Homework 1 Essay

413 WordsSep 30, 20142 Pages
Mathematical Physiology Homework #1 Bob Forder 1. Consider the single channel model for opening and closing C −− O. − k k+ Let fo denote the concentration of open gates and fc denote the concentration of closed gates. Then the model can be expressed as the differential equation dfo /dt = k + − (k + + k − )fo by noting that fo + fc = 1. (a) For initial conditions fo (0) = 0.0, fo (0) = 0.5 and fo (0) = 1.0. 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 (b) For parameters i. k + = 10, k − = 0.1, ii. k + = 0.1, k − = 10 and iii. k + = 1, k − = 1. 1 0.8 0.6 0.4 0.2 0 0 1 2 3 4 5 1 (c) For case i. we have τc = and f∞ = For case ii. we have τc = and f∞ For case iii. we have τc = and f∞ = k− k− k− 1 1 ≈ 0.099 = + +k 10.1 k+ 10 ≈ 0.99. = k− + k+ 10.1 1 1 ≈ 0.099 = + +k 10.1 k+ 1 = − ≈ 0.0099. = + k +k 10.1 1 1 = = 0.5 + +k 2 k+ 1 = = 0.5. − + k+ k 2 2. Consider the extension of the closed and open channel to one with an inactive state that recycles the closed state 2 3 1 C −− O, O −− I, I −− C. − − − k+ k1 k+ k2 k+ k3 The three differential equation system for populations of such channels is dfo + − − + = k1 fc + k2 fi − (k1 + k2 )fo , dt dfi + − + − = k2 fo + k3 fc − (k3 + k2 )fi , dt dfc − + + − = k1 fo + k3 fi − (k1 + k3 )fc . dt By noting that fo + fi + fc = 1 we can eliminate one of the differential equations (in this case we will eliminate the equation for dfi /dt) dfo − + − + − − = −(k1 + k2 + k2 )fo + (k1 − k2 )fc + k2 , dt dfc + + − + + − = (k1 − k3 )fo − (k1 + k3 + k3 )fc + k3 . dt As a matrix equation this is x = Ax + b where ˙ x= − − + fo k− + k+ + k− k1 − k2 k2 ,A = 1 − 2 + 2 − + + ,b = + . fc k1 − k3 k1 + k3 + k3 k3 2 Finally, this is a plot of the number of open channels from t = 0 to t = 10 with initial − − conditions fo (0) = 0.0, fc (0) = 1.0 for k2 = 0.1 and k2 = 10. 1 0.8 0.6 0.4 0.2 0

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