Answer these questions on the whiteboard (or paper). You may work in a group of 2 or 3. Be sure to verify your solution with a TA/instructor before leaving.
Basic (✓):
Consider the differential equations describing two-state kinetics:
$$
\frac{dP_A}{dt} = k_{BA}P_B(t) - k_{AB}P_A(t)
$$
$$
\frac{dP_B}{dt} = k_{AB}P_A(t) - k_{BA}P_B(t)
$$
- To eliminate the second equation, substitute $P_B(t) = 1 - P_A(t)$ into the first equation.
- If you've taken differential equations (and remember it!) you should be able to solve this first-order separable equation. As a shortcut, assume (correctly) that $P_A(t) = P_A^{eq} + a e^{-\lambda t}$, where $a$ and $\lambda$ are constants to be determined and $P_A^{eq}$ is the probability of A at equilibrium ($t \rightarrow \infty$).
- Solve for $P_A(0)$ using our assumed form of $P_A(t)$. What is the role of $a$?
- Substitute $P_A(t) = P_A^{eq} + a e^{-\lambda t}$ into your differential equation.
- Solve for $P_A^{eq}$ (Hint: what is the rate of change at equilibrium?)
- Solve for $\lambda$
- Solve for $P_B(t)$
Using this solution, determine the transition matrix in terms of $\tau$ for $k_{AB} = 6$ and $k_{BA} = 2$.
Apply your transition matrix with $\tau = 1$ to initial conditions $[1.0, 0]$ (calculate the actual numbers).
Upload your solution to GradeScope.
Extra (✓+):
Plot $P_A$ with respect to time as determined by solving the kinetic equations exactly and by applying transition matrices calculated for $\tau = 0.1, 0.2, 0.5$. Use initial conditions of $[0.5, 0.5]$. Plot the MSM values with markers at each point (e.g. using the 'o-' matplotlib style).
Upload your figure to GradeScope.