Workflow for Model Construction

We developed a workflow for systematically converting metabolic reconstructions into large-scale kinetic models of metabolism (see Fig. 1). It is designed to take sparse or full data sets, performs a thorough analysis of parameter constraints, and then generates the kinetic model using massive data integration. In particular, it addresses steps (ii) and (iii) of model construction, and their associated issues. The resulting models meet the following requirements: the kinetic constants are close to measured values or, where data are not available, in biochemically plausible ranges; the model reproduces a steady state with realistic stationary fluxes and concentrations; model parameters and stationary fluxes are consistent with thermodynamic laws, ensuring the existence of a consistent equilibrium state.

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Figure 1. Stages of data addition (+) and method runs ([>]) in the workflow.

Letters and numbers in parentheses refer, respectively, to input data and to the stages of the workflow. (1) Geometric FBA [32] is run, yielding a thermodynamically feasible, stationary flux distribution matching the flux data; (2) a sub-network of flux-carrying reactions is exported, (3) consistent values of all metabolite concentrations and equilibrium constants are determined, (4) kinetic constants appearing in the sub-network are balanced, (5) kinetic rate laws and associated kinetic constants are inserted into the model, and (6) the maximal reaction rates are adjusted to reproduce the steady-state fluxes calculated before. The laboratory data feeding into “Flux” can include quantitative metabolomic measurements of metabolites, and also dynamically calculated flux values for the network. The lower half of the diagram shows a cycle of experimentation and modelling: here the result of the MCA can be used to target which rate laws should be measured in vitro after measurement, this rate law can be substituted into the model, with a view to better fit the observed perturbation behaviour. All grey arrows refer to aspects of the workflow where additional data can be added in as knowledge increases. The pathway shown is a truncated version of the trehalose pathway. Abbreviations: T6P = trehalose 6-phosphate; aaT = - trehalase; Glu = Glucose; G6P = glucose 6-phosphate.

https://doi.org/10.1371/journal.pone.0079195.g001

We use the metabolic network as a frame in which certain quantities are predefined (see Fig. 2). All quantities involved in more than one reaction – fluxes, metabolite concentrations, and equilibrium constants – are chosen in agreement with the network constraints. Then, consistent Michaelis constants and catalytic constants are selected, for every reaction, using parameter balancing. The resulting model parameters satisfy all Wegscheider conditions [28], Haldane relationships [29], and observed flux directions, and after a simple rescaling of enzyme levels, actualise the predefined steady state. Enzyme re-scaling is based on strategies outlined in Kholodenko et al (1998) [30]. The resulting models could immediately be used to direct experimentation and accurately predict certain biological behaviours. The workflow is extendable to all organisms for which there is a genome-scale metabolic reconstruction established (something which is becoming more accessible for many organisms [31]), and which can be cultured under steady-state conditions. This addresses the needs of systems biology and biotechnology by bridging the void between kinetic models, stoichiometric models, and experimental investigation.

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Figure 2. Consistent parameter sets for large kinetic models.

(A) The metabolic network provides a frame for formulating parameter dependencies. Stationary fluxes , concentrations c, and equilibrium constants must be thermodynamically consistent. (B) For each reaction, the Michaelis constants and need to agree with the predefined quantities , , and . (C) Given a consistent parameter set, enzymatic reactions can be safely connected. The resulting model will actualise a predefined steady state, with rate laws satisfying the following conditions: (i) Quantities shared by several reactions – for instance, metabolite concentrations – have the same values in each of them. (ii) For internal metabolites, incoming and outgoing fluxes are balanced. (iii) Quantities that arise from differences along reactions satisfy the Wegscheider conditions: for instance, their sums over closed loops vanish. (iv) The kinetic constants satisfy the Haldane relationships, which relate the kinetic constants of a rate law to the equilibrium constant of the reaction. (v) Flux directions agree with thermodynamic forces, given by the negative differences of chemical potentials.

https://doi.org/10.1371/journal.pone.0079195.g002

The basic workflow and associated methods are illustrated in Fig. 1, where letters represent data input or output and numbers represent methods to process the data. First, we shall introduce each of the pieces of data and methods contained within the workflow and explain them in a general way. Then, we present a kinetic model of yeast metabolism, built using this workflow, and discuss some of its dynamic and control properties. The model construction is described in detail in the Methods section.

Large-scale Yeast Model

As an example case for our workflow, we generated a large-scale kinetic model of yeast metabolism. Sacharomyces cerevisiae is one of the most studied eukaryotic model organisms, and a comparatively large amount of data is available for fluxes, concentrations, and equilibrium constants. The model generation process is described in detail in the Methods section. The final model contains 285 flux-carrying reactions and 294 metabolites. As expected, it displays a large flux through the glycolytic pathway and a large production of ethanol (see Fig. 3 (b)). This behaviour is primarily defined by the flux input from the kinetic models used and reflects the expected behaviour when growing the organism in the laboratory. A detailed list of flux values is given in S4.

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Figure 3. Fluxes and control coefficients in the yeast metabolic model.

(A) Fluxes obtained from Geometric FBA. Only selected reactions with large fluxes are depicted, co-substrates are not shown (flux directions and magnitudes shown by arrows). (B) Flux control coefficients. Top: Control exerted by the glucose transporter (GluT). Unscaled flux control coefficients are shown in shades of blue (positive values) and red (negative values). Bottom: control exerted by the biomass production reaction. High-flux reactions respond most strongly: an increased glucose import increases the glycolytic flux, while increased biomass production directs fluxes to other pathways and thereby decreases the glycolytic flux. Flux control coefficients for a model with allosteric regulation are shown in Figures G and H in File S1.

https://doi.org/10.1371/journal.pone.0079195.g003

Being able to direct the model into the desired steady state is a significant improvement over previous methods [21], [23] and to the basic implementation of parameter balancing [25]. However, our construction does not guarantee that the steady state is stable. A model in a stable steady state will return to that state after a small perturbation in a system variable, whilst a model with an unstable steady state, when perturbed, will show increasing divergence or oscillatory behaviour and is therefore fragile against molecular noise. In this instance, the model demonstrates a stable steady state with strictly negative Jacobian eigenvalues. We confirmed this by perturbing the model using a ten-fold increase of extracellular glucose and mapping the relaxation of the system back to the original state. The stability is a clear improvement over the large-scale model presented in Smallbone 2007 [22], which showed an unstable steady state with 17% of the eigenvalues having a positive real part, the largest value being around . We believe that ensuring thermodynamic consistency and introducing the saturable rate laws have been key contributors to the stability of the system.

To study the model response to changes in enzyme concentrations, we performed metabolic control analysis (MCA, see Methods section for details). The resulting control coefficients describe the relative flux or concentration changes caused by relative changes in enzyme levels. The results in Fig. 3 show that overall flux control (see Methods) is heavily dominated by reactions that balance and produce co-factors. In addition, high control is exerted by some early reactions of the citric acid cycle. A study by Thomas and Fell (1998) [33] demonstrates that ATP-utilising reactions show a high control over flux changes within glycolysis. However, transport reactions also show high control over metabolic fluxes in a number of organisms [34]–[36]. Others have shown that the control is distributed between ATP-utilising reactions and glucose transport [37]. Whilst there is literature to support the control patterns we have identified, more experimental data and evidence will have to be collected to be certain of the accuracy.

We compared the flux control in glycolytic enzymes of the large yeast model with the flux control in each of the small yeast models used to construct it (see Table 1). Even though the large model contains input data from the smaller models, we expected to see changes in the control distribution because control is a systemic property and will be affected by extending the system. Within the large model, linear chains of reactions from glycolysis are now involved in branch-points between other major pathways, including pentose phosphate pathway, trehalose biosynthesis, and the citric acid cycle. This has led to a clear shift in the control distribution in glycolysis.

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Table 1. Overall flux control exerted by different enzymes in glycolysis.

https://doi.org/10.1371/journal.pone.0079195.t001

Many of the models used to construct the large model contained enzymatic regulation by allosteric effectors. To ensure that the shift in control behaviour was not due to the lack of regulation in the large model, we generated a version of the model with known regulation included. Whilst there were no dramatic changes in the pattern of overall control, the control within the glycolytic pathway was largely shifted towards trends seen within the smaller models. High control was observed for ATPase, glycerol-3-phosphate dehydrogenase, hexokinase, alcohol dehydrogenase, and glyceraldehyde-3-phosphate dehydrogenase. Low control was seen for phosphoglycerate kinase, phosphoglycerate mutase, enolase, phosphofructokinase, and fructose bisphosphate aldolase. The control for glucose transport decreased further, showing the lowest control out of all the enzymes in the pathway under conditions where regulation is included. We also noted that the model with allosteric regulation is not stable at the constructed steady state and, when integrated, moves to a close-by stable steady state. We believe this is related to the resolution of the steady state calculation being set to therefore small numerical differences during the rate scaling, particularly in the smaller fluxes (), will cause perturbations away from the scaled state. The changes caused by allosteric regulation are interesting to note because our knowledge of systemic enzymatic regulation is poor. Given these findings, we expect the control distribution from the model to be subject to major changes when new regulation is discovered and included.

Using a 30% increase in extracellular glucose, we simulated the dynamic response to perturbations in both the standard model and the model including regulation. All plots can be seen in Figures A–F in File S1. Both the standard model and the regulation model demonstrate very similar behaviour during the perturbation. Both models qualitatively reproduced expected behaviours such as increase in extracellular glucose causing increased flux in glucose transport, glucose-6-phosphate isomerase, and ethanol transport, pushing the model towards increased fermentative metabolism. There is also a small increase in biomass production. The converse behaviours are true during decreased levels of extracellular glucose.

The changes in metabolite concentrations around key metabolite pools, where flux is diverted into different pathways, also primarily conform to what we would expect: there is a general increase in concentrations leading through to fermentation. Amino acid pools show a short delay before they respond to the changes in extracellular glucose. Since it takes some time for the increased glucose to proliferate through the interim metabolic pools, this is something we would expect, unless there are regulators that alter biomass production to respond to changes in extracellular glucose more rapidly. Co-factor balances within the model generally behave as expected, showing a conservation between their concentrations. We noted that ATP concentration decreased upon addition of extracellular glucose. This seemed counter-intuitive, given that ATP regeneration depends on energy provided by glucose, but this paradoxical phenomenon has been observed in laboratory as a normal physiological behaviour of the cell [38], [39]. It is demonstrative of the turbo effect where ATP is first rapidly consumed in upper glycolysis, leaving a lag before it is later produced through aerobic metabolism. This behaviour is not captured by the small models, which were used in contraction of the large model. It is therefore an emergent dynamic property of the large-scale system, and highlights the utility and need for creating large kinetic models.

Tools and File Formats

To generate a large-scale kinetic model of yeast metabolism, a set of tools and scripts were used to process different file formats. For the model itself, the standard format SBML (Systems Biology Markup Language [40]) was used throughout the workflow. Flux data and metabolite concentrations were integrated into the process via normal spreadsheets, whilst the parameter list for parameter balancing was given in the table format SBtab (Systems Biology Table, http://www.sbtab.net). MATLAB and COPASI [41] were used for numerical analyses and data generation. The software for parameter balancing is written in Python [42] and available from SourceForge (https://sourceforge.net/projects/parbalancing). The final SBML models produced for yeast can be found in BioModels database (http://www.ebi.ac.uk/biomodels/). The original model is MODEL1204270000 (see File S2), and the regulation model is MODEL1307040000 (see File S3).

Network [Data a]

The Yeast 4.0 model [3] was used to define the stoichiometric matrix. This is a comprehensive reconstruction of the yeast metabolism, which demonstrates a large improvement on lipid metabolism compared to previous reconstructions, and also shows a higher connectivity between the reactions.

Flux Data [Data b]

To obtain a set of flux data, we selected a group of metabolic models from BioModels Database [1] that are yeast-specific and contain glucose as the primary carbon source (models 61 [43], 64 [44], 172 [45], 176 [46], and 177 [46]; see Supporting Table 2 in File S1). Each model was run to steady state from its operating state and the resulting flux for each reaction was noted. Where more than one model provided flux values for the same reaction, the median value was used. This provides an approximation of the flux when yeast is grown using extracellular glucose as the sole carbon source. The directions of allowed flux through the transport reactions were kept as described in the original Yeast 4.0 model.

Flux Balance Analysis [Step 1]

A flux distribution was computed by Geometric FBA [32], an algorithm that iteratively reduces the solution space based on the principle of minimal sum of fluxes. The resulting fluxes are at steady state and thermodynamically feasible, unless the imposed flux constraints force futile cycles, and should yield a more biologically realistic result because of the extra constraints used for computation. The flux data (see Data b) were used to constrain the lower and upper flux bounds of the associated reactions during Geometric FBA calculation. The Geometric FBA solution has to adhere as closely as possible to the constrained fluxes, whilst also satisfying the chosen objective function of the system, which was to maximise growth. The confidence in a flux value can be expressed by allowing smaller or larger upper and lower bounds around the given value. Larger bounds should be allowed for reactions showing a significant change in pathway position (i.e. when a reaction previously located in a linear chain is now a branching point).

Export Network of Interest [Step 2]

From the resulting flux distribution, we removed all reactions with vanishing fluxes. The pathways represented within the model can be seen in Supporting Table 1 in File S1. Reactions with negative fluxes were reversed with respect to sign of rate and stoichiometric coefficients to produce positive fluxes for all reactions. The reversal is not strictly necessary but simplifies the computations used in further steps of the workflow. This step reduced the network to a central subnetwork of flux-carrying reactions. As more data are obtained, the network can be expanded. To further reduce the network in size, reactions may also be omitted using a finite cut-off value for the fluxes (e.g. mM/s). In these instances, the Geometric FBA needs to be rerun, using the reduced network as an input, to ensure there is still a solution to the system. The reduced flux solution should also be retested for thermodynamic feasibility, which is vital to ensure that the flux scaling in Step [5] will be possibly.

Kinetic and Thermodynamic Constants [Data c]

Where appropriate, the equilibrium constants were taken from models available in BioModels Database that use glucose as their primary carbon source (BioModels 61 [43], 64 [44], 172 [45], 176 [46], and 177 [46]; see Supporting Table 5 in File S1). These values were used as fixed data points. All transport reactions were set to have equilibrium constants of 1. The Michaelis constants were initially assigned values equal to the steady-state concentration of their corresponding metabolite. This helped generate sensible upper and lower constraint bounds, these were used during parameter balancing. Constants for allosteric regulators were taken as averages of those found in the BRENDA database [47], the values used can be found in Supporting Table 7 in File S1.

Metabolite Concentrations [Data d]

Intracellular and extracellular metabolite concentrations were taken from yeast-specific models in BioModels Database that use glucose as the primary carbon source (see Supporting Table 3 in File S1). Unknown intracellular concentrations were set equal to the median value of known intracellular concentrations, 0.549 mM. These are listed in Supporting Table 3 in File S1. For the extracellular concentrations, the median value of all extracellular concentrations in Supporting Table 4 in File S1 was used (24.5 mM). Some extracellular concentrations were manually adjusted (see Supporting Table 6 in File S1)). Protons and phosphate were assumed to have fixed concentrations within the system.

Balancing the Kinetic Constants [Step 4]

For creating a complete and feasible set of kinetic constants and their uncertainty ranges, we used parameter balancing [25]. The kinetic constants are mutually dependent due to thermodynamic constraints (e.g. the ratio of catalytic constants of a reaction can be calculated from the equilibrium constant and the Michaelis constants through the Haldane relationship). Thus, on the one hand, the parameter set can be complemented for missing values. On the other, different data values may be in conflict and require adjustment. Parameter balancing exploits these dependencies and searches for a most plausible set of kinetic constants, based on Bayesian parameter estimation and on the current and limited collection of kinetic constants available. To resolve underdetermined values, it uses prior distributions of every type of parameter (such as Michaelis constants or catalytic rate constants; for the exact values, see [25]). Since independent priors can capture only some types of kinetic constants (the mutually independent ones), we use pseudo values [13], in addition, to represent prior knowledge on other, dependent kinetic constants. Although priors and pseudo values are provided with large standard variations, they can still raise the reliability of the estimate in comparison to not using any values at all. Eventually, because of the constraints between parameters, the estimate of a parameter – e.g. some Michaelis constant in the model – will not only depend on data for this Michaelis constant, or on the prior of Michaelis constants in general, but also on known equilibrium constants, catalytic constants, and their priors. The estimation is performed within a Bayesian framework and results in a posterior distribution, from which the mean and median values - as well as the corresponding standard deviations - can be obtained as point estimates for each of the kinetic constants.

The central step in parameter balancing, finding the posterior mode of the kinetic constants vector within predefined boundaries is a convex optimisation problem, so gradient-based optimisation algorithms can be used. Nevertheless, since all parameters of a model are coupled by constraints, the calculation can become demanding for larger networks. To reduce the computational effort, we broke down the network into single reactions and balanced the Michaelis constants and catalytic constants for each of them individually. As shown in Fig. 2, a network can be split into single reactions and plugged together again, without losing its thermodynamic feasibility, but only if the shared quantities are held fixed. Therefore, the equilibrium constants needed to be predetermined (in Step [3]) and accepted as fixed values during parameter balancing.

Assigning Kinetic Rate Laws [Step 5]

The values obtained from parameter balancing can be used as kinetic constants in the rate laws of our model. For this task, we chose the common modular rate law [20], a generalised form of the reversible Michaelis-Menten kinetics applicable to any reaction stoichiometry. Regarding a reaction , for instance, we can calculate the rate in mM/s as(2)where

 = maximal reaction velocity     = concentrations (mM)

 = dissociation constant of A     = reactant constants of A

 = equilibrium constant

This rate law represents a random-order enzyme mechanism, the terms of the denominator, when expanded, each represent one possible binding state of the enzyme. The common modular rate law resembles the convenience kinetics [18] with only slight modifications. As an exception, the rate laws for biomass, growth, and lipid production reactions (all operating at the same flux rate) were replaced by linlog rate laws, or, whenever the rate law yielded negative values, by a value of 0 (i.e. a rate law akin to ).

Adjusting the Maximal Velocities to Steady-state Fluxes [Step 6]

The desired flux through each reaction at steady state had already been calculated in Step [1], and the later steps of the methodology have ensured that the directions of each reaction rate is the same as the pre-calculated fluxes. Therefore, to match the rates to the steady-state fluxes, the maximal velocities can simply be rescaled by positive factors. This ensures that the model is at steady state at the point of construction.

Metabolic Control Analysis (MCA)

Metabolic control analysis is a technique for studying how system variables such as fluxes and concentrations are affected by small changes in system parameters. It uses control coefficients, of which there are two types: flux control coefficients and concentration control coefficients . They measure how local perturbations of reaction rates affect the steady-state flux () or steady-state concentration () of the system and are defined as follows:(3)(4)where is a parameter affecting exclusively the reaction, is the rate as a function of the concentrations and system parameters, and and are steady-state fluxes and concentrations as functions of the system parameters. Control is a distributed systemic property. If there is a change in control at one point, it will be compensated for elsewhere in the network. The reactions that exert the largest control can be expected to define the behaviour of the system most heavily. Further information can be found in Heinrich and Schuster [39]. We define the general flux control exerted by reaction by the root sum of squares of all its flux control coefficients

(5)The 5% of reactions with the highest were considered high-control reactions.