44
OutlineforToday
• Courseorganizationandlogistics• “Electricity101”• TheElectricalPropertiesofNeurons• TheDeterminantsofMembranePotential
Courseorganization
• 201ACo-Directors:– PeterSargent– JenniferWhistler
• Partof200,201A,201B,201Csequence• Overallcoursedirector,EricHuang• Meets3-4morningsaweekforlectureordiscussion• Coursematerialsare(shouldbe)onthewebsite
Week Date Day Room Time Topic Leader
week$1 9/19/16 Mon Neuroscience$Asilomar$Retreat9/20/16 Tue Neuroscience$Asilomar$Retreat9/21/16 Wed no$class9/22/16 Thu GH@S261 9a@11a Membrane$Potential Peter$Sargent9/23/16 Fri GH@S261 9a@11a Action$Potentials Peter$Sargent
week$2 9/26/16 Mon GH@S261 9a@11a Single$Channels Peter$Sargent9/27/16 Tue GH@S261 8a@11a Biophysics$Problem$Set Peter$Sargent9/28/16 Wed no$class9/29/16 Thu GH@S261 9a@11a Structure@Function$1 Yuriy$Kirichok9/30/16 Fri BH@209 9a@11a Structure@Function$2 Yuriy$Kirichok
week$3 10/3/16 Mon Rosh$Hashana10/4/16 Tue Rosh$Hashana10/5/16 Wed no$class10/6/16 Thu GH@S261 8a@11a Potassium$Channels/Paper$Discussion Lily$Jan10/7/16 Fri BH@209 9a@11a Synaptic$Transmission$1 Peter$Sargent
week$4 10/10/16 Mon GH@S261 9a@11a Synaptic$Transmission$2 Peter$Sargent10/11/16 Tue no$class10/12/16 Wed Yom$Kippur10/13/16 Thur GH@S261 8a@11a Synaptic$Transmission$Problem$Set$&$Paper$Disc. Peter$Sargent10/14/16 Fri BH@209 9a@11a Integration$1 Kevin$Bender
week$5 10/17/16 Mon GH@S261 9a@12n exam$1,$in$class10/18/16 Tue GH@S261 9a@11a Integration$2 Kevin$Bender10/19/16 Wed no$class10/20/16 Thu GH@S261 8a@11a Integration$Paper$Discussion Kevin$Bender10/21/16 Fri BH@209 9a@11a Glutmate$Receptors Roger$Nicoll
week$6 10/24/16 Mon BH@261 9a@11a Plasticity Roger$Nicoll10/25/16 Tue GH@S261 8a@11a Plasticity$Paper$Discussion Roger$Nicoll10/26/16 Wed no$class10/27/16 Thu GH@S261 9a@11a Neurotransmitter$Release Rob$Edwards10/28/16 Fri GH@S261 9a@11a Neurotransmitter$Reuptake Rob$Edwards
week$7 10/31/16 Mon GH@S261 8a@11a Release/Reuptake$Paper$Discussion Rob$Edwards11/1/16 Tue GH@S261 9a@11a Receptor$Pharmacology$1 Jennifer$Whistler11/2/16 Wed no$class11/3/16 Thu GH@S261 9a@11a Receptor$Pharmacology$2 Jennifer$Whistler11/4/16 Fri BH@209 8a@11a Receptor$Pharmacology$Paper$Discussion Jennifer$Whistler11/4/16 Fri BH@209 11a@12n In$class$review
week$8 11/7/16 Mon GH@S261 9a@12n exam$2,$in$class11/8/16 Tue 9a@11a11/9/16 Wed 9a@11a11/10/16 Thu 9a@11a11/11/16 Fri 9a@11a Veteran's$Day,$no$class
SfN$starts$November$12
Schedule(2016)
Exam#1:45%
Exam#2:55%
Assessment
• Twoexams:October17,November7– Firstexam:inclass.3hours.Openbook,butnowebaccess.
– Secondexam:inclassor take-home.Classdecides.
• Thecourseisletter-graded(A,B,C,etc.)• YoumustgetaBorbettertopassthecourse(programspecific)
• Historically,thecoursehasbeengraded“B-modal”
Resources• Texts– onreserve(RutterCenterlibrary)
– FainG(2014)MolecularandCellularPhysiologyofNeurons,2nd ed.,HarvardUniversityPress.(1-dayreserve)
– Hille B(1984)IonChannelsofExcitableMembranes,2nd edition(currenteditionisthe3rd),Sinauer.(1-dayreserve)
– JohnstonD,WuSM(1995)FoundationsofCellularNeurophysiology.MITPress.(1-dayreserve)
– Kandel ERetal.(2013)PrinciplesofNeuralScience,4th edition(currenteditionisthe5th),McGraw-Hill.(1-dayreserve)
– NichollsJetal.(2001)FromNeurontoBrain,2nd edition(currenteditionisthe5th),Sinauer.(1-dayreserve)
– Purves Detal.(2001)Neuroscience,2nd edition(currenteditionisthe5th),Sinauer.(1-dayreserve).
• ReviewArticles• Me!8am– 9amsessions(bringquestions!)
– Friday,September23– Monday,September26– Thursday,September29– Friday,October7– Monday,October10
• Yourpeers.
I.Electricity101
• Ourexpectations
Whatwascoveredinbootcamp?
• ElectricityBasics?• Electricalpropertiesofplasmamembranes:capacitanceandconductance?
• Neuronsaspassiveconductorsofelectricity?• Electricalequivalentofanexcitablecell(fromHodgkinHuxley)?
SlideswithGreenBackgroundaresupplemental:• Provide“Color”• Providebackground• Provideexpectationsoffamiliarity
(“youshouldknowthis”)
Units• ofvoltageorpotential(potentialdifference)V orE –volts(V)
• ofchargeQ – coulombs(C)• ofcurrentI – amperes(A)• ofcapacitanceC – Farads(F)• ofresistanceR – ohms(Ω)• ofconductanceG – Siemens(S)
-
+
CathodeAnode
SymbolsUnits
FromCollegePhysics
• =aresistororaconductor;R=1/G;G=1/R•
• Voltageacrossaresistor:V=IR(Ohm’slaw)•
• Ifthinkingofasaconductor:I=GV(Ohm’slaw)•
• Voltageacrossacapacitor:V=Q/C• Inseries:resistorsadd,conductorsaddinversely,
capacitorsaddinversely• Inparallel:resistorsaddinversely,conductorsadd,
capacitorsadd
Fromcollegephysics(cont.)
• Kirchoff’s currentlaw(KCL)– thesumofcurrentsatanodeiszero.
• Kirchoff’s voltagelaw(KVL)– sumofpotentialdifferencesaroundaclosednetworkiszero.
• “Voltagedivider”“Currentdivider”
EquivalenciesintheSIsystemofunits
• Ampere(A)=1C/s• Volt(V)=unitofelectromotiveforcerequiredtodrive1Aofcurrentacross1ohmofresistance.
• Farad(F)=1C/1V• Ohm(Ω)=1V/1A• Faraday(F)– chargepermoleofunitaryions.=elementarychargexAvogadro'snumber=~1.6x10-19 Cx~6.0x1023 =9.6x104 C.
II.ElectricalPropertiesofNeurons
• Propertiesofbiologicalmembranes• Theconsequencesofcapacitanceontime-variantpropertiesofmembranepotential
• Whencellsare“isopotential”• Whencellsarenot“isopotential”
Membraneproperties• Thehydrocarboninteriorofmembranesisaninsulator
sandwichedbetweenconductors(saline):acapacitor• Thedialectric constant,e,ofhydrocarbonislow(≈2);evacuum=1;ewater=78)
• Ionsandpolarmoleculescrossvery rarely.
H2O,salts
(CH2)n
H2O,salts
Dialectric constant(EncyclopediaBritannica):propertyofanelectricalinsulatingmaterial(adielectric)equaltotheratioofthecapacitanceofacapacitorfilledwiththegivenmaterialtothecapacitanceofanidenticalcapacitorinavacuumwithoutthedielectricmaterial
Transportmechanisms
Ionmovementacrossmembranesismediatedbyproteins.
Consequencesofaddingchannelstoalipid bilayer
• Resistanceofapurelipidbilayer:≈109 ohmcm2(high!)• Q:Whatistheconductanceofa1µm2 patchofbilayer?• Q:Whatistheconsequenceofadding100channels,each
withconductanceof3pS,tothemembrane?– Calculatethis!– A:Itincreasesbyafactorof≈107.
• Q:Whathappenstothecapacitanceofthemembranewhenyouaddthese100channels?– A:verylittle!– Q:Why?– A:becausetheproteinoccupiesonlyabout1%ofthesurfacearea.
Rm =specificmembraneresistanceΩ·cm2
Cm =specificmembranecapacitanceF/cm2
Plasmamembrane=G(R)andCinparallel• >99%hydrocarbon– capacitance
<1%channels– conductance
http://www.tutorhelpdesk.com/
Extracellular
Intracellular
Anioncarryingoutwardcurrentmayeither passthroughachannelor itmaypartiallydischarge(orcharge)thepotentialacrossthemembrane
++++++++++
_____ _____
+
+
+
+ionic capacitative
RC
Eachpopulationofchannelsisequivalenttoaconductor(resistor)andabatteryinseries
• Theconductor representsthechannelsandtheirsummedconductance(butrememberthatchannelsareneverentirelyselectiveforindividualions)
• Thebattery representstheelectrochemicalgradientactingontheion
Thepresenceofcapacitanceinthemembraneresultsintime-varyingresponses
YoucancalculateVR(t),VC(t),fromI.
ChargingofthecapacitorE
E
CE
CE
Thegeometryofthecelldeterminesitspassiveelectricalproperties:(1)sphere
Spheres(cellbodies)areisopotential,sincetheresistancebetweenanytwopointswithinthesphereissmallcomparedtothemembraneresistance.Thus,themembranepotentialiseverywherethesameandtheactionpotentialoccurseverywhereatonce
cellbodyt =ReqCeq
))/exp(1()1( tt tRIeRIV mm
t
mmm --=-=-
))/(exp()( tt tRIeRIV mm
t
mmm -==-
Risingphase
Fallingphase
Whydoesfarad*ohm=second(unitsoft)?Faradºcoulomb/volt,butVoltºampere*ohm=coulomb*ohm/secThereforefaradºsec/ohm,andfarad*ohmºsec
Thegeometryofthecelldeterminesitspassiveelectricalproperties:(2)axonordendrite
exponentialdecay
Whatdeterminestherateofdecayoflocalpotentialswithdistancealongcylinders?Therelativevalueofthe“escape” andaxialresistances.
Axonsanddendritesarenotisopotential.Thereissignificantaxialresistance.
The“lengthconstant”isameasureofthedegreetowhichapotentialdecayswithdistancealonganaxon
• Lengthconstant,l,=distanceoverwhichthesteadystatesignaldecaysto1/e(37%)ofitsoriginalvalue
rm
ri
ro
i
m
oi
mr
rrr
r»
+=
)(l sinceusuallyri >>ro
Thelengthconstantisdefinedatsteadystate,whendV/dt=0
• It’smoreprecisetocalculatewhatwouldhappenasafunctionofdistancetotransient signals(likeEPSPs)
• Thecableequation (http://en.wikipedia.org/wiki/Cable_theory)
•
•
Usefulforcalculatinghowsignalswilldecayasafunctionoftimeanddistancefromtheirpointoforigin.
VtV
xV
+¶¶
=¶¶ tl 2
22
mm
i rV
tVc
xV
r+
¶¶
=¶¶
2
21
Substitutinginlengthconstant(l)andtimeconstant(t)
TheCableEquation
Actualdistance/electrotonic distance
Ifd=0.2l?Ifd=2l?
d
Electronicdistance,expressedasafractionofl
Synapticresponsesareslowerandsmallerthegreatertheirelectronicdistancefromtherecordingsite
FromBekkers andStevens(1996)
filtering
III.TheDeterminantsofMembranePotential
• Twofactorsdeterminethemembranepotentialofacell:1. Thepermeability(conductance)ratiosofitsmembrane
topermeantions,and2. Theconcentrationgradientsofthoseionsacrossthe
membrane.
• Thesodiumpumpdetermines,directlyorindirectly,theconcentrationgradientsofions(#2above)andtheyalsomakeasmallcontributiontomembranebybeingelectrogenic.
RestingPotentialandtheDeterminantsofMembranePotential
• Neurons(andmostothercells)haveastanding/restingpotentialthatisnegativeinsidewithrespecttotheoutside(definedas0mV).Thispotentialistypicallyintherangeof-40mVto-90mV.
• Thenegativepotentialarisesbecauserestingcellshaveahighpermeabilitytopotassium,whichismoreconcentratedinsidethecellthanout,andalowpermeabilitytosodium,whichismoreconcentratedoutsidethecellthaninside.
Ionsareasymmetricallydistributedacrossthemembrane
Ion Intracellular(mM) Extracellular(mM)
Potassium 135 4
Sodium 18 145
Chloride 8 105
Calcium 0.00005 1.5
Magnesium 0.2 2
Bicarbonateandotherorganicanions
10 25
Largeanions 135
Na+K+
Ca2+
Cl-
HCO3-
A- Cl-Na+K+
Buildingacell,part1:StartwithK+ channelsonly
• Potassiumcontinuestoleavethecelluntiltheconcentration“force”isbalancedbytheopposingelectricalpotential.
i
o
i
oK K
KmVKK
zFRTE
][][log58
][][ln ==
Thecell,andpotassium,areatequilibrium.
Movementofionsinsolution(Hille,2011)
• Diffusionflux,Fick(1855)• Ds isthediffusioncoefficient,whichhastheunitsofcm2/s.
• Meansquaredisplacement (onedimension)
MS = −DSdcSdx
r2 = 2Dt
1.
t ≈ r2
2D
For2and3dimensions,thedenominatoris4Dand6D,respectively.
Time Distance
0.01µm 100ns
0.1µm 10µs
1µm 1ms
10µm 100ms
100µm 10s
1mm 1000s
ForglucoseD=0.5x10-5 cm2/s
Movementofionsinsolution(cont.)
• Undertheinfluenceofanelectricfield,
MS = −uScSdψdx
Convertingtocurrent IS = zSFMS = −zSFuScSdψdx
Molarflux
Replacingdy/dx withE/d … IS =−zSFuScS
dE
uS electricalmobilitycS concentrationzS valenceΨ potential
Whatisthis?
2.
Nernst-Planckequation
• Nernst(1888)andPlanck(1890)realizedthatthetwotermsdescribingtheinfluenceofelectricfieldsandBrownianmotiononionscouldbecombinedintoasingleexpression.
• Weneedtoconvertmolarfluxintocurrent,where• Multiplyequation#3bytoproduce
• SubstituteforfromtheEinsteinrelation(1905),relatingdiffusionandelectricalmobility,toget:
MS = −DSdcSdx
−uScSdψdx
Combining1. and2.
IS = zSFMS
3.
uS
IS = −zSFDSdcSdx
− zSFuScSdψdx
DS =uSRTzSF
IS = −zSFDSdcSdx
− zSFDszsFRT
cSdψdx
zSF
Nernst-Planck
• Rearrangingterms…
• WhenIs =0,
• Integratingoverx,
÷øö
çèæ +-=
dxd
RTcFz
dxdcfDzI SSS
SSSy
( )SSSS
cdxd
FzRT
dxd
cFzRT
dxd ln1
-=-=yy
1
221 ][
][lnSS
FzRTEEES
=D=-
NPE:NernstPlanckequation
Forourpurposes,2is“outside”andE2 =0.
Modelcell,permeableonlytoK+
• IfKo is4mMandKi is135mM,Vm willgoto~-89mV. i
o
i
oK K
KmVKK
zFRTE
][][log58
][][ln ==
• Butwait!IsitOKtoassumethattheconcentrationshaveremainedconstant?
• Consideracellthatisaspherewitha25µmdiameter.• vol.=8x10-12 l.;surfacearea=2x10-5 cm2
• Whatisthecapacitanceofthemembrane?• Q=CV,V=Q/C• Capacitanceofbiologicalmembranes@ 10-2 F/m2 =1µF/cm2;
thereforeC=2x10-11 F• Calculatecharge(Q)neededtochargethemembraneto90mV
• Q=CV=2x10-11 F*0.09V=1.8x10-12 coulombs• Thismuchchargecarriedby1.8x10-17 molesofmonovalention,which,in
thiscell,correspondstoabout2µM (1partin70,000).
Buildingacell,part2:AddNa+ channels
• Neitherpotassiumnorsodiumisatequilibrium.Althoughnetcurrentiszero,thissituationisnotsustainable.
• Thesodiumpump(Na+/K+-dependentATPase)comestotherescue.
Sodium-potassiumATPase
• Coupled K+ entry)• Drivenbymass
action underphysiologicalconditions,meaningthatsubstrateavailabilityislimiting
• Electrogenic,meaningthatitproducesacurrent
GoldmanHodgkinKatz(GHK)equation
• Thisappliestonon-equilibriumsituationswhentotalmembranecurrentiszero.It’sthusamoregeneralexpressionthantheNernstequation,whichreferstoanequilibriumsituation.
• Goldman(1943)andHodgkinandKatz(1949)• Assumptions:
– ionfluxisinfluencedbothbytheconcentrationgradientandbytheelectricfield,accordingtotheNernst-PlanckEquation
– ionscrossthemembraneindependentlyofoneanother(nophysicalorelectrostaticinteraction)
– theelectricfieldisconstant(constantfield).
oCliNaiK
iCloNaoKrev ClPNaPKP
ClPNaPKPFRTE
][][][][][][ln
++++
=
• Ionsdon’tinteractwithoneanother.• Nosaturation• Nointeractionoftheionwiththechannel(whichwouldviolatetheconstantfieldassumption)
• Nochannelblock
• Nonethelesstheequationhasproventobefairlyaccurate.Someoftheassumptions(e.g.,block,saturation)havelittleeffectontheestimateofreversalpotential.
SomeoftheassumptionsusedtoderivetheGHK:
arewrong!
Anelectricalapproach(g,notP)
0=++= ClKNai IIII
)( imii EVgI -= Ohm’slaw:current=conductance*drivingforce
0)()()( =-+-+- ClmClKmKNamNa EVgEVgEVg
SolveforVm
Cli
ClK
i
KNa
i
Nam E
ggE
ggE
ggV
S+
S+
S=
• NotasgeneralasGHK• If youchangetheconcentrationorpermeantionsoutsidethecell,gis
likelytochange,butnotP• Nonetheless useful• BoththisexpressionandGHKreducetoNernst whenonlyoneionis
considered,assumingthatgandPareequivalent.
drivingforce =extenttowhichanionisoutofequilibrium
ElectricalEquivalentofaNeuron
Convention:directionoftravelofpositiveions=directionofcurrentflowConvention:inwardcurrentisnegativeinsign(FranklinB)
1. Bygeneratinggradientsofpermeantions2. Bypassingcurrent.GHKdoesnot accountforthis
contribution.
• Inamodelwithonlysodiumandpotassiumcurrents,whatistheratioofiNa andiK fluxthroughchannelsintheabsenceofthepump?
• Inthepresenceof thepump?
Howdoesthepumpcontributetotherestingpotential?
Leakcurrentandtherestingcell
• Whatarethechannelsthataccountfortherestingcell’s“K-like”potential?Arethesesimplyafewofthevoltage-dependentpotassiumchannelsthathappentobeopenatrest?
• No.– HCN(hyperpolarizationactivatedcyclicnucleotidegated),cation
channels,whichproducetheIh current– 2PK channels– Mcurrent(K)– BK andSK channels(calciumactivated)– NALCNchannel(NaLeakChannelNon-selective),cation
Thepropertiesofleakchannels,actingtogether
• Whatistheinfluenceofleakchannelsonsignaling?• Whatisthedifferencebetweentheinfluenceofleakchannels
andthatofchloridechannels,whenchlorideisatequilibrium?
• Endofday1
