User:Johnsy/Lipoprotein Modelling/Biochemical Mathematics

Michaelis-Menten Kinetics
The Michaelis-Menten equation is a well known equation in biochemistry relating the rate at which an enzymatic reaction occurs and the concentration of the substrate available. It is derived from first considering the reaction below whereby the substrate and enzyme come together to from the enzyme-substrate complex. The enzyme-substrate complex is then able to react forming the product. After the product is formed, it is released yielding the free enzyme again which is able to further react with more substrate.

$$ E+S \xrightarrow[k_{-1}]{k_{1}} ES \xrightarrow{k_{2}} E + P $$

From the law of mass action, we can come up with a four dimensional model describing the rate equations governing the above system:

$$ \begin{alignat}{2} \frac{d[E]}{dt} & = k_{-1}[ES] + k_{2}[ES] - k_{1}[E][S] \\ \frac{d[S]}{dt} & = k_{-1}[ES] - k_{1}[E][S] \\ \frac{d[ES]}{dt} & = k_{1}[E][S] - k_{-1}[ES] - k_{2}[ES] \\ \frac{d[P]}{dt} & = k_{2}[ES] \end{alignat} $$

One crucial assumption we make is the quasistatic approximation, such that the concentration of the enzyme remains constant and does not change over time. By making this assumption, we also assume that the complex is at equilibrium. We notice that if we add the frist and thrid equations, they sum to zero. This occurs also if we add the second, thrid, and fourth equations. By algebraic substitution and manipulation, it is easy to see that we can minimise this four dimensional model into a one dimensional non-linear model. The substrate concentration (and likewise the product concentration) is governed by the following rate equation shown below.

$$ \frac{d[P]}{dt} = - \frac{d[S]}{dt} = V_{max}\frac{[S]}{K_{m} + [S]} $$

In the above equation, Vmax is the maximum rate of turnover of the enzyme and is dependent on the initial enzyme concentration as well as the rate at which the enzyme-substrate complex dissociates into the enzyme and product.

$$ V_{max} = k_{2}[E]_{0} $$

Km is known as the Michaelis-Menten constant for the enzyme and is graphically interpreted as the concentration at which the reaction rate is half the maximum rate. In terms of enzyme properties, the Michaelis-Menten constant is a measure of how tight the enzyme binds the substrate. Tighter, more specific binding to the enzyme would be reflected as a lower value of Km, however, this only applies when k2 is much larger than k1.

When is it possible to use the Michaelis-Menten equation? The quasi-steady-state approximation only holds when the following relation is true:

$$ \frac{[E]_{0}}{K_{m} + [S]_{0}} \ll 1 $$

This means that there is a negligible decrease in substrate concentration for the duration of the brief transient time where the enzyme substrate complex goes into equilibrium.

For a more complete picture of Michaelis-Menten kinetics, see the Wikipedia article.

MWC Model (Voet & Voet, Biochemistry)
The MWC model was developed by Jacques Monod, Jeffries Wyman, and Jean-Pierre Changeux to describe allosteric interactions for proteins. Some rules of the model include:
 * 1) An allosteric protein is an oligomer of protomers that are symmetrically related.
 * 2) Each protomer can exist in (at least) two conformational states (designated T - tensed and R - relaxed). These states are in equilibrium whether or not ligand is bound to the oligomer.
 * 3) The ligand can bind to the protomer in either conformation (ie conformation change alters the affinity of the protomer for the ligand)
 * 4) Molecular symmetry of the protein is conserved during conformation change. Thus, there are no oligomers that simultaneously contain both R and T state protomers.

The model then defines a few constants that are used:
 * $$L = \frac{[T]_{0}}{[R]_{0}}$$ - the equilibrium constant for the conformation interconversion between the two states (T and R) of the oligomeric protein in the absence of any ligand
 * kR - the microscopic dissociation constant for the R state, dependent on the number of ligands bound to R
 * kT - the microscopic dissociation constant for the T state
 * Ys - the fractional saturation for ligand binding
 * $$\alpha = \frac{[S]}{k_{R}}$$ - the normalized ligand concentration
 * $$c = \frac{k_{R}}{k_{T}}$$ - the ratio of ligand binding constants, c increases with the ligand binding affinity of the T state relative to the R state.

For homotropic interactions, the MWC model predicts that the fractional saturation for ligand binding is given by the equation below: $$ Y_{s} = \frac{\alpha(1+\alpha)^{n-1} + Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n + L(1+c\alpha)^n} $$

For heterotropic interactions, we must make a few more assumptions based upon the interaction between the oligomer and substrate, activators, and inhibitors.
 * 1) The substrate only binds to the R state, and thus c = 0.
 * 2) The activator (A) binds only to the R state
 * 3) The inhibitor (I) binds only to the T state

And the constants that are used in addition to the one's above:
 * $$\alpha = \frac{[S]}{k_{R}}$$ - normalized ligand concentration
 * $$\beta = \frac{[I]}{k_{I}}$$ - normalized inhibitor concentration
 * $$\gamma = \frac{[A]}{k_{A}}$$ - normalized activator concentration
 * kI - inhibitor dissociation constant
 * kA - activator dissociation constant

$$ Y_{s} = \frac{\alpha(1+\alpha)^{n-1}}{(1+\alpha)^n + \frac{L(1+\beta)^n}{(1+\gamma)^n}} $$

Michaelis-Menten Inhibition
The Michaelis-Menten equation can be modified slightly to take into account inhibitors. First, we deal with competitive inhibitors which alter the Michaelis-Menten constant, Km by a factor of &alpha;, where &alpha; is a function of the inhibitor concentration which must always be greater than or equal to 1. In reversible competitive inhibition, the inhibitor binds to the active site, but can also dissociate from the active site, making it available for substrate to bind. The rate equation with competitive inhibition taken into account is the following:

$$ \frac{d[P]}{dt} = V_{max}\frac{[S]}{\alpha K_{m} + [S]} $$

The same assumptions apply to this altered Michaelis-Menten equation that were valid for the original case without inhibition. Furthermore, through similar analysis, we can also consider uncompetitive inhibition, where the inhibitor binds to the enzyme-substrate complex but not to the free enzyme with a function &alpha;'.

$$ \frac{d[P]}{dt} = V_{max}\frac{[S]}{K_{m} + \alpha '[S]} $$

Similarly, we can consider mixed inhibition where the inhibitor binds to enzyme sites that participate in both substrate binding and catalysis (non-active sites).

$$ \frac{d[P]}{dt} = V_{max}\frac{[S]}{\alpha K_{m} + \alpha '[S]} $$

A competitive inhibitor could be modelled more detailed by knowing the dissociation constant of the inhibitor and the concentration/amount of inhibitor. (Source: Wikipedia)

$$ \frac{d[P]}{dt} = \frac{V_{max}[S]}{K_m + [S] + \frac{K_m}{K_i}[I]} $$