Modeling background
Regulatory mechanism of compatible solutes.
The accumulation of osmotic active compound in cells is highly harmonious with cells’ function. This phenomenon is called compatible solute.
While the surrounding osmotic pressure bumps up, K+ in cells is accumulated a lot, with the increase of Glu synthesis. Then K+ and Glu become the secondary messengers, leading the compatible solute’s accumulation. The compatible solute opens the channels which K+ out flows through, in order to balance the osmotic pressure.
The osmotic pressure out of cells rises, cells expel compatible solute like K+ from intracellular environment to adjust and control the osmotic pressure be balanced. The emission of K+ leads the change of membrane potential.

Modeling assumption
1.We presume the Kcsa protein we bring in is not co-regulated by compatible solute.
2.We presume the osmotic pressure doesn’t change after adjustment.
3.We presume the environment out of cells is large enough that any matter or physicochemical property won’t change.
Modeling building
1.The osmotic pressure doesn’t change, being kept in a normal level. The figure of osmotic pressure is 709.12kPa in reference.
2.We set the osmotic pressure out of cells P
3.We set the concentration of K+ in cells c_1, and the concentration of K+ out of cells c_2.
4.We build the connection between ionic concentration osmotic pressure and potential by introduce Nernst equation.

In a certain limit, the quantity of osmotic pressure increase and K+ production is proportional: ∆n=k∆P.
We use the outflow rate v to express cells’ impermeability. In this phase v=dn/dt.
While if the osmotic pressure is higher than this range, cells will occur the regulatory mechanisms mentioned above. We can use Nernst equation to measure the concentration of K+ indirectly. While it is intervened by compatibility regulation mechanism, there is no linearity connection. Method of variation of constant is ∆n=k(x)∆P.