Linear biochemical pathway

The molecules progress through the pathway sequentially from the starting substrate to the final product.

Each step in the pathway is usually facilitated by a different specific enzyme that catalyzes the chemical transformation.

An example includes DNA replication, which connects the starting substrate and the end product in a straightforward sequence.

Biological cells consume nutrients to sustain life.

Some of the molecules are used in the cells for various biological functions, and others are reassembled into more complex structures required for life.

An individual cell contains thousands of different kinds of small molecules, such as sugars, lipids, and amino acids.

For example, the most widely studied bacterium, E. coli strain K-12, is able to produce about 2,338 metabolic enzymes.

[1] These enzymes collectively form a complex web of reactions comprising pathways by which substrates (including nutients and intermediates) are converted to products (other intermediates and end-products).

Linear pathways follow a step-by-step sequence, where each enzymatic reaction results in the transformation of a substrate into an intermediate product.

This intermediate is processed by subsequent enzymes until the final product is synthesized.

Multiple computer simulations can be run to try to understand the pathway's behavior.

Another way to understand the properties of a linear pathway is to take a more analytical approach.

Analytical solutions can be derived for the steady-state if simple mass-action kinetics are assumed.

[2][3][4] Analytical solutions for the steady-state when assuming Michaelis-Menten kinetics can be obtained[5][6] but are quite often avoided.

The three approaches that are usually used are therefore: It is possible to build a computer simulation of a linear biochemical pathway.

This can be done by building a simple model that describes each intermediate through a differential equation.

If mass-action kinetics are assumed for the reaction rates, then the differential equation can be written as:

A generally more powerful way to understand the properties of a model is to solve the differential equations analytically.

Analytical solutions are possible if simple mass-action kinetics on each reaction step are assumed: where

If the equilibrium constant for this reaction is: The mass-action kinetic equation can be modified to be: Given the reaction rates, the differential equations describing the rates of change of the species can be described.

will equal: By setting the differential equations to zero, the steady-state concentration for the species can be derived.

into one of the rate laws will give the steady-state pathway flux,

Note that the pathway flux is a function of all the kinetic and thermodynamic parameters.

This result yields two corollaries: For the three-step linear chain, the flux control coefficients are given by:

Given these results, there are some patterns: With more moderate equilibrium constants, perturbations can travel upstream as well as downstream.

, is better able to influence the reaction rates upstream, which results in an alteration in the steady-state flux.

If it is assumed that the equilibrium constants are all greater than 1.0, as earlier steps have more

terms, it must mean that earlier steps will, in general, have high larger flux control coefficients.

In a linear chain of reaction steps, flux control will tend to be biased towards the front of the pathway.

Note that this rule only applies to pathways without negative feedback loops.