# Category Archives: Biology

## Binding #2: The Michaelis-Menton Equation

The Michaelis-Menton Equation has a very similar form to the Hill Equation but the key difference is that it deals with enzyme rates not ligand/receptor or drug/target interactions per se. Basically, it describes how fast an enzyme (E) makes its product (P) as a function of the total concentration of substrate ([S]t). This rate of production formation (d[P]/dt) is proportional to the kcat and the amount of complex ([ES]) which is exact what the Michaelis-Menton equation models. The Michaelis-constant (Km = (koff+kcat) / kon) describes how tightly the substrate binds the enzyme and the kcat is a rate-constant that describes how quickly the enzyme can make the product. Here,brackets denote concentrations and a t subscript indicates “total concentrations.”

## Binding#1: Understanding the Hill Equation

If you want to understand how drugs work/behave (or just how 2 molecules interact with each other) then you need to understand one of the oldest and most useful theories in biochemistry: the Hill Equation. The Hill Equation mathematically models how Ligands(L)/Drugs interaction with their Receptors(R)/targets and generally assumes that the amount of complex that forms (RL) is proportional to some biological response . Here the Kd represents the “dissociation constant” (which is a measure of the strength of the interaction between R and L) and brackets denote concentrations/dosages (with a t subscript indicating total concentrations).

## Timescales, Kinetics, Rates, Half-lives in Biology

Understanding interactions between processes with different rates can made significantly easier when you know the approximate timescales or half-lives for those processes.   Comparing two processes (e.g. alpha helix folding of a peptide chain (micro seconds (10-6s)) and translation of a peptide chain ( milliseconds(10-3s))) if the timescales differ by more than a factor of 100 then you can reasonably assume that as the faster process is occurring, the slower process is standing still (i.e. not occurring). Equivalently you can assume that as the slower process is occuring the faster process occurs instantly. What this means for the example cited above is that the folding of a peptide chain can and does occur before translation of that peptide chain has finished.1-3

## Understanding Ligand-Receptor Dose Response Curves

Figure 1 A.
Most Ligand (L)–Receptor (R) dose–response curves are characterized by formation of a complex followed by biological effect. The underlying assumption is that biological effect is proportional to the amount of complex that forms. B. Assuming the biological effect and amount of complex formed are proportional, dose–response curves can be envisioned on two equivalent axes: amount complex (in red) and percentage biological response (in blue). The key to intuitively understanding these curves is to understand the Effective Concentration 50% or EC50 which tells you the dose of ligand necessary to elicit a 50% response. The EC50 is also known as the ligand’s potency.

Biological phenomena are often caused by the binding of a ligand (L (e.g. drug, hormone, etc.)) to a target receptor (R), which induces a biological response (Figure 1A). Typically, the biological response is proportional to the amount of ligand-receptor complex (RL) that forms (Figure 1A); as a result, the dose–response curve can have two equivalent y-axes: amount of complex (in red) or biological response (in blue). These curves have a sigmoidal (S-like) shape, where low doses of ligand have no effect and high doses plateau at maximal (100%) complex formation/biological response (Figure 1B).