Branched pathways

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:

: $\backslash frac\{ds\_1\}\{dt\}\; =\; v\_1\; -\; (v\_2\; +\; v\_3)$

At steady-state the rate of change of $S\_1$ is zero. This gives rise to a steady-state constraint among the branch reaction rates:

: $v\_1\; =\; v\_2\; +\; v\_3$

Such constraints are key to computational methods such as flux balance analysis.

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 $e\_1$, $e\_2$, and $e\_3$ 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, $J\_2$ are given by:

: $C^\{J\_2\}\_\{e\_1\}\; =\; \backslash frac\{\backslash varepsilon\_2\}\{\backslash varepsilon\_2\; \backslash alpha\; +\; \backslash varepsilon\_3\; (1-\backslash alpha)\; -\; \backslash varepsilon\_1\}$

: $C^\{J\_2\}\_\{e\_2\}\; =\; \backslash frac\{\backslash varepsilon\_3\; (1-\backslash alpha)\; -\; \backslash varepsilon\_1\}\{\backslash varepsilon\_2\; \backslash alpha\; +\; \backslash varepsilon\_3\; (1-\backslash alpha)\; -\; \backslash varepsilon\_1\}$

: $C^\{J\_2\}\_\{e\_3\}\; =\; \backslash frac\{-\; \backslash varepsilon\_2\; (1-\backslash alpha)\}\{\backslash varepsilon\_2\; \backslash alpha\; +\; \backslash varepsilon\_3\; (1-\backslash alpha)\; -\; \backslash varepsilon\_1\}$

where $\backslash alpha$ is the fraction of flux going through the upper arm, $J\_2$, and $1-\backslash alpha$ the fraction going through the lower arm, $J\_3$. $\backslash varepsilon\_1,\; \backslash varepsilon\_2,$ and $\backslash varepsilon\_3$ are the elasticities for $s\_1$ with respect to $v\_1,\; v\_2,$ and $v\_3$ respectively.

For the following analysis, the flux $J\_2$ 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 $J\_2$ or most of the flux goes through the lower branch, $J\_3$. 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 $J\_3$

If most of the flux goes through $J\_3$, then $\backslash alpha\; \backslash rightarrow\; 0$ and $1\; -\; \backslash alpha\; \backslash rightarrow\; 1$ (condition (b) in the figure), the flux control coefficients for $J\_2$ with respect to $e\_2$ and $e\_3$ can be written:

: $C^\{J\_2\}\_\{e\_2\}\; \backslash rightarrow\; 1$

: $C^\{J\_2\}\_\{e\_3\}\; \backslash rightarrow\; \backslash frac\{\backslash varepsilon\_2\}\{\backslash varepsilon\_1\; -\; \backslash varepsilon\_3\}$

File:BranchPointEffect.png

That is, $e\_2$ acquires proportional influence over its own flux, $J\_2$. Since $J\_2$ only carries a very small amount of flux, any changes in $e\_2$ will have little effect on $S$. Hence the flux through $e\_2$ is almost entirely governed by the activity of $e\_2$. Because of the flux summation theorem and the fact that $C^\{J\_2\}\_\{e\_2\}\; =\; 1$, it means that the remaining two coefficients must be equal and opposite in value. Since $C^\{J\_2\}\_\{e\_1\}$ is positive, $C^\{J\_2\}\_\{e\_3\}$ 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 $C^\{J\_2\}\_\{e\_3\}$and $C^\{J\_2\}\_\{e\_1\}$ 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.{{cite journal |last1=Kacser |first1=H. |title=The control of enzyme systems in vivo : Elasticity analysis of the steady state |journal=Biochemical Society Transactions |date=1 January 1983 |volume=11 |issue=1 |pages=35–40 |doi=10.1042/bst0110035|pmid=6825913 }} This has been termed the branchpoint effect by some in the literature.{{Cite journal |last=LaPorte |first=D C |last2=Walsh |first2=K |last3=Koshland |first3=D E |date=November 1984 |title=The branch point effect. Ultrasensitivity and subsensitivity to metabolic control. |url=https://linkinghub.elsevier.com/retrieve/pii/S002192581889857X |journal=Journal of Biological Chemistry |language=en |volume=259 |issue=22 |pages=14068–14075 |doi=10.1016/S0021-9258(18)89857-X|doi-access=free }}

File:Branch ccf.png

The following branch pathway model (in antimony format) illustrates the case $J\_1$ and $J\_3$ have very high flux control and step J2 has proportional control.

J1: $Xo -> S1; e1*k1*Xo

J2: S1 ->; e2*k3*S1/(Km1 + S1)

J3: S1 ->; e3*k4*S1/(Km2 + S1)

k1 = 2.5;

k3 = 5.9; k4 = 20.75

Km1 = 4; Km2 = 0.02

Xo =5;

e1 = 1; e2 = 1; e3 = 1

A simulation of this model yields the following values for the flux control coefficients with respect to flux $J\_2$

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.

1. Define the fractional flux through $J\_2$ and $J\_3$ as $\backslash alpha\; =\; J\_2/J\_1$ and $1\; -\; \backslash alpha\; =\; J\_3/J\_1$ respectively.
2. Increase $e\_2$ by $\backslash delta\; e\_2$. This will decrease $S\_1$ and increase $J\_1$ through relief of product inhibition.{{Cite journal |last1=Liu |first1=Yan |last2=Zhang |first2=Fan |last3=Jiang |first3=Ling |last4=Perry |first4=J. Jefferson P. |last5=Zhao |first5=Zhihe |last6=Liao |first6=Jiayu |date=2021-12-15 |title=Product inhibition kinetics determinations - Substrate interaction affinity and enzymatic kinetics using one quantitative FRET assay |url=https://www.sciencedirect.com/science/article/pii/S014181302102376X |journal=International Journal of Biological Macromolecules |language=en |volume=193 |issue=Pt B |pages=1481–1487 |doi=10.1016/j.ijbiomac.2021.10.211 |pmid=34780893 |s2cid=244107621 |issn=0141-8130|url-access=subscription }}
3. Make a compensatory change in $J\_1$ by decreasing $e\_1$ such that $S\_1$ is restored to its original concentration (hence $\backslash delta\; S\_1\; =\; 0$).
4. Since $e\_1$ and $S\_1$ have not changed, $\backslash delta\; J\_1\; =\; 0$.

Following these assumptions two sets of equations can be derived: the flux branch point theorems and the concentration branch point theorems.{{Cite book |last=Sauro |first=Herbert |title=Systems Biology: An Introduction to Metabolic Control Analysis |publisher=Ambrosius Publishing |year=2018 |isbn=978-0-9824773-6-6 |edition=1st |pages=115–122}}

From these assumptions, the following system equation can be produced:

: $C^\{J\_1\}\_\{e\_2\}\; \backslash frac\{\backslash delta\; e\_2\}\{e\_2\}\; +\; C^\{J\_1\}\_\{e\_3\}\; \backslash frac\{\backslash delta\; e\_3\}\{e\_3\}\; =\; \backslash frac\{\backslash delta\; J\_1\}\{J\_1\}\; =\; 0$

Because $\backslash delta\; S\_1\; =\; 0$ and, assuming that the flux rates are directly related to the enzyme concentration thus, the elasticities, $\backslash varepsilon^\{v\}\_\{e\_i\}$, equal one, the local equations are:

: $\backslash frac\{\backslash delta\; v\_2\}\{v\_2\}\; =\; \backslash frac\{\backslash delta\; e\_2\}\{e\_2\}$

: $\backslash frac\{\backslash delta\; v\_3\}\{v\_3\}\; =\; \backslash frac\{\backslash delta\; e\_3\}\{e\_3\}$

Substituting $\backslash frac\{\backslash delta\; v\_i\}\{v\_i\}$ for $\backslash frac\{\backslash delta\; e\_i\}\{e\_i\}$ in the system equation results in:

: $C^\{J\_1\}\_\{e\_2\}\; \backslash frac\{\backslash delta\; v\_2\}\{v\_2\}\; +\; C^\{J\_1\}\_\{e\_3\}\; \backslash frac\{\backslash delta\; v\_3\}\{v\_3\}\; =\; 0$

Conservation of mass dictates $\backslash delta\; J\_1\; =\; \backslash delta\; J\_2\; +\; \backslash delta\; J\_3$ since $\backslash delta\; J\_1\; =\; 0$ then $\backslash delta\; v\_2\; =\; -\; \backslash delta\; v\_3$. Substitution eliminates the $\backslash delta\; v\_3$ term from the system equation:

: $C^\{J\_1\}\_\{e\_2\}\; \backslash frac\{\backslash delta\; v\_2\}\{v\_2\}\; -\; C^\{J\_1\}\_\{e\_3\}\; \backslash frac\{\backslash delta\; v\_2\}\{v\_3\}\; =\; 0$

Dividing out $\backslash frac\{\backslash delta\; v\_2\}\{v\_2\}$ results in:

: $C^\{J\_1\}\_\{e\_2\}\; -\; C^\{J\_1\}\_\{e\_3\}\; \backslash frac\{v\_2\}\{v\_3\}\; =\; 0$

: $v\_2$ and $v\_3$ can be substituted by the fractional rates giving:

: $C^\{J\_1\}\_\{e\_2\}\; -\; C^\{J\_1\}\_\{e\_3\}\; \backslash frac\{\backslash alpha\}\{1-\backslash alpha\}\; =\; 0$

Rearrangement yields the final form of the first flux branch point theorem:

: $C^\{J\_1\}\_\{e\_2\}(1-\backslash alpha)\; -\; C^\{J\_1\}\_\{e\_3\}\; \{\backslash alpha\}\; =\; 0$

Similar derivations result in two more flux branch point theorems and the three concentration branch point theorems.

: $C^\{J\_1\}\_\{e\_2\}\; (1-\backslash alpha)\; -\; C^\{J\_1\}\_\{e\_3\}\; (\backslash alpha)\; =\; 0$

: $C^\{J\_2\}\_\{e\_1\}\; (1-\backslash alpha)\; +\; C^\{J\_2\}\_\{e\_3\}\; (\backslash alpha)\; =\; 0$

: $C^\{J\_3\}\_\{e\_1\}\; (\backslash alpha)\; +\; C^\{J\_3\}\_\{e\_2\}\; =\; 0$

: $C^\{S\_1\}\_\{e\_2\}\; (1-\backslash alpha)\; +\; C^\{S\_1\}\_\{e\_3\}\; (\backslash alpha)\; =\; 0$

: $C^\{S\_1\}\_\{e\_1\}\; (1-\backslash alpha)\; +\; C^\{S\_1\}\_\{e\_3\}\; =\; 0$

: $C^\{S\_1\}\_\{e\_1\}\; (\backslash alpha)\; +\; C^\{S\_1\}\_\{e\_2\}\; =\; 0$

Following the flux summation theorem{{Cite journal |last=Agutter |first=Paul S. |date=2008-10-21 |title=The flux-summation theorem and the 'evolution of dominance' |url=https://www.sciencedirect.com/science/article/pii/S0022519308003937 |journal=Journal of Theoretical Biology |language=en |volume=254 |issue=4 |pages=821–825 |doi=10.1016/j.jtbi.2008.07.027 |issn=0022-5193 |pmid=18706429|url-access=subscription }} and the connectivity theorem{{Cite journal |last1=Kacser |first1=H. |last2=Burns |first2=J. A. |date=1973 |title=The control of flux |url=https://pubmed.ncbi.nlm.nih.gov/4148886 |journal=Symposia of the Society for Experimental Biology |volume=27 |pages=65–104 |issn=0081-1386 |pmid=4148886}} the following system of equations can be produced for the simple pathway.{{Cite journal |last1=Fell |first1=David A. |last2=Sauro |first2=Herbert M. |year=1985 |title=Metabolic control and its analysis. Additional relationships between elasticities and control coefficients |url=https://onlinelibrary.wiley.com/doi/10.1111/j.1432-1033.1985.tb08876.x |journal=European Journal of Biochemistry |language=en |publication-date=May 1985 |volume=148 |issue=3 |pages=555–561 |doi=10.1111/j.1432-1033.1985.tb08876.x |issn=0014-2956 |pmid=3996393 |doi-access=free}}

: $C^\{J\_1\}\_\{e\_1\}\; +\; C^\{J\_1\}\_\{e\_2\}\; +\; C^\{J\_1\}\_\{e\_3\}\; =\; 1$

: $C^\{J\_2\}\_\{e\_1\}\; +\; C^\{J\_2\}\_\{e\_2\}\; +\; C^\{J\_2\}\_\{e\_3\}\; =\; 1$

: $C^\{J\_3\}\_\{e\_1\}\; +\; C^\{J\_3\}\_\{e\_2\}\; +\; C^\{J\_3\}\_\{e\_3\}\; =\; 1$

: $C^\{J\_1\}\_\{e\_1\}\; \backslash varepsilon^\{v\_1\}\_s\; +\; C^\{J\_1\}\_\{e\_2\}\; \backslash varepsilon^\{v\_2\}\_s\; +$

C^{J_1}_{e_3} \varepsilon^{v_3}_s = 0

: $C^\{J\_2\}\_\{e\_1\}\; \backslash varepsilon^\{v\_1\}\_s\; +\; C^\{J\_2\}\_\{e\_2\}\; \backslash varepsilon^\{v\_2\}\_s\; +$

C^{J_2}_{e_3} \varepsilon^{v_3}_s = 0

: $C^\{J\_3\}\_\{e\_1\}\; \backslash varepsilon^\{v\_1\}\_s\; +\; C^\{J\_3\}\_\{e\_2\}\; \backslash varepsilon^\{v\_2\}\_s\; +$

C^{J_3}_{e_3} \varepsilon^{v_3}_s = 0

Using these theorems plus flux summation and connectivity theorems values for the concentration and flux control coefficients can be determined using linear algebra.

: $C^\{J\_2\}\_\{e\_1\}\; =\; \backslash frac\{\backslash varepsilon\_2\}\{\backslash varepsilon\_2\; \backslash alpha\; +\; \backslash varepsilon\_3\; (1-\backslash alpha)\; -\; \backslash varepsilon\_1\}$

: $C^\{J\_2\}\_\{e\_2\}\; =\; \backslash frac\{\backslash varepsilon\_3\; (1-\backslash alpha)\; -\; \backslash varepsilon\_1\}\{\backslash varepsilon\_2\; \backslash alpha\; +\; \backslash varepsilon\_3\; (1-\backslash alpha)\; -\; \backslash varepsilon\_1\}$

: $C^\{J\_2\}\_\{e\_3\}\; =\; \backslash frac\{-\; \backslash varepsilon\_2\; (1-\backslash alpha)\}\{\backslash varepsilon\_2\; \backslash alpha\; +\; \backslash varepsilon\_3\; (1-\backslash alpha)\; -\; \backslash varepsilon\_1\}$

: $C^\{S\_1\}\_\{e\_1\}\; =\; \backslash frac\{1\}\{\backslash varepsilon\_2\; \backslash alpha\; +\; \backslash varepsilon\_3\; (1-\backslash alpha)\; -\; \backslash varepsilon\_1\}$

: $C^\{S\_1\}\_\{e\_1\}\; =\; \backslash frac\{-\; \backslash alpha\}\{\backslash varepsilon\_2\; \backslash alpha\; +\; \backslash varepsilon\_3\; (1-\backslash alpha)\; -\; \backslash varepsilon\_1\}$

: $C^\{S\_1\}\_\{e\_3\}\; =\; \backslash frac\{-(1-\backslash alpha)\}\{\backslash varepsilon\_2\; \backslash alpha\; +\; \backslash varepsilon\_3\; (1-\backslash alpha)\; -\; \backslash varepsilon\_1\}$

• Control coefficient (biochemistry)
• Elasticity coefficient
• Metabolic control analysis

Category:Metabolic pathways