ScrumPy: metabolic modelling with Python

ScrumPy is a software package used for the definition and analysis of metabolic models. It is written using the Python programming language that is also used as a user interface. ScrumPy has features for both kinetic and structural modelling, but the emphasis is on structural modelling and those features of most relevance to analysis of large (genome-scale) models. The aim is at describing ScrumPy's functionality to readers with some knowledge of metabolic modelling, but implementation, programming and other computational details are omitted. ScrumPy is released under the Gnu Public Licence, and available for download from http://mudshark.brookes.ac.uk/ ScrumPy.     

Mathematical modelling of dynamics and control in metabolic networks: VI. Dynamic bifurcations in single biochemical control loops

A dynamic stability analysis of an extended form of the Goodwin equations is presented. The Goodwin equations are extended to include Michaelis-Menten kinetics for the removal of the end-product. Inclusion of saturation kinetic behavior substantially increases the likelihood of dynamic instability in this model control loop. Oscillations are found for reaction chains of low order, as low as second order, and low degrees of co-operativity, as low as v = 2, simultaneously, thus indicating that dynamic instability in this system exists for physiologically realistic parameter values. The branches of bifurcated solutions are computed numerically and unstable Hopf bifurcations are found. Further, solution branches from stable Hopf bifurcation points are found to "fold back", i.e. have periodic limit points, producing situations where multiple stable limit cycles exist.     

Kinetic metabolic modelling for the control of plant cells cytoplasmic phosphate

A previously developed kinetic metabolic model for plant metabolism was used in a context of identification and control of intracellular phosphate (Pi) dynamics. Experimental data from batch flask cultures of Eschscholtiza californica cells was used to calibrate the model parameters for the slow dynamics (growth, nutrition, anabolic pathways, etc.). Perturbation experiments were performed using a perfusion small-scale bioreactor monitored by in vivo³¹P NMR. Parameter identification for Pi metabolism was done by measuring the cells dynamic response to different inputs for extracellular Pi (two pulse-response experiments and a step-response experiment). The calibrated model can describe Pi translocation between the cellular pools (vacuole and cytoplasm). The effect of intracellular Pi management on ATP/ADP and phosphomonoesters concentrations is also described by the model. The calibrated model is then used to develop a control strategy on the cytoplasmic Pi pool. From the identification of the systems dynamics, a proportional-integral controller was designed and tuned. The closed-loop control was implemented in the small-scale NMR bioreactor and experimental results were in accordance with model predictions. Thus, the calibrated model is able to predict cellular behaviour for phosphate metabolism and it was demonstrated that it is possible to control the intracellular level of cytoplasmic Pi in plant cells.     

METAMOD: software for steady-state modelling and control analysis of metabolic pathways on the BBC microcomputer

METAMOD, a BBC microcomputer-based software package for steady-statemodelling and control analysis of model metabolic pathways,is described, The package consists of two programs. METADEFallows the user to define the pathway in terms of reactions,rate equations and initial concentrations of metabolites. METACALuses one of two algorithms to calculate the steady-state concentrationsand fluxes. One algorithm uses the current ratio of productionand consumption rates of variable metabolites to adjust iterativelytheir concentrations in such a way that they converge towardsthe steady state. The other algorithm solves the roots of thesystem equations by means of a quasi-Newtonian procedure. Controlanalysis allows the calculation of elasticity, control and responsecoefficients, by means of finite difference approximation. METAMODis interactive and easy to use, and suitable for teaching andresearch purposes. Received on January 17, 1986; accepted on June 2, 1986     

Plant genome-scale metabolic reconstruction and modelling

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Highlights► Recent advances have been made on plant genome-scale metabolic reconstruction. ► Cellular compartmentation makes plant genome-scale reconstruction challenging. ► Current reconstructions capture important features of plant metabolism. ► The models have been used to study isolated tissues and tissue interaction. ► We review the challenges and potential of plant reconstruction and modelling.     

A structured approach to the study of metabolic control principles in intact and impaired mitochondria

We devised an approach to extract control principles of cellular bioenergetics for intact and impaired mitochondria from ODE-based models and applied it to a recently established bioenergetic model of cancer cells. The approach used two methods for varying ODE model parameters to determine those model components that, either alone or in combination with other components, most decisively regulated bioenergetic state variables. We found that, while polarisation of the mitochondrial membrane potential (ΔΨ(m)) and, therefore, the protomotive force were critically determined by respiratory complex I activity in healthy mitochondria, complex III activity was dominant for ΔΨ(m) during conditions of cytochrome-c deficiency. As a further important result, cellular bioenergetics in healthy, ATP-producing mitochondria was regulated by three parameter clusters that describe (1) mitochondrial respiration, (2) ATP production and consumption and (3) coupling of ATP-production and respiration. These parameter clusters resembled metabolic blocks and their intermediaries from top-down control analyses. However, parameter clusters changed significantly when cells changed from low to high ATP levels or when mitochondria were considered to be impaired by loss of cytochrome-c. This change suggests that the assumption of static metabolic blocks by conventional top-down control analyses is not valid under these conditions. Our approach is complementary to both ODE and top-down control analysis approaches and allows a better insight into cellular bioenergetics and its pathological alterations.     

Mathematical modelling of dynamics and control in metabolic networks. II. Simple dimeric enzymes

The dynamics of enzyme cooperativity are examined by studying a homotropic dimeric enzyme with identical reaction sites, both of which follow irreversible Michaelis-Menten kinetics. The problem is approached via scaling and linearization of the governing mass action kinetic equations. Homotropic interaction between the two sites are found to depend on three dimensionless groups, two for the substrate binding step and one for the chemical transformation. The interaction between the two reaction sites is shown capable of producing dynamic behavior qualitatively different from that of a simple Michaelis-Menten system; when the two sites interact to increase enzymatic activity over that of two independent monomeric enzymes (positive cooperativity) damped oscillatory behavior is possible, and for negative cooperativity in the chemical transformation step a multiplicity of steady states can occur, with one state unstable and leading to runaway behavior. Linear analysis gives significant insight into system dynamics, and their parametric sensitivity, and a way to identify regions of the parameter space where the approximate quasi-stationary and quasi-equilibrium analyses are appropriate.     

Mathematical modelling of dynamics and control in metabolic networks. I. On Michaelis-Menten kinetics

As a starting point for modeling of metabolic networks this paper considers the simple Michaelis-Menten reaction mechanism. After the elimination of diffusional effects a mathematically intractable mass action kinetic model is obtained. The properties of this model are explored via scaling and linearization. The scaling is carried out such that kinetic properties, concentration parameters and external influences are clearly separated. We then try to obtain reasonable estimates for values of the dimensionless groups and examine the dynamic properties of the model over this part of the parameter space. Linear analysis is found to give excellent insight into reaction dynamics and it also gives a forum for understanding and justifying the two commonly used quasi-stationary and quasi-equilibrium analyses. The first finding is that there are two separate time scales inherent in the model existing over most of the parameter space, and in particular over the regions of importance here. Full modal analysis gives a new interpretation of quasi-stationary analysis, and its extension via singular perturbation theory, and a rationalization of the quasi-equilibrium approximation. The new interpretation of the quasi-steady state assumption is that the applicability is intimately related to dynamic interactions between the concentration variables rather than the traditional notion that a quasi-stationary state is reached, after a short transient period, where the rates of formation and decomposition of the enzyme intermediate are approximately equal. The modal analysis reveals that the generally used criterion for the applicability of quasi-stationary analysis that total enzyme concentration must be much less than total substrate concentration, et much less than St, is incomplete and that the criterion et much less than Km much less than St (Km is the well known Michaelis constant) is the appropriate one. The first inequality (et much less than Km) guarantees agreement over the longer time scale leading to quasi-stationary behavior or the applicability of the zeroth order outer singular perturbation solution but the second half of the criterion (Km much less than St) justifies zeroth order inner singular perturbation solution where the substrate concentration is assumed to be invariant. Furthermore linear analysis shows that when a fast mode representing the binding of substrate to the enzyme is fast it can be relaxed leading to the quasi-equilibrium assumption. The influence of the dimensionless groups is ascertained by integrating the equations numerically, and the predictions made by the linear analysis are found to be accurate.(ABSTRACT TRUNCATED AT 400 WORDS)     

Mathematical modelling of dynamics and control in metabolic networks. V. Static bifurcations in single biochemical control loops

Here we expand an earlier study of feedback activation in simple linear reaction sequences by searching the parameter space of biologically realistic rate laws for multiple stable steady states. The impetus for this work is to seek the origin of decision making strategies at the metabolic level, with particular emphasis on the switching between the operating conditions needed to meet changing substrate availability and organism requirements. The control loop considered herein is a linear reaction chain in which the end product of the reaction sequence feedback activates the first reaction in the sequence to produce feedback control. It has been found that the criteria for the existence of multiple steady state solutions in such loops involve only the kinetics of the regulatory enzyme controlling the first reaction and that of end product removal. The effects of these kinetics are examined here using two representative models for the regulatory enzyme: the lumped controller, based on Hill-type kinetics, and the symmetry model. The behavior of these two models is qualitatively similar, and both show the characteristics needed for switching between low and high substrate utilization. The removal rate is assumed to be of the Michaelis-Menten type. Judicious scaling of the governing equations permits separation of genetically determined kinetic parameters from concentration dependent ones. This allows us to conclude that, for a fixed set of kinetic parameters, the steady state flux through the loop can be switched between stable steady states by merely varying metabolite or enzyme concentrations. In particular, when the initial substrate exceeds a certain critical level, the loop can be "switched on" (by a discontinuous increase in the flux through the chain), and similarly, when it falls below a critical level, the pathway is shut down. Similar effects can be realized by varying the ratios of enzyme concentrations. It is proposed that by identifying these critical points one can gain significant insight into the objectives of decision making at the metabolic level.     

Kinetic modelling of plant metabolic pathways

This paper provides a review of kinetic modelling of plant metabolic pathways as a tool for analysing their control and regulation. An overview of different modelling strategies is presented, starting with those approaches that only require a knowledge of the network stoichiometry; these are referred to as structural. Flux-balance analysis, metabolic flux analysis using isotope labelling, and elementary mode analysis are briefly mentioned as three representative examples. The main focus of this paper, however, is a discussion of kinetic modelling, which requires, in addition to the stoichiometry, a knowledge of the kinetic properties of the constituent pathway enzymes. The different types of kinetic modelling analysis, namely time-course simulation, steady-state analysis, and metabolic control analysis, are explained in some detail. An overview is presented of strategies for obtaining model parameters, as well as software tools available for simulation of such models. The kinetic modelling approach is exemplified with discussion of three models from the general plant physiology literature. With the aid of kinetic modelling it is possible to perform a control analysis of a plant metabolic system, to identify potential targets for biotechnological manipulation, as well as to ascertain the regulatory importance of different enzymes (including isoforms of the same enzyme) in a pathway. Finally, a framework is presented for extending metabolic models to the whole-plant scale by linking biochemical reactions with diffusion and advective flow through the phloem. Future challenges include explicit modelling of subcellular compartments, as well as the integration of kinetic models on the different levels of the cellular and organizational hierarchy.     

Characterisation of the control of respiration in potato tuber mitochondria using the top-down approach of metabolic control analysis.

Control over oxidative phosphorylation by purified potato mitochondria was determined using the top-down approach of metabolic control analysis. The control over the respiration rate, phosphorylation rate, proton-leak rate and proton motive force exerted by the respiratory chain, phosphorylation reactions and the proton leak were measured over a range of phosphorylation rates from resting (state 4) to maximal (state 3). These rates were obtained by adding different amounts of hexokinase in the presence of glucose, or different amounts of oligomycin in the presence of ADP. The respiratory substrate was NADH or succinate, both of which feed electrons directly to ubiquinone. The rate of oxygen consumption by the alternative oxidase pathway was negligible with NADH as substrate but was measurable with succinate and was subtracted. Control over the respiration rate in potato mitochondria was predominantly exerted by the respiratory chain at all rates except close to state 4, where control by the proton leak was equally or more important. For oxidation of NADH, the flux control coefficient over the respiration rate exerted by the respiratory chain in state 3 was between 0.8 and 1.0, while in state 4, control over the respiration rate was shared about equally between the chain and the proton leak. The control over the phosphorylation rate was predominantly exerted by the respiratory chain, although at low rates control by the phosphorylation system was also important. For oxidation of NADH, the flux control coefficient over the phosphorylation rate exerted by the respiratory chain in state 3 was 0.8-1.0, while near state 4 the flux control coefficients over the phosphorylation rate were about 0.8 for the phosphorylation system and 0.25 for the chain. Control over the proton leak rate was shared between the respiratory chain and the proton leak; the phosphorylation system had negative control. For oxidation of NADH, the flux control coefficients over the leak rate in state 3 were 1.0 for the leak, 0.4 for the chain and -0.4 for the phosphorylation system, while in state 4 the flux control coefficients over leak rate were about 0.5 for the leak and 0.5 for the chain. Control over the magnitude of the protonmotive force was small, between -0.2 and +0.2, reflecting the way the system operates to keep the protonmotive force fairly constant; the respiratory chain and the phosphorylation system had equal and opposite control and there was very little control by the proton leak except near state 4.     

Ultrastructure and metabolic activity of pea mitochondria under clinorotation

Experimental data on the mitochondrial ultrastructure and tissue respiration in root apex as well as metabolic activity of the organelles isolated from pea seedling roots after 5-day of clinorotation are presented. It was shown that mitochondrial condensation in the distal elongation zone correlated with an increased rate of oxygen uptake on 7%. We also observed increase in rate of malate oxidation and respiratory control ratio increased simultaneously with a decreased in efficiency of oxidative phosphorylation. Such character of mitochondrial rearrangements in simulated microgravity is assumed to be a consequence of adaptation to these conditions.     

DEVS modelling and simulation of the cellular metabolism by mitochondria

We present a method for modelling and simulating metabolic pathways in the cells (namely, the glycolysis and Krebs' cycle), using the discrete event system specification (DEVS) formalism. The hierarchical nature of DEVS makes it ideal for describing naturally hierarchical systems as the Cell, while its discrete-event approach improves performance due to the asynchronous nature of the events involved. DEVS time-based nature can adequately represent the timing of the chemical reactions. We show how this methodology enables creating a precise and easy way to model and simulate biological systems, including advanced visualisation of the experiments. The results presented, which focus on the simulation of the cellular metabolism pathways in mitochondria, show the potential of our approach.     

Long range control circuits within mitochondria and between nucleus and mitochondria

Summary In the preceding paper of this series (Dujardin et al. 1980a) we described general methods of selecting and genetically characterizing suppressor mutations that restore the respiratory capacity of mit ^- mitochondrial mutations. Two dominant nuclear (NAM1-1 and NAM2-1) and one mitochondrial (mim2-1) suppressors are more extensively studied in this paper. We have analysed the action spectrum of these suppressors on 433 mit ^- mutations located in various mitochondrial genes and found that they preferentially alleviate the effects of mutations located within intron open reading frames of the cob-box gene. We conclude that these suppressors permit the maturation of cytochrome b mRNA by restoring the synthesis of intron encoded protein(s) catalytically involved in splicing i.e. mRNA-maturase(s) (cf. Lazowska et al. 1980). NAM1-1 is allele specific and gene non-specific: it suppresses mutations located within different introns. NAM2-1 and mim2-1 are intron-specific: they suppress mutations all located in the same (box7) intron of the cobbox gene. Analyses of cytochrome absorption spectra and mitochondrial translation products of cells in which the suppressors are associated with various other mit ^- mutations show that the suppressors restore cytochrome b and/or cytochrome oxidase (cox 1) synthesis, as expected from their growth phenotype. This suppression is, however, only partial: some new polypeptides characteristic of the mit ^- mutations can be still detected in the presence of suppressor. Interestingly enough when box7 specific suppressors NAM2-1 and mim2-1 are associated with a complete cob-box deletion (leading to a total deficiency of cytochrome b and oxidase) partial restoration of cox I synthesis is observed while cytochrome b is still totally absent. These results show that in strains carrying NAM2-1 or mim2-1 the presence of cytochrome b gene is no longer required for the expression of the oxi3 gene pointing out to the possibility of a mutational switch-on of silent genes, whether mitochondrial, mim2-1, or nuclear, NAM2-1. This switch-on would permit the synthesis of an active maturase acting as a substitute for the box7 maturase in order to splice the cytochrome b and oxidase mRNAs.     

Long range control circuits within mitochondria and between nucleus and mitochondria

Summary To uncover the functional circuitry both within the mitochondrial genome and between the mitochondrial and the nuclear genome, we have developed a general method for selecting and characterizing genetically suppressor mutations that restore the respiratory capacity of mit ^- mitochondrial mutants.Several hundreds of pseudo-wild type revertants due to a second unlinked mutation which suppresses a target mit ^- mutation were isolated. The suppressor mutations were found located either in the nuclear (abbreviated NAM for nuclear accommodation of mitochondria) or in the mitochondrial genome (abbreviated MIM for mitochondrial-mitochondrial interaction).The specificity of action of various suppressors upon some 250 different mit ^- mutations located in several genes was tested. According to this specificity of action, suppressors were subdivided into two major classes: allele specific or gene specific suppressors. Because the cob-box mitochondrial gene has a mosaic organization, we were able to find a novel third class of extragenic suppressors specific for mit ^- mutations within the introns of this gene.Four examples of suppressors showing various specificities of action illustrate our approach. (1) a nuclear gene controlling specific alleles of different mitochondrial genes; (2) a nuclear gene controlling selectively one intron of a split mitochondrial gene; (3) a mitochondrial gene controlling specific alleles of different mitochondrial genes; (4) a region in one complex mitochondrial gene which controls selectively one intron of another split mitochondrial gene.Different mechanisms of suppression are discussed stressing the alleviation of splicing deficiencies of intron mutations.     

Chemical modification of mitochondria. Uncoupler binding by mitochondria in different metabolic states

At low uncoupler concentrations the binding of carbonyl-cyanide-m-chlorophenyl-hydrazone to mitochondria was found to depend sensitively on the metabolic state of mitochondria. The binding data are consistent with the assumption that at low concentrations and pH 7.4 the uncoupler is bound mainly in anionic form to the inner mitochondrial membrane and that upon energization the inner membrane undergoes conformation change, exposes buried ionizable groups and hence acquires a negative net membrane charge. Deenergization of the inner membrane by a small amount of uncoupler removes the negative net membrane charge and consequently increases the apparent binding constants. Based upon the present results on uncoupler binding and previous observations on the physiological properties of alkylating uncouplers, a possible molecular mechanism involving electron carriers and coupling factors is suggested for coupling electron transport to phosphorylation.