Multistationarity in Sequential Distributed Multisite Phosphorylation Networks

Multisite phosphorylation networks are encountered in many intracellular processes like signal transduction, cell-cycle control, or nuclear signal integration. In this contribution, networks describing the phosphorylation and dephosphorylation of a protein at n sites in a sequential distributive mechanism are considered. Multistationarity (i.e., the existence of at least two positive steady state solutions of the associated polynomial dynamical system) has been analyzed and established in several contributions. It is, for example, known that there exist values for the rate constants where multistationarity occurs. However, nothing else is known about these rate constants. Here, we present a sign condition that is necessary and sufficient for multistationarity in n-site sequential, distributive phosphorylation. We express this sign condition in terms of linear systems, and show that solutions of these systems define rate constants where multistationarity is possible. We then present, for n≥2, a collection of feasible linear systems, and hence give a new and independent proof that multistationarity is possible for n≥2. Moreover, our results allow to explicitly obtain values for the rate constants where multistationarity is possible. Hence, we believe that, for the first time, a systematic exploration of the region in parameter space where multistationarity occurs has become possible. One consequence of our work is that, for any pair of steady states, the ratio of the steady state concentrations of kinase-substrate complexes equals that of phosphatase-substrate complexes.     

Wide range of metabolic adaptations to the acquisition of the Calvin cycle revealed by comparison of microbial genomes

Knowledge of the genetic basis for autotrophic metabolism is valuable since it relates to both the emergence of life and to the metabolic engineering challenge of incorporating CO₂ as a potential substrate for biorefining. The most common CO₂ fixation pathway is the Calvin cycle, which utilizes Rubisco and phosphoribulokinase enzymes. We searched thousands of microbial genomes and found that 6.0% contained the Calvin cycle. We then contrasted the genomes of Calvin cycle-positive, non-cyanobacterial microbes and their closest relatives by enrichment analysis, ancestral character estimation, and random forest machine learning, to explore genetic adaptations associated with acquisition of the Calvin cycle. The Calvin cycle overlaps with the pentose phosphate pathway and glycolysis, and we could confirm positive associations with fructose-1,6-bisphosphatase, aldolase, and transketolase, constituting a conserved operon, as well as ribulose-phosphate 3-epimerase, ribose-5-phosphate isomerase, and phosphoglycerate kinase. Additionally, carbohydrate storage enzymes, carboxysome proteins (that raise CO₂ concentration around Rubisco), and Rubisco activases CbbQ and CbbX accompanied the Calvin cycle. Photorespiration did not appear to be adapted specifically for the Calvin cycle in the non-cyanobacterial microbes under study. Our results suggest that chemoautotrophy in Calvin cycle-positive organisms was commonly enabled by hydrogenase, and less commonly ammonia monooxygenase (nitrification). The enrichment of specific DNA-binding domains indicated Calvin-cycle associated genetic regulation. Metabolic regulatory adaptations were illustrated by negative correlation to AraC and the enzyme arabinose-5-phosphate isomerase, which suggests a downregulation of the metabolite arabinose-5-phosphate, which may interfere with the Calvin cycle through enzyme inhibition and substrate competition. Certain domains of unknown function that were found to be important in the analysis may indicate yet unknown regulatory mechanisms in Calvin cycle-utilizing microbes. Our gene ranking provides targets for experiments seeking to improve CO₂ fixation, or engineer novel CO₂-fixing organisms.     

Comparative evolution of photosynthetic genes in response to polyploid and nonpolyploid duplication

The likelihood of duplicate gene retention following polyploidy varies by functional properties (e.g. gene ontologies or protein family domains), but little is known about the effects of whole-genome duplication on gene networks related by a common physiological process. Here, we examined the effects of both polyploid and nonpolyploid duplications on genes encoding the major functional groups of photosynthesis (photosystem I, photosystem II, the light-harvesting complex, and the Calvin cycle) in the cultivated soybean (Glycine max), which has experienced two rounds of whole-genome duplication. Photosystem gene families exhibit retention patterns consistent with dosage sensitivity (preferential retention of polyploid duplicates and elimination of nonpolyploid duplicates), whereas Calvin cycle and light-harvesting complex gene families do not. We observed similar patterns in barrel medic (Medicago truncatula), which shared the older genome duplication with soybean but has evolved independently for approximately 50 million years, and in Arabidopsis (Arabidopsis thaliana), which experienced two nested polyploidy events independent from the legume duplications. In both soybean and Arabidopsis, Calvin cycle gene duplicates exhibit a greater capacity for functional differentiation than do duplicates within the photosystems, which likely explains the greater retention of ancient, nonpolyploid duplicates and larger average gene family size for the Calvin cycle relative to the photosystems.     

Multistationarity in mass action networks with applications to ERK activation

Ordinary Differential Equations (ODEs) are an important tool in many areas of Quantitative Biology. For many ODE systems multistationarity (i.e. the existence of at least two positive steady states) is a desired feature. In general establishing multistationarity is a difficult task as realistic biological models are large in terms of states and (unknown) parameters and in most cases poorly parameterized (because of noisy measurement data of few components, a very small number of data points and only a limited number of repetitions). For mass action networks establishing multistationarity hence is equivalent to establishing the existence of at least two positive solutions of a large polynomial system with unknown coefficients. For mass action networks with certain structural properties, expressed in terms of the stoichiometric matrix and the reaction rate-exponent matrix, we present necessary and sufficient conditions for multistationarity that take the form of linear inequality systems. Solutions of these inequality systems define pairs of steady states and parameter values. We also present a sufficient condition to identify networks where the aforementioned conditions hold. To show the applicability of our results we analyse an ODE system that is defined by the mass action network describing the extracellular signal-regulated kinase (ERK) cascade (i.e. ERK-activation).     

On the regulatory significance of inhibitors acting on non-equilibrium enzymes in the Calvin photosynthesis cycle

Control analyses and kinetic model studies have been performed in order to obtain quantitative information on the regulatory significance of 12 experimentally well-documented inhibitory interactions of Calvin cycle intermediates with the four non-equilibrium cycle enzymes. Evidence is presented to show that none of these interactions contributes significantly to the cycle flux control over the range of external orthophosphate concentrations where the reaction cycle shows close to optimal activity. Contrary to what has been generally supposed, the examined inhibitions appear to be of little interest for our understanding of the biological regulation of the Calvin photosynthesis cycle under conditions of light and carbon dioxide saturation.     

Dynamical Behaviour of Viral Cycle and Identification of Steady States

The molecular biology of viruses can be effectively described by kinetic logic as several feedback loops are implicated in all viral cycles and as viral proteins generally display several functions. We applied this method to the study of the rhabdovirus cycle.Formally, the dynamics of the model are explored on the basis of a discrete caricature (kinetic logic), with special emphasis on the role of the constitutive feedback loops to determine the essential dynamical behaviour of the viral cycle. From a biological point of view, our model accounts for several stable regimes or attractors: healthy cells, acute infection and different kinds of persistent infections, a multistationarity in good agreement with the existence of several positive feedback loops in our system.     

The Calvin cycle revisited

The sequence of reactions in the Calvin cycle, and the biochemical characteristics of the enzymes involved, have been known for some time. However, the extent to which any individual enzyme controls the rate of carbon fixation has been a long standing question. Over the last 10 years, antisense transgenic plants have been used as tools to address this and have revealed some unexpected findings about the Calvin cycle. It was shown that under a range of environmental conditions, the level of Rubisco protein had little impact on the control of carbon fixation. In addition, three of the four thioredoxin regulated enzymes, FBPase, PRKase and GAPDH, had negligible control of the cycle. Unexpectedly, non-regulated enzymes catalysing reversible reactions, aldolase and transketolase, both exerted significant control over carbon flux. Furthermore, under a range of growth conditions SBPase was shown to have a significant level of control over the Calvin cycle. These data led to the hypothesis that increasing the amounts of these enzymes may lead to an increase in photosynthetic carbon assimilation. Remarkably, photosynthetic capacity and growth were increased in tobacco plants expressing a bifunctional SBPase/FBPase enzyme. Future work is discussed which will further our understanding of this complex and important pathway, particularly in relation to the mechanisms that regulate and co-ordinate enzyme activity.This revised version was published online in October 2005 with corrections to the Cover Date.     

Evolutionary History of the Enzymes Involved in the Calvin–Benson Cycle in Euglenids

Euglenids are an ancient lineage that may have existed as early as 2 billion years ago. A mere 65 years ago, Melvin Calvin and Andrew A. Benson performed experiments on Euglena gracilis and elucidated the series of reactions by which carbon was fixed and reduced during photosynthesis. However, the evolutionary history of this pathway (Calvin–Benson cycle) in euglenids was more complex than Calvin and Benson could have imagined. The chloroplast present today in euglenophytes arose from a secondary endosymbiosis between a phagotrophic euglenid and a prasinophyte green alga. A long period of evolutionary time existed before this secondary endosymbiotic event took place, which allowed for other endosymbiotic events or gene transfers to occur prior to the establishment of the green chloroplast. This research revealed the evolutionary history of the major enzymes of the Calvin–Benson cycle throughout the euglenid lineage and showed that the majority of genes for Calvin–Benson cycle enzymes shared an ancestry with red algae and/or chromophytes suggesting they may have been transferred to the nucleus prior to the acquisition of the green chloroplast.     

Interruption of the Calvin cycle inhibits the repair of Photosystem II from photodamage

In photosynthetic organisms, impairment of the activities of enzymes in the Calvin cycle enhances the extent of photoinactivation of Photosystem II (PSII). We investigated the molecular mechanism responsible for this phenomenon in the unicellular green alga Chlamydomonas reinhardtii. When the Calvin cycle was interrupted by glycolaldehyde, which is known to inhibit phosphoribulokinase, the extent of photoinactivation of PSII was enhanced. The effect of glycolaldehyde was very similar to that of chloramphenicol, which inhibits protein synthesis de novo in chloroplasts. The interruption of the Calvin cycle by the introduction of a missense mutation into the gene for the large subunit of ribulose-1,5-bisphosphate carboxylase/oxygenase (Rubisco) also enhanced the extent of photoinactivation of PSII. In such mutant 10-6C cells, neither glycolaldehyde nor chloramphenicol has any additional effect on photoinactivation. When wild-type cells were incubated under weak light after photodamage to PSII, the activity of PSII recovered gradually and reached a level close to the initial level. However, recovery was inhibited in wild-type cells by glycolaldehyde and was also inhibited in 10-6C cells. Radioactive labelling and Northern blotting demonstrated that the interruption of the Calvin cycle suppressed the synthesis de novo of chloroplast proteins, such as the D1 and D2 proteins, but did not affect the levels of psbA and psbD mRNAs. Our results suggest that the photoinactivation of PSII that is associated with the interruption of the Calvin cycle is attributable primarily to the inhibition of the protein synthesis-dependent repair of PSII at the level of translation in chloroplasts.     

Replacing the Calvin cycle with the reductive glycine pathway in Cupriavidus necator

Formate can be directly produced from CO₂ and renewable electricity, making it a promising microbial feedstock for sustainable bioproduction. Cupriavidus necator is one of the few biotechnologically-relevant hosts that can grow on formate, but it uses the Calvin cycle, the high ATP cost of which limits biomass and product yields. Here, we redesign C. necator metabolism for formate assimilation via the synthetic, highly ATP-efficient reductive glycine pathway. First, we demonstrate that the upper pathway segment supports glycine biosynthesis from formate. Next, we explore the endogenous route for glycine assimilation and discover a wasteful oxidation-dependent pathway. By integrating glycine biosynthesis and assimilation we are able to replace C. necator's Calvin cycle with the synthetic pathway and achieve formatotrophic growth. We then engineer more efficient glycine metabolism and use short-term evolution to optimize pathway activity. The final growth yield we achieve (2.6 gCDW/mole-formate) nearly matches that of the WT strain using the Calvin Cycle (2.9 gCDW/mole-formate). We expect that further rational and evolutionary optimization will result in a superior formatotrophic C. necator strain, paving the way towards realizing the formate bio-economy.     

Discovery of the canonical Calvin–Benson cycle

It has been 65 years since the Calvin–Benson cycle was first formulated. In this paper, the development of the concepts that are critical to the cycle is traced and the contributions of Calvin, Benson, and Bassham are discussed. Some simplified views often found in text books such as ascending paper chromatography and the use of the “lollipop” for short labeling are discussed and further details given. Key discoveries that underpinned elucidation of the cycle such as the importance of sedoheptulose phosphate and ribulose 1,5-bisphosphate are described. The interchange of ideas between other researchers working on what is now called the pentose phosphate pathway and the development of the ideas of Calvin and Benson are explored while the gluconeogenic aspects of the cycle are emphasized. Concerns raised about anomalies of label distribution in glucose are considered. Other carbon metabolism pathways associated with the Calvin–Benson cycle are also described. Finally, there is a section describing the rift between Calvin and Benson.

     

Effects of the Calvin cycle on nicotinamide adenine dinucleotide concentrations and redox balances of Xanthobacter flavus

The levels of reduced and oxidized nicotinamide adenine dinucleotides were determined in Xanthobacter flavus during a transition from heterotrophic to autotrophic growth. Excess reducing equivalents are rapidly dissipated following induction of the Calvin cycle, indicating that the Calvin cycle serves as a sink for excess reducing equivalents. The physiological data support the conclusion previously derived from molecular studies in that expression of the Calvin cycle genes is controlled by the intracellular concentration of NADPH.     

Interruption of the Calvin cycle inhibits the repair of Photosystem II from photodamage

In photosynthetic organisms, impairment of the activities of enzymes in the Calvin cycle enhances the extent of photoinactivation of Photosystem II (PSII). We investigated the molecular mechanism responsible for this phenomenon in the unicellular green alga Chlamydomonas reinhardtii. When the Calvin cycle was interrupted by glycolaldehyde, which is known to inhibit phosphoribulokinase, the extent of photoinactivation of PSII was enhanced. The effect of glycolaldehyde was very similar to that of chloramphenicol, which inhibits protein synthesis de novo in chloroplasts. The interruption of the Calvin cycle by the introduction of a missense mutation into the gene for the large subunit of ribulose-1,5-bisphosphate carboxylase/oxygenase (Rubisco) also enhanced the extent of photoinactivation of PSII. In such mutant 10-6C cells, neither glycolaldehyde nor chloramphenicol has any additional effect on photoinactivation. When wild-type cells were incubated under weak light after photodamage to PSII, the activity of PSII recovered gradually and reached a level close to the initial level. However, recovery was inhibited in wild-type cells by glycolaldehyde and was also inhibited in 10-6C cells. Radioactive labelling and Northern blotting demonstrated that the interruption of the Calvin cycle suppressed the synthesis de novo of chloroplast proteins, such as the D1 and D2 proteins, but did not affect the levels of psbA and psbD mRNAs. Our results suggest that the photoinactivation of PSII that is associated with the interruption of the Calvin cycle is attributable primarily to the inhibition of the protein synthesis-dependent repair of PSII at the level of translation in chloroplasts.     

Computer modelling and experimental evidence for two steady states in the photosynthetic Calvin cycle.

We present observations of photosynthetic carbon dioxide assimilation, and leaf starch content from genetically modified tobacco (Nicotiana tabacum) plants in which the activity of the Calvin cycle enzyme, sedoheptulose-1,7-bisphosphatase, is reduced by an antisense construct. The measurements were made on leaves of varying ages and used to calculate the flux control coefficients of sedoheptulose-1,7-bisphosphatase over photosynthetic assimilation and starch synthesis. These calculations suggest that control coefficients for both are negative in young leaves, and positive in mature leaves. This behaviour is compared to control coefficients obtained from a detailed computer model of the Calvin cycle. The comparison demonstrates that the experimental observations are consistent with bistable behaviour exhibited by the model, and provides the first experimental evidence that such behaviour in the Calvin cycle occurs in vivo as well as in silico.     

Cyclic electron flow in C3 plants

This paper summarized our present view on the mechanism of cyclic electron flow in C3 plants. We propose that cyclic and linear pathways are in competition for the reoxidation of the soluble primary PSI acceptor, Ferredoxin (Fd), that freely diffuses in the stromal compartment. In the linear mode, Fd binds ferredoxin-NADP-reductase and electrons are transferred to NADP+ and then to the Benson and Calvin cycle. In the cyclic mode, Fd binds a site localized on the stromal side of the cytochrome b6f complex and electrons are transferred to P700 via a mechanism derived from the Q-cycle. In dark-adapted leaves, the cyclic flow operates at maximum rate, owing to the partial inactivation of the Benson and Calvin cycle. For increasing time of illumination, the activation of the Benson and Calvin cycle, and thus, that of the linear flow, is associated with a subsequent decrease in the rate of the cyclic flow. Under steady-state conditions of illumination, the contribution of cyclic flow to PSI turnover increases as a function of the light intensity (from 0 to approximately 50% for weak to saturating light, respectively). Lack of CO2 is associated with an increase in the efficiency of the cyclic flow. ATP concentration could be one of the parameters that control the transition between linear and cyclic modes.     

Positive circuits and d-dimensional spatial differentiation: application to the formation of sense organs in Drosophila

We discuss a rule proposed by the biologist Thomas according to which the possibility for a genetic network (represented by a signed directed graph called a regulatory graph) to have several stable states implies the existence of a positive circuit. This result is already known for different models, differential or discrete formalism, but always with a network of genes contained in a single cell. Thus, we can ask about the validity of this rule for a system containing several cells and with intercellular genetic interactions. In this paper, we consider the genetic interactions between several cells located on a d-dimensional lattice, i.e., each point of lattice represents a cell to which we associate the expression level of n genes contained in this cell. With this configuration, we show that the existence of a positive circuit is a necessary condition for a specific form of multistationarity, which naturally corresponds to spatial differentiation. We then illustrate this theorem through the example of the formation of sense organs in Drosophila.     

Review article. New insights into the structure and function of sedoheptulose-1,7-bisphosphatase; an important but neglected Calvin cycle enzyme

The photosynthetic carbon reduction (Calvin) cycle is the primary pathway for carbon fixation and the enzyme sedoheptulose-1,7-bisphosphatase functions in the regenerative phase of this cycle where it catalyses the dephosphorylation of sedoheptulose-1,7-bisphosphate. This enzyme is unique to the Calvin cycle and has no counterpart in non-photosynthetic organisms. The isolation and sequence analysis of an SBPase clone has led to a number of investigations which have yielded interesting and novel information on this enzyme and in this paper the biochemistry and molecular biology of SBPase are reviewed. Some recent exciting developments are also reported, including the analysis of transgenic plants with reduced levels of SBPase which has shown that SBPase is a key regulator of carbon flux and mutagenesis studies which have resulted in the identification of the redox active cysteines responsible for the regulation by light of SBPase catalytic activity.     

Regulation of the Calvin cycle for CO2 fixation as an example for general control mechanisms in metabolic cycles.

The theory of a metabolic cycle with the main portion of its intermediates remaining inside the cycle during one turnover has been developed. On this basis, the regulation of the Calvin cycle is analyzed. It is demonstrated that not only the reactions of non-equilibrium enzymes, as the carboxylation of ribulose 1,5-bisphosphate, but reactions that operate close to a thermodynamic equilibrium, especially the reduction of 3-phosphoglycerate and the transketolase reaction can significantly influence the total turnover period in the Calvin cycle. The role of compensating mechanisms in the maintenance of the photosynthesis rate upon changes of environmental conditions and of enzyme contents is analyzed for the Calvin cycle. It is shown that the change of the total quantity of the metabolites is one of the main self-regulated mechanisms in the Calvin cycle. A change of the ATP/ADP ratio can be used by the cell to maintain the CO2 assimilation rate, when the total quantity of the metabolites is changed. The developed analysis permits to explain some experimental data obtained with transgenic plants with restricted efflux of carbon from the chloroplasts.     

Chemical Reaction Systems with Toric Steady States

Mass-action chemical reaction systems are frequently used in computational biology. The corresponding polynomial dynamical systems are often large (consisting of tens or even hundreds of ordinary differential equations) and poorly parameterized (due to noisy measurement data and a small number of data points and repetitions). Therefore, it is often difficult to establish the existence of (positive) steady states or to determine whether more complicated phenomena such as multistationarity exist. If, however, the steady state ideal of the system is a binomial ideal, then we show that these questions can be answered easily. The focus of this work is on systems with this property, and we say that such systems have toric steady states. Our main result gives sufficient conditions for a chemical reaction system to have toric steady states. Furthermore, we analyze the capacity of such a system to exhibit positive steady states and multistationarity. Examples of systems with toric steady states include weakly-reversible zero-deficiency chemical reaction systems. An important application of our work concerns the networks that describe the multisite phosphorylation of a protein by a kinase/phosphatase pair in a sequential and distributive mechanism.