Euler integration to solve for LR(t) based on the free parameters kon, koff and Rtot. The...

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Euler integration to solve for LR(t) based on the free parameters kon, koff and Rtot. The input is a step increase in [L]. Go through the code and make sure you understand what every step is doing. It may seem complicated at first, but most of the lines are comments to help you understand what's happening. How does each line relate to your steps in Problem 3? Start with Kon=1 M s, koff = 1 s and Rtot = 1 M. a) The final [LR] is governed by the fractional occupancy equation. First, get the analytical solution to steady-state [LR] across the following [L] concentrations: 0.3, 1, 3, 10, and 30 M. Note that fractional occupancy solves for LR/Rtot. Now, run simulations for each concentration and using the Data Cursor tool, find the steadystate [LR]. Plot the theoretical and the computed solutions of [LR] versus [L] on separate figures. Do they agree? b) For the differential equation that includes both forward and reverse reactions, the approach to steady state is exponential, but it incorporates both rates. For all [L], the rate constant is kobs = Kon[L]+koff. Use simulations to show that this is the case. Begin by determining the kobs for the values of [L] used for 4a), above. To do this, simulate the reaction for a value of [L], and then use the cursor tool (Data Tips under the Tools pulldown menu) to identify the time constant (recall that the time constant is the time it takes for to reach 63% completion - don't forget that ligand is introduced at t=1s!). Use the time constant to calculate the rate constant, which gives you your kobs. Repeat for more values of [L], making note of the kobs each time. Compare these values to kobs predic by the above equation. How do they compare? c) Now let's look at more realistic time scales. Change the parameters: kon=10 Ms and Koff=10s. (Thought experiment: what's the Kd?) To capture short-term dynamics, update the time step to 1 microsecond (us) and the duration of the simulation to 10 milliseconds (ms). Change the input ligand concentration such that L=10 M for after t=1ms. Plot the result LR vs time and make sure it agrees with your intuition. Use plots and the cursor tool to answer the following questions: What is the steady-state value after ligand is added? What is the time constant? Decrease both rate constants, each by a factor of 10 (note that this doesn't change the affinity, only the kinetics). How does the steady-state value change? The time constant? Now return koff to 104s and leave kon unchanged. Does this increase or decrease the affinity? How does the steady-state value change? The time constant? Euler integration to solve for LR(t) based on the free parameters kon, koff and Rtot. The input is a step increase in [L]. Go through the code and make sure you understand what every step is doing. It may seem complicated at first, but most of the lines are comments to help you understand what's happening. How does each line relate to your steps in Problem 3? Start with Kon=1 M s, koff = 1 s and Rtot = 1 M. a) The final [LR] is governed by the fractional occupancy equation. First, get the analytical solution to steady-state [LR] across the following [L] concentrations: 0.3, 1, 3, 10, and 30 M. Note that fractional occupancy solves for LR/Rtot. Now, run simulations for each concentration and using the Data Cursor tool, find the steadystate [LR]. Plot the theoretical and the computed solutions of [LR] versus [L] on separate figures. Do they agree? b) For the differential equation that includes both forward and reverse reactions, the approach to steady state is exponential, but it incorporates both rates. For all [L], the rate constant is kobs = Kon[L]+koff. Use simulations to show that this is the case. Begin by determining the kobs for the values of [L] used for 4a), above. To do this, simulate the reaction for a value of [L], and then use the cursor tool (Data Tips under the Tools pulldown menu) to identify the time constant (recall that the time constant is the time it takes for to reach 63% completion - don't forget that ligand is introduced at t=1s!). Use the time constant to calculate the rate constant, which gives you your kobs. Repeat for more values of [L], making note of the kobs each time. Compare these values to kobs predic by the above equation. How do they compare? c) Now let's look at more realistic time scales. Change the parameters: kon=10 Ms and Koff=10s. (Thought experiment: what's the Kd?) To capture short-term dynamics, update the time step to 1 microsecond (us) and the duration of the simulation to 10 milliseconds (ms). Change the input ligand concentration such that L=10 M for after t=1ms. Plot the result LR vs time and make sure it agrees with your intuition. Use plots and the cursor tool to answer the following questions: What is the steady-state value after ligand is added? What is the time constant? Decrease both rate constants, each by a factor of 10 (note that this doesn't change the affinity, only the kinetics). How does the steady-state value change? The time constant? Now return koff to 104s and leave kon unchanged. Does this increase or decrease the affinity? How does the steady-state value change? The time constant?