Branched pathways, also known as branch points (not to be confused with the mathematical branch point), are a common pattern found in metabolism. This is where an intermediate species is chemically made or transformed by multiple enzymatic processes. linear pathways only have one enzymatic reaction producing a species and one enzymatic reaction consuming the species.
Simple Branch Pathway. and are the reaction rates for each arm of the branch.
In general, a single branch may have producing branches and consuming branches. If the intermediate at the branch point is given by , then the rate of change of is given by:
At steady-state when the consumption and production rates must be equal:
A simple branched pathway has one key property related to the conservation of mass. In general, the rate of change of the branch species based on the above figure is given by:
At steady-state the rate of change of is zero. This gives rise to a steady-state constraint among the branch reaction rates:
Branched pathways have unique control properties compared to simple linear chain or cyclic pathways. These properties can be investigated using metabolic control analysis. The fluxes can be controlled by enzyme concentrations , , and respectively, described by the corresponding flux control coefficients. To do this the flux control coefficients with respect to one of the branch fluxes can be derived. The derivation is shown in a subsequent section. The flux control coefficient with respect to the upper branch flux, are given by:
where is the fraction of flux going through the upper arm, , and the fraction going through the lower arm, . and are the elasticities for with respect to and respectively.
For the following analysis, the flux will be the observed variable in response to changes in enzyme concentrations.
There are two possible extremes to consider, either most of the flux goes through the upper branch or most of the flux goes through the lower branch, . The former, depicted in panel a), is the least interesting as it converts the branch in to a simple linear pathway. Of more interest is when most of the flux goes through
If most of the flux goes through , then and (condition (b) in the figure), the flux control coefficients for with respect to and can be written:
Changes in flux control depending on whether the flux goes through the upper or lower branches. The system output is J2. If most of the flux goes through J2(a) then the pathway behaves like a simple linear change, where flux control on J3 is negligible and control is shared between J1 and J2. The other extreme is when most of the flux goes through J3 (b). This makes J2 highly sensitive to changes in J1 and J3 resulting is very high flux control, often greater than 1.0. Under these conditions the flux control at J3 is also negative since J3 can siphon off flux from J2.
That is, acquires proportional influence over its own flux, . Since only carries a very small amount of flux, any changes in will have little effect on . Hence the flux through is almost entirely governed by the activity of . Because of the flux summation theorem and the fact that , it means that the remaining two coefficients must be equal and opposite in value. Since is positive, must be negative. This also means that in this situation, there can be more than one Rate-limiting step (biochemistry) in a pathway.
Unlike a linear pathway, values for and are not bounded between zero and one. Depending on the values of the elasticities, it is possible for the control coefficients in a branched system to greatly exceed one.[3] This has been termed the branchpoint effect by some in the literature.[4]
Flux control coefficients in a branched pathway where most flux goes through . Note that step 2 has almost proportional control over while steps 1 and 3 show greater than proportional control over .
The following branch pathway model (in antimony format) illustrates the case and have very high flux control and step J2 has proportional control.
In a linear pathway, only two sets of theorems exist, the summation and connectivity theorems. Branched pathways have an additional set of branch centric summation theorems. When combined with the connectivity theorems and the summation theorem, it is possible to derive the control equations shown in the previous section. The deviation of the branch point theorems is as follows.
Define the fractional flux through and as and respectively.
Increase by . This will decrease and increase through relief of product inhibition.[5]
Make a compensatory change in by decreasing such that is restored to its original concentration (hence ).
Since and have not changed, .
Following these assumptions two sets of equations can be derived: the flux branch point theorems and the concentration branch point theorems.[6]
From these assumptions, the following system equation can be produced:
Because and, assuming that the flux rates are directly related to the enzyme concentration thus, the elasticities, , equal one, the local equations are:
Substituting for in the system equation results in:
Conservation of mass dictates since then . Substitution eliminates the term from the system equation:
Dividing out results in:
and can be substituted by the fractional rates giving:
Rearrangement yields the final form of the first flux branch point theorem:[6]
Similar derivations result in two more flux branch point theorems and the three concentration branch point theorems.
Using these theorems plus flux summation and connectivity theorems values for the concentration and flux control coefficients can be determined using linear algebra.[6]
^Kacser, H. (1 January 1983). "The control of enzyme systems in vivo : Elasticity analysis of the steady state". Biochemical Society Transactions. 11 (1): 35–40. doi:10.1042/bst0110035. PMID6825913.
^ abcSauro, Herbert (2018). Systems Biology: An Introduction to Metabolic Control Analysis (1st ed.). Ambrosius Publishing. pp. 115–122. ISBN978-0-9824773-6-6.