Introduction


Biology is different because everything in biology is inherited. A blueprint called the genome made of the chemical DNA carries all the information needed to make a new animal or plant. The DNA can only code proteins. Proteins are a one dimensional sequence of amino acids. DNA is a one dimensional sequence of nucleotides (like adenine and guanine). The genome controls the properties of proteins by controlling the sequence of amino acids that make up the protein.

Some amino acids have permanent negative charge and are called acids. Others have permanent negative charge and are called bases. The genome can put negative and positive charges in different places by controlling the location of the amino acids (chiefly negative glutamate E, negative aspartate D, positive lysine K, and positive arginine R).

The permanent charges of protein determine many the properties of the protein and most of the
properties of proteins with a hole down their middle, called ion channels. The permanent charges are present in extraordinary concentrations along the walls of ion channels, more than ten molar,. (Solid crystal of sodium chloride NaCl is 37 molar!) These very large charge densities of permanent charge require large concentrations of mobile charges in the hole in the channel to maintain approximate electroneutrality and avoid forces that would explode the protein. Nearby mobile charges ('ions') are very crowded and so the dominant forces determining ion movement are those of crowded charges, namely electrical and steric forces preventing overlap.

In summary, the forces controlling ion movement are determined by the density of permanent charge in the wall of the channel, which is detemined by the genome by the sequence of acid and base amino acid side chains (mostly E,D,K, or R) that make up the wall of the channel.

The genome controls function by controlling the location of negative and positive charges.

Thus, a main mathematical issue should be how does the distribution of permanent charge control function. Function of a channel is the current through the channel, so we can write this in pure
math language. 

A main mathematical issue should be how does the distribution of permanent charge control
the current through a channel. 

Current is influenced by other variables so this statement can be extended. 
The current is carried by ions through the protein moving from one
reservoir (extracellular space outside cells) to another (intracellular space inside cells). The current
depends on the concentrations of ions on both sides of the channel and on the voltage across
the channel. 

Thus, a main mathematical issue, central to the biology of channels, is how does the distribution of
permanent charge change the current vs. voltage, and current vs. concentration behavior of the open channel?