Abstract
Background
The set of extreme pathways (ExPa), {p_{i}}, defines the convex basis vectors used for the mathematical characterization of the null space of the stoichiometric matrix for biochemical reaction networks. ExPa analysis has been used for a number of studies to determine properties of metabolic networks as well as to obtain insight into their physiological and functional states in silico. However, the number of ExPas, p = {p_{i}}, grows with the size and complexity of the network being studied, and this poses a computational challenge. For this study, we investigated the relationship between the number of extreme pathways and simple network properties.
Results
We established an estimating function for the number of ExPas using these easily obtainable
network measurements. In particular, it was found that log [p] had an exponential relationship with
Conclusion
This relationship typically gave an estimate of the number of extreme pathways to within a factor of 10 of the true number. Such a function providing an estimate for the total number of ExPas for a given system will enable researchers to decide whether ExPas analysis is an appropriate investigative tool.
Background
Extreme pathways (ExPas) of a metabolic network are the irreducible set of vectors that define the basis of the nullspace of the network's stoichiometric matrix. Every allowable solution to the flux balance equations of a reaction network in steady state, S·ν = 0, can be represented as a nonnegative linear combination of the extreme pathway vectors. ExPas are biochemically and thermodynamically feasible pathways that transform a selection of the given substrates to a selection of allowable products. ExPas have been extensively used for the analyses of metabolic networks (see, for example, [15]). Typically, such analyses used ExPas to define possible phenotypic states of metabolic networks under different simulation conditions, to identify network redundancy, and to reveal eigenpathways that effectively characterize all relevant physiological states of a metabolic network. Modified versions of ExPa analyses have also been applied to regulatory networks [6,7] and signaling pathways [8]. Such applications are still in their infancy and are important research topics. However, as the size of a network increases, the redundancy of the network, that is, the number of different pathways that transform given substrate(s) to given product(s) [9,10], becomes more apparent, and the number of ExPas increases rapidly. Redundancy also exists in small systems but this can be easily handled and even provide insights to legitimate alternative pathways. As the number of ExPas increases at a drastic rate, performing insightful analyses using ExPas become increasingly difficult.
The fact that the set of ExPas of a biochemical reaction network defines the boundaries of its convex steadystate solution space makes them a valuable tool for metabolic network analysis. Further, they emphasize alternative pathways that exist in a network, which may otherwise be overlooked, and that can enrich the understanding of its possible physiological states. However, the increasing details included in reconstructed metabolic networks lead to the combinatorial explosion of the number of ExPas and their computation time. A method providing a good estimate for the final number of ExPas for a given system will enable researchers to decide whether ExPas analysis is a appropriate tool for their objectives.
Another method often used for characterizing the steadystate solution space for a biochemical reaction network is known as Elementary Modes (EMs) [11]. Both ExPa and EM analyses require the resulting solution vectors to be nondecomposable and unique. In addition, ExPa vectors are required to be systemically independent [12]. As a result, ExPas for a system are a minimal set of EMs, and the number of ExPas is less than or equal to the number of EMs. Since both ExPas and EMs are biochemically and thermodynamically feasible pathways, the number of these pathways cannot be estimated using traditional graph theoretical algorithms, such as the Dijkstra's algorithm [13], for finding all shortest paths.
The combinatorial complexity of Elementary Modes of a network was previously described
by Klamt et al. [14] by providing an upperbound for the number of EMs. In their work, the authors considered
the following combinatoric problem: given a network with n reactions and m metabolites, the maximal number of independent pathways occurs when each possible
subset of the reactions consisting of the m metabolites are independent. This maximal number was found to be
In this study, we investigated the relationship between the number of ExPas for a
given network, p = {p_{i}}, and its basic network measurements. Several network measurements are commonly
used in describing the topological features of a network and include connectivity,
clustering coefficient, network diameter, and degree distribution [15]. The number of ExPas for a network can vary dramatically under different simulation
conditions, that is, different environmental constraints. Consequently, basic network
information such as the numbers of reactions and metabolites of the network cannot
be solely used to provide a meaningful estimate. Since ExPas are connected reactions,
we hypothesized that the higher the reaction connections, the larger the number of
ExPas. Based on this hypothesis, we demonstrated an exponential relationship between
log [p] and
Results
The number of extreme pathways (ExPa), p = {p_{i}}, for a metabolic network increases drastically with the complexity and size of
the network. An estimate for p for a given network can help one decide whether ExPa analysis is a feasible tool for
one's research objective. In this study, we investigated the relationship between
p and a number of factors, θ_{i}, formed by simple network measurements such as the incoming and outgoing degree of
reactions, d_{∓}(r_{i}) =
Identification of Significant Contributing Factors
We aimed to identify factors that can be used for establishing appropriate estimating
functions. Desirable factors must be i) easily obtained and ii) specific for a given model. For example, network measurements such as the incoming
and outgoing degrees of each reaction, d_{∓}(r_{i}) =
A number of potential factors for the estimating functions were formed using the aforementioned
network measurements (Table 1). The correlations of these potential factors and p were evaluated using the Pearson's productmoment correlation, r, and Spearmanrank correlation, ρ. The Pearson's correlation is generally used as an indicator for the strength and
direction of a linear relationship and is considered to be robust enough to handle nonparametric data.
On the other hand, the Spearman's correlation describes the monotonic relationship
between two variables without making any assumptions about the frequency distribution
of the variables. We used both of these correlation coefficients on the original and
logged data to avoid misinterpretation due to the wide ranges of data (Table 1). For the original data, the factor with the highest Pearson correlation before was
found for
Table 1. Identification of Potential Contributing Factors
Single Factor Estimate
The factors identified in the previous section, namely
Figure 1. Relationship between the Number of ExPas and Factor
Figure 2. Relationship between the Number of ExPas and Factor
where
Estimation Using θ_{1}
Using factor
and with ω_{2}:
The fitted curves given by Equations (2) and (3) are shown in Figure 1. The Pearson's correlation was 0.883439 for the function given by Equation (2), whereas
that given by Equation(3) resulted in a better fit, with correlation being 0.900704
and reduced mean absolute and rootmeansquare errors (Table 2). The overall performance of this estimator was evaluated. It was found that the
number of ExPas for most of the models (47 out of 52) could be described to within
a factor of 10 using the estimating functions, while those that could not tended to
be overestimated (Table 2). The inclusion of the factor
Table 2. Fit of Training Data to Estimating Functions
Estimation Using θ_{2}
Using the second factor, θ_{2}, the estimating functions with and without scaling had the respective forms
and
The relationships between log [p] and Equations (4) and (5) are displayed in Figure 2. In this case, the Pearson's correlation before the inclusion of R^{ω }was 0.887057, and was improved to 0.898332 after scaling.
Similar to the case for θ_{1}, the errors were reduced after scaling (Table 2). The unscaled estimating function, Equation (4), again, described most of the models (48 out of 52) to within a a factor of 10 with respect to the actual ExPa numbers (Table 2). The inclusion of the scaling factor resulted in less outliers, a better correlation and reduced errors (Table 2).
Performance of Estimations Functions
The performance of the estimating functions (2–5) was tested using an additional 16 models. These models were reduced but functional models of the central metabolism derived from reconstructed metabolic networks of 3 different organisms, namely H. pylori [3], M. barkeri [16], and H. influenzae [1]. All four estimating functions successfully predicted 9 out of the 16 test models (56%) to within a factor of 10 (Figure 3). For all four estimating functions, the number of ExPas of seven models were overestimated by a factor greater than 10 while none were underestimated beyond that factor. In particular, the estimating function f_{2}(θ_{1}) yielded the smallest error range for all models. The 16 test data points had the highest correlation with f_{2}(θ_{1}), as did the 52 data points used for its formulation (Table 3). We concluded that the estimating function f_{2}(θ_{1}) can typically successfully estimate the number of ExPas of a metabolic network to within a factor of 10.
Figure 3. Comparison of Test Models to the Estimating Functions. Figures displaying the relationships amongst the test data points and the four estimation functions given by equations (2), (3), (4) and (5). These are shown in (i), (ii), (iii) and (iv) respectively. The red lines in each case are given by f_{i}(θ_{j}) ± 1 and indicate the boundaries of the regions that are within a factor of 10 of the respective estimations.
Table 3. Fit of Test Data to Estimating Functions
Consideration of Other Network Measurements
During the development of these estimating functions, other factors such as the degrees
of exchange metabolites were considered. In the case of exchange metabolites, the
correlation of the sum of the degrees of all input metabolites,
Discussion
The goal for this study was to produce an estimating function using basic network measurements. Specifically, we aimed to obtain a function such that only a single factor is used for estimation. In principle, it was possible to use a multivariate (polynomial) regression method using a number of the factors described in the section 'Identification of Significant Contributing Factors'. However, the independence assumption upon which this method is based was not applicable as the factors themselves tend to be highly correlated. Furthermore, it would have been difficult to interpret which factors were truly responsible for the increase in p, and would probably lead to inaccurate estimations in test models. Here, the most descriptive factor was θ_{1}, which includes the clustering coefficients. The interpretation of the clustering coefficient used in this study is also often used in sociology and biochemical networks (see, for example, [15,17]). There are other interpretations of the clustering coefficient, such as that described by Soffer et al. [18]. Their definition eliminates degreecorrelation biases, thus, quantifying the connectivity amongst the neighbors of a vertex independent to its degree and the degree of its neighbors. It would be interesting to use a similar definition for directed graphs and investigate its effects on ExPa estimation. Additionally, it is possible that other factors may provide a more accurate estimation for the number of ExPas. However, these factors may only be found by detailed analyses of network structures.
The estimating function given by Equation (3) typically estimated the number of ExPas of the test models to within a factor of 10. In cases where it failed, it did not underestimate the number of ExPas. The version of the E. coli reconstructed network used by Klamt et al. [14] was not elementally and chargebalanced and has since been replaced by updated versions [19,20]. We used a revised version iJE660a, which was found to be the closest to what they used, and is publicly available [19,21], to compare our method with Klamt's. When the estimating function was applied to this version, assuming that all reactions were active concurrently, 7 × 10^{12 }ExPas were estimated with our method, whereas Klamt's method yielded an upperbound of 5 × 10^{13 }after disregarding inactive reactions in the unbalanced and smaller model. Given that iJE660a has 41 more reactions and all the reactions are elementally and chargebalanced, we are confident that our estimating function can also serve as a conservative upperbound of the number of ExPas after some adjustments. For larger networks such as the latest published reconstruction of E. coli consisting of 904 cited reactions [20], we estimate 3 × 10^{18 }ExPas, The Human reconstructed network with 3311 reactions [22] is predicted to have 10^{29 }ExPas when all reactions were active concurrently.
Conclusion
In this study, we investigated the possibility of estimating with confidence the number
of extreme pathways (ExPa), p, for metabolic networks. Our effort concentrated on the use of simple network measurements,
namely the incoming and outgoing degrees,
The set of extreme pathways is the convex basis used for biochemical characterization of the nullspace of the stoichiometric matrix for a biochemical reaction network. ExPa analyses have typically been used to characterize phenotypic states of metabolic networks and identify network redundancy. Beyond these uses, the singular value decomposition of the extreme pathway matrix has been used to identify eigenpathways that are capable of characterizing phenotypic states of a system [23,24]. Nevertheless, applications such as these require ExPas to be calculated prior to any analysis. The number of ExPas is set to increase dramatically with network size and complexity. In particular, with the increase in details of metabolic network reconstructions and the emergence of reconstruction of global transcription/translation networks, new techniques for calculating and analyzing ExPas are much needed. Since the goal of systems biology is to study an organism as a whole, different types of biochemical networks will eventually be combined so that the system can be studied in its entirety. To overcome future computational challenges as well as being equipped with the necessary analytical techniques should become our immediate goal.
Methods
Basic Concepts and Notations
Hypergraph
We introduce some basic concepts and notations that will assist us in describing the
measurements needed. We first note that a metabolic network can be described as a
directedhypergraph, where a node represents a metabolite and an edge a reaction.
The stoichiometric matrix, S, can thus be seen as a nodeedge incidence matrix. A directedhypergraph
Reaction Adjacency and Neighbourhood Matrices Â,
Δ
^
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiqacuWFuoargaqcaaaa@2E27@
The adjacency matrix contains information about whether one reaction 'goes into' another.
Using the notation introduced in the 'Hypergraph' section, two reactions r_{i }and r_{j }are adjacent if H(r_{i}) ∩ T(r_{j}) ≠ ∅ or H(r_{j}) ∩ T(r_{i}) ≠ ∅, that is, the intersection of the set of outputs of reaction r_{i }and the set of inputs for reaction r_{j }is nonempty, or vice versa. In particular, we say r_{i }'goes into' to r_{j }if H(r_{i}) ∩ T(r_{j}) ≠ ∅. The adjacency matrix Â is constructed from the stoichiometric matrix S by partitioning S into two digitized components Ŝ_{+ }and Ŝ_{}, where
The elements of the adjacency matrix
Reactions r_{i }and r_{j }are said to be connected if any of H(r_{i}) ∩ T(r_{j}), T(r_{i}) ∩ H(r_{i}), H(r_{i}) ∩ H(r_{j}) or T(r_{i}) ∩ T(r_{j}) is nonempty. The neighbourhood matrix is given in the form
where
Network Measurements
Effective Number of Reactions R = R_{eff}
For any models of a reconstructed network, redundancy in terms of reactions that are not utilized is often expected. This is due to the fact that, for any specific model, there is a set of reactions that is not used under the specific simulation conditions, and therefore can be removed from the network without affecting the model's function. Extreme pathways are classified into 3 types, with TypeI being those that have exchange fluxes across the system boundaries that correspond to noncurrency metabolites [25]. Here, we denote the set of reactions that are present in at least one TypeI ExPas by R = R_{eff}. This number can be obtained by optimization techniques such as FluxBalance Analysis [26] using tools such as SimPheny by Genomatica or FluxAnalyzer [27].
Reaction Connectivity d_{±}(r_{i}) =
d
±
i
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGKbazdaWgaaWcbaGaeyySae7aaSbaaWqaaiabdMgaPbqabaaaleqaaaaa@31AA@
Having identified the set of reactions from a reconstruction that are active in a
model, the stoichiometric matrix S can then be reduced by removing inactive reactions and metabolites. The connectivity
(degree) of each active reaction can then be calculated. Since S can be considered the nodeedge incidence matrix for a directedhypergraph, it is
more appropriate to consider the incoming and outgoing metabolites separately. The
adjacency and connectivity between each pair of reactions can then be described in
terms of the definition given in the section titled 'Reaction Adjacency and Neighbourhood
Matrices Â,
From Figure 4, it can be seen that the number of possible pathways through a given reaction r_{i }is given by
Figure 4. Connectivity of Reactions. Diagram describing different types of connectivities. Reaction r_{i }utilizes metabolite A, which is produced by three reactions, and produces metabolite B, which is consumed by three reactions. Reaction r_{i }then has an incoming degree of d_{}(r_{i}) = 3 due to metabolite A, and outgoing degree of d_{+}(r_{i}) = 3.
Reaction Clustering Coefficient c_{i }= c(r_{i})
A metabolic network is described by the stoichiometric matrix S. This Smatrix can be seen as a nodeedge incidence matrix for a directed hypergraph.
However, the clustering coefficient for a hypergraph is not well defined. Since we
are interested in how the reactions are connected, we can use the adjacency matrix,
Â, which contains the nodenode (reactionreaction) information of the network, where
Figure 5. Projection from Directed Hypergraph to Onemode Graph. Projection from directed hypergraph to onemode graph, where the hyperedges on the lefthand side become the nodes of the the graph on the righthand side. A thick black arrow in the graph on the right signifies an edge r_{i }is adjacent to r_{j}, whereas a thin blue line signifies two edges connected that are not adjacent.
where k_{i }is the number of reactions that r_{i }is connected to, i.e., k_{i }is the number of nonzero elements of the vector
Figure 6. Relationship between Reactions with Nonzero Clusteringcoefficient and Alternative Routes. Diagram showing relationship between nonzero clustering coefficients and alternative pathways. (i) shows three possible routes for a simple system; (ii) is the nondirected representation of this system using the above projection. The system has nonzero clustering coefficients, emphasizing alternative routes are possible; (iii) is the projection conforming to that shown in Figure 5, where nonzero clustering coefficient is found for 5 of the reactions that are involved in the branching points of alterative routes.
Reconstructed Networks and Simulation Conditions
Reconstructed metabolic networks of H. pylori [3], Human Cardiac Mitochondria [4], the Human Red Blood Cell [5], and the core E. coli [28] network were used in this study. These networks are all mass and chargebalanced, and were all tested on SimPheny (Genomatica) for the ability to produce biomass constituents. Furthermore, older versions of published, available networks that were used for extreme pathway and fluxbalance analysis, which may not be completely balanced, were included in this study. This selection of networks represented a spectrum of complexity. Elementary network measurements, the number of generated models, and the source of each of these networks are detailed in Table 4.
Table 4. Basic Information of Models Used
A total of 52 models were used from these networks. Most of these models were used to test for production of products given a specified substrate along with core exchange metabolites. The remaining 9 models included singlereaction deletions and/or the request for specific demand metabolites given a combination of primary substrates. The specific environmental conditions of these models are listed in Table 5, along with the names of abbreviated metabolites in Table 6.
Table 5. Environmental Conditions of Models. Table detailing the models used for estimation formulation, along with their environmental conditions.
Table 6. External Metabolites Abbreviation
A total of 16 functional models from 3 different organisms were used to test the validity of estimations formulated in the section 'Single Factor Estimate'. The networks used were 2 reduced versions of H. pylori, one consisting of 168 reactions and 170 metabolites, the other with 48 reactions and 65 metabolites. Reduced versions of the networks of M. barkeri [16] and H. influenzae [1] were also used. These consisted of 84 and 61 reactions and 121 and 83 metabolites, respectively. These networks are similar to the core E. coli model, with each reaction being mass and chargebalanced. They were also tested for the production of biomass using SimPheny and hence are functional systems.
The number of data points may seem unorthodox. Although in theory it was possible to automatically generate networks of different input/output combinations, leading to a larger number of training and validation data points, in practice, only a few of such combinations would have resulted in models that produce nonTypeIII extreme pathways as well as biomass constituents. For the validation stage, models producing biomass constituents as well as nonTypeIII pathways that could be calculated quickly were desired. By drastically reducing networks, it was difficult to construct models that maintained biomass production.
In the case of H. pylori, three models were produced using the larger network in its entirety. Since the computation of ExPas is a timeconsuming process, a smaller network was created to facilitate this process. This smaller network was subjected to random reactiondeletion while ensuring that the subsequent modified models could still produce equal amount of biomass. Five such models with random deletion were produced for this study. In addition, 5 models for M. bakeri and 3 for H. influenzae were generated in a similar fashion so that ExPa computation could be done within a reasonable computational time and effort. These are listed in Table 7.
Table 7. Environmental Conditions of Test Models
Calculation of Extreme Pathways and Network Measurements
The extreme pathways of all models were computed using an implementation of the algorithm given in [29]. This implementation includes the C++ STL and the number theory library NTL. Algorithms for calculating the greatest common factor of a set of integers of arbitrary size and for sparsematrix operations were also implemented. Network properties, including incoming and outgoing degrees and clustering coefficients of reactions, were calculated using a C++ implementation of the methods described in 'Basic Concepts and Notations' and 'Network Measurements' using sparsematrix algorithms and bitwise operations.
Correlation Coeffcients
Both Pearson's productmoment and Spearman's rank correlation coefficients were used as a guide to help identify important factors that contribute to the number of extreme pathways. The former is defined as
for a series of n measurements x_{i }and y_{i}. It was used in this study as a guide to detect linear relationships amongst the data and estimating functions. The nonparametric correlation coefficient used in this study is defined as
where D_{i }is the difference in the ranks of the corresponding values of the n pairs (x_{i}, y_{i}). This was used to decide whether a factor increased monotonically with the number of ExPas p.
Authors' contributions
MY and BØP designed the study. MY performed and developed the programs used for the calculation of extreme pathways and network measurements, and analyzed and interpreted the data. IT provided all data of reconstructed models used in this study. MY drafted the manuscript while IT and BØP provided critical edits and important intellectual content. MY, IT & BØP have read and approved the final version of this manuscript.
Acknowledgements
This work was funded by National Institute of General Medical Sciences (NIGMS, grant number GM68837). BØP serves on the Scientific Advisory Board of Genomatica Inc. The authors would like to thank Marc Abrams for critical reading of this manuscript.
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