Bioelectricity and Biophotonics Engineering - Loughborough University PDF

WSC331 Bioelectricity and Biophotonics Engineering Felipe Iza P 3G Plasma and Pulsed Power Group Loughborough University, U.K. [email protected]...

WSC331 Bioelectricity and Biophotonics Engineering Felipe Iza P 3G Plasma and Pulsed Power Group Loughborough University, U.K. [email protected] Slide Set Bioelec 8 http://www.lboro.ac.uk/departments/meme/staff/felipe-iza 1  Voltage clamp  Technique to measure ion currents while holding the transmembrane potential constant  Insights into the channels conductivity  The basic idea: Interior / Cytoplasm Im 𝑑𝑉 𝑚 +∑ 𝐼𝑖𝑜𝑛 + IC Ik gk INa gNa ICl gCl 𝐼 𝑚=𝐶𝑚 Vm Cm Ek ENa ECl 𝑑𝑡 - Extracellular medium Voltage clamp measurement: Measure Im at different Vm 2 Voltage Clamp (IV)  The current required to keep Vm equal to Vc is measured & recorded.  This current is the ion flux across ion channels as voltage-gated channels open & close. 3  Patch clamp  Limitation of voltage clamp technique: we control many channels and of different types at the same time. Not all the channels experience the same transmembrane potential unless special measures are in place.  Can we look at one channel at a time?  Patch clamp  Patch clamp looks a small area of the membrane (m2) rather than the whole cell.  Challenge: Smaller currents!!! 4  Patch clamp: Current traces  Patch-clamp current recordings have discontinuities that reflect the opening and closing of individual channels.  Channels have typically 2 states: open and close  Time in each state varies randomly with a given probability of being open and close. Registration of the flow of current through a single ion channel at the neuromuscular endplate of frog muscle fiber with patch clamp method. (From Sakmann and Neher, 1984.) 5 Today’s lecture  Ion Channels  Link between microscopic and macroscopic quantities  Macroscopic model: Dynamics of a First order system  Hodgkin-Huxley model:  K and Na channels 6  Micro to Macro (I)  Let p be the probability of a single channel being open.  Probability of a channel being close q=1-p.  Let’s assume we have N such channels. Note: This is a Binomial distribution  The expected (or mean) value of open channels is ⟨ 𝑁 𝑜 ⟩ =𝑝𝑁  The standard deviation of the distribution is: 𝜎 𝑁 =√ 𝑝(1−𝑝)𝑁 𝑜 7  Micro to Macro (II)  Matlab demo 8  Micro to Macro (III)  Current through a single open channel 𝑖𝑝 =𝛾 𝑝 (𝑉 𝑚 − 𝐸 𝑝 )  Current through the membrane 𝐼 𝑝= ∑ 𝑖𝑝 =𝑁 𝑜 𝛾 𝑝 (𝑉 𝑚 − 𝐸𝑝 ) ¿𝑎𝑙𝑙 ¿ 𝑐h𝑎𝑛𝑛𝑒𝑙𝑠  Macroscopic membrane conductance ⟨ 𝑔𝑝 ⟩=𝑝𝑁 𝛾𝑝 9  Micro to Macro (IV) 𝑔 𝑝 =𝑝𝑁 𝛾 𝑝  We can measure p using the patch clamp technique  We can measure gp for a membrane using the voltage clamp technique  We can determine pN from the above relation and try to make a physical meaning of it (historical approach – Hodgkin & Huxley)  However, here we are going to assume pN as a means to determine gp. 10 Ion Channels – Macroscopic Kinetics A simplified model Consider:  N channels of a given ion type  Channels are independent but governed by the same statistics  Channels are bi-stable, i.e. they are open or closed  Transition between open and closed states is stochastic Question:  How can we describe the number of open/close channels?  And how does that number change with transmembrane voltage? Membrane 11  Macroscopic Channel Kinetics 𝑁 =𝑁 𝑐 (𝑡)+𝑁 𝑜 (𝑡) N: # of channels No: # of open channels at time t Nc: # of closed channels at time t 𝛼 𝑁𝑐 → ← 𝑁 𝑜 (𝑡 ) : Rate constant for switching from 𝛽 closed to open : Rate constant for switching from open to closed 𝑑𝑁 𝑐 Note:  and  are assumed to depend only on the =𝛽 𝑁𝑜 −𝛼𝑁𝑐 transmembrane voltage Vm, i.e. for 𝑑𝑡 a given Vm,  and  are constant. 12  Macroscopic Channel Kinetics 𝑁 =𝑁 𝑐 (𝑡)+𝑁 𝑜 (𝑡) 𝑑 𝑁𝑜 + ( 𝛼+ 𝛽 ) 𝑁 𝑜=𝛼 𝑁 𝑑𝑡 𝑑𝑁 𝑐 =𝛽 𝑁𝑜 −𝛼𝑁𝑐 𝑁 𝑜 (𝑡)= 𝐴 exp [ − ( 𝛼+ 𝛽 ) 𝑡 ]+ 𝛼 𝑁 𝑑𝑡 𝛼 +𝛽 The above equation gives the evolution of the number of open channels in time. Say V changes   and  change the number of open channels will evolve in time reaching a final number of : 𝛼 𝑁 𝑜 (𝑡 →∞)= 𝑁 𝛼 +𝛽 13  Steady state 𝛼 𝑁 𝑜 (𝑡 →∞)= 𝑁 𝛼 +𝛽 After a sufficient long time (t ):  The average number of open channels is constant. Channels will fluctuate between the open and closed states but the number of channels that close and the number of channels that open are equal.  The average number of open channels depend on  and  at present time, i.e. after a sufficient long time, any history is lost. 14  Exercise  How does the number of open channels vary when =0.02ms-1 and =0.1ms-1 in the following cases: 1. All channels are initially closed 2. All channels are initially open 3. Half the channels are initially open 15  Solution 𝛼 𝑁 𝑜 (𝑡)= 𝐴 exp [ − ( 𝛼+ 𝛽 ) 𝑡 ]+ 𝑁 𝛼 +𝛽 Determine A based on the initial condition: 𝛼 𝛼 1. 0= 𝐴+ 𝛼 +𝛽 𝑁 ⇒ 𝐴=− 𝛼+ 𝛽 𝑁 𝛼 𝛽 2. 𝑁 = 𝐴+ 𝛼+ 𝛽 𝑁 ⇒ 𝐴= 𝛼+ 𝛽 𝑁 𝑁 𝛼 𝛽 −𝛼 = 𝐴+ 𝑁 ⇒ 𝐴= 𝑁 3. 2 𝛼+ 𝛽 2(𝛼 +𝛽 ) 16  Solution Substitute A in the general solution and plot: 1 𝛼 0.9 𝑁 𝑜 (𝑡)= 𝐴 exp [ − ( 𝛼+ 𝛽 ) 𝑡 ]+ 𝑁 0.8 𝛼 +𝛽 Average number of open channels 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 20 40 60 80 100 Time (ms) 17  Reminder… 𝑑𝑥 1 + 𝑥=𝐾 𝑑𝑡 𝜏 Solution:  xf xi to 18  Alternatively… 𝑑𝑥 1 𝑑 𝑁𝑜 + 𝑥=𝐾 𝑑𝑡 𝜏 + ( 𝛼+ 𝛽 ) 𝑁 𝑜=𝛼 𝑁 𝑑𝑡 19  Exercise  Compare the time evolution of No and No4 when =1ms- 1 , =0.2ms-1 and all channels are initially closed. 20  Solution 𝑑 𝑁𝑜 + ( 𝛼+ 𝛽 ) 𝑁 𝑜=𝛼 𝑁 𝑑𝑡 𝛼 𝑁 𝑜 (𝑡)= 𝐴 exp [ − ( 𝛼+ 𝛽 ) 𝑡 ]+ 𝑁 𝛼 +𝛽 𝛼 𝛼 0= 𝐴+ 𝑁 ⇒ 𝐴=− 𝑁 𝛼 +𝛽 𝛼+ 𝛽 𝛼 𝑁 𝑜 (𝑡)= 𝑁 {1−exp [ − ( 𝛼 +𝛽 ) 𝑡 ] } 𝛼+ 𝛽 ( ) 4 𝛼 4 4 𝑁 𝑜 (𝑡)= 𝑁 { 1− exp [ − ( 𝛼+𝛽 ) 𝑡 ] } 4 𝛼 +𝛽 21  Solution 1 0.8 Note: Fraction of open channels 0.6  No/N = Probability of one channel being open 0.4  No4/N4 = Probability of four 0.2 channels being open simultaneously 0 0 1 2 3 4 5 Time (ms) close open close close open close open close close open close close open close close close 22  Potassium channels  The potassium channel can be thought as consisting of 4 equal subunits (n) that need to undergo a conformational change in order for the channel to open. 𝑔𝐾 =𝑝𝐾 𝑁𝛾𝐾 Membrane conductance gk Probability of a channel being open pk Probability of a subunit being open no  Assuming first order kinetics for the subunits: 𝛼𝑛 → 𝑑 𝑛𝑜 𝑛𝑐 ← 𝑛 𝑜 (𝑡 ) =𝛼𝑛 (𝑛−𝑛𝑜 )− 𝛽𝑛 𝑛𝑜 𝛽𝑛 𝑑𝑡 23  Potassium channels 𝛼𝑛 → 𝑑 𝑛𝑜 𝑛𝑐 ← 𝑛 𝑜 (𝑡 ) =𝛼𝑛 (𝑛−𝑛𝑜 )− 𝛽𝑛 𝑛𝑜 𝛽𝑛 𝑑𝑡  Probability of a subunit being open = no/n = number of subunits open / total number of subunits 𝛼𝑛 𝑛𝑜(𝑡)=𝐴exp(¿−𝑡/𝜏𝑛)+ 𝑛¿ 𝛼𝑛+𝛽𝑛 24  Sodium channel  The sodium channel can be thought as consisting of 4 subunits that need to undergo a conformational change in order for the channel to open.  Three subunits are identical (m) and the fourth one is different (h) 𝑔𝑁𝑎=𝑝𝑁𝑎 𝑁𝛾𝑁𝑎 Membrane conductance gNa Probability of a channel being open pNa 25  Overview  Channel composed of various subunits  For each subunit, the rate constant for switching from closed to open and from open to closed depend on the applied voltage: Potassium: n and n are function of Vm Sodium: m, m, h, h, are function of Vm  As ’s, ’s change, the number of open subunits follows first order kinetics, e.g.: 𝑛𝑜 (𝑡) 𝑛∞ (𝑛∞ −𝑛 𝑜𝑖𝑛𝑖𝑡𝑖𝑎𝑙) 1 𝛼𝑛 = − exp(¿ −𝑡/𝜏 𝑛 )with 𝜏 𝑛 = and 𝑛∞= 𝑛¿ 𝑛 𝑛 𝑛 𝛼 𝑛 + 𝛽𝑛 𝛼 𝑛 +𝛽 𝑛  The probability of a channel being open is then the multiplication of the probability of each subunit being open: Potassium: pK=no4 Sodium: pNa=mo3ho  The membrane conductance is then the product of the number of open channels (total number of channels times the probability of a channel being open) and the conductance of a single open channel: 𝑔𝑝=𝑝𝑝 𝑁 𝑝 𝛾𝑝 26 Today’s lecture  Ion Channels  Link between microscopic and macroscopic quantities  Macroscopic model: Dynamics of a First order system  Hodgkin-Huxley model:  K and Na channels 27 Next Lecture  Hodgkin-Huxley model:  Voltage dependence of rate processes  Subthreshold excitation  Action potentials 28