3. Fast & Slow Timescales If the time scale of activation is much faster than that of inactivation we can use these two different timescales to simplify the problem. k 1 >>k 2 Scale time by the slow reaction constant: Substitute into equations and look for “small parameter”  <<1 ratio of slow to fast timescales

4. Slow Timescale Analysis continued The second equation becomes: Now allow   0 From either equation: “ rapid equilibrium” between the open and closed channels quasi-steady state

5. Slow timescale continued Rapid equilibrium between open and closed, now we need a differential equation independent of  Add the two equations: Let Combined fractions of open and closed channels Have to eliminate x o from this

6. Slow Timescale Ion Channel From algebraic equation: y(0)=1 Differential Equation for y:

7. Fast Timescale Fast Timescale Differential Equations

9. Another Example T o T R T wo T wi m w C pw m R C pR k w Heat Transfer across a reactor wall State Variables, T W and T R , note T W already lumped. Q R

10. Heat Transfer Continued Rearrange the Equations for T w and T R [time] -1 Slow time scale Transfer Rate across wall small compared to the thermal mass of reactor

11. Heat Transfer Continued The Thermal Mass of Reactor is large compared to the wall,  is ratio of wall to reactor thermal mass . The wall temperature is just the average at slow time scale.

12. Heat Transfer Continued Eliminate T W by using the relationship from the previous result Now solve the ODE

13. Fast Timescale Allow   0 Ratio of rate of heat transfer to thermal mass of the wall is large. Now solve the ODE and substitute into AE Implicitly Q R must also be slow