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MBW:The (Right) Null Space of the Stoichiometric Matrix

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This wiki page is a summary of Chapter 9 in the textbook Systems Biology – Properties of Reconstructed Networks [1].

Cells have the ability to convert one chemical compound to another through a series of reaction steps. These reactions, when linked together, form metabolic pathways that enable the cell to perform a variety of functions, from energy production to routine maintenance. A set of pathways forms a network, which can be studied to uncover system behavior and guide genome engineering efforts.

The chapter focuses on an analysis of the right null space of the stoichiometric matrix, here forth referred to as the “null space” and the “S matrix”, respectively. The null space gives important insight into the steady state behavior of a biochemical network, and is an integral part of common flux analysis methods, like Flux Balance Analysis, Extreme Pathway Analysis, or Elementary Mode Analysis.

Please refer to the textbook for more in-depth discussions regarding other fundamental subspaces of S, properties of reconstructed networks, and interpretation of solutions.

Key Points

  • The null space of the stoichiometric matrix can be described by a set of linear basis vectors, r_{i}. The number of linear basis required to span a space is equal to the number of reactions in the network minus the rank of the stoichiometric matrix.
  • The flux solution given by the linear basis might not always be biologically relevant. It is important to pay attention to reaction reversibility when analyzing the linear basis. Negative values in the vector correspond to a flux in the reverse direction.
  • The null space can be described by convex basis vectors, p_{i}, which consider non-negative fluxes, or elementary reactions, only. The convex basis forms the boundaries of a flux cone, inside which all possible steady state flux solutions are contained. The number of convex basis vectors can be greater than or equal to the number of linear basis vectors.
  • The p_{i}s can be mapped on to a metabolic network, and represent the extreme pathways corresponding to specific cellular function.
  • Extreme pathways can be classified as Type I, II, or III, based on the function that the pathway serves.
  • Elementary mode analysis is related to extreme pathway analysis, though elementary modes do not need to be linearly independent. There may be many more elementary modes than extreme pathway.
  • Software is available to assist with extreme pathway analysis.

Introduction to S and the Null Space

The Stoichiometric Matrix


The S matrix has one row for each metabolite (chemical species) participating in a reconstructed biochemical network, and one column for every reaction (or flux).

By constructing an S matrix, a system-wide mass balance can be constructed to describe the relative change in metabolite concentration versus time:



  • dX/dt = column vector describing the time rate of change of each metabolite concentration. If dX/dt = zeros(size(dX/dt)), a steady state mass balance is obtained.
  • S = stoichiometric matrix
  • v = column vector describing the flux through each reaction

The Null Space of S

The null space of S contains the steady state flux distributions that the network could achieve, so all vectors vss for which:


The null space contains a set of basis vectors (ri) that, if concatenated to form the columns of a matrix R, will satisfy the expression:


A weighted sum of the basis vectors will give a steady state flux distribution:

v_{{ss}}=\sum w_{i}*r_{i}

Each basis vector is orthogonal to all rows of the S matrix, so the system is said to be balanced. This steady state solution is of interest, since cells tend to maintain states of homeostasis.

The size of the null space of S is called the nullity of S, and is equal to the difference between the number of fluxes in the network (n) and the rank of S (r).


Choosing a Basis

Linear Basis

A linear system will have infinite basis vectors, so the challenge is in realizing which basis are biologically relevant. For instance, reactions that are irreversible should not achieve negative flux values, which indicate that the reaction is proceeding in a thermodynamically unfavorable direction. To illustrate these concepts, consider this network with six reactions and four metabolites:


The stoichiometric matrix for this system is:


The null space can then be found by solving for a flux distribution vector, v that satisfies:


Linear basis vectors can be found using Gaussian elimination or Singular Value Decomposition (SVD), among other techniques. Section 9.2 of the textbook demonstrates one method, and here MATLAB’s null() function is used. The null function returns the basis calculated from the SVD of S.

There number of basis vectors is equal to the dimension of the null space. For this example, there are six fluxes and the rank(S) is four, so we will find 6 - 4 = 2 basis vectors.

%Define the S matrix:
S = [1 -1 0 0 -1 0;0 1 -1 0 0 0;0 0 1 -1 0 1;0 0 0 0 1 -1];
%Use the null function to find basis:
R = null(S,'r');
%Any steady state flux is a combination of these basis vectors.
%v = w1*r1 + w2*r2, 
%where w1 and w2 are constant weights, assigned arbitrarily
w = [2, 1];
vss = w(1)*R(:,1) + w(2)*R(:,2);
solution = S*vss;

disp('S*vss = 0 for basis vectors.')
disp('Steady state solution:')

Output shows that the basis found by null() does satisfy S*vss = 0. This hold for any weighing function (w) where -∞ < w < ∞ . Modifying w results in unique flux distribution values. There are many methods for determining the weight vector, including those based on thermodynamic properties and those derived from experimentally determined flux values [2]. For this sample network, using the code above, the steady state flux distribution (vss) and basis vectors (columns in R) are calculated to be:

v={\begin{bmatrix}2\\1\\1\\2\\1\\1\end{bmatrix}} , R={\begin{bmatrix}1&0\\1&-1\\1&-1\\1&0\\0&1\\0&1\end{bmatrix}}

Though the criteria that S*vss = 0 holds, note that one of the basis vectors contains negative values for v2 and v3. If these reactions are irreversible, this solution would not be biologically relevant.

Convex Basis

Convex analysis can be used to avoid the issue of negative flux values for irreversible reactions. In addition to the requirement that S*vss = 0, upper and lower bounds are imposed on the flux solutions:

0\leq v_{i}\leq v_{{i,max}}

A steady state flux vector solution can be obtained from:

v_{{ss}}=\sum _{{i=1}}\alpha _{i}*p_{i}

Where \alpha , the weights, are positive and finite, and the p_{i} are the convex basis vectors.. In contrast to the linear basis, where the number of ri is equal to the dim(Nul S), the number of convex basis vectors can be greater than (n-r). This indicates that there are multiple combinations of p_{i}s that will lead to the same flux solution.

The relationship between allowable flux values can be depicted as a hyperbox[3], where the S*v = 0 scenario is one hyperplane intersecting the box. . The image below depicts a hyperbox in a network with three reactions, v1, v2, and v3. The null space is constrained by the maximum values for these fluxes.


Extreme Pathways

Flux Cone


The convex basis vectors (p_{i}s) can be thought of as the boundaries of a flux cone, the extreme values that form the edges of the null space. The figure on the right shows a flux cone for a network with four reactions, where the point A illustrates a feasible solution inside the cone.

The p_{i}s are unique, and their values correspond to the amount of flux through a particular reaction. If the convex basis values are drawn on a metabolic network map, they are referred to as extreme pathways.

Every biologically achievable steady state flux solution can be obtained from a linear combination of these basis vectors, so all biologically achievable steady state flux solutions fall on or inside the cone. To mathematically represent all points inside of the flux cone of a given network, use the equation:

C=v:v=\sum _{{i=1}}^{{p}}\alpha _{i}*p_{i}\geq 0 for all i


  • C = convex cone defining the boundaries of the steady state flux vectors
  • v = flux vector
  • p = number of extreme pathways
  • \alpha _{i}*p_{i} = weights multiplied by the extreme pathways

Classification of Extreme Pathways

A classification system for extreme pathways has been defined, which groups the pathways based on the type of flux vectors contained in the pathway[4].

  • Type I: Primary Systemic Pathways, the major contributors to metabolism. Exchange fluxes for both primary metabolites and currency metabolites in these pathways are nonzero. These pathways are of the greatest interest, as they can incorporate transport of metabolites into and out of the cell, cofactor requirements, and balances between cellular functions.
  • Type II: Futile Cycles, involve only the exchange of currency (or carrier) metabolites, like ATP, NADH, and NADPH. These pathways relate to cellular energy and redox needs.
  • Type III: Internal Cycles, where all of the reactions take place solely inside the cell, without any contribution from external metabolites. The net flux through a Type III pathway is always zero, and in a closed system, all pathways are Type III.

Computation of Extreme Pathways

ExPa is one open source software that is available to assist with the computation of extreme pathways for small networks. Briefly, the iterative algorithm utilized by ExPa uses elementary row operations to determine convex basis vectors[5]. The software tends to work well for smaller networks, but as the size of the problem increases, the number of extreme pathways increases faster, since p\geq n-r.

If the number of extreme pathways is larger than n – r, the p_{i} do not point to unique solutions, and it is possible to study only a subset of the extreme pathways.

Elementary Modes vs Extreme Pathways

Elementary Mode Analysis and Extreme Pathway Analysis are closely related methods for uncovering important pathways in cellular metabolism. When reactions in a network are reversible, there will be more extreme pathways than elementary modes, as only linearly independent pathways are included in the extreme pathway subset. This concept is illustrated in the figure below, adapted from Figure 9.13 in the textbook. For this network, there are four elementary modes, but only three extreme pathways, since EM4 is a linear combination of EM1 and EM3. Emvseps.png


Convex analysis has been used to study flux in reaction networks for more than twenty years. Some important years in extreme pathway history include:

  • 1988: B.L. Clarke presents stoichiometric network analysis (SNA) and extreme currents to describe stability in system of inorganic chemical reactions[6].
  • 1994: S. Schuster and C. Hilgetag defined elementary modes, sets of flux vectors that can be transformed into one another simply by positive scalar multiplication. Different elementary modes were said to represent unique cellular functions. [7].
  • 2000: Schilling et. al. introduce extreme pathways, the irreducible set of elementary modes. [4]
  • 2002: Papin et al. apply extreme pathway analysis is applied to genome scale problems [8]
  • 2011: Extreme pathway analysis and elementary mode analysis continue to be relevant. Orman et. al study liver metabolism using pathway analysis techniques[9].


  1. Palsson, Bernhard Ø. (2006). Systems Biology - Properties of Reconstructed Networks. Cambridge University Press.
  2. Cong T. Trinh, Aaron Wlaschin, Friedrich Srienc. Elementary Mode Analysis: A Useful Metabolic Pathway Analysis Tool for Characterizing Cellular Metabolism. Appl Microbiol Biotechnol. 2009 January; 81(5): 813–826.
  3. Nathan D. Price, Jan Schellenberger, Bernhard O. Palsson, Uniform Sampling of Steady-State Flux Spaces: Means to Design Experiments and to Interpret Enzymopathies, Biophysical Journal, Volume 87, Issue 4, October 2004, Pages 2172-2186, ISSN 0006-3495, 10.1529/biophysj.104.043000. (
  4. 4.0 4.1 Schilling CH, Letscher D, Palsson BØ. 2000. Theory for the systemic definition of metabolic pathways and their use in interpreting metabolic function from a pathway-oriented perspective. J Theor Biol 203:229–248.
  5. S.L. Bell and B.O. Palsson. ExPa: A program for calculating extreme pathways in biochemical reaction networks. Bioinformatics, 21:1739–1740, 2005.
  6. Clarke BL. Stoichiometric network analysis. Cell Biophys. 1988;12:237.
  7. S. Schuster and C. Hilgetag. On elementary flux modes in biochemical reaction systems at steady state. Journal of Biologicle Systems, 2:165–182, 1994.
  8. J.A. Papin, N.D. Price, and B.O. Palsson. Extreme pathway lengths and reaction participation in genome-scale metabolic networks. Genome Research, 12:1889–1900, 2002.
  9. Mehmet A. Orman, Francois Berthiaume, Ioannis P. Androulakis, Marianthi G. Ierapetritou, Pathway analysis of liver metabolism under stressed condition, Journal of Theoretical Biology, Volume 272, Issue 1, 7 March 2011, Pages 131-140, ISSN 0022-5193, 10.1016/j.jtbi.2010.11.042. (