
where is the proportionality constant for binding and is the proportionality constant for unbinding. The rate of change of occupied receptors is thus, If we convert Equation 2.8.1 to represent the change in the fraction of bound receptors by dividing by , and rearrange the terms a little bit, then we can solve it by using what is called an integration factor.^{2.3}That is, if we let , we can rewrite Equation 2.8.1 as Rearranging the terms gives us: or With Equation 2.8.4 in the exact same form as Equation 2.10.15, the form required for our general solution, we can easily solve for . First, we will determine the integrating factor. That is, from Equations 2.8.4 and 2.10.17
We can now plug the integrating factor, along with bits from Equation 2.8.4 into Equation 2.10.16 and solve for . Thus,
Since we know that at time that some fraction of receptors are occupied, , we can solve for . That is, and simple rearrangement gives us, Thus, our general solution for is
Phase Plane Analysisno_title
Example 2.8.1.4 (Phase Plane Analysis)
If we return to Equation 2.8.3, we can determine the asymptotic behavior of the solution without having to solve for it.^{2.4}First, we will determine any points where the system is at equilibrium. That is, determine where the derivative is zero. Thus, when , for all , we are at an equilibrium and will not move from it. When , then the slope for all is positive (to see this, try plugging in ), and thus, as , approaches from below. If , then the slope for all will be negative and as , approaches the equilibrium from above. Note that when . This is called the equilibrium dissociation constant and is the concentration of drug where the fraction of bound receptors is 1/2. Thus, from the kinetics of binding and unbinding, you can directly get a feel for the concentration of drug required to have half the receptors occupied.
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