# What is the meaning and significance of extreme pathways

Can someone please explain me what extreme pathways are? I found this definition in this article:

Extreme pathways are a unique and minimal set of vectors that completely characterize the steady-state capabilities of genome-scale metabolic networks.

Now, frankly speaking, I did not understand the head or tail of this definition? What is steady state? What does it mean by steady state capabilities? Can someone please explain me in a simpler way? PS: I am a computer Science student

## 1 Answer

For any dynamic system defined by different components, the steady state is the state of the system in which the components remain constant over time. If you consider the example of growth (by cell division) the point at which the total number of cells remain constant would be the steady state. This is a point at which birth rate = death rate. Similarly for a multi-component system such as a gene and its activator, the steady state would be the point where the mRNAs and the proteins of the gene and its activator are constant. If only few components are constant then the system is said to be in a quasi-steady state.

A point to be noted that in chemical (or biochemical) systems steady state, as a term, is different from equilibrium (in physics these terms are used interchangeably). Equilibrium is a condition when, for a single reversible reaction, the forward rate = backward rate.

In mathematical terms steady state is a point where the rate of change of components = 0. In an ordinary differential equations based model:

$$\frac{d\bar{X}}{dt}=0 \\$$ where $\bar{X}$ is a vector in which each element is a component of the system. In case of the transcription example, $X(1),X(2)..$ would be mRNA, protein, activator mRNA and so on. In other words $\dfrac{dX(i)}{dt}=0$ for all $i$.

Steady state capabilities should mean the properties of the system at its steady state.

The kind of study described in the question is called metabolic flux balance analysis in which different metabolic reactions are described in the form of linear equations represented by $Sv=0$ where $v$ is the vector of all fluxes (metabolic reaction rates) $S$ is the stoichiometry matrix (which has the stoichiometry of all components in different metabolic reactions). The RHS is zero because we are evaluating the system at its steady state i.e. we are interested in finding out the condition in which the net metabolic reaction flux is 0. In other words all metabolic reactions, balance each other.

You need to read more about this. There are a lot of books on this topic. You can start with this review.

In flux balance analysis, the linear equations are overdetermined i.e. there are many solutions. People generally use linear programming (simplex algorithm) to find optimal solutions. In the space of the optimal solutions, the vertices represent extreme reactions. The entire optimal solution space can be described as a linear combination of these extreme reactions. Schematic representation of a convex cone characterized by five extreme pathways. Extreme Pathways 1–5 (EP1, EP2, EP3, EP4, and EP5) circumscribe the solution space for the three fluxes indicated (vA, vB, vC). EP4 lies in the plane formed by fluxes vA and vB. Consequently, flux vC does not participate in that extreme pathway. EP3, EP4, and EP5 are all close and represent different uses of a network to achieve a similar overall result. All points within the convex cone can be described as a non-negative linear combination of the extreme pathways .

• What is RHS? The RHS is zero  – aaaaaa Jul 16 '15 at 6:39
• @aaaaaa in S.v=0 the zero represents no net flux. – WYSIWYG Jul 16 '15 at 6:47