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UCSD PHYS 171 - Action Potentials

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Physics 171/271 - Chapter 5 - David Kleinfeld - Fall 20055 Action Potentials - Conductanc e EquationsFIGURE - chapt-8-hh-currents.epsThe experiments of Hill, Katz, Hogkin, and Huxley laid out the ionic basis ofspike generation. We have already considered some of the fundamental physics thatgoes into this:• Lipid membranes are the means to form cellular compartments. This, bydefinition, provides a means to develop and maintain concentration differences.the voltage dro p is confined to the membrane.• Electrochemistry, via ionic concentration gradients, is the basis for potentialsacross a cell membrane. The alternative - the movement o f charge that isconfined to a transmembrane protein - is not observed.• Pumps for Na+and K+, with Cl−as the dominant counter ion, ar e the basisfor the concentration gradient. The dominant pump is Na- K-ATPase, aka,the Na+/K+exchanger. Suffice it to say that the pump is sufficiently slow sothat it, and other pumps, do not compete with the spike generations. On theother hand, the pump rate is sufficiently high so that the ion concentrationgradients are maintained for reasonable spike rates.• Conservation of current, via K irchoff’s Law, as a means to describe cables isused as the basis for a description of the transmembrane voltages.• Permeabilities that can switch with voltage according to a Boltzman relation.We considered an extreme version o f this relation in the past.At the time of the original experiments the field of electrical circuits and elec-trochemistry were pretty mature, so there was a theoretical framework in place forthe planning of experiments and interpretations. But our discussion certainly hasmore structure built into it than is suggested by the historical record.5.1 Circuit Equation with Active CurrentsLet’s develop the framework for the physics and electrochemistry of the action po-tential in a cell that is electrically compact. This allows one to form a plan, a ndthus put the experiments in a context.τ∂V (t)∂t= −rm2πaIm(t) (5.1)where Im(x, t) now includes all membrane currents, including the V2πarmcurrentdue to passive flow. The sign convention is that current flows out.1The value of the currents Im(x, t) are given by the Nernst-Plank relation, so thatfor the Na+currentINa+(t) = eA DNa+(V, t)L!Na+eV (t)kBT[Na+]in− [Na+]oute−eV (t)kBT1 − e−eV (t)kBT(5.2)where the possible transient properties of the current are set by the temporal de-pendence of D, and the po ssible switching of the current with volta ge is set by thevo ltage dependence of D, so that D = D(V, x, t). We can rearrange t he equationinto a more suitable formINa+(t) ="AL·e2DNa+(V, t)[Na+]inkBT#· V (t)1 − e−e[V (t) − VN a+]kBT1 − e−eV (t)kBT(5.3)which is in the f orm of a conductance times a voltage, i.e.,INa+(t) = gNa+(V, t) · V (t)1 − e−e[V (t) − VN a+]kBT1 − e−eV (t)kBT(5.4)FIGURE - chapt-8-i-v.epsFor changes in potential that a r e |V | <kBTe, the current can be approximated bya linear relation (at least it often is approximated by a linear relation; it dependson the definition of is)INa+(x, t) ≈ gNa+(V, t) · (V (t) − VNa+) (5.5)The essential nonlinearities of the membrane as well as the channel are thenincorporated into gNa+→ gNa+(t, V ), are detailed below. While this preserves theessential nonlinearities for action potential generation, it obfuscates the biophysics.The total current, Im(t), incorporates both voltage dependent, i.e., gNa+(V, t) andgK+(V, t) and voltage independent, i.e., gCl−, terms. By tradition, all the voltageindependent terms are lumped and called gLeak. To jump to the chase, we writeτ∂V (t)∂t= −rm2πaIo(t) −rmgNa+(V, t)2πa· V (t)1 − e−e[V (t) − VN a+]kBT1 − e−eVkBT(5.6)−rmgK+(V, x, t)2πa· V (t)1 − e−e[V (t) − VK+]kBT1 − e−eVkBT− −rmgleak2πa[V (t) − Vleak)]25.2 Functional Form of the ConductancesThe business end is the form of the conductances gion(V, t), although in the labora-tory one measures the current. The expectation is that the conductance is in theform of a scale factor times a voltage (and time) dependent term for the opening ofchannels, denoted Popen(V, t). This probability is itself the product of any number ofvo ltage and time dependent terms that sense the membrane voltage and either acti-vate the or inactivate the current, denoted Pact(V, t) and Pinact(V, t) = 1 − Pact(V, t),respectively.Thus for each channel we can writegion(V, t) = ¯gPopen(V, t) (5.7)= ¯g (Pact(V, t)P′act(V, t) · · · (1 − P′′act(V, t)) (1 − P′′′act(V, t)) · ··)= ¯g (Pact(V, t)P′act(V, t) · · · P′′inact(V, t)P′′′inact(V, t) · ··)In practice, channels t hat have been identified to date have identical activatingand identical inactivating terms. For example, we will see that the sodium currentis of the formgNa+(V, t) = ¯gNa+P3act(V, t)P′inact(V, t) (5.8)In general, the activation and inactivation terms are governed by a first orderequation that describes their dynamic. We havePact(V, t) + Pinact(V, t) = 1 (5.9)anddPact(V, t)dt=1τact(V )Pinact(V, t) −1τinact(V )Pact(V, t) (5.10)= − 1τact(V )+1τinact(V )!Pact(V, t) +1τinact(V )= − 1τact(V )+1τinact(V )!Pact(V, t) + 1τact(V )+1τinact(V )!Pact(V, ∞)= − 1τact(V )+1τinact(V )!(Pact(V, t) − Pact(V, ∞))where Pact(V, ∞) is the steady value of the a ctivation. ThusdPact(V, t)dt= −Pact(V, t) − Pact(V, ∞)τobs(V )(5.11)In total, there are two inherently voltage dependent terms, the previous steadystate va lue and the observed time constant. We consider the steady-state behaviorand kinetics of a two-state system as a means of understanding and parameterizing3the basic physics of these terms in the current. The idea is that a thermal average ora population of two-state systems is a reasonable portrayal of ionic currents. In fact,the decomposition of macroscopic currents in terms of cha nnels is a justification forthis.FIGURE - chapt-8-channel-summary.epsFor sake of argument, lets say that the activation sensor wo rks by having a dipoleinteract with the transmembrane potential. This interaction isEnergy = −~µ ·~E = zed∂V∂x≈ zdL!eV (5.12)≡ z′eVThe steady state extent of activat io n to inactivation is given by the usual Boltz-man relationPact(V, ∞)Pinact(V, ∞)= ez′e(Vss−Vth)kBT(5.13)where Vthis the internal potential drop across the activation sensor, and, asbefore, Vssis the steady-state level of the voltage. ThusPact(V,


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UCSD PHYS 171 - Action Potentials

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