9.16 Problem Set #2 In this assignment you will buil a simulation of the presynapti terminal. The simulation an be broken own into three parts: simulation of the arriving ation potential (base on the Hogkin-Huxley equations from the last assignment); simulation of the alium urrent influx; an, finally, simulation of the vesile release ynamis. Part 1 Simulation of the Ation Potential For simpliity, you may assume that the alium urrent oes not hange the shape of the ation potential ramatially; hene, you an use the ation potential waveform from the previous assignment as the starting point. Look for the solution to the previous assignment poste on the web. You will nee to take the voltage output from this simulation an fee it as an input to the next part of the assignment. For all subsequent simulations assume a onstant urrent of 2 na injete for 1 ms. Plot the resulting ation potentials here. Part 2 Simulation of the Calium Current Influx Sine the Ca 2+ onentration insie the ell varies, you will nee to use the Golmann- Hogkin-Katz (GHK) equation to alulate the Ca 2+ urrent: I Ca = P Ca e V εv [ Ca] [ Ca] e in V ε 1 out ε =.7788 mv -1 To moel the Ca 2+ hannel kinetis, you may aopt a simplifie moel (from Borst an Sakmann, J. Phys., 1998), whih onsoliates the effets of ifferent alium hannels present in the terminal into a single ativation parameter (): α = 1.78e β =.14 V / 23.3 /15 e V = α ( 1 ) β t P = P 2 Ca max P max = 8.265 1-5 Simultaneously, you will nee to solve for [Ca] in, whih hanges ue to the influx of Ca 2+ ions into the terminal through the Ca 2+ urrent. The synapti terminal an be moele as a single ompartment with a alium buffer (B) an iffusion of free Ca 2+ away from the release site (iffusion of the alium buffer is a relatively slow proess an an therefore be omitte from the moel). The ynamis of the Ca 2+ buffering an iffusion an be summarize in the following kineti sheme:
I Ca Aumulation (A) [Ca] + [B] k 1 [Ca][B] k 2 [BCa] Diffusion (D) Cytoplasm This orrespons to the following equations: [ B] = [ BTot ] [ BCa] t t [ Ca] I A + k [ BCa] k [ Ca][ B] D[ Ca] = Ca 2 1 [ BCa] = k [ Ca][ B] k [ BCa] 1 The relevant onstants are: 2 [Ca] out = 2 µm; [B Tot ] = 5 µm; k 1 =.1 ms -1 ; k 2 =.1 ms -1 ; A = 1; D = 1 a) Run the simulation an plot the alium urrent influx in the same graph with the ation potentials. Observe that the peak alium influx oinies with the repolarizing phase of the ation potential. Why? 6 4 Timing of Calium Current vs Ation Potentials ICa (x1) Ation Potentials 2-2 -4-6 -8
Start the simulation with the following initial onitions: =.17; [Ca] in =.24; [BCa] = b) Plot the free [Ca] in in the same graph with the buffere alium [BCa]. Note that as the buffer saturates, the onentration of free Ca 2+ inreases. Aumulation of free Ca 2+ in the presynapti terminal results in the enhanement of vesile fusion, an auses a form of short-term synapti plastiity alle paire-pulse failitation. (This will be explore further in Part 3). 45 Internal Free Ca an Buffere Ca 4 35 Conentration (um) 3 25 2 15 [Ca]in [BCa] 1 5 ) Explore the effets of buffer onentration [B Tot ] on the extent of paire-pulse failitation. Part 3- Simulation of the Vesile Release In this part of the assignment, you will take the internal alium onentration ompute in Part 2, an use it to rive the vesile release mahinery. First, you will have to simulate the bining of Ca ++ to a Ca ++ sensor (CS) loate on the oke vesiles. For simpliity, you may assume that bining to the sensor is instantaneous, in whih ase the fration of sensors bining 4 Ca ++ moleules (remember that vesile release obeys a 4 th orer Ca ++ epeneny) is given by the ose response relationship: [ Ca] [ Ca] + 4 CS = K D = 1 µm K D
To moel the vesile population ynamis, you may assume a onstant soure of vesiles insie the ell, a rate onstant α of vesile oking, an a rate onstant β of vesile unoking. The rate of fusion of oke vesiles epens on CS. The setup for the problem is the following: Constant Vesile Soure The orresponing moel equation is: t α RP = β RP + α RP CS Fusion Rate = RP CS β Releasable Vesile Pool (RP) CS Release Vesile τ = 5; (vesile relaxation time onstant) m = 8; (mean number of vesiles in the reaily releasable pool) β = 1/τ; α = m/τ; a) Run the simulation with the initial onition RP = 8, an plot the size of the reaily releasable vesile pool as a funtion of time. Observe that the vesile pool epletes fast. Explore the effets of the vesile relaxation time onstant on the extent of vesile pool epletion. 8 Vesile Pool Depletion 75 Releasable Vesile Pool Size 7 65 6 55 5
b) Finally, ombine the effets of internal alium onentration an the vesile pool size on the rate of vesile release. Plot the vesile fusion rate as a funtion of time. 8 Vesile Fusion 7 6 Fusion Rate (vesiles/ms) 5 4 3 2 1 Comment on what you see.