Branched pathways

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. This is contrasted by linear pathways, which only have one enzymatic reaction producing a species and one enzymatic reaction consuming the species.

Branched pathways are present in numerous metabolic reactions, including glycolysis, and the synthesis of lysine, glutamine, and penicillin.[1] Further examples of branched pathways can be found in the production of aromatic amino acids.[2]

Simple Branch Pathway

In general, a single branch may have $b$ producing branches and $d$ consuming branches. If the intermediate at the branch point is given by $s_{i}$ , then the rate of change of $s_{i}$ is given by:

$\sum _{i=1}^{b}v_{i}-\sum _{j=1}^{d}v_{j}={\frac {ds_{i}}{dt}}$

At steady-state when $ds_{i}/dt=0$ the consumption and production rates must be equal:

$\sum _{i=1}^{b}v_{i}=\sum _{j=1}^{d}v_{j}$

One way to investigate the properties of a branched pathway is to carry out computer simulations or to look at the control coefficients for flux and species concentrations using metabolic control analysis.

Simple branch pathway

Shown to the right is a simple branched pathway with one producing reaction and two consuming reactions. If we assume the pathway is at steady state then conservation of mass means that:

$J_{1}=J_{2}+J_{3}$

The fluxes can be controlled by enzyme concentrations $e_{1}$ , $e_{2}$ , and $e_{3}$ respectively, defining the flux control coefficients.

Following the flux summation theorem[3] and the connectivity theorem[4] the following system of equations can be produced for the simple pathway.[5]

$C_{e_{1}}^{J_{1}}+C_{e_{2}}^{J_{1}}+C_{e_{3}}^{J_{1}}=1$

$C_{e_{1}}^{J_{2}}+C_{e_{2}}^{J_{2}}+C_{e_{3}}^{J_{2}}=1$

$C_{e_{1}}^{J_{3}}+C_{e_{2}}^{J_{3}}+C_{e_{3}}^{J_{3}}=1$

$C_{e_{1}}^{J_{1}}\varepsilon _{s}^{v_{1}}+C_{e_{2}}^{J_{1}}\varepsilon _{s}^{v_{2}}+C_{e_{3}}^{J_{1}}\varepsilon _{s}^{v_{3}}=0$

$C_{e_{1}}^{J_{2}}\varepsilon _{s}^{v_{1}}+C_{e_{2}}^{J_{2}}\varepsilon _{s}^{v_{2}}+C_{e_{3}}^{J_{2}}\varepsilon _{s}^{v_{3}}=0$

$C_{e_{1}}^{J_{3}}\varepsilon _{s}^{v_{1}}+C_{e_{2}}^{J_{3}}\varepsilon _{s}^{v_{2}}+C_{e_{3}}^{J_{3}}\varepsilon _{s}^{v_{3}}=0$

Equations can be paired off, but with three unknown flux control coefficients the system of equations cannot be solved. To produce the third equation the steady state concentration thought experiment is used.

$C_{e_{1}}^{J_{1}}+C_{e_{2}}^{J_{1}}+C_{e_{3}}^{J_{1}}=1$

$C_{e_{1}}^{J_{1}}\varepsilon _{s}^{v_{1}}+C_{e_{2}}^{J_{1}}\varepsilon _{s}^{v_{2}}+C_{e_{3}}^{J_{1}}\varepsilon _{s}^{v_{3}}=0$

Steady State Concentration Thought Experiment

The steady-state concentration thought experiment follows a series of steps.

Define the fractional flux through $J_{2}$ and $J_{3}$ as $\alpha =J_{2}/J_{1}$ and $1-\alpha =J_{3}/J_{1}$ respectively.
Increase $e_{2}$ by $\delta e_{2}$ . This will decrease $S_{1}$ and increase $J_{1}$ through relief of product inhibition.[6]
Make a compensatory change in $J_{1}$ by decreasing $e_{1}$ such that $S_{1}$ is restored to its original concentration ( $\delta S_{1}=0$ ).
Since $e_{1}$ and $S_{1}$ have not changed, $\delta J_{1}=0$ .

Following these assumptions two sets of equations are produced. The flux branch point theorems and the concentration branch point theorems.[7]

Derivation

From these assumptions the following system equation can be produced:

$C_{e_{2}}^{J1}{\frac {\delta e_{2}}{e_{2}}}+C_{e_{3}}^{J1}{\frac {\delta e_{3}}{e_{3}}}={\frac {\delta J_{1}}{J_{1}}}=0$

Because $\delta S_{1}=0$ and, assuming that the flux rates are directly related to the enzyme concentration thus, the elasticities, $\varepsilon _{e_{i}}^{v}$ , equal one, the local equations are:

${\frac {\delta v_{2}}{v_{2}}}={\frac {\delta e_{2}}{e_{2}}}$

${\frac {\delta v_{3}}{v_{3}}}={\frac {\delta e_{3}}{e_{3}}}$

Substituting ${\frac {\delta v_{i}}{v_{i}}}$ for ${\frac {\delta e_{i}}{e_{i}}}$ in the system equation results in:

$C_{e_{2}}^{J1}{\frac {\delta v_{2}}{v_{2}}}+C_{e_{3}}^{J1}{\frac {\delta v_{3}}{v_{3}}}=0$

Conservation of mass dictates $\delta J_{1}=\delta J_{2}+\delta J_{3}$ since $\delta J_{1}=0$ then $\delta v_{2}=-\delta v_{3}$ . Substitution eliminates the $\delta v_{3}$ term from the system equation:

$C_{e_{2}}^{J1}{\frac {\delta v_{2}}{v_{2}}}-C_{e_{3}}^{J1}{\frac {\delta v_{2}}{v_{3}}}=0$

Dividing out ${\frac {\delta v_{2}}{v_{2}}}$ results in:

$C_{e_{2}}^{J1}-C_{e_{3}}^{J1}{\frac {v_{2}}{v_{3}}}=0$

$v_{2}$ and $v_{3}$ can be substituted by the fractional rates giving:

$C_{e_{2}}^{J1}-C_{e_{3}}^{J1}{\frac {\alpha }{1-\alpha }}=0$

Rearrangement yields the final form of the first flux branch point theorem:[7]

$C_{e_{2}}^{J1}(1-\alpha )-C_{e_{3}}^{J1}{\alpha }=0$

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

Flux branch point theorems

$C_{e_{2}}^{J_{1}}(1-\alpha )-C_{e_{3}}^{J_{1}}(\alpha )=0$

$C_{e_{1}}^{J_{2}}(1-\alpha )+C_{e_{3}}^{J_{2}}(\alpha )=0$

$C_{e_{1}}^{J_{3}}(\alpha )+C_{e_{2}}^{J_{3}}=0$

Concentration branch point theorems

$C_{e_{2}}^{S1}(1-\alpha )+C_{e_{3}}^{S1}(\alpha )=0$

$C_{e_{1}}^{S1}(1-\alpha )+C_{e_{3}}^{S1}=0$

$C_{e_{1}}^{S1}(\alpha )+C_{e_{2}}^{S1}=0$

Using these theorems plus flux summation and connectivity theorems values for the concentration and flux control coefficients can be determined using linear algebra.[7]

$C_{e_{1}}^{J2}={\frac {\varepsilon _{2}}{\varepsilon _{2}\alpha +\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}}$

$C_{e_{2}}^{J2}={\frac {\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}{\varepsilon _{2}\alpha +\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}}$

$C_{e_{3}}^{J2}={\frac {-\varepsilon _{2}(1-\alpha )}{\varepsilon _{2}\alpha +\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}}$

$C_{e_{1}}^{S_{1}}={\frac {1}{\varepsilon _{2}\alpha +\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}}$

$C_{e_{1}}^{S_{1}}={\frac {-\alpha }{\varepsilon _{2}\alpha +\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}}$

$C_{e_{3}}^{S_{1}}={\frac {-(1-\alpha )}{\varepsilon _{2}\alpha +\varepsilon _{3}(1-\alpha )-\varepsilon _{1}}}$

Control properties of a branch pathway

The derivations above allow the properties of a simple branch to be investigated. For example, if most of the flux goes through $J_{3}$ , then $\alpha \rightarrow 0$ and $1-\alpha \rightarrow 1$ . Under these conditions, the flux control coefficients for $J_{3}$ with respect to $e_{2}$ and $e_{3}$ can be written:

$C_{e_{2}}^{J_{2}}\rightarrow 1$

$C_{e_{3}}^{J_{2}}\rightarrow {\frac {\varepsilon _{2}}{\varepsilon _{1}-\varepsilon _{3}}}$

That is, $e_{2}$ acquires proportional influences 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 $C_{e_{2}}^{J_{2}}=1$ , it means that the remaining two coefficients must be equal and opposite in value. Since $C_{e_{3}}^{J_{2}}$ is negative, $C_{e_{1}}^{J_{2}}$ must be positive. 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 $C_{e_{3}}^{J_{2}}$ and $C_{e_{1}}^{J_{2}}$ 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.[8][9]

References

Heijnen, J. J.; van Gulik, W. M.; Shimizu, H.; Stephanopoulos, G. (2004-10-01). "Metabolic flux control analysis of branch points: an improved approach to obtain flux control coefficients from large perturbation data". Metabolic Engineering. 6 (4): 391–400. doi:10.1016/j.ymben.2004.07.002. ISSN 1096-7176. PMID 15491867.
"W_2022_Bis2a_Igo_Reading_15". Biology LibreTexts. 2021-12-13. Retrieved 2022-12-15.
Agutter, Paul S. (2008-10-21). "The flux-summation theorem and the 'evolution of dominance'". Journal of Theoretical Biology. 254 (4): 821–825. doi:10.1016/j.jtbi.2008.07.027. ISSN 0022-5193. PMID 18706429.
Kacser, H.; Burns, J. A. (1973). "The control of flux". Symposia of the Society for Experimental Biology. 27: 65–104. ISSN 0081-1386. PMID 4148886.
Fell, David A.; Sauro, Herbert M. (1985). "Metabolic control and its analysis. Additional relationships between elasticities and control coefficients". European Journal of Biochemistry (published May 1985). 148 (3): 555–561. doi:10.1111/j.1432-1033.1985.tb08876.x. ISSN 0014-2956. PMID 3996393.
Liu, Yan; Zhang, Fan; Jiang, Ling; Perry, J. Jefferson P.; Zhao, Zhihe; Liao, Jiayu (2021-12-15). "Product inhibition kinetics determinations - Substrate interaction affinity and enzymatic kinetics using one quantitative FRET assay". International Journal of Biological Macromolecules. 193 (Pt B): 1481–1487. doi:10.1016/j.ijbiomac.2021.10.211. ISSN 0141-8130. PMID 34780893. S2CID 244107621.
Sauro, Herbert (2018). Systems Biology: An Introduction to Metabolic Control Analysis (1st ed.). Ambrosius Publishing. pp. 115–122. ISBN 978-0-9824773-6-6.
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. PMID 6825913.
LaPorte, D. C.; Walsh, K.; Koshland, D. E. (25 November 1984). "The branch point effect. Ultrasensitivity and subsensitivity to metabolic control". The Journal of Biological Chemistry. 259 (22): 14068–14075. doi:10.1016/S0021-9258(18)89857-X. ISSN 0021-9258. PMID 6389540.

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[1] Heijnen, J. J.; van Gulik, W. M.; Shimizu, H.; Stephanopoulos, G. (2004-10-01). "Metabolic flux control analysis of branch points: an improved approach to obtain flux control coefficients from large perturbation data". Metabolic Engineering. 6 (4): 391–400. doi:10.1016/j.ymben.2004.07.002. ISSN 1096-7176. PMID 15491867.

[2] "W_2022_Bis2a_Igo_Reading_15". Biology LibreTexts. 2021-12-13. Retrieved 2022-12-15.

[3] Agutter, Paul S. (2008-10-21). "The flux-summation theorem and the 'evolution of dominance'". Journal of Theoretical Biology. 254 (4): 821–825. doi:10.1016/j.jtbi.2008.07.027. ISSN 0022-5193. PMID 18706429.

[4] Kacser, H.; Burns, J. A. (1973). "The control of flux". Symposia of the Society for Experimental Biology. 27: 65–104. ISSN 0081-1386. PMID 4148886.

[5] Fell, David A.; Sauro, Herbert M. (1985). "Metabolic control and its analysis. Additional relationships between elasticities and control coefficients". European Journal of Biochemistry (published May 1985). 148 (3): 555–561. doi:10.1111/j.1432-1033.1985.tb08876.x. ISSN 0014-2956. PMID 3996393.

[6] Liu, Yan; Zhang, Fan; Jiang, Ling; Perry, J. Jefferson P.; Zhao, Zhihe; Liao, Jiayu (2021-12-15). "Product inhibition kinetics determinations - Substrate interaction affinity and enzymatic kinetics using one quantitative FRET assay". International Journal of Biological Macromolecules. 193 (Pt B): 1481–1487. doi:10.1016/j.ijbiomac.2021.10.211. ISSN 0141-8130. PMID 34780893. S2CID 244107621.

[:0-7] Sauro, Herbert (2018). Systems Biology: An Introduction to Metabolic Control Analysis (1st ed.). Ambrosius Publishing. pp. 115–122. ISBN 978-0-9824773-6-6.

[8] 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. PMID 6825913.

[9] LaPorte, D. C.; Walsh, K.; Koshland, D. E. (25 November 1984). "The branch point effect. Ultrasensitivity and subsensitivity to metabolic control". The Journal of Biological Chemistry. 259 (22): 14068–14075. doi:10.1016/S0021-9258(18)89857-X. ISSN 0021-9258. PMID 6389540.