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Biophysics Notes Membranes and Transport Membrane Biophysics A. Nernst Equation Gconc = kBT ln(Co/Ci) Gvoltage = qV, at equilibrium, qV = kBT ln(Co/Ci) V = kBT/q ln(Co/Ci) *** = RT/F ln(Co/Ci), with F = 96,400 Coulomb/mole (kB = 1.38 x 10-23 J/K, R = 8.31 J/mol.K) B. Goldman-Hodgkin-Katz equation P C i Pi C i 1 k B T i i i 2 , [P] = cm/s, for example: ln V = q P C i 2 Pi C i i i 1 i V = k B T Pk K o PNa Na o PCl Cl i ln P K P Na P Cl q Na Cl i i o k Example: Approximate Neuron: K b Na o o , with b = P /P Na K K b Na i i V = 58 mV log b = 0.02 for many neurons (at rest). [K]i = 125 mM [K]o = 5 mM [Na]i = 12 mM [Na]o = 120 mM V = -71 mV Can define Vk = 58 mV log([K]o/[K]i) = -80 mV (this is at 298 K) VNa = + 58 mV. The value of b Vmembrane closer to Vk. 1 Straight Line PNa = 0 (Nernst), Curved fit is using Goldman-HodgkinKatz, vary Ko and measure Vm. Do Soma 1 (Nernst) , C. Electrical Model Mammalian Cells Squid Axon VNa (mV) +67 +55 VK (mV) -84 -75 VCl (mV) -60 -60 VCa (mV) +125 +125 Analyze Circuit with kirchoff’s laws (or Ohm’s law) 1) Iin = Iout 2) V around loop = 0 Let input current, I = 0 (No net current) ICl + ICa + INa + IK = 0 For each loop have Ii = 1/Ri (Vm – Vi) w/ Vm = membrane potential difference Let 1/R = g (like conductance) Eg. IK = gK (Vm – VK), squid axon, Vm = -60 mV IK = gK (-60 – (-75)) mV = gK(+15 mV). g always positive. [K]in =125mM V = V – V , positive current = positive ions flowing out of the cell. in out Vm = -60 VK = -75 Vm not sufficient to hold off K flow so ions flow out. When Vm = Vk then no flow. [K]out = 5 mM Kirchoff’s rules Ii = 0 gi (Vm – Vi) = 0 2 g i Vi g Cl VCl g K VK g Na VNa g Ca VCa , which is equivalent to g Cl g K g Na g Ca gi Goldman-Hodgkin-Katz equation. Vm = Do Soma 3 D. Donnan Equilibrium Say have two equally permeable ions: K+ and Cl- and have A with charge Z inside cell w/ no permeability Vk = 58 mV log([K]o/[K]i) VCl = -58 mV log([Cl]o/[Cl]i), At equilibrium, Vm = VK = VCl log([K]o/[K]i) = -log([Cl]o/[Cl]i) KoClo = KiCli Donnan Rule (Dropping []) [product permeable inons outside] = [product permeable ions inside] Other conditions: electroneutrality osmolarity Goldman- Hodgkin-Katz For our simple system, electroneutrality Ko = Clo, Ki = Cli + ZA Putting this is the Donnan rule Ko2 = KiCli = Ki(Ki – ZA) Ki2 – ZAKi = Ko2 [complete square] (Ki – AZ/2)2 – (AZ/2)2 = Ko2 Ki = (Ko2 + (AZ/2)2)1/2 +AZ/2 2Ki = (4Ko2 + A2Z2)1/2 + AZ 2K o Vm = 58 mV log([K]o/[K]i) = 58 mV log 1/ 2 2 4K o A 2 Z 2 AZ + Both K and Cl are at equilibrium at this unique potential. E. Animal Cell Model 1. Model Ci (mM)* Co (mM) P>0? K 125 5 Y Na+ 12 120 N** Cl5 125 Y A108 0 N H2O 55,000 55,000 Y * Should really use Molality (moles solute/ kg solvent) instead of per liter – accounts for how molecules displace water (non-ideality). + 3 ** More on this later 2. Maintenance of Cell Volume. Membrane evolved to keep stuff in. a) Osmolarity used to define [H2O] – add sugar to H20 and [H2O] goes down since V increases. Solution of 1 mole/liter of dissolved particles is 1 osmolar As osmolarity increases, [H2O] decreases 1 M NaCl = 2 Osmolar Osmosis – diffusion of H2O down concentration gradient N.B. In preparing Hb, put RBC in dI water, H2O goes in , not everything permeable lysis. b) Cell Volume. Ex - Keeping P inside osmolarity inside Not same as that outside H2O flows in Lysis. Si = So (Diffusion – equate chemical potential) Si + Pi = So (Osmolarity) How solve this paradox? Note (this same thing happens in dialyzing proteins). 4 c) How keep P in and not lyse cells? 1) Pwater = 0 – hard to do but some epithelial cells do it. 2) Cell wall – no swelling = plants and bacteria 3) Make membrane exclude extracellular solutes Tonicity: isotonic – no effect of volume hypotonic – cause swelling hypertonic – cause shrinking Consider if in the diagram above we now made the cell also impermeable to S, then the problem would be solved when Si = So. 3. Impermeable Sodium Model Apply electroneutrality (for permeable ions) Donnan Equilibrium Osmotic balance Electroneutrality Clo = 125mM Donnan KoClo = KiCli (5)(125) = Ki5 Ki = 125 mM Osmolarity outside – 250 mOsm need inside = 250 mOsm 250 = 125 + Nai + 108 + 5 Nai = 12 mOsm VK = -25 mV ln(125/5) = -81 mV VCl = -25 mV ln(125/5) = -81 mV Vm = -25mV ln[(125+125)/10] = -81 mV F. Diffusion and Active Transport 1. Pi = (ikBT)/d = Di/d, with D – diffusion constant [cm2/sec] 2. In reality, cell is permeable to sodium, but pumps it out. Had (Goldman-Hodgkin-Katz) 5 k BT Pk K o PNa Na o Vm = ln P K P Na q Na i i k Now, with pump K o Na o , with = (n/m)(PNa/PK), Vm = Vo ln K i Na i n/m = 2/3 Vm VK. 3. Na-K Pump - Two sets of two membrane spanning subunits Phosphorylation by ATP induces a conformational change in the protein allowing pumping Each conformation has different ion affinities. Binding of ion triggers phosphorylation. Shift of a couple of angstroms shifts affinity. Exhibits enzymatic behavior such as saturation. 4. Electrical Model Now include pump: g Cl VCl g K VK g Na VNa g Ca VCa I p I Vm = g Cl g K g Na g Ca Do Soma exercises 4 and 5 (start). G. Patch Clamping 1. Technique - Measure individual ion channels - Elongate Pipette - Can pull patch away or not detach it at all - Usually maintain V fixed across membrane – See if channel open or closed observe current is quantized: channel is either open or closed: easy to get conductivity, I = gV. 6 Vh = holding potential, with equal concentration of permeable ion on both sides, get g When have not equal concentration of permeable ions on both sides get Nernst potential, Vx Ix = gx (Vh – Vx) 2. Voltage Gated Channels - there exist randomness in the opening and closings - For some channels, the proportion of time the channel is open is dependent on the membrane potential (eg. Na, K channels of neurons). - Average of many channels is predictable. 3. - Reversal potential for voltage gated channels Current depends on [ions] in addition to Vm (eg. [ions] low analogy to a resistor breaking down) When ions not too low, get VNernst I = gx*(Vh – Vx) When Vh = Vx No I “reversal potential” Can reverse I with Vh Can measure Vh Can test selectivity of ions. 4. Multiple channels Can get several on a patch I t Parallel Resistors: 1/Req = 1/Ri geq = gi Series: 1/geq = 1/gi 5. Ligand gated channels n acetylcholine (nAchR) – closed until opened by binding Ach bind let both Na and K pass Not sensitive to Vm 7 I [Ach]2 Do Patch 1,2,4,5 F. Carrier Transport – example of facilitated transport. 1. Lowers activation energy 2. Obeys saturation kinetics 3. Highly selective Simple Model: K Co Yo Ci Yi CYo CYi Solute Flux Co Ci J = Jmax K C o K Ci Jmax = NYDY/d2 NY = number of carriers DY = diffusion constant of Carrier d = membrane thickness For Ci = 0, get J = Jmax [Co/(K + Co)] Jmax Co 8 Biophysics Notes Nerve Excitaion Nerve Excitation A. General Background 1. Basics a) Nerve Signals communicate stimulus-response mechanism b) Nerve signals = electrical signals i.e. modulations of membrane potential c) Patellar reflex – Knee jerk reflex. Tap Knee stretch thigh muscle sensory neuron spinal cord motor neuron muscle contraction. d) Motor vs sensory neurons Motor – cell body = soma (nucleus contained) Small – 20 –30 mm, has branched (dendrites) which receive signals Sensory – no dendrites 9 2. Historical Perspective a) Couldn’t measure electrical signals of individual cells – too small b) J.Z. Young rediscovers Giant Squid Axon – 1 mm diameter c) Already Known: there exist action potentials – electrical pulses that don’t travel like electrical current (much slower: 1 – 100 m/s vs 3 x 108 m/s) K+, Na+ concentrations play an important role, esp. Na+ Hodgkin and Katz: as [Na+] decreases the velocity of the action potential also decreases. H & H, 1963 – membrane potential reverses during impulse, goes to + 100 mV Electrical conductance of membrane increases 40x – H&H propose nerve impulses are due to transient changes in Na and K conductance: Nobel Prize in 1963. 3. Voltage Clamp a) They used voltage clamp developed by KS Cole 1940, FBA keeps V constant b) Supply I to keep V constant c) Provides experimental control Vo = K(Vc – Vm), K = amplifier gain, Vc = control voltage, Vo = output voltage, Vm = membrane voltage B. Membrane Analog - Space clamped axon – two silver wires inserted all the way: radial currents only Generally (w/o a clamp) Cm = membrane capacitance gNa, gk are now variable gL = leak current – not specific 10 INa = gNa(Vm – VNa), IK = gK(Vm -VK), IL = gL(Vm – Vk) For Cm have diplacement current (rearrangement of charges) IC = Cm dV/dt - VNa, VK, VL all determined by [ ]i and [ ]o, which along with Cm are fixed gNa and gK are variable. Itot = INa + IK + IL + Cm dV/dt; Voltage clamp dV/dt = 0 Actual clamping is complicated – often need to step potential via conditioning steps. Notes: Negative current: + ions into axon Positive current: + ions out of axon VK = -72 mV, VNa = + 55 mV V = Vin - Vout Vm = Vh = 1/gtot (gKVK + gNaVNa + gLVL + I), clamp and look at I C. Voltage Clamp experiments 1. gL, H&H found gNa, gK ~ 0 at resting potential and they are only activated when the axon is depolarized (becomes less negative). They found gL by hyperpolarizing (making more negative). I = IL = gL (Vm – VL), VL < Vresting (-60 mV) 2. Voltage steps – depolarizing V Do overview at end briefly first Depolarization early negative current late positive current As V increases The amplitide of the negative current decreases At V = 117, (Vm = 57 mV), I neg = 0 If V > 117 mV get early positive current As V increases Late positive current increases monotonically As V increases Rate of current development increases (+ and -) As V increases Switch from negative to positive current gets earlier (channels open earlier) Dynamics Do Axon 1 and 2&3 and Voltage clamp currents. 11 3. Separating INa and IK a. Observe VK = -72 mV, VNa = +55 mV, Vresting = -60mV INa = gNa(Vm – VNa), IK = gK(Vm – VK) Near Resting IK - always positive, INa – negative, for small, medium depolarizations decreasing as V increases, becoming positive for large V b. Separation I reversal could be due to larger later (positive) current or cessation of early (negative) current. To solve this, H&H make Vm = VNa so all I is IK They then varied [Na]o, set Vm = VNa and hence studied the voltage dependence of IK Then deduced voltage dependence of INa since Itot = INa + IK Do Axon 4 – Inactivation. 4. Na inactivation and de-inactivation 1. V activation , time inactivation deinactivation deinactivation is also voltage dependent 2. H&H use conditioning steps: Brief conditioning depolarization reduced INa during 2nd step As t for conditioning step increases INa 2nd step decreases (more sodium channels inactivated during conditioning step) H&H find time and voltage dependence of inactivation ex – conditioning step + 29 mV inactivation = 2 ms (nearly complete) step + 8 mV inactivation > 8 ms (less inactivation) They found that even at resting potential, many sodium channel are inactive. Do Axon 5 conductances 3. H&H used long conditioning steps to study de-inactivation This turned off (closed) Na channels then set 2nd voltage to recovery voltage – vary time to 3rd voltage to look at current. D. Empirical Equations 1. Generally, rate of change of membrane conductance g: dg/dt = (1-g) - g, = opening (fwd rate), = closing rewrite: 12 dg g g , g = g = steady state dt value of g. let u = g - g, dg = -du ln( g - g) = -t/ + c g = g - ( g - go) exp(-t/ go = g at t = 0 2. Potassium Emprically, H&H found they needed g g4, they normalized maximum conductance gk = gkmaxn4, gkmax = max conductance (all open) n = activation; n g in above equations n = n - ( n - no) exp(-t/ n4 = fraction of channels open (between 0 and 1). 3. Sodium gNa = gNamaxm3h, m is like n for K, h describes inactivation 0 m 1, m3 = fraction activated, 0 h 1: 1= fully recovered and 0 = none recovered m3h = fraction of channels open 4. Current equation Cm dV/dt = gNamaxm3h (Vm – VNa) - gkmaxn4 (Vm – Vk) – gL(Vm – VL) + I n = n - ( n - no) exp(-t/and same for m,h. These give fractions open and are between 1 and 0. Do Axon 6 Impulse conductance E. Nerve Impulse Properties 1. Resting Potential Vm = -60 mV (includes effect of ion leak), Na and K channels closed, 40% Na channels are inactivated 2. Membrane depolarized w/ brief pulse some Na channels open, Na flows in, more depolarization more Na flows in negative current 3. Vm approaches VNa (momentarily) (VNa = 55 mV) but Na channels inactivate and K channels open positive current Vm becomes less positive 4. Na channels close, K still open hyperpolarize. 5. Na channels de-inactivate, K channels close F. Impulse, Threshold and Refractory Period. 1. If apply depolarization to resting neuron, you get depolarization, but no impulse unless depolarization > threshold 2. Threshold depends on duration of pulse (A increases as duration decreases) and time after last impulse 13 3. For some duration after impulse, threshold = infinity: cannot get impulse – duration here is due to refractory period (Most sodium channels are inactive). Do Axon 7 and 8 Threshold and refractory. G. Channel Perspective 1. When muscle stretches action potential. 2. All or none 3. Na channel: Depolariztion m gate opens Both gates respond to depolarization but speed is different h gate closes = inactivation meanwhile n (K+) opens while h closed, m open is ineffective go to picture on page 8 14 15 16 H. Spread of Action Potential 1. Basic Picture depolarization at one spot causes depolarization at neighboring spots action potential there Na coming in diffuses along axon causing depolarization Depolarization can be bi-directional but doesn’t go back on itself due to refractory period. 2. Spread of V Depends on internal conductance vs that across membrane Invertebrates: large radius small R Vertebrates: insulate with glial cells (myelin sheath), breaks = nodes of Rainier = access for ions. 17 I. Synaptic Transmission 18 19 20