1) (2 pts) Let's think about the kinetics of the ligands coming off of receptors. Think of
the presynaptic cell releasing a bolus of neurotransmitter, and then having the free neurotransmitter in the synaptic cleft instantaneously fall to zero due to reuptake by the presynaptic cell and breakdown of neurotransmitter by enzymes. Call this time t=0. Any ligand that subsequently unbinds from its receptor on the postsynaptic cell is also taken up or broken down, so effectively kon=0 at t ³ 0.
a) At t ³ 0, what is the differential equation describing the rate of ligand dissociation,
otherwise known as the change in LR with time?
b) Define the fraction of receptors occupied with ligand at time zero to be LR0. Solve the differential equation above to get LR as a function of time, LR(t).
c) If all of the receptors were occupied with ligand at t=0 and koff=5 s-1, what fraction of the receptors have ligand bound after 1 second?
d) Imagine this neuron is stimulating the ocular muscle that can move your eye at very fast rates and hence fires at the rate of ~100 times per second. Does this off rate make sense for this system? Why?
2) (3 pts) Euler's integration is an important numerical technique for solving differential equations, especially when they're too complicated for analytical solutions. A tutorial handout is available on canvas. If you know the derivative of a function and the value of the function at some starting point a , you can approximate the function . You can do this by choosing a small step forward in (we'll call this step ) and use what we know about the derivative to estimate . We can iterate this process - now that we have an estimate for , that becomes our new "known" value of the function , and we can now figure out . If we repeat this process, we can trace out an estimate for over the range of that we care about.
a) Use the formal definition of the derivative (i.e. the first line given to you below) to come up with a step-by-step proof for Euler's method. Begin with the assumption that for small :
And come up with an expression for what you want to know (i.e. ) in terms of what you do know (i.e. , , and ).
b) Let's see how Euler's method for coming up with an approximate solution compares to a known, exact solution. Consider the dissociation system:
Use Euler's method to estimate , i.e. the concentration at 0.3 seconds. Start with the known starting point . Iterate through by choosing . Next, find the analytical solution of . That is, find as a function of time. How does your approximation of compare with the true value (given by the analytical solution) for each step? It might be helpful to organize everything into a table:
Note that the blue boxes should have the same value, and the orange boxes should have the same value.
3) (3 pts) On Canvas you will find Matlab code BME201_HW2_2020.m. This code carries out 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 µM -1s-1, koff = 1 s-1 and Rtot = 1 µM.
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 µM. Note that fractional occupancy solves for LR/Rtot. Now, run simulations for each concentration and using the Data Cursor tool, find the steady-state [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 4-5 values of [L] (good idea to use the same values as 4a). To do this, simulate the reaction for a value of [L], and then use the cursor tool 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 predicted by the above equation. How do they compare?
c) Now let's look at more realistic time scales. Change the parameters: kon=103 µM -1s-1 and koff=104 s-1. To capture short-term dynamics, update the time step to 1 microsecond (µs) and the duration of the simulation to 10 milliseconds (ms). Change the input ligand concentration such that L=10 µM 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 to104 s-1 and leave kon unchanged. Does this increase or decrease the affinity? How does the steady-state value change? The time constant?
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