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The study of biological systems at the genome scale helps us understand fundamental biological processes that govern the activity of living organisms and regulate their interactions with the environment. Genome-scale metabolic models are usually analysed using constraint-based methods, since detailed rate equations and kinetic parameters are often missing. However, constraint-based analysis is limited in capturing the dynamics of cellular processes. In this paper, we present an approach to build a genome-scale kinetic model of

Genome-scale metabolic models are essential to bridge the gap between metabolic phenotypes and genome-derived biochemical information, as they provide a platform for the interpretation of experimental data related to metabolic states and enable the

The availability of genome-scale metabolic networks has accelerated the development of methods to analyse system-wide metabolic properties. A fundamental aim of systems biology is to predict cellular behaviours

Reverse engineering is often used in systems biology to reconstruct biological interactions and constrain kinetic parameter values from experimental data [

The model building approach presented in Adiamah

We used the GRaPe software [

The generic rate equations provided in [_{1}, _{2}, ...) and the concentrations of products are represented by a vector _{1}, _{2}, ...). The flux

where _{Ai}^{th} substrate and _{Bj}^{th} product of the reaction, _{0} is the concentration of enzyme, ^{+}^{-}

Kinetic models have been shown to produce accurate and testable results [

Schematic overview of the model development process.

One of the issues relating to parameter estimation is that of mathematical redundancy. The redundancy results in multiple sets of parameter values that can fit equally well to an experimental data set. A simple example of redundancy is when two parameters,

Once a model has been constructed or uploaded in GRaPe, PVA can be performed using the same data required to estimate parameter values for the model. The PVA algorithm works by repeating the estimation of kinetic parameters for the model multiple times using a genetic algorithm (GA). The GA works by populating a set of random initial parameter values; this is why results may differ after each run of the algorithm when there is redundancy. These estimated values are then optimised in an iterative manner until the maximum number of iterations is reached or a suitable solution is found. In GRaPe, GA uses flux and metabolic data to constrain parameters as illustrated in

There are now several genome-scale metabolic reconstructions of

Using GRaPe [

We obtained flux distributions for three steady-states with glycerol being the only carbon source using the interactive web-based tool developed by Beste

We performed three separate parameter estimations for each of the three glycerol consumption rates. The kinetic parameters for each reaction in the model were estimated using GRaPe’s genetic algorithm. Model 1, with a glycerol consumption rate at 0 mmol/gDW/h, had 2297 kinetic parameters after parameter estimation; Model 2 had 2537 parameters with a glycerol consumption rate at 0.5 mmol/gDW/h, and Model 3 had 2931 parameters after parameter estimation with a glycerol consumption rate at 1 mmol/gDW/h. The difference in the number of parameters after each estimation was due to different numbers of reactions having a zero flux in each case. Furthermore, reactions with negative fluxes had their substrates and products swapped around to prevent having negative kinetic parameter values. The three models are provided in SBML format in

We performed a steady-state analysis for Model 1, 2 and 3 using COPASI. The results were then compared with the FBA flux distribution obtained from the Beste model under the same experimental conditions. Our verification analysis showed a near-perfect agreement between the results obtained from our models and the respective FBA simulation.

We identified reactions that showed the greatest change in flux with respect to change in glycerol consumption rate (

Main response of

To determine the degree of redundancy in the values of our estimated kinetic parameters, we performed PVA by repeating our estimation algorithm 100 times for Model 2. It is well known that different sets of parameters values can fit to experimental time-series data resulting in mathematical redundancy [

In order to facilitate their interpretation, the results of PVA were split into five different categories based on the stoichiometry of the reaction (uni-uni, uni-bi, bi-uni, bi-bi and reactions of more than two substrates or products). The results of PVA show that many parameters are not strongly constrained (_{f}_{f}

Relative changes in flux with changes in glycerol consumption rate. The response of selected reactions when the glycerol uptake rate is at 0 and 0.5 mmol/gDW/h is compared in Experiment A, between 0 and 1 mmol/gDW/h in Experiment B, and between 0.5 and 1 mmol/gDW/h in Experiment C. Only reactions with an absolute change greater than 10 % (0.1) and an absolute flux greater than 0.01 are shown. Green indicates a decrease and red indicates an increase in flux. Reactions identifiers are the same as in the Beste model.

Average parameter values and standard deviations of estimated kinetic parameters after repeating the genetic algorithm 100 times. The parameters were classified into five reaction types: uni-uni (black), uni-bi (red), bi-uni (blue), bi-bi (purple) and convenience kinetics (green). Both axes of the graph are in logarithmic scale.

Average parameter values and standard deviation (Stdev) for the most constrained parameters in logarithmic scale over 100 iterations of parameter variability analysis (PVA). Reactions are of type uni-uni, uni-bi, bi-uni, bi-bi, or convenience kinetics (CK).

Most constrained parameters | |||
---|---|---|---|

Parameter | Reaction type | Average | Stdev |

_{f} |
CK | -0.78 | 2.35 |

_{f} |
uni-bi | -1.39 | 1.41 |

_{f} |
bi-uni | -1.71 | 1.31 |

_{f} |
bi-bi | -1.31 | 1.75 |

The high degree of redundancy in the parameter values as indicated by PVA comes in support of our underlying assumption that accurate rate equations and kinetic parameters are not necessarily crucial in constraining the behaviour of the biological system. Nevertheless, the integration of genomic and proteomic data, together with metabolic and flux data, is expected to reduce mathematical redundancy as shown by previous studies [

The computation of 100 sets of parameters for each reaction in Model 2 (with 739 metabolites, 856 metabolic reactions, 856 enzymes and 2537 kinetic parameters) for PVA took over 5 hours and 40 minutes. The relatively fast computing time was a result of reducing the objective function to 10^{-4} and limiting the data points to three decimal places in the input dataset. The objective function is the summed squared mean distance measured between the simulated data and input data. Reducing the objective function increased computing time but improves the quality of the parameter fit to input data. We performed an experiment to determine the relationship between the value of the objective function and the time taken to compute PVA for one reaction with two substrates, two products, one enzyme and six kinetic parameters (^{-10} and beyond. The relationship that is observed between the objective function and computing time appears to be linear (PVA was computed on a desktop computer with a quad CPU having 3.00 GHz, 2.99 GHz processor speed and 4 GB of RAM).

Computing times of parameter variability analysis (PVA) against changes in objective function. PVA was performed for a reaction with two substrates, two products, one enzyme and six kinetic parameters. For each PVA run, the summed squared mean distance measured between the simulated data and input data, known as the objective function, was set and the time taken to compute PVA results (running the genetic algorithm 100 times) was recorded. The results indicate a linear relation between the objective function and the computing time until the limits of computational precision are reached. Both axes of the graph are in logarithmic scale.

Another variable that can increase computing time in parameter estimation is the number of data points in the experimental dataset. To examine how the number of data points influences computing time, we performed parameter estimation for a single reaction with two substrates, two products, one enzyme and six kinetic parameters (

Relationship between number of input data points and computing time. PVA was performed for a single reaction of two substrates, two products, one enzyme and six kinetic parameters. PVA was repeated six times and for each iteration the number of data points in the input dataset for parameter estimation was increased from 3 to 30. The results show a rising curve in a non-linear shape.

We tested the predictive capability of our ^{-4}. The range of objective functions observed for individual reactions was between 10^{-8} and 10^{‑20}. After parameter estimation, three steady-state analyses were performed with glycerol uptake at 0, 0.5 and 1 mmol/gDW/h using COPASI.

The resulting model was only able to predict the steady-state when glycerol update was at 0.5 mmol/gDW/h. Changing the glycerol level in this model resulted in simulation errors. A possible explanation for this unexpected observation is that combining three separate steady states is a fundamentally different experiment from having a dynamic change in glycerol level. Input flux distributions obtained from separate FBA simulations may be inappropriate to reproduce the dynamics of metabolic adjustment. To create a suitable training data set for dynamic modelling, intermediate data points covering the transition between steady states would be needed, but these data points cannot be obtained by FBA and require detailed experimental measurements. Another possible approach is that forward and backward reaction velocities (_{f}_{r}

In this paper, we present the first genome-scale kinetic model of

Predicting cellular behaviours

Beste _{f}^{-8}. The compromise between computing time and more precise parameter values must always be considered when performing parameter estimation. Furthermore, our results also show that computing time increases non-linearly with the number of data points in the parameter estimation training data. When parameter estimation is being carried out for a system in steady-state, the number of data points can be reduced to lower the computing time.

An attempt was also made to constrain our kinetic parameters by training them with data based on three distinct experimental conditions. However, our model was able to predict only one state revealing the limits of using FBA steady states to constrain a dynamic model. Optimisation techniques can be used to estimate kinetic parameters based on simulated or experimental data [

In this article, we developed a genome-scale kinetic model of

D.A.A. is supported by a studentship from the Biotechnology and Biological Sciences Research Council (BBSRC), UK.

The authors declare no conflict of interest.

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