NPB 110B Fall 2024 Lecture 7 Resting Membrane Potential II PDF

Lecture 7: Resting membrane potential II 10/07/2024 Sources: Quantitative Physiology for Engineers (Feher) Medical Physiology (Boron and Boulpaep) Human Physiology (Widmaier, Raff and Strang) Human Physiology (Silverthorn) Topics Goldman-Hodgki...

Lecture 7: Resting membrane potential II 10/07/2024 Sources: Quantitative Physiology for Engineers (Feher) Medical Physiology (Boron and Boulpaep) Human Physiology (Widmaier, Raff and Strang) Human Physiology (Silverthorn) Topics Goldman-Hodgkin-Kat (GHK) Equation Equivalent Circuit Representation Sodium calcium exchanger revisited Sodium Potassium ATPase revisited Goldman-Hodgkin-Kat (GHK) Equation In reality many charged molecules contribute to the overall electrical properties of the cell membrane. For a given concentration gradient, the greater the membrane permeability to an ion, the greater the contribution that ion will make to the membrane potential. The GHK equation is an expansion of the Nernst Equation that gives the resting membrane potential in terms of ion permeabilities and concentrations inside and outside the cell. 𝑃𝐾+ ∙ 𝐾 + 𝑜𝑢𝑡 + 𝑃𝑁𝑎+ ∙ 𝑁𝑎+ 𝑜𝑢𝑡 + 𝑃𝐶𝑙− ∙ 𝐶𝑙 − 𝑖𝑛 𝑉𝑀 = 58𝑚𝑉 ∙ 𝑙𝑜𝑔10 𝑃𝐾+ ∙ 𝐾 + 𝑖𝑛 + 𝑃𝑁𝑎+ ∙ 𝑁𝑎+ 𝑖𝑛 + 𝑃𝐶𝑙− ∙ 𝐶𝑙 − 𝑜𝑢𝑡 𝑅𝑇 in the ln version can be treated as a constant, 58 mV at 20oC and 62 mV at 37oC 𝐹 Note that Cl- concentrations are reversed because Cl- is an anion and its movement has the opposite effect on membrane potential. Goldman-Hodgkin-Kat (GHK) Equation Takes into account the relative permeabilities of the main three ions contributing to the resting potential and also their concentrations to obtain a more accurate estimate. This model to describe ionic flow across the membrane makes several assumptions: Ion movement within the membrane obeys the Nernst Equation Ions move across the membrane independently (without interacting with each other) The electric field across the membrane is constant Thus this model falls short in describing membrane current when these assumptions are not valid. Equivalent Circuit Representation of a Membrane Biological membranes exhibit properties similar to electrical circuits: voltages (across the membrane and the equilibrium potentials), ion (current) flow and resistance, 𝑅, to this flow Conductance is the reciprocal of resistance: 𝐺 = 𝑅1 Each ion carries a current that can be written as the product of the conductance to that ion and the difference between the membrane potential and the equilibrium potential. 𝑉 𝐼 = = 𝐺 ∙ 𝑉 = 𝐺 ∙ 𝑉𝑀 − 𝐸 𝑅 Assuming independence, total current through the membrane is the sum of inward and outward currents. 𝐼 = 𝐼𝐶 + 𝐼1 + 𝐼2 + 𝐼3 + ⋯ + 𝐼𝑁 Equivalent Circuit Representation of a Membrane Total current through the membrane is the sum of inward and outward currents 𝐼𝑇𝑜𝑡𝑎𝑙 = 𝐼𝐶 + 𝐼𝐾+ + 𝐼𝑁𝑎+ + 𝐼𝐶𝑎2+ + 𝐼𝐶𝑙− Each ion current, 𝐼𝑖 , is the product of the conductance, 𝐺𝑖 , and the 𝐼𝑖 = 𝐺𝑖 ∙ 𝑉𝑀 − 𝐸𝑖 difference between 𝑉𝑀 and 𝐸𝑖. 𝑑𝑉 𝐼𝑇𝑜𝑡𝑎𝑙 =𝐶 + 𝐺𝐾+ ∙ 𝑉𝑀 − 𝐸𝐾+ + 𝐺𝑁𝑎+ ∙ 𝑉𝑀 − 𝐸𝑁𝑎+ 𝑑𝑡 +𝐺𝐶𝑎2+ ∙ 𝑉𝑀 − 𝐸𝐶𝑎2+ + 𝐺𝐶𝑙− ∙ 𝑉𝑀 − 𝐸𝐶𝑙− Equivalent Circuit Representation of a Membrane At steady state: 𝐼𝑇𝑜𝑡𝑎𝑙 = 0 𝐺𝐾+ ∙ 𝐸𝐾+ + 𝐺𝑁𝑎+ ∙ 𝐸𝑁𝑎+ + 𝐺𝐶𝑎2+ ∙ 𝐸𝐶𝑎2+ + 𝐺𝐶𝑙− ∙ 𝐸𝐶𝑙− 𝑉𝑀 = 𝑑𝑉 𝐺𝐾+ + 𝐺𝑁𝑎 + 𝐺𝐶𝑎2+ + 𝐺𝐶𝑙− =0 𝑑𝑡 This gives the resting membrane potential in terms of ionic conductances and equilibrium potentials. Equivalent Circuit Representation of a Membrane The equilibrium potential resembles the electromotive force: the difference from the membrane potential will determine in which direction (into the cell or out) the ion will flow. The equilibrium potential is the reversal potential—voltage at which the direction of current reverses  Sodium calcium exchanger Sodium calcium exchanger moves one 𝑪𝒂𝟐+ out of the cell against its electrochemical gradient for every three 𝑵𝒂+ molecules that it moves into the cell down their electrochemical gradient. The exchanger can run backwards (making 𝑁𝑎+ leave the cell and 𝐶𝑎2+ enter the cell) under certain physiological conditions or by altering one or more of the ionic gradients involved in the exchanger. The direction of transport is determined by whether the energy provided by the entry of three 𝑁𝑎 + is greater than or less than the energy required to extrude one 𝐶𝑎2+.  Sodium calcium exchanger The energy released by 𝑁𝑎+ moving into the cell depends on the driving force on 𝑁𝑎+ ions: 𝐸𝑁𝑎 − 𝑉𝑀. The energy needed to move 𝐶𝑎2+ out of the cell depends on the driving force on the 𝐶𝑎2+ ion: 𝐸𝐶𝑎 − 𝑉𝑀 If 3 ∙ 𝐸𝑁𝑎 − 𝑉𝑀 > 2 ∙ 𝐸𝐶𝑎 − 𝑉𝑀 𝑁𝑎+ ions move into the cell, while 𝐶𝑎2+ ions move out of the cells. If 3 ∙ 𝐸𝑁𝑎 − 𝑉𝑀 < 2 ∙ 𝐸𝐶𝑎 − 𝑉𝑀 , 𝑁𝑎+ ions move out of the cell, while 𝐶𝑎2+ ions move into the cells.  Sodium calcium exchanger The energy released by 𝑁𝑎+ moving into the cell depends on the driving force on 𝑁𝑎+ ions: 𝐸𝑁𝑎 − 𝑉𝑀. The energy needed to move 𝐶𝑎2+ out of the cell depends on the driving force on the 𝐶𝑎2+ ion: 𝐸𝐶𝑎 − 𝑉𝑀 At some 𝑉𝑀 , the energy for moving three 𝑁𝑎+ ions and the energy for moving one 𝐶𝑎2+ ion will be equal. If we call this the reversal potential (𝑉𝑅 ), then 3 ∙ 𝐸𝑁𝑎 − 𝑉𝑅 = 2 ∙ 𝐸𝐶𝑎 − 𝑉𝑅 or 𝑉𝑅 = 3 ∙ 𝐸𝑁𝑎 − 2 ∙ 𝐸𝐶𝑎 At membrane potentials, 𝑉𝑀 , more negative than the reversal potential, 𝑉𝑅 , 𝑁𝑎+ moves in and 𝐶𝑎2+ moves out. At membrane potentials more positive, 𝐶𝑎2+ moves in and 𝑁𝑎+ moves out.  Sodium potassium ATPase The sodium potassium ATPase (or sodium potassium exchange pump) transports 𝑵𝒂+ out of the cell and 𝑲+ back into the cell. For every molecule of ATP hydrolyzed, three 𝑵𝒂+ ions are transported out and two 𝑲+ ions are transported in. Because of the unequal number of ions being transported into and out of the cell, the sodium potassium ATPase causes a small change in membrane potential.  Sodium potassium ATPase The sodium potassium ATPase (or sodium potassium exchange pump) transports 𝑵𝒂+ out of the cell and 𝑲+ back into the cell. With an active pump and assuming that all other permeating ions are in steady state, the GHK equation for 𝑉𝑀 becomes: 𝑟 ∙ 𝑃𝐾 ∙ 𝐾 + 𝑜𝑢𝑡 + 𝑃𝑁𝑎 ∙ 𝑁𝑎+ 𝑜𝑢𝑡 𝑉𝑀 = 58 𝑚𝑉 ∙ 𝑙𝑜𝑔10 𝑟 ∙ 𝑃𝐾 ∙ 𝐾 + 𝑖𝑛 + 𝑃𝑁𝑎 ∙ 𝑁𝑎+ 𝑖𝑛 The permeability of 𝐾 + is modified (𝑟 ∙ 𝑃𝐾 ∙ 𝐾 + ) to account for the action of the pump. The value of 𝑟 = 1.5 reflects this activity. Summary When the constant field assumption holds, membrane voltage can be described by ionic permeabilities and concentrations inside and outside the cell (the GHK equation). When the cell membrane is represented by an electrical circuit, membrane voltage can be described by ionic conductances and equilibrium potentials. The sodium calcium exchanger can run backwards depending on the membrane potential. This sodium potassium pump is electrogenic (provides a very small contribution to the cell’s resting membrane potential). This influence can be represented in the GHK equation.