Comparison of GJE and SVD

Five large metabolic networks of increasing complexity were selected to test the performance of symbolic GJE to that of numerical SVD. These metabolic networks were formulated in a closed form as described by Famili and Palsson [12]. To obtain non-trivial solutions to the steady state equations, the metabolic networks need to be converted to open form where the boundary conditions are specified via transport fluxes into the network rather than via external metabolites. For simplicity, we convert the metabolic networks to open form by introducing transport fluxes across the network boundary to metabolites that either appear in exactly one reaction or are products of polymerization reactions (see Methods).

The kernel of five stoichiometric matrices were solved for using both numerical SVD and the symbolic GJE routine with the results given in Table 1. The computation time for both methods was found to be almost the same with SVD being slightly faster. However, we noted that the numerical SVD routine used effectively two CPUs (see Methods for details about the test computer system) while the symbolic GJE routine used only one. Hence for a number of parallel kernel calculations that would consume all computer CPUs, the symbolic GJE routine would be more productive. Figure 1 (top) illustrates how the kernel computation time depends on the size of the network. The computational time increases exponentially with the size. It should be noted that the ratio of these exponents depends on a computer system and underlying computational libraries. Also note that the complexity of both SVD and GJE algorithms are O(mn²), that is, increasing the network size by a factor of 10, the complexity should increase by 1000 times. The actual complexity increase (about 400 for SVD and 640 for GJE) is smaller because of using threaded libraries for the SVD routine and because of computing with high sparsity for the GJE routine. The numerical errors introduced when using SVD were found to be insignificant for the purpose of biological flux calculations and confirm the fact of numerical robustness of the SVD routine. This assessment was made by calculating the maximum relative flux error ε_SVD using Equation (11). Note that this loss of accuracy is in agreement with the condition number calculated for V_indep in Equation (9); the number of inaccurate digits is approximately equal to the order of magnitude of ε_SVD.

Model	Publ.	Species	Reactions		Flux variables		CPU time (s)		ε SVD ×10-12	Condition number
			Orig.	Open	Dep.	Indep.	SVD	GJE
Example		118	129	156	118	39	0.02	0.03	0.003	15
iPS189	[15]	433	350	482	413	69	0.3	0.4	0.07	31000
iND750	[16]	1177	1266	1561	1162	399	6.2	8.0	6.37	68000
AraGEM	[17]	1767	1625	2361	1720	641	19.7	34.2	12.65	3000
iAF1260	[18]	1972	2382	2773	1960	813	30.5	34.3	1.43	2800
Recon 1	[10]	3188	3742	4480	3169	1311	123.5	145.6	32.63	71000

Kernel computation times for numerical SVD and symbolic GJE for the example yeast network given in Figure 3 and five genome-scale metabolic networks. All techniques are described in Methods. The condition number was calculated for V_indep from Equation (9). The inversion of V_indep is required to directly compare SVD results with the solution found from GJE. The difference between the results is given by ε _SVD in Equation (11).

With our test computer system both numerical SVD and symbolic GJE routines can easily cope with 4000+ reaction networks. To test the limits of these routines, we repeatedly doubled the sizes of considered networks by repeating given stoichiometric matrix diagonally within a doubled stoichiometric matrix and then randomly shuffling the columns. The doubled stoichiometric matrix would then correspond to two independent but identical metabolic networks. The shuffling is needed for modeling the structure of actual metabolic network models where the order of columns is arbitrary. The process of increasing the sizes of networks was repeated with doubled stoichiometric matrices until applying our routines were close to exceeding the resources of our computer system. Figure 1 (bottom) shows the dependence of the memory usage on the size of the network. The memory usage for computing the kernels increases exponentially with the size. The two times smaller memory increase when using the symbolic GJE routine compared to the numerical SVD routine is explained by the fact that symbolic GJE routine preserves sparsity while the result of numerical SVD routine is generally non-sparse. This is illustrated in Figure 2 where the corresponding kernels from SVD and GJE algorithms are shown for the example yeast network (see next Section). For other tested networks the sparsity of GJE kernels varied in the range 95-99.9% and the sparsity of SVD kernels in 1-25%.

Application of constraints to the example yeast network

Often one needs to constrain the flux values that are physiologically meaningful, that is, either they have been experimentally measured or they must be non-negative due to the irreversibility of some reactions. We demonstrate the application of constraints by calculating a flux distribution for an example yeast network. The metabolic network is given as an SBML file in additional file 1: yeast_example.xml, and is laid out in Figure 3. This network contains 129 reactions and 118 metabolites, including 62 metabolites in the cytosol, 29 metabolites in mitochondria, and 27 metabolites that are external to the network. Because the list of external metabolites is known in this example then the system can be converted to open form by removing those rows from the stoichiometric matrix that correspond to external metabolites. Note that this is our alternative method of opening metabolic networks (see Methods).

The symbolic GJE of the stoichiometric matrix for the open system provides 91 relations for the dependent fluxes expressed in terms of 39 independent fluxes. A full list of reactions, metabolites, and steady state flux relations is given in additional file 2: yeast_example.pdf. The corresponding kernel matrix is shown in Figure 2. The relations are formed from the rows of this matrix.

The independent fluxes can be selected prior to performing GJE. We compose the set of independent fluxes from biomass production rates that have been experimentally measured. In total, 26 measured biomass fluxes taken from Cortassa et al [13] were used to constrain this network. The table of such exchange fluxes and their values is given in additional file 2: yeast_example.pdf. The remaining 13 independent flux variables are left unspecified which means that the symbolic GJE routine will choose a viable set of independent fluxes. In this example these are all internal to the network.

After substituting the biomass production values into the steady state flux relations, 27 dependent fluxes become fully specified with 64 relations described by the 13 internal variables. Inspection of this system of equations (also given in additional file 2: yeast_example.pdf) immediately reveals which part of the metabolism each of these 13 variables controls. Each internal variable is connected to dependent variables via nonzero entries in the corresponding column of the kernel matrix. The set of these dependent variables share metabolites and thus can be considered as one connected sub-network of the original system. Five of these sub-networks determine the split between cytosolic and mitochondrial valine, leucine, alanine, and aspartate biosynthesis via BAT1 and BAT2, the split between LEU4 and LEU9, ALT1 and ALT2, and AAT1 and AAT2. Two determine the interconversion and transport of glutamine, glutamate, and oxoglutarate via the split between GLT and GDH. The remaining six determine: (1) urea cycle flux, (2) relative production of glycine from either serine or threonine, (3) the flux of D-Glucose 6-phosphate directed towards D-Ribulose 5-phosphate, (4) production of pyruvate by the malic enzyme MAE 1, (5) the production of phosphoenolpyruvate by PCK 1, and (6) the relative production of acetaldehyde to acetyl-CoA from pyruvate. Figure 3 gives one flux distribution calculated by specifying the values for the 26 biomass fluxes and 13 internal fluxes. The values chosen to substitute into the flux relations are highlighted on the figure.

In addition to constraining the measured independent variables directly, knowledge about the dependent fluxes in the example yeast network was used to constrain the network. We specified the net flux direction for reactions that involved the production of carbon dioxide. The constraints of measured flux values and the specified net flux direction of reactions, can be written as a system of 91 flux relations, 26 measured independent fluxes, and 17 inequalities. Following this all redundancies were removed using computational geometry techniques described in Methods. The result is a set of five upper and lower bound conditions for 5 independent fluxes, given in additional file 2: yeast_example.pdf.

Symbolic relationships give an informative representation of metabolic network structure

There are several technical issues that complicate analysis of large metabolic networks. Among them are numerical robustness of the algorithm and presentation of solution to the researcher [5, 6]. Those problems are resolved when using symbolic GJE presented in this work. While GJE and SVD provide mathematically equivalent methods of solving for the steady state flux relations of metabolic networks, there is a difference in how the solutions are formed. In SVD, steady state solution is given through a combination of eigenvectors that often span the entire metabolic network [12]. Those eigenvectors contain information about the metabolic network, however extracting and interpreting this information is not always trivial and has inspired the creation of a diverse set of tools and techniques [14]. In contrast, symbolic GJE gives the researcher an opportunity to find the set of independent fluxes and relationships between independent and dependent fluxes. Through such relationships it is easy to see which dependent fluxes are influenced by any particular independent flux and gain insight into the operation of the metabolic network.

In the example yeast network given in Figure 3, many different sets of independent fluxes can be used to find a steady state solution. The GJE routine allows the researcher to specify which independent fluxes will be used to form the solution. By choosing biomass production rates, one can constrain the operation of the metabolic network to any given set of biomass measurements.

In our example, application of biomass constraints leaves 13 independent variables that are internal to the network and define all steady state flux distributions. We found that these 13 independent fluxes influence only a specific portion of the metabolism. Each independent variable only influences those dependent fluxes that have non-zero values in its column of the GJE kernel matrix. This property has potentially far reaching implications for the physical interpretation of steady state metabolism in large networks. All nonzero entries in each column of the GJE kernel define a set of dependent variables. These variables share metabolites and thus form a sub-network. Sub-networks that share common dependent variables can be combined into a larger sub-network. For example, it allows one to identify sub-networks within the metabolic network that are linked with shared metabolites and are controlled by sets of independent fluxes. In the example yeast network two fluxes are needed to describe glutamine, glutamate, and oxoglutarate transport and interconversion while five fluxes control the split between cytosolic and mitochondrial production of valine, leucine, alanine, and aspartate. The loops within these sub-networks are determined solely by independent fluxes that occur within each sub-network.

Applicability of symbolic GJE and technical issues

We found that the computational time of applying symbolic GJE and numerical SVD routines to be similar for all networks considered. The memory usage of numerical SVD routine for networks with 6000+ reactions became close to exceeding memory resources of our test computer system. With the same memory usage level GJE routine would be able to analyze a network with 10⁶ reactions, however, this calculation is estimated to take one year. Even when memory usage will be optimized in the SVD routine, the doubling network size will quadruple SVD memory usage while GJE memory usage would only double. This is because GJE algorithm preserves sparsity.

We did not observe the phenomena of coefficient explosion that would be typical for GJE algorithm using rational arithmetics on large matrices. This is explained because genome-scale stoichiometric matrices are inherently sparse and majority of elements are small integers such as 1 or -1. In addition, SympyCore [11] minimizes the number of operations by its pivot element selection rule (see Methods) to reduce computational time and this has added benefit of reducing the chance of coefficient explosion.

The reduced row echelon form of the stoichiometric matrix is formed by elementary row operations. The sequence of elementary row operations typically depends on the original ordering of the rows and columns, which is arbitrary. However, if one chooses the set of independent flux variables, i.e. columns to be skipped in the reduction process, the same reduced row echelon form of the matrix is found irregardless of the original ordering of the rows and columns. For this to be true, the columns corresponding to the chosen set must be linearly independent. When a viable set of independent flux variables is unknown or only partially known beforehand, the GJE routine implemented in SympyCore will choose the remaining independent flux variables to complete the matrix reduction process.

Flux analysis in vivo

One of the most challenging tasks for the analysis of fluxes in vivo is intracellular compartmentation. There are several levels of compartmentation that ought to be taken into account in a large scale metabolic model. They range from the organ level to the sub-cellular level. The genome-scale metabolic models used in this text [10] are typical in that they are compartmentalized into standard intracellular compartments separated by membrane barriers, such as mitochondria. However, even smaller compartmental units exist such as submembrane space leading to the coupling between the K-ATP sensitive channel and creatine kinase [19], or intracellular diffusion barriers grouping ATPases and mitochondrial oxidative phosphorylation in cardiomyocytes [20–23], and the compartmentation of metabolites within enzyme systems [24]. These forms of compartmentation are often excluded from metabolic models. A genome-scale model that includes all such smaller compartmental units has yet to be formulated and will be larger. The symbolic GJE routine developed in this paper would be a suitable tool to analyze such large networks due to its efficiency.

Frequently, compartmentation can be analyzed by fully or partially decoupling the links between metabolites and reactions in the stoichiometric matrix. However, concentration gradients within the cell cannot be incorporated into a stoichiometric model. This form of compartmentation requires the use of reaction-diffusion models that take into account the three dimensional organization of the cell [25, 26], and the development and application of specialized techniques such as the measurement of diffusion coefficient in the cell [27] and the use of kinetic measurements to estimate the diffusion restrictions partitioning the cell into compartments [22]. Thus the concentration gradients limit the application of stoichiometric modeling to the thermodynamic level.

Even without resorting to spatial modeling, the analysis of compartmentation remains challenging since more data is required to constrain the extra degrees of freedom introduced when splitting up metabolic pools. A recent organ level study of human brain [28] discusses the challenges of both composing an organ level compartmentalized model and obtaining the data required to constrain it. Our analysis of the example yeast network shows that each degree of freedom controls a local sub-networks of fluxes. By specifying intercompartmental fluxes to be part of the set of independent fluxes the influence of compartmentation may be characterized by a subset of variables making the analysis of compartmentation more straight forward.

Functional coupling within enzyme systems is often neglected in large scale metabolic models. When studying enzyme kinetics, it is often assumed that the distribution of the states of the enzyme remains stationary and is determined by the availability of metabolites. This assumption has been applied to study coupled enzyme systems [29] whose steady state is non-trivial since they may contain hundreds of transformations. When this assumption is made, individual mechanistic transformations can be treated in the same way as chemical reactions. The ability to choose some of these mechanistic transformations to be part of the set of independent fluxes would aid in the constraint process. It would also help one to incorporate enzyme mechanisms into larger stoichiometric models since the fluxes through the branches in the enzyme mechanism would be controlled by a subset of the independent variables and this subset would not influence remote regions of the metabolism.

Several approaches have been developed to study flux distributions in vivo without perturbing enzyme function. Notably, isotope labeling [30] and magnetization transfer [31]. The dynamic component of the labeling can be used to reveal compartmental effects such as the identification of barriers to metabolite transport. However this approach requires the use of optimization tools that must scan a high-dimensional space [30]. Recently, an improved optimization approach was developed that makes use of a flux coordinate system found using GJE [32]. Our GJE routine allows for the pre-selection of the independent variables, and it is anticipated that a well chosen flux coordinate system would further improve the application of this optimization procedure.

Different representations of steady state solutions

The goals of constraint based flux analysis are currently pursued using an increasing number of complimentary approaches including extreme currents [33], extreme pathways [34], elementary modes [35, 36], minimal generators [37], minimal metabolic behaviors [38], and other techniques [39]. In this paper we only applied symbolic GJE algorithm to carry out Metabolic Flux Analysis (MFA).

SympyCore can be extended by implementing the double description method [40] which is an integral part of Elementary Flux Mode Analysis (EFMA).

Although both MFA and EFMA provide solutions to the same steady state problem, comparing these solutions must take into account differences in the representations of the solutions and underlying assumptions in these methods. While MFA defines a subspace of steady state flux distributions then EFMA restricts this subspace by taking into account of irreversibility of certain reactions.

Within MFA, to represent a point in such a flux subspace, it is convenient to use a linear combination of the columns of the kernel of the stoichiometric matrix. Note that such a kernel is not unique: in the SVD approach the kernel depends on the ordering of reactions as they are used to compose the stoichiometric matrix; and in the symbolic GJE approach, the kernel depends on the initial choice of independent and dependent flux variables. Reaction irreversibilities convert to constraints on the coefficients of the linear combination. In the case of the SVD kernel, these constraints are difficult to interpret because of the convolved nature of the SVD coefficients: change of one coefficient will have effect to all fluxes. In the case of the GJE kernel, the coefficients are fluxes themselves (independent fluxes) and hence the constraints on the coefficients have a straightforward interpretation.

Within EFMA, it is mathematically more convenient to use convex polytope to represent the restricted part of the flux subspace because the conditions of reaction irreversibilities directly define the representation. This approach has given rise to the now widely used notation of elementary flux modes [41] and extreme pathways [34] that mathematically speaking are extreme rays of the convex polytope of thermodynamically feasible steady state flux distributions. It is interesting to note that in the case of pointed polytope the steady state flux distribution can be represented as a conical combination of elementary flux modes. While the elementary flux modes are uniquely determined then different combinations of elementary flux modes may define the same steady state solutions. This is orthogonal to kernel based representations: steady state solutions can be represented via different kernels but when fixing a kernel then the linear combination of its columns uniquely defines the flux distribution.

Statement of the steady state problem

Every chemical reaction and thus reaction system has the strict requirement of conservation of mass. A system of mass balances around each species has the form:

(1)

where is a length m vector of the time derivative for each mass density of metabolic species, N is the m × n stoichiometric matrix that links metabolites to their reactions via stoichiometry, and ν is a length n vector that describes the flux through each reaction. For a system at steady state with n reactions and m species, the system of chemical reactions becomes:

(2)

The number of flux variables that need to be specified to calculate a viable steady state is f = n - r where r is the rank of N. Let us denote the vectors of dependent and independent flux variables as ν_dep and ν_indep of length r and f, respectively. Then with a n × n permutation matrix P that reorders the columns of N such that columns corresponding to dependent flux variables appear earliest, the steady state Equation (2) reads

(3)

where and N = [N₁N₂] P^T. Clearly, when the m × r matrix N₁ is regular (m = r and det N₁ ≠ 0), the relation between ν_dep and ν_indep vectors can be computed directly:

(4)

However, for many metabolic networks the stoichiometric matrix N may contain linearly dependent rows (r < m). In addition, ν_indep or P are not known in advance.

In the following we consider two methods based on GJE and SVD procedures that solve Equation (2) for the relation between dependent and independent flux variables: The general solution is written as:

(5a)

(5b)

We identify R as a kernel of the steady state solution where the columns are flux basis vectors and ν_indep are flux coordinates.

Solving the steady state problem via GJE

Solving the steady state problem via GJE is based on transforming the stoichiometric matrix N to a row-echelon form N^GJE where all columns corresponding to dependent flux variables would have exactly one nonzero element and Equation (5) can be easily composed ( is identity matrix and hence ). The column permutation matrix P is constructed during the GJE process while applying the leading row and column selection rules (pivot element selection). One of the advantage of using GJE is that it allows one to influence the pivot element selection rules so that a preferred flux basis for the system will be obtained. If the selected flux variables cannot form a basis, the routine will move one or more of the preselected independent variables to become dependent.

Note that in numerical GJE algorithms the typical leading row and column selection rule consists of choosing a pivot element with largest absolute value for maximal numerical stability. Symbolic GJE algorithms that calculate in fractions avoid numerical rounding errors and can implement more optimal selection rules that take into account the sparsity of the system. In SympyCore [11] the leading row and column selection rule consists of choosing such a pivot element that minimizes the number of row operations for minimal computation time.

Solving the steady state problem via SVD

Solving the steady state problem via SVD is based on decomposing the stoichiometric matrix N into a dot product of three matrices:

(6)

where u, v = [V_imV_ker] are orthogonal matrices and σ is a diagonal matrix with nonzero values on the diagonal. The solution to the steady state Equation (2) is

(7)

where α is a f vector of arbitrary parameters. Note that the SVD approach does not provide a numerically reliable and efficient way to determine the vectors of dependent and independent flux variables and in the following we use these in the form of the permutation matrix P found from the GJE approach:

(8)

which gives Equation (5b):

(9)

where V_indep is a regular f × f matrix.

Processing and analysis of metabolic networks

SBML models of metabolic networks were obtained from the BiGG database [42]. During the parsing all floating point numbers were converted to fractional numbers. All species that did not participate in any reactions were excluded. Species that are appear as both a reactant and product, i.e. in polymerization reactions, were removed from the list of reactants, and an additional reaction transporting this species across the system boundary was added.

Each metabolic network was transformed into open form using the following rule: if a species participated in exactly one reaction, a reaction transporting this species across the system boundary was added. As an alternative rule used in the example yeast network, if all transport reactions out of the system are known, then transformation to open form is accomplished by removing rows for the species that are external to the system.

Both of these approaches result in equivalent steady state solutions because adding extra reactions extends linear pathways that each contain a species that exits the system. Both approaches were applied to the example yeast network: external species were removed to calculate the flux distribution in Figure 3 and additional file 2: yeast_example.pdf while the algorithm to add extra transport reactions was used to calculate the values in Table 1.

Composing the example yeast network

The example yeast network given in Figure 3 was manually composed for analyzing carbon isotope dynamics, and thus excludes metabolites that do not participate in carbon rearrangement, i.e. cofactors. To simplify the model, Carbon 3 of histidine (by InChI carbon number) was assumed to come from bicarbonate, and not Carbon 2 of ATP. Similarly, Carbon 1 of methionine was also assumed to come from bicarbonate, and not 5-Methyltetrahydropteroyltri-L-glutamate. In addition, the glyoxylate cycle and thus the third pathway for producing glycine was removed. All relevant details of the network including metabolite abbreviations, reaction definitions, the steady state solution, and substituted flux values used to constrain the system are given in additional file 2: yeast_example.pdf.

The example yeast network makes use of fictitious metabolites that link the stoichiometry of coupled reactions. The three pentose phosphate pathway reactions are broken into two parts each linked with a fictitious metabolite that represents the carbon skeleton that is broken off of one metabolite in the first step and transferred to the next. In the additional file 2: yeast_example.pdf fictitious metabolite names start with either a capital X, Y, or Z, followed by a lower case Greek number indicating the number of carbons they contain followed by a section indicating their use. This latter section is either the yeast enzyme they participate in, the code for the metabolite they are derived from, or GOG indicating the transfer of glutamate to 2-Oxoglutarate.

Applying constraints to the steady state solution

The GJE routine provides a flux based coordinate system to describe the steady state flux space while SVD provides an orthogonal coordinate system. When specifying a flux value that is part of a flux coordinate system, one dimension from the steady state flux space is removed.

Many different sets of independent flux variables can form a coordinate system for the steady state flux space. The GJE routine allows the researcher to specify which flux variables forms a flux coordinate system and thus can match the choice of coordinate system with the experimental data available. The basis vectors formed from a flux coordinate system are often sparse and tend to span connected portions of the metabolism.

Let us assume a relation between dependent and independent flux variables as given in Equation (5). In addition to that, let us assume some constraining knowledge about the dependent variables, for example, the flux positivity for irreversible reactions: for some i ∈ [1; r]. The problem being solved is how the constraints on ν_dep constrain the independent flux variables ν_indep. The goal is to determine how the steady state flux space is bounded. This is useful for many techniques used to analyze the properties of metabolic networks, for example in optimization procedures that must scan the steady state flux space while avoiding regions that are not feasible [32].

To find the constraints for ν_indep, we set up the following system:

(10a)

(10b)

(10c)

where g × f matrix G and g vector b define g measured data constraints for ν_indep; q × r matrix Q that defines q positivity constraints for ν_dep. The system in Equation (10) defines a convex polytope and due to the constraining parts it is redundant. The redundancy can be removed by using the following geometric computation algorithm: solve the vertex enumeration problem for the convex polytope defined by Equation (10) and then using the obtained vertexes and rays solve the facet enumeration problem. The solution to the facet enumeration problem is a set of inequalities that has no redundancies and defines the same convex polytope as Equation (10). Note that the intermediate result of the vertex enumeration problem (polytope vertexes and rays) provides convenient information to volume scanning applications.

Computational software and error analysis

The GJE results of this paper are obtained using a Python package SympyCore [11] that implements both memory and processor efficient sparse matrix structures and manipulation algorithms. For solving the steady state problem we are using the symbolic matrix object method get_gauss_jordan_elimination_operations that allows one to specify the list of preferred leading columns (that is, the preferred list of dependent flux variables) for the GJE algorithm and after applying the GJE process the method returns a matrix object that is in row-echelon form. In addition to that, the method returns also a list of all applied row operations that can be later efficiently applied to other matrix objects. This feature is especially useful for adding extra columns to a stoichiometric matrix and then applying GJE process without the need to recompute the row-echelon form of the original matrix. One could use this to add transport reactions to a metabolic network during the constraint process.

The SVD results of this paper were obtained using a Python package NumPy [43] that provides a function numpy.linalg.svd for computing SVD of an array object. NumPy was built with LAPACK and ATLAS (version 3.8.3) libraries that provide a state-of-the-art routine (dgesdd) for computing SVD.

Since the results obtained with the symbolic GJE routine are correct and the results of the numerical SVD routine contain numerical rounding errors then in the error analysis we are using maximal relative flux error

(11)

where and are matrix elements in Equation (5) obtained with GJE and SVD routines, respectively. Note that ε_SVD characterizes relative errors in dependent flux variables introduced by the numerical SVD routine.

For solving vertex and facet enumeration problems we use a Python package pycddlib [44], a wrapper of the cddlib (version 094g) that implements the double description method [40].

The Python scripts used for computing the results are available in SympyCore [45]. The performance timings were obtained on a Ubuntu Linux dual-core (AMD Phenom(tm) II X2 550) computer with 4GB RAM.