1.3 The Goldman Equation and how to use it.

How to use the goldman equation.

1. Dr. Veronica Campanucci 1.3 The Goldman Equation

2. Objectives for Section 1.3: To understand how Na and K contribute to the neuron’s “steady state”. To understand the Goldman Equation and its relationship with the membrane potential. Readings: From Boron: The resting membrane potential and the Nernst Equation Chapter 5 Chapter 6 From Kandel: Membrane potential and the passive electrical properties of the neuron Chapter 9

3. Why do ions move across a semi- permeable membrane to generate a membrane potential an ionic concentration gradient results in a net diffusion of ions toward the compartment of lower concentration. ions will tend to flow toward a compartment that has an opposite charge . Thus, ions move in response to concentration and charge to reach equilibrium. ?

4. E Na = +67 mV E K = -95 mV E Ca = +123 mV E Cl = varies… V m = -80 mV DF = ( V m – E x ) Figure 6-10 (Boron) The DF acts on the ion, causing its net movement across the membrane down its own electrochemical gradient. (- DF) = inward (+ DF) = outward For cations only!!! Electrochemical driving force

5. The main contributors to the resting potential of a cell are Na + and K + because at rest the cell is permeable only to these two ion. There are sodium and potassium background channels (open at resting potential) In general, the contribution of chloride are calcium to the resting potential is ignored because Cl- and Ca2+ channels are not open at rest. So, which of these ions and how they contribute to the resting potential

6. A, In a resting cell, in which only K+ channels are present, K+ is at equilibrium, therefore: Vm = EK Figure 9-4 B, If background Na+ channels are present, Na+ ions diffuse into the cell depolarizing Vm . The Vm settles at a new level, influx of Na+ is balanced by the eflux of K+. Taken from Kandel, ER. et al. Principles of Neural Science, 6th ed.

7. C, The K+ conductance is much greater than that of Na+. Thus, a relatively small net driving force for K+ drives a current equal and opposite to the Na+ current driven by the much larger net driving force for Na+. This is a steady-state condition, in which neither Na+ nor K+ is at equilibrium but the net flux of charge is null. Figure 9-4 Reaching the neuronal “steady-state” Taken from Kandel, ER. et al. Principles of Neural Science, 6th ed.

8. The electrochemical gradient of Na + , K + and Ca 2+ are established by active transport of ions…. Separation of charges MUST be mantianed constant The Na-K pump PREVENTS DISIPATION OF IONIC GRADIE NTS!!! Figure 9-5 Taken from Kandel, ER. et al. Principles of Neural Science, 6th ed.

9. What is the effect of blocking the Na/K pump Krishnan et al. J Neurophysiol 2015;113:3356-3374 ©2015 by American Physiological Society Na/K pump blocker: ouabain Loss of ion gradients Induction of seizures in mice Seizure-like activity ouabain -75 mV

10. Calculation of the Nernst potential when [K] o increased by 7mM: E K = 58 mV x log 10 = -91 mV 4 mM 155 mM E K = 58 mV x log 10 = -65 mV 11 mM 148 mM

11. How can we calculate the potential of a cell when there is more than one type of channel open The Goldman equation enables us to calculate how the contribution of multiple currents determines the resting membrane potential. ?

12. The Goldman Equation P K [K] o + P na [Na] o + P Cl [Cl] i P K [K] I + P na [Na] I + P Cl [Cl] o _____ _______________ V m = 58 mV x log 10 Resting values P K = 1 P Na = 0.035 P Cl = 0.001 V m = -64 mV The permeabilities and concentration gradients for all ions must be considered when channels for more than one ion are open in the membrane. The greater the permeability for a given ion, the greater its contribution to the membrane potential. This is the resting potential

13. The Goldman Equation [K] o + a [Na] o [K] i + a [Na] i __ __ ____ V m = 58 mV x log 10 This is a simplified form that assumes that P Cl is very small and defines a as being equal to P Na /P K : Using the values given above, a would be equal to 0.035/1 or 0.035. This simplified version of the equation is useful for resting potential.

14. Therefore… The Goldman equation shows us that the membrane potential of a cell could be changed by either changing the gradient (i.e. Nernst potential) or the relative permeability for an ion . For example, increasing the ionic gradient for K + would make E K more negative which would in turn make the resting membrane potential more negative. Increasing the permeability to K + would also make the resting membrane potential more negative by pulling the membrane towards E K .

15. Key points from 1.2 and 1.3 High external Na + and low internal Na + means that the equilibrium potential for Na + is very positive and that Na + ions tend to flow inward at physiological potentials High internal K + and low external K + means that the equilibrium potential for K + is very negative and that K + ions tend to flow outward at physiological potentials The membrane potential of a cell ( V m ) is dependent primarily on its permeability to these two ions and thus the cell potential will always fall between the two equilibrium potential s (except, as we shall see, during electrophysiological experiments!). The number of ions that need to pass into a cell to create a membrane potential is tiny and the original internal and external concentrations can generally be considered to be unchanged.