
MBW:The (Right) Null Space of the Stoichiometric MatrixFrom MathBioThis 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 indepth discussions regarding other fundamental subspaces of S, properties of reconstructed networks, and interpretation of solutions. ContentsKey Points
Introduction to S and the Null SpaceThe Stoichiometric Matrix
By constructing an S matrix, a systemwide mass balance can be constructed to describe the relative change in metabolite concentration versus time:
Where:
The Null Space of SThe null space of S contains the steady state flux distributions that the network could achieve, so all vectors v_{ss} for which:
The null space contains a set of basis vectors (r_{i}) 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:
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 BasisLinear BasisA 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, 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. %MATLAB: DETERMINE BASIS %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:') disp(round(solution*1000)/1000) Output shows that the basis found by null() does satisfy S*v_{ss} = 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 (v_{ss}) and basis vectors (columns in R) are calculated to be: Though the criteria that S*v_{ss} = 0 holds, note that one of the basis vectors contains negative values for v_{2} and v_{3}. If these reactions are irreversible, this solution would not be biologically relevant. Convex BasisConvex analysis can be used to avoid the issue of negative flux values for irreversible reactions. In addition to the requirement that S*v_{ss} = 0, upper and lower bounds are imposed on the flux solutions: A steady state flux vector solution can be obtained from: Where , the weights, are positive and finite, and the are the convex basis vectors.. In contrast to the linear basis, where the number of r_{i} is equal to the dim(Nul S), the number of convex basis vectors can be greater than (nr). This indicates that there are multiple combinations of 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, v_{1}, v_{2}, and v_{3}. The null space is constrained by the maximum values for these fluxes. Extreme PathwaysFlux ConeThe convex basis vectors (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 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: for all Where:
Classification of Extreme PathwaysA classification system for extreme pathways has been defined, which groups the pathways based on the type of flux vectors contained in the pathway^{[4]}.
Computation of Extreme PathwaysExPa 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 . If the number of extreme pathways is larger than n – r, the do not point to unique solutions, and it is possible to study only a subset of the extreme pathways. Elementary Modes vs Extreme PathwaysElementary 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. HistoryConvex analysis has been used to study flux in reaction networks for more than twenty years. Some important years in extreme pathway history include:
References
