Towards a genomescale kinetic model of cellular metabolism
 Kieran Smallbone†^{1, 2},
 Evangelos Simeonidis†^{1, 3}Email author,
 Neil Swainston^{1, 4} and
 Pedro Mendes^{1, 4, 5}
DOI: 10.1186/1752050946
© Smallbone et al; licensee BioMed Central Ltd. 2010
Received: 27 July 2009
Accepted: 28 January 2010
Published: 28 January 2010
Abstract
Background
Advances in bioinformatic techniques and analyses have led to the availability of genomescale metabolic reconstructions. The size and complexity of such networks often means that their potential behaviour can only be analysed with constraintbased methods. Whilst requiring minimal experimental data, such methods are unable to give insight into cellular substrate concentrations. Instead, the longterm goal of systems biology is to use kinetic modelling to characterize fully the mechanics of each enzymatic reaction, and to combine such knowledge to predict system behaviour.
Results
We describe a method for building a parameterized genomescale kinetic model of a metabolic network. Simplified linlog kinetics are used and the parameters are extracted from a kinetic model repository. We demonstrate our methodology by applying it to yeast metabolism. The resultant model has 956 metabolic reactions involving 820 metabolites, and, whilst approximative, has considerably broader remit than any existing models of its type. Control analysis is used to identify key steps within the system.
Conclusions
Our modelling framework may be considered a steppingstone toward the longterm goal of a fullyparameterized model of yeast metabolism. The model is available in SBML format from the BioModels database (BioModels ID: MODEL1001200000) and at http://www.mcisb.org/resources/genomescale/.
Background
Recent advances in genome sequencing techniques and bioinformatic analyses have led to an explosion of systemswide biological data. In turn, the reconstruction of genomescale networks for microorganisms has become possible. Whilst the first stoichiometric models were limited to the central metabolic pathways, later efforts such as iFF708 [1] and iND750 [2] were much more comprehensive. A recent communitydriven reaction network for S. cerevisiae (bakers' yeast) consists of 1761 reactions and 1168 metabolites [3].
The ability to analyse, interpret and ultimately predict cellular behaviour is a long soughtafter goal. The genome sequencing projects are defining the molecular components within the cell, but describing their integrated function will be a challenging task. Ideally, one would like to use enzyme kinetics to characterize fully the mechanics of each reaction, in terms of how changes in metabolite concentrations affect local reaction rates. However, a considerable amount of data and effort is required to parameterize even a small mechanistic model; the determination of such parameters is costly and timeconsuming, and moreover much of the required information may be difficult or impossible to determine experimentally. Instead, genomescale metabolic modelling has relied on constraintbased analysis [4], which uses physicochemical constraints such as mass balance, energy balance, thermodynamics and flux limitations to describe the potential behaviour of an organism. Such methods, however, ignore much of the dynamic nature of the system and are unable to give insight into cellular substrate concentrations. These methods are more suitable for defining the wider limits of systems behaviour than making reliable and accurate predictions about metabolism.
In a previous paper, we presented a method for constructing a kinetic model for a metabolic pathway based only on the knowledge of its stoichiometry [5]. Here, we present a first attempt at the creation of a parameterized, genomescale kinetic model of metabolic networks, through appending existing kinetic models of constituent metabolic pathways from the BioModels database [6] to a stoichiometric model of yeast metabolism [3]. The results (see Additional file 1) are presented in SBML (Systems Biology Markup Language; http://sbml.org/) [7], using MIRIAMcompliant annotations (Minimal Information Requested In the Annotation of Models; http://www.ebi.ac.uk/miriam/) [8]. Critically, such markup allows automated reasoning about the model's assumptions and provenance.
Results and Discussion
Algorithm
Model construction
A number of reconstructions of the metabolic network of yeast based on genomic and literature data have been published. However, due to different approaches utilized in the reconstruction, as well as different interpretations of the literature, the earlier reconstructions differ significantly. A community effort resulted in a consensus network model of yeast metabolism, combining results from previous models ([3], available from http://www.compsysbio.org/yeastnet). In all, the resulting consensus network consists of 1857 reactions (of which 1761 are metabolic) involving 2153 chemical species (of which 1168 are metabolites). Species in the model are annotated using both databasedependent (e.g. ChEBI [9]) and databaseindependent (e.g. InChI [10]) references, generating for the first time a representation that allows computational comparisons to be performed.
That is, we define an objective function Z, a linear combination of the fluxes v_{ j }, that we maximize over all possible steady state fluxes (N v = 0; where N is the m × n stoichiometric matrix) satisfying certain constraints. In many genome scale metabolic models, a biomass production reaction is defined explicitly that may be taken as a natural form for the objective function. The metabolic reconstruction used here [3] lacks such a sink for metabolism. We accomplish this by adding a pseudoreaction representing cellular growth (sometimes referred to as "biomass production"). The biomass composition used here is taken from the iND750 model [2].
In a previous paper [5], we defined a method for the generation of kinetic models of cellular metabolism, based solely on the knowledge of reaction stoichiometries. This modelling framework requires little experimental data regarding variables and no knowledge of the underlying mechanisms for each enzyme; nonetheless it allows inference of the dynamics of cellular metabolite concentrations. The fluxes found through FBA are allowed to vary dynamically [13]. To create a kinetic model (of minimal complexity), four sets of information are required:

Network stoichiometry (N).

Reference fluxes (v*) through the network.

Reference metabolite concentrations (x*).

Elasticities (ε)  changes in reaction rates with effector levels.
Flux estimation
Selected reaction fluxes used in the model
Reaction  Flux (mM/s) 

acetaldehyde transport  0.00141 
adenylate kinase  0 
alcohol dehydrogenase, reverse rxn (acetaldehyde → ethanol)  1.17 
ATPase, cytosolic  0.595 
enolase  1.76 
ethanol transport  0.0134 
fructosebisphosphate aldolase  0.733 
glycerol3phosphate dehydrogenase (NAD)  0.149 
glycerol3phosphatase  0.051 
glyceraldehyde3phosphate dehydrogenase  1.06 
glucose transport (uniport)  0.59 
glycerol transport via channel  0.00141 
hexokinase (Dglucose:ATP)  0.866 
phosphofructokinase  0.606 
glucose6phosphate isomerase  0.733 
phosphoglycerate kinase  0.875 
phosphoglycerate mutase  1.76 
pyruvate kinase  1.06 
pyruvate decarboxylase  1.25 
triosephosphate isomerase  0.395 
alpha, alphatrehalosephosphate synthase (UDPforming)  0.04 
where BM denotes the subset of j that includes all the reactions with fluxes defined in BioModels. A unique reference flux (see additional file 2) is chosen from the space of all solutions to the above problem, by finding the box that defines the maximum and minimum values attainable by each v_{ j }, then choosing a flux as close as possible to the centre of the box. Iterating, the method minimizes and centres the flux through the network and, in this case, fixes all 956 fluxes to unique values. The algorithm [14] that produces the unique solution from the available flux space is described briefly below.
A simple FBA formulation is solved, in order to identify the maximum achievable growth rate, Z*. For the first iteration, we minimize the total flux required to achieve Z*. This assumption (i.e. that the cell minimizes its total flux. [15]) may be posed as a LP problem by decomposing fluxes v_{ j }into their positive and negative parts. The solution of this first iteration provides the minimal total flux through the network (Z_{1}). We then find the bounds on each reaction flux, subject to the new constraint that the total flux through the network cannot be larger than Z_{1}. The bounds are calculated by solving an optimisation problem for maximizing and minimizing the flux of each reaction iteratively. These limits are set as the new upper and lower bounds for the fluxes. The "centre" for each flux is the mean of the new bounds, as the most representative value of all solutions.
In the second iteration, we place a box around the hull (defining new bounds), before minimizing the distance between the flux of each reaction and the centre value, subject to the constraint that the total network flux cannot exceed Z_{1}, as found in the first iteration. In turn, this leads to new bounds and a corresponding centre. Each iteration of the algorithm adds an additional constraint, and the flux is drawn towards the centre of the bounds. After a finite number of iterations, the bounds converge to a single solution, within a specified tolerance.
The algorithm is explained in detail in a previous paper [14], which described a method for finding a unique solution within the space of all possible flux distributions in FBA. In that paper, the algorithm is used on four recent genomescale metabolic reconstructions. Using an iteration of linear programs, unique flux solutions are found in the available flux space for each organism.
Concentrations
Selected intracellular metabolite concentrations used in the model
Metabolite  Concentration (mM) 

3PhosphoDglyceroyl phosphate  2.75 × 10^{4} 
DGlycerate 2phosphate  0.0371 
3PhosphoDglycerate  0.278 
Acetaldehyde  0.17 
ADP  1.63 
AMP  0.796 
ATP  1.13 
CO2  1 
Dihydroxyacetone phosphate  0.59 
Ethanol  50 
DFructose 2,6bisphosphate  0.02 
DFructose 6phosphate  0.112 
DFructose 1,6bisphosphate  2.82 
Glyceraldehyde 3phosphate  0.069 
DGlucose 6phosphate  1.02 
DGlucose  0.0906 
Glycerol  2.27 
Glycerol 3phosphate  0.457 
Nicotinamide adenine dinucleotide  1.5 
Nicotinamide adenine dinucleotide  reduced  0.0861 
Phosphoenolpyruvate  0.0302 
Pyruvate  8.36 
Extracellular metabolite concentrations used in the model
Metabolite  Concentration (mM) 

4Aminobenzoate  0.0015 
LArginine  1 
LAspartate  1 
Biotin  8.2 × 10^{5} 
Citrate  1 
Fumarate  1 
DGlucose  11.1 
LGlutamate  1 
LHistidine  1 
myoInositol  0.055 
potassium  7.11 
LLeucine  1 
LLysine  1 
LMalate  1 
LMethionine  1 
Sodium  1.71 
Ammonium  38 
(R)Pantothenate  0.0042 
Pyridoxine  0.0019 
Pyruvate  1 
Riboflavin  5.3 × 10^{4} 
LSerine  1 
Sulfate  42.2 
Succinate  1 
Thiamin  0.0012 
LThreonine  1 
LTryptophan  1 
LValine  1 
Elasticities
An assumption of irreversible massaction kinetics would lead to reaction rate v = k A^{2}B and hence elasticity , the negative of its stoichiometry (2).
Linlog kinetics
where c denotes the compartment volumes. The benefit of this approximation lies in the existence of analytic forms for steady states and their stability matrix [5], thus avoiding computational problems associated with models of this size [18]. In a recent investigation, the linlog approximation was proved better than its alternatives (linear, power laws, generic and convenience) at describing E. coli sugar metabolism [19].
Testing
Control analysis
To test the resultant genomescale model, and to try and indentify key steps in the metabolic network of yeast, we calculate the flux control coefficients for reactions, as defined by metabolic control analysis (MCA). MCA studies how the control of fluxes and intermediate concentrations in a metabolic pathway is distributed among the different enzymes that constitute the pathway. Developed independently by Kacser and Burns [20] and Heinrich and Rapoport [21], the main theorems of MCA were given rigorous theoretical backing by Reder [22]. Of particular interest is the connectivity theorem, highlighting the close relationship between the local properties of individual reactions (elasticities) and global properties of the system (control coefficients). This theorem links the properties of the individual reactions (elasticities) to the properties of the system (control coefficients).
Whilst Reder's formula is often used in computational applications, it assumes that a certain matrix is invertible; this may not be true, especially if some reference reaction rates are zero. For example, the number of independent metabolites is often defined solely in terms of stoichiometry as rank(N) (here = 616). However, once kinetics are taken into account, this number drops drastically to rank(N·diag(v*)·ε) = 205. Reder's method only holds if these two values are identical. Thus, in Methods, we derive again the main results of MCA without relying on such an assumption.
Reactions exerting most control over glucose transport
Reaction  C ^{ J } 

glucose transport (uniport)  1.149 
glucosamine6phosphate deaminase  0.787 
glutaminefructose6phosphate transaminase  0.655 
glutamine synthetase  0.520 
inorganic diphosphatase  0.421 
Lasparaginase  0.323 
ATPase, cytosolic  0.250 
phosphofructokinase  0.235 
glycerol3phosphate dehydrogenase (NAD)  0.233 
adenylate kinase (GTP)  0.231 
Reactions exerting most control over biomass production
Reaction  C ^{ J } 

glucosamine6phosphate deaminase  0.532 
glutaminefructose6phosphate transaminase  0.441 
glutamine synthetase  0.358 
H2O transport via diffusion  0.212 
inorganic diphosphatase  0.193 
glycerol3phosphate dehydrogenase (NAD)  0.189 
Lasparaginase  0.146 
adenylate kinase (GTP)  0.142 
glucose transport (uniport)  0.132 
ribonucleosidetriphosphate reductase (UTP)  0.104 
Implementation
The systems biology approach often involves the development of mechanistic models, such as the reconstruction of dynamic systems from the quantitative properties of their elementary building blocks. Typically, this is performed in a 'bottomup' manner, whereby models built as individual elements are experimentallydetermined. Here we propose an alternative, 'topdown' mechanism, whereby an approximative model of the whole system is built initially; this model can then be used to guide experimental design and can subsequently be updated as specific knowledge becomes available from experimental results, following the iterative 'cycle of knowledge' approach [23]. At any point of this iterative approach, detailed kinetic rate laws can be included if they become available, in which case the approach is then a hybrid topdown and bottomup approach.
The genomescale model that is produced with the presented methodology is offered in SBML format, with MIRIAMcompliant annotations. Such markup allows automated reasoning about the model's assumptions and provenance [24]. A variety of software programs (e.g. COPASI [25]) have been designed to interface with SBML, but do not generally encounter models of this size. Indeed, the kinetic model produced here has over an order of magnitude more metabolites and reactions than any other kinetic model found in the BioModels repository. As the field develops, so larger models will be built, and software programs will be required to interface with models of at least this size. Thus, this methodology also allows software testing and advancement. The presence of analytic solutions facilitates validation of new tools, and avoids the usual problems with the high demands on computational power that models of this size have.
Conclusions
In this paper, we present a novel methodology that can be used to create a parameterized, genomescale kinetic model of the metabolic network of an organism. The methodology is demonstrated by its application on yeast metabolism, through appending existing kinetic submodels from the BioModels database to a stoichiometric model of yeast. The final model has 956 metabolic reactions involving 820 metabolites and, to our knowledge has significantly wider scope than any previous models of comparable type. We demonstrate the usefulness of such a model, by applying the principles of metabolic control analysis to identify key steps within the network.
Critically, both the original stoichiometric model, and the kinetic model that constitutes the endresult of the method are available in SBML, using MIRIAMcompliant annotations. Models in BioModels are annotated with computerreadable references such as ChEBI [9] or InChI [10], which made it possible to curate the mapping to the stoichiometric model in a semiautomated manner. While fullyautomated mapping of BioModels reactions to those in our stoichiometric model would be preferable, inconsistencies such as unbalanced reactions in either data resource prevent this at the current time. As systems biology is still a new and emerging field, it should be expected that discrepancies and other annotation issues will improve considerably. This, combined with greater availability of kinetic models for reactions and pathways in model repositories such as BioModels in the future, would mean that our methodology could be used to provide an increasingly more accurate and detailed genomescale, kinetic model for an organism, in an efficient and automated manner. Furthermore, the approach should benefit from expanding its scope in order to exploit other resources containing kinetic data, such as SABIORK [26] and BRENDA [27].
Our methodology clearly has limitations, in that the linlog framework is only valid in a region near the chosen reference state. Moreover, due to the vast lack of information, many of the parameters used in building the model are unknown and must be estimated through techniques such as flux balance analysis. Nonetheless, our modelling framework is a necessary stepping stone at creation of a genomescale kinetic model, and may thus be considered the first step in the deductiveinductive 'cycle of knowledge' crucial for systems biology [23]. We have demonstrated that this first model can be used to pinpoint, through sensitivity analysis, reactions that have the most control over the network, or reactions for which small perturbations of the values of their kinetic parameters lead to significant changes in the predictions of the model. Subsequent experimental work, such as kinetic assays may be used to improve the model's resolution. In the present case this includes glucosamine6phosphate deaminase, glutaminefructose6phosphate transaminase and glutamine synthetase. The model (see additional file 1) is publically available for download in SBML format from the BioModels database (BioModels ID: MODEL1001200000) and at http://www.mcisb.org/resources/genomescale/.
Methods
Control analysis
where ε' is the n × m unscaled elasticity matrix.
In general, the rank(N ε') = m_{0} <m and the system defined above will display moiety conservations  certain metabolites can be expressed as linear combinations of other metabolites in the system. Note that the number of independent metabolites is not given simply by rank(N), as is generally (and erroneously) suggested; rather the local dynamics of the system must also be taken into account via the elasticity matrix. The conservations may be removed through matrix decomposition, using a m × m_{0} link matrix L that relates the complete vector of internal metabolites to the vector of independent metabolites [28]. Writing A = N ε' and letting A_{ r }denote a m_{0} × m matrix composed of linearly independent rows of A, the corresponding link matrix is defined as , where '^{+}' denotes the MoorePenrose pseudoinverse [29]; hence A = L·A_{ r }.
where the m_{0} × m_{0} matrix (N_{r}·ε'. L) is invertible through introduction of the link matrix L.
respectively. If we compare our expressions to those given in Reder [22], we see that they are identical, save in her case r' is defined as the independent rows of N, leading to . If r = r' (i.e. if rank(N ε') = rank(N)), then L = L' and the two results are equivalent.
As such, we may see that we have extended Reder's work to encompass the possibility that rank(N ε') < rank(N), as is the case for our model (rank(N ε') = 205, whilst rank(N) = 616). From Equations (10) and (11), one may trivially deduce the summation and connectivity theorems.
Nomenclature
Nomenclature
Index  Description  Size 

i  species/metabolites  m 
j  reactions  n 
BM  subset of j: reactions with fluxes defined in BioModels  55 
r  subset of i: all independent metabolites  m _{0} 
Variable  Description  Dimensions 
A  N·ε'  m × m 
c  compartment volumes  m × 1 
C ^{ J }  scaled flux control coefficients  n × n 
C ^{ J } '  unscaled flux control coefficients  n × n 
C ^{ S } '  unscaled concentration control coefficients  m × n 
e _{ j }  denotes the j^{th} standard basis vector  n × 1 
f  vector specifying the optimized fluxes  n × 1 
N  stoichiometric matrix  m × n 
L  link matrix  m × m _{0} 
t  time  
x  metabolite concentrations  m × 1 
x*  reference metabolite concentrations  m × 1 
x _{ r }  independent metabolite concentrations  m_{0} × 1 
v  flux vector  n × 1 
v*  reference flux vector  n × 1 
v ^{min}  lower bounds vector  n × 1 
v ^{max}  upper bounds vector  n × 1 
v ^{ T }  fluxes defined in the Biomodels database  55 × 1 
Z  optimization objective  
Z*  maximum achievable growth rate  
Z _{1}  minimal total flux through the network  
δ  perturbation  
ε  elasticity  m × n 
ε'  unscaled elasticity matrix  m × n 
Notes
Declarations
Acknowledgements
We are grateful for the financial support of the BBSRC and EPSRC through grant BB/C008219/1 "The Manchester Centre for Integrative Systems Biology (MCISB)". We also thank Michael Howard for invaluable discussions, and our MCISB colleagues.
Authors’ Affiliations
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