ps3_solutions

ps3_solutions - Harvard-MIT Division of Health Sciences and...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Departments of Electrical Engineering, Mechanical Engineering, and the Harvard-MIT Division of Health Sciences and Technology 6.022J/2.792J/BEH.371J/HST542J: Quantitative Physiology: Organ Transport Systems PROBLEM SET 3 SOLUTIONS February 26, 2004 Harvard-MIT Division of Health Sciences and Technology HST.542J: Quantitative Physiology: Organ Transport Systems Instructors: Roger Mark and Jose Venegas
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Problem 1 A microelectrode is inserted into a cardiac pacemaker cell, and the (schematized) potential recorded is shown in Figure 1. The intra-cellular and extra-cellular concentrations of sodium and potassium are shown in Table 1. Figure 1: Table 1: [K + ] [Na + ] Inside 100 7 Outside 5 140 6.022j—2004: Solutions to Problem Set 3 2
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Questions: A. Assume that the membrane potential is determined only by the concentrations of Na + and K + and the membrane conductances to these ions, G Na and G K . Show a simple electrical model of the membrane, neglecting membrane capacitance. Label carefully, including polarity conventions and the inside and the outside of the membrane. Express V m , the membrance potential, in terms of Nernst potentials ( V Na , V K ) and membrane conductances. V m = G K G K + G Na V K + G Na G Na + G K V Na (assuming J m = 0 ) G Na G K V Na V K V m outside inside + B. What are the equilibrium potentials for sodium and potassium? Assume RT F log e 60 mV V Na = 60 log p [Na] out [Na] in P = 60 log p 140 7 P = 60 log 20 = ( 60 )( 1 . 3 ) = + 78mV V K = 60 log p [K] out [K] in P = 60 log p 5 100 P = 60 log . 05 = ( 60 )( - 1 . 3 ) = - 78mV C. Let α be the ratio of conductances of the membrane to potassium and sodium. α G K G Na Express the membrane potential V m in terms of α . α = G K G Na 3 6.022j—2004: Solutions to Problem Set 3
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K + G Na = G K + G K α = G K p 1 + 1 α P = ( 1 + α) α G K V m = G K ( 1 + α α ) G K V K + G K α ( 1 + α α ) G K V Na = α ( 1 + α) V K + 1 ( 1 + α) V Na = p α 1 + α P ( - 78 ) + p 1 1 + α P ( 78 ) = 78 1 + α ( 1 - α) = 78 ( 1 - α) ( 1 + α) D. Sketch α vs. time for the cell. Solve for α in terms of V m . V m = 78 ( 1 - α) ( 1 + α) V m + V m α = 78 - 78 α ( V m + 78 = ( 78 - V m ) α = 78 - V m 78 + V m V m = - 60 α = 78 + 60 78 - 60 = 138 18 = 7 . 67 V m = - 40 α = 78 + 40 78 - 40 = 118 38 = 3 . 1 V m = + 20 α = 78 - 20 78 + 20 = 58 98 = 0 . 59 See sketch. 6.022j—2004: Solutions to Problem Set 3
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This note was uploaded on 11/11/2011 for the course BIO 2.797j taught by Professor Matthewlang during the Fall '06 term at MIT.

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ps3_solutions - Harvard-MIT Division of Health Sciences and...

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