroides depends nonmonotonically on temperature. The time of this reaction is approximately 100 ms at 270-300 K and decreases as the temperature both increases and decreases beyond this temperature range. It is known that the dome-shaped dependence of the thermodynamic stability on temperature is an intrinsic feature of many proteins in solution. The experimental results on the nonmonotonous temperature dependence of P+ and QA- recombination rate constant are discussed in terms of general thermodynamic approaches. The dynamic properties of the network of hydrogen bonds that are involved in the relaxation processes accompanying the electron transport are considered as a regulatory factor of the efficiency of electron transfer.     

The pH-dependent redox inactivation of amicyanin from Paracoccus versutus as studied by rapid protein-film voltammetry

The redox properties of the blue copper protein amicyanin have been studied with slow and fast scan protein-film cyclic voltammetry. At slow scan rates, which reveal the thermodynamics of the redox reactions, the reduction potential of amicyanin depends on pH in a sigmoidal manner, and the data can be analysed in terms of electron transfer being coupled to a single protonatable group with pKa(red)=6.3 and pKa(ox) < or = 3.2 at 22 degrees C. Voltammetry at higher scan rates reveals the kinetics and shows that the low-pH reduced form of amicyanin is not oxidised directly; instead, oxidation occurs only after conversion to the high-pH form. Simulations show that this conversion, which gates the electron transfer, occurs with a rate constant >750 s-1 at 25 degrees C. In order to decrease the rate of the coupled reaction, the experiments were performed at 0 degrees C, at which the rate constant for this conversion was determined to be 35 +/- 20 s-1. Together with evidence from NMR, the results lead to a mechanism involving protonation and dissociation of the copper coordinating histidine-96 in the reduced form.     

Stopped flow and steady state kinetic studies of the effects of metabolites on the soluble form of NADP+:isocitrate dehydrogenase

The cytosolic form of NADP+:isocitrate dehydrogenase, a primary source of the NADPH required for de novo fatty acid synthesis in lactating bovine mammary gland, was studied to determine possible mechanisms of regulation by metabolites. Stopped flow kinetics showed a distinct lag time, followed by attainment of an apparently linear final velocity. Direct nonlinear regression analyses of the reaction progress curves allowed for the calculation of the rate constant (kappa) for the transition of the enzyme from an inactive to an active form; this transition is best catalyzed by its metal-substrate complex. Preincubation with metal-substrate or metal-citrate nearly abolished the lag by increasing kappa 10-fold. In steady state experiments, analyses of velocity versus metal-citrate complex as a binding isotherm, following the assumptions of Wyman's theory of thermodynamic linkage, showed that binding of metal-citrate complex could both activate and inhibit the enzyme. This analysis suggested: (a) activation by binding to sites with an average dissociation constant of 0.25 mM; (b) inhibition by binding to sites with an average dissociation constant of 3.83 mM; and (c) modulation (reactivation) by binding to sites with an average dissociation constant of 1.54 mM. Concentration ranges observed for these transitions are compatible with physiological conditions, suggesting that complexes of metal-citrate and metal-isocitrate serve to modulate the activity of NADP+:isocitrate dehydrogenase.     

Effects of solvent viscosity and different guanidine salts on the kinetics of ribonuclease A chain folding

The kinetics of folding of the two forms of unfolded ribonuclease A have been measured as a function of solvent viscosity by adding either glycerol or sucrose. The aim is to find out if either reaction is rate limited by segmental motion whose rate depends on external friction. The fast folding reaction (U₂ ? N) is known to be the direct folding process, and the slow folding reaction (U₁ ? N) is known to be rate limited by an interconversion between two forms (U₁ ? U₂) which are present after unfolding in strongly denaturing conditions. No dependence on solvent viscosity is found, either for the direct folding reaction or for the interconversion reaction. Each folding reaction has also been tested to see if its rate depends on the concentration of one or more partly folded intermediates, by adding denaturants destabilize any partly folded structures. Different guanidine salts are used as denaturants to vary the denaturing effectiveness of the salt while holding the guanidinium ion concentration constant. The rates both of the direct folding reaction and of the interconversion reaction decrease in relation to the denaturing effectiveness of the salt. However, there is a basic difference between the responses of the fast and slow folding reactions to low concentrations of denaturants. Although each folding reaction produces native protein, there is an 800-fold decrease in the rate of the fast folding reaction in 1M guanidine thiocyanate and only a 13-fold decrease in the rate of the slow folding reaction. This is consistent with the fast reaction being the direct folding process and the slow reaction being rate limited by the initial conversion of the slowrefolding to the fast-refolding form. Both the lack of viscosity dependence and the effects of denaturants indicate that the formation of structure is rate limiting in the direct folding reaction, U₂ ? N. The failure to find a viscosity dependence for the interconversion reaction, U₁ ? U₂, indicates that in this reaction also friction-limited segmental motion is not the rate-limiting process. Since the U₁ ? U₂ interconversion still occurs when the polypeptide chain is completely unfolded, the surprising result is that its rate in refolding conditions depends significantly on a reaction intermediate which is “denatured” by guanidine salts.     

Kinetic modeling and fitting software for interconnected reaction schemes: VisKin

Reaction kinetics for complex, highly interconnected kinetic schemes are modeled using analytical solutions to a system of ordinary differential equations. The algorithm employs standard linear algebra methods that are implemented using MatLab functions in a Visual Basic interface. A graphical user interface for simple entry of reaction schemes facilitates comparison of a variety of reaction schemes. To ensure microscopic balance, graph theory algorithms are used to determine violations of thermodynamic cycle constraints. Analytical solutions based on linear differential equations result in fast comparisons of first order kinetic rates and amplitudes as a function of changing ligand concentrations. For analysis of higher order kinetics, we also implemented a solution using numerical integration. To determine rate constants from experimental data, fitting algorithms that adjust rate constants to fit the model to imported data were implemented using the Levenberg-Marquardt algorithm or using Broyden-Fletcher-Goldfarb-Shanno methods. We have included the ability to carry out global fitting of data sets obtained at varying ligand concentrations. These tools are combined in a single package, which we have dubbed VisKin, to guide and analyze kinetic experiments. The software is available online for use on PCs.     

The kinetics of the reduction of cytochrome c by the superoxide anion radical.

1. At neutral pH ferricytochrome c is reduced by the superoxide anion radical (O2-), without loss of enzymatic activity, by a second order process in which no intermediates are observed. The yield of ferrocytochrome c (82-104%), as related to the amount of O2- produced, is slightly dependent on the concentration of sodium formate in the matrix solution. 2. The reaction (k1 equals (1.1+/-0.1) - 10(6) M-1 - s-1 at pH 7.2, I equals 4 mM and 21 degrees C) can be inhibited by superoxide dismutase and trace amounts of copper ions. The inhibition by copper ions is removed by EDTA without interference in the O2- reduction reaction. 3. The second-order rate constant for the reaction of O2- with ferricytochrome c depends on the pH of the matrix solution, decreasing rapidly at pH greater than 8. The dependence of the rate constant on the pH can be explained by assuming that only the neutral form of ferricytochrome c reacts with O2- and that the alkaline form of the hemoprotein is unreactive. From studies at pH 8.9, the rate for the transition from the alkaline to the neutral form of ferricytochrome c can be estimated to be 0.3 s-1 (at 21 degrees C and I equals 4 mM). 4. The second-order rate constant for the reaction of O2- with ferricytochrome c is also dependent on the ionic strength of the medium. From a plot of log k1 versus I1/2-(I + alphaI1/2)-1 we determined the effective charge on the ferricytochrome c molecule as +6.3 and the rate constant at I equals 0 as (3.1+/-0.1) - 10(6) M-1 - s-1 (pH 7.1, 21 degrees C). 5. The possibility that singlet oxygen is formed as a product of the reaction of O2- with ferricytochrome c can be ruled out on thermodynamic grounds.     

Calculation of thermodynamic properties of species from binding of a ligand by a macromolecule

In the absence of experimental methods for determining concentrations of species in protein-ligand binding, it is not possible to determine the thermodynamic properties of species directly. However, this article on a simple reaction system shows that measurements of the average number of oxygen molecules bound at various T, pH and concentrations of molecular oxygen can be used to calculate thermodynamic properties of species. The simple system considered has some of the characteristics of the binding of oxygen by hemoglobin, but it has been simplified so that the method for obtaining thermodynamic information can be clarified. A table of standard thermodynamic properties of species is the most efficient way to store thermodynamic information on a reaction system. All the standard further transformed thermodynamic properties at specified T, pH and concentrations of molecular oxygen, all the standard transformed thermodynamic properties at specified T and pH, and all the standard thermodynamic properties of species at a specified temperature can be calculated. These calculations are based on the fact that the mathematical function for the standard further transformed Gibbs energy of the system contains all the thermodynamic information on the system. These properties are all interrelated by Maxwell equations.     

Laser excitation studies of the product release steps in the catalytic cycle of the light-driven enzyme, protochlorophyllide oxidoreductase

The latter stages of the catalytic cycle of the light-driven enzyme, protochlorophyllide oxidoreductase, have been investigated using novel laser photoexcitation methods. The formation of the ternary product complex was initiated with a 6-ns laser pulse, which allowed the product release steps to be kinetically accessed for the first time. Subsequent absorbance changes associated with the release of the NADP+ and chlorophyllide products from the enzyme could be followed on a millisecond timescale. This has facilitated a detailed kinetic and thermodynamic characterization for the interconversion of all the various bound and unbound product species. Initially, NADP+ is released from the enzyme in a biphasic process with rate constants of 1210 and 237 s(-1). The rates of both phases show a significant dependence on the viscosity of the solvent and become considerably slower at higher glycerol concentrations. The fast phase of this process exhibits no dependence on NADP+ concentration, suggesting that conformational changes are required prior to NADP+ release. Following NADP+ release, the NADPH rebinds to the enzyme with a maximum rate constant of approximately 72 s(-1). At elevated temperatures (>298 K) chlorophyllide is released from the enzyme to yield the free product with a maximum rate constant of 20 s(-1). The temperature dependencies of the rates of each of these steps were measured, and enthalpies and entropies of activation were calculated using the Eyring equation. A comprehensive kinetic and thermodynamic scheme for these final stages of the reaction mechanism is presented.     

The reaction of trimethylamine dehydrogenase with trimethylamine

The reductive half-reaction of trimethylamine dehydrogenase with its physiological substrate trimethylamine has been examined by stopped-flow spectroscopy over the pH range 6.0-11.0, with attention focusing on the fastest of the three kinetic phases of the reaction, the flavin reduction/substrate oxidation process. As in previous work with the slow substrate diethylmethylamine, the reaction is found to consist of three well resolved kinetic phases. The observed rate constant for the fast phase exhibits hyperbolic dependence on the substrate concentration with an extrapolated limiting rate constant (klim) greater than 1000 s-1 at pH above 8.5, 10 degrees C. The kinetic parameter klim/Kd for the fast phase exhibits a bell-shaped pH dependence, with two pKa values of 9.3 +/- 0.1 and 10. 0 +/- 0.1 attributed to a basic residue in the enzyme active site and the ionization of the free substrate, respectively. The sigmoidal pH profile for klim gives a single pKa value of 7.1 +/- 0. 2. The observed rate constants for both the intermediate and slow phases are found to decrease as the substrate concentration is increased. The steady-state kinetic behavior of trimethylamine dehydrogenase with trimethylamine has also been examined, and is found to be adequately described without invoking a second, inhibitory substrate-binding site. The present results demonstrate that: (a) substrate must be protonated in order to bind to the enzyme; (b) an ionization group on the enzyme is involved in substrate binding; (c) an active site general base is involved, but not strictly required, in the oxidation of substrate; (d) the fast phase of the reaction with native enzyme is considerably faster than observed with enzyme isolated from Methylophilus methylotrophus that has been grown up on dimethylamine; and (e) a discrete inhibitory substrate-binding site is not required to account for excess substrate inhibition, the kinetic behavior of trimethylamine dehydrogenase can be readily explained in the context of the known properties of the enzyme.     

Thermodynamics and kinetics of protein folding: an evolutionary perspective

This article appeals to an evolutionary model which postulates that primordial proteins were described by small polypeptide chains which (i) lack disulfide bridges, and (ii) display slow folding rates with multi-state kinetics, to determine relations between structural properties of proteins and their folding kinetics. We parameterize the energy landscape of proteins in terms of thermodynamic activation variables. The model studies evolutionary changes in these thermodynamic parameters, and we invoke relations between these activation variables and structural properties of the protein to predict the following correspondence between protein structure and folding kinetics. 1. Proteins with inter- and intra-chain disulfide bridges: large variability in both folding rates and stability of intermediates, multi-state kinetics. 2. Proteins which lack inter and intra-chain disulfide bridges. 2.1 Single-domain chains: fast folding rates; unstable intermediates; two-state kinetics. 2.2 Multi-domain monomers: intermediate rates; metastable intermediates; multi-state kinetics. 2.3 Multi-domain oligomers: slow rates; metastable intermediates; multi-state kinetics. The evolutionary model thus provides a kinetic characterization of one important subfamily of proteins which we describe by the following properties: Folding dynamics of single-domain proteins which lack disulfide bridges are described by two-state kinetics. Folding rate of this class of proteins is positively correlated with the thermodynamic stability of the folded state.     

Counteraction of trehalose on urea-induced protein unfolding: Thermodynamic and kinetic studies

Urea-induced protein denaturation can be effectively inhibited by trehalose, but the thermodynamic and kinetic behaviors are still unclear. Herein, the counteraction of trehalose on urea-induced unfolding of ferricytochrome c was studied. Thermodynamic parameters for the counteraction of trehalose were derived based on fluorescence spectroscopic data. Then the kinetics was emphatically investigated by stopped-flow fluorescence spectroscopy. Urea-induced unfolding of ferricytochrome c in 8.00 mol/L urea solution reveals two observable phases, including fast and slow phases following a burst phase. Trehalose has little influence on the burst phase amplitude. Nevertheless, the observable unfolding pathway is significantly affected by trehalose. At lower trehalose concentrations (<0.20 mol/L) in 8.00 mol/L urea, the unfolding pathways still keep to show two phases. However, the rate constant and amplitude for the fast phase diminish with increasing trehalose concentration. In contrast, the rate constant for the slow phase shows only a slight change with a significant increase of the amplitude. At higher trehalose concentrations (>0.30 mol/L), the unfolding pathway is transformed into a single slow phase. The rate constant and amplitude for the single phase also decrease with increasing trehalose concentration. The studies are expected to help our understanding of trehalose effects on protein stability.     

Kinetic analysis on nitric oxide binding of recombinant Prolixin-S, a nitric oxide transport protein from the bloodsucking bug, Rhodnius prolixus.

Kinetics of the NO binding and removal reaction of recombinant Prolixin-S (rProlixin-S) were analyzed using stopped-flow spectrophotometry. The reaction was observed as a biphasic process. The rate constant of the fast phase increased linearly as NO concentration increased. The rate constant at the slow phase increased as NO concentrations increased at low NO concentration, then reached a plateau at high NO concentration. These NO dependencies of the reaction are characteristic of a bimolecular two-step consecutive reaction. The reaction consisted of the fast NO binding reaction of rProlixin-S and the following slow structural change of NO-protein complex. Kinetic studies revealed that the NO binding rate constant was independent of pH, but the rate constant of the NO removal reaction increased as pH increased. The apparent NO dissociation constant (Kd) of rProlixin-S was also calculated from the values of the kinetic parameters obtained in this work. The Kd value increased as pH and temperature increased. The Kd value of rProlixin-S and NO was 10-300 nM in regular physiological condition, which is 103 higher and 103 lower than those of the other ferric and ferrous hemoproteins and NO, respectively. These results indicate that Prolixin-S is one of NO transport proteins regulating blood pressure.     

A new rate law describing microbial respiration

The rate of microbial respiration can be described by a rate law that gives the respiration rate as the product of a rate constant, biomass concentration, and three terms: one describing the kinetics of the electron-donating reaction, one for the kinetics of the electron-accepting reaction, and a thermodynamic term accounting for the energy available in the microbe's environment. The rate law, derived on the basis of chemiosmotic theory and nonlinear thermodynamics, is unique in that it accounts for both forward and reverse fluxes through the electron transport chain. Our analysis demonstrates how a microbe's respiration rate depends on the thermodynamic driving force, i.e., the net difference between the energy available from the environment and energy conserved as ATP. The rate laws commonly applied in microbiology, such as the Monod equation, are specific simplifications of the general law presented. The new rate law is significant because it affords the possibility of extrapolating in a rigorous manner from laboratory experiment to a broad range of natural conditions, including microbial growth where only limited energy is available. The rate law also provides a new explanation of threshold phenomena, which may reflect a thermodynamic equilibrium where the energy released by electron transfer balances that conserved by ADP phosphorylation.     

A New Rate Law Describing Microbial Respiration

The rate of microbial respiration can be described by a rate law that gives the respiration rate as the product of a rate constant, biomass concentration, and three terms: one describing the kinetics of the electron-donating reaction, one for the kinetics of the electron-accepting reaction, and a thermodynamic term accounting for the energy available in the microbe's environment. The rate law, derived on the basis of chemiosmotic theory and nonlinear thermodynamics, is unique in that it accounts for both forward and reverse fluxes through the electron transport chain. Our analysis demonstrates how a microbe's respiration rate depends on the thermodynamic driving force, i.e., the net difference between the energy available from the environment and energy conserved as ATP. The rate laws commonly applied in microbiology, such as the Monod equation, are specific simplifications of the general law presented. The new rate law is significant because it affords the possibility of extrapolating in a rigorous manner from laboratory experiment to a broad range of natural conditions, including microbial growth where only limited energy is available. The rate law also provides a new explanation of threshold phenomena, which may reflect a thermodynamic equilibrium where the energy released by electron transfer balances that conserved by ADP phosphorylation.     

Thermodynamics of Random Reaction Networks

Reaction networks are useful for analyzing reaction systems occurring in chemistry, systems biology, or Earth system science. Despite the importance of thermodynamic disequilibrium for many of those systems, the general thermodynamic properties of reaction networks are poorly understood. To circumvent the problem of sparse thermodynamic data, we generate artificial reaction networks and investigate their non-equilibrium steady state for various boundary fluxes. We generate linear and nonlinear networks using four different complex network models (Erdős-Rényi, Barabási-Albert, Watts-Strogatz, Pan-Sinha) and compare their topological properties with real reaction networks. For similar boundary conditions the steady state flow through the linear networks is about one order of magnitude higher than the flow through comparable nonlinear networks. In all networks, the flow decreases with the distance between the inflow and outflow boundary species, with Watts-Strogatz networks showing a significantly smaller slope compared to the three other network types. The distribution of entropy production of the individual reactions inside the network follows a power law in the intermediate region with an exponent of circa −1.5 for linear and −1.66 for nonlinear networks. An elevated entropy production rate is found in reactions associated with weakly connected species. This effect is stronger in nonlinear networks than in the linear ones. Increasing the flow through the nonlinear networks also increases the number of cycles and leads to a narrower distribution of chemical potentials. We conclude that the relation between distribution of dissipation, network topology and strength of disequilibrium is nontrivial and can be studied systematically by artificial reaction networks.