1. A mathematical model is presented for photosynthetic carbohydrate formation in C3 plants under conditions of light and carbon dioxide saturation. The model considers reactions of the Calvin cycle with triose phosphate export and starch production as main output processes, and treats concentrations of NADPH, NAD+, C 0 2 , and H + as fixed parameters of the system. Using equilibrium approximations for all reaction steps close to equilibrium, steady-state and transient-state relationships are derived which may be used for calculation of reaction fluxes and concentrations of the 13 carbohydrate cycle intermediates, glucose 6-phosphate, glucose I-phosphate, ATP, ADP, and inorganic (0rtho)phosphate.2. Predictions of the model were examined with the assumption that photosynthate export from the chloroplast occurs to a medium containing orthophosphate as the only exchangeable metabolite. The results indicate that the Calvin cycle may operate in a single dynamically stable steady state when the external concentration of orthophosphate does not exceed 1.9 mM. At higher concentrations of the external metabolite, the reaction system exhibits overload breakdown; the excessive rate of photosynthate export deprives the system of cycle intermediates such that the cycle activity progressively approaches zero.3 . Reactant concentrations calculated for the stable steady state that may obtain are in satisfactory agreement with those observed experimentally, and the model accounts with surprising accuracy for experimentally observed effects of external orthophosphate on the steady-state cycle activity and rate of starch production.4. Control analyses are reported which show that most of the non-equilibrium enzymes in the system have a strong regulatory influence on the steady-state level of all of the cycle intermediates. Substrate concentration control coefficients for cycle enzymes may be positive, such that an increase in activity of an enzyme may raise the steady-state concentration of the substrate is consumes.5. Under optimal external conditions (0.15-0.5 mM orthophosphate), reaction flux in the Calvin cycle is controlled mainly by ATP synthetase and sedoheptulose bisphosphatase; the cycle activity approaches the maximum velocity that can be supported by the latter enzyme. At lower concentrations of external orthophosphate the cycle activity is controlled almost exclusively by the phosphate translocator. At high external orthophosphate concentrations the phosphate translocator resumes predominant control, but also other non-equilibrium enzymes gain strong flux control with one notable exception : ribulosebisphosphate carboxylase has no significant regulatory influence on the cycle activity under conditions of light and C 0 2 saturation, nor does it control the concentration of any cycle intermediate other than its substrate.

The catalytic interaction of glyceraldehyde-3-phosphate dehydrogenase with glyceraldehyde 3-phosphate has been examined by transient-state kinetic methods. The results confirm previous reports that the apparent K , for oxidative phosphorylation of glyceraldehyde 3-phosphate decreases at least 50-fold when the substrate is generated in a coupled reaction system through the action of aldolase on fructose 1,6-bisphosphate, but lend no support to the proposal that glyceraldehyde 3-phosphate is directly transferred between the two enzymes without prior release to the reaction medium. A theoretical analysis is presented which shows that the kinetic behaviour of the coupled two-enzme system is compatible in all respects tested with a free-diffusion mechanism for the transfer of glyceraldehyde 3-phosphate from the producing enzyme to the consuming one.Ovadi and Keleti in 1978 reported that the apparent K , for the catalytic interaction of glyceraldehyde-3-phosphate dehydrogenase with glyceraldehyde 3-phosphate decreases about 50-fold when the substrate is generated in a coupled reaction system through the action of aldolase on fructose 1,6-bisphosphate 111. This effect was attributed to reversible complex formation between the two enzymes, permitting the substrate to be directly transferred from aldolase to glyceraldehyde-3-phosphate dehydrogenase without prior release to the reaction medium. Kinetic evidence claimed to be indicative of a direct transfer of metabolites has later been presented also for coupled two-enzyme reactions involving aldolase and glycerol-3-phosphate dehydrogenase [2], as well as glyceraldehyde-3-phosphate dehydrogenase and phosphoglycerate kinase [3], alcohol dehydrogenase [4], or various other dehydrogenases [5].The above reports are of outstanding enzymological interest because they suggest that enzymes in the glycolytic and ancillary pathways might be structurally designed to interact with a protein-substrate complex rather than with a free substrate. They also bring into question inferences provided by the application of modern control theory [6, 71 for analysis of the dynamic behaviour and regulation of the glycolytic sequence of reactions ; such analyses invariably have been based on the tacit assumption that metabolite transfer between consecutive enzymes in the glycolytic pathway occurs via the reaction medium through free diffusion. It seems most important, therefore, to ascertain that unambiguous kinetic evidence has been presented to support the proposed existence of mechanisms for direct metabolite transfer in reaction systems involving glycolytic enzymes.

The regulatory implications of the interaction of ribulosebisphosphate carboxylase with metabolites participating in the Calvin photosynthesis cycle has been examined by control analysis based on our recently described kinetic model for photosynthetic carbohydrate formation in the chloroplast of C3 plants. The results provide clear evidence that the Calvin cycle activity under conditions of light and CO2 saturation is insignificantly affected by the inhibition of ribulosebisphosphate carboxylase caused by metabolites such as 3‐phosphoglycerate, fructose 1,6‐bisphosphate, sedoheptulose 1,7‐bisphosphate, NADPH, and inorganic orthophosphate. Due to the exceptionally high stromal concentration of the carboxylase, metabolite binding to the enzyme affects the Calvin cycle activity indirectly by reducing the pool of free orthophosphate and phosphorylated metabolites available for the cyclic reactions. This pool reduction corresponds typically to about 5 mM total phosphate and derives mainly from the binding of ribulose bisphosphate and orthophosphate. Substantial amounts of the metabolites interacting with ribulosebisphosphate carboxylase are present in an enzyme‐bound form. The bound form of the Calvin cycle intermediates sedoheptulose bisphosphate, fructose bisphosphate, and ribulose bisphosphate typically accounts for about 70, 80, and 90%, respectively, of the total stromal concentration of the intermediate.

An extension of the available kinetic theory for reactions in the transient state is presented which establishes that single-enzyme reactions may exhibit damped oscillations under the conditions of standard kinetic experiments performed by stopped-flow techniques. Such oscillations may occur for reasonable magnitudcs of rlite constants in the enzymic reaction mechanism and at physiological concentrations of enzyme and substrate. In the simplest reaction systems, the oscillations will be strongly damped and lead to progress curves resembling those of a reaction governed by standard exponential transients; statistical rcgression methods may then have to be applied for their detection and characterization. The observation that single-enzyme reactions may exhibit oscillatory behaviour points to a previously unrecognized possible source of the damped oscillations observed in metabolic systems such as the pathways of glycolysis or photosynthesis.Present mechanistic knowledge about enzymic catalysis is based to a large extent on results obtained by kinetic mcthodology. The corresponding theory has been the subject of extensive investigations and several generalized treatments are now available to show how experimentally applicable rate equations can be derived for enzymc reactions in steady as well as transient states [I -51. Since the time dependence of enzymic reactions is governed by non-lincar differcntial equations lacking a general analytical solution, the applicability of analytically derived rate equations has to be justified by a proper choice of (linearizing) experimental conditions. In thc case of steady-state kinetic studies, one examines initial reaction velocities using substrate in large excess to enzyme. Similar precautions havc to bc takcn in transient state kinetic experiments perrormed by stopped-flow techniques to ensure that reactions proceed under approximately linear (pseudo first-order) conditions.Assuming that linearizing pseudo first-ordcr conditions do occur, the initial time course of an enzymic reaction will be governed by one or several exponential transients [4]. Previous dctailcd theoretical treatments have considered only the case that rate parameters for the exponential transients are real. The possibility cannot be excluded, however, that the transient ratc parameters under certain conditions inay be complex, such that thc time course exhibits thc characteristics of a damped oscillation rather than conforming to an exponential first-order process in the standard kinetic sense.The occurrence of damped oscillations in biological reaction systems has received much attention during the last two decades [6 ~ 81. Considering the attempts to interpret these oscillations in terms of a kinetic coupling between distinct en7ymes in metabolic pathways, it is of obvious interest to examine if a single-cnzymc reaction may exhibit an oscillatory behaviour. The theoretical analysis now prescnted cstablishes that such may well be the case. Generalized and specific re- The non-linear differential equat...

Kinetic model studies and control analyses of the Calvin photosynthesis cycle have been performed to characterize the dependence of the cycle activity on maximum velocities and K,values for the intcraction of thc non-equilibrium cycle enzymes and ATP synthetase with their substrates under conditions of light and carbon dioxide saturation. The results show that K, values have no major influence on the cycle activity at optimal concentrations of external orthophosphate. The maximum cycle activity is controlled mainly by the catalytic capacities of ATP synthetase and sedoheptulose-bisphosphatase, and is close to the maximum cycle flux that can be supported by these two enzymes.The Calvin cycle for photosynthetic carbohydrate formation plays a critical role in agricultural productivity and is of outstanding importance for life on earth in general. Considerable research, therefore, has been directed towards the biological regulation of this metabolic pathway [l -41. Such research has provided valuable information on a variety of factors that may control and ultimately limit the cycle activity, but a deeper insight into the regulatory mechanisms that actually apply has been hampered by the extreme kinetic complexity of the reaction system ; the Calvin cycle involves 13 enzymes acting on 16 metabolites in an intricate network of reactions, and is dcpendent on input processes providing the system with ATP and NADPH as well as on output steps withdrawing photosynthetic products from the reaction cycle. The dynamic and regulatory properties of such a complcx system cannot be reliably established by intuitive reasoning, but require detailed analysis and characterization by mathematical modelling.We have recently presented a kinetic model for the Calvin photosynthesis cycle and ancillary pathway of starch production in the chloroplast of C 3 plants under conditions of light and carbon dioxide saturation [5]. This model is based on experimentally documcnted rate equations for all enzyinically catalysed non-equilibrium steps in the reaction system and has been shown to provide a most satisfactory description of certain experimentally observed stcady-state characteristics of the photosynthetic process of carbohydrate formation. Despite extensive use of simplifying approximations, the model is rather complex and describes the rate behaviour of the Calvin cycle as a function of more than 50 different reaction parameters. Such complexity may be a drawback in certain contexts, but increases the reliability and utility of the model from an analytical point of view. Examination of the parameter dependence of a model that is sufficiently elaborated to account for all major factors of potential regulatory interest should lead to a substantially deepened understanding of the actual operation and control of the modelled biological system.Most of the parameters considered in our model represent kinetic or equilibrium constants for enzymically catalysed steps of the examined reaction system. In particular, the model includes kinetic param...