麻省理工学院：《电气工程与计算机科学》学习资料_Week6

Week 6 review What was covered Ion transport Active transport Steady state, rest, quasi-equilibrium, and equilibrium Indirect and direct effect of active transport Circuit review Ion transport inside outside The membrane is only permeable to ion n that has a valence, Zn (Valence is the amount of charge on one ion. So if there is a diffe concentration between the inside and the outside the ions are going to move down the concentration gradient because of diffusion. But this will begin to make charge build up on the membrane, that effect, drift, will cause the ions to flow in he opposite direction. Therefore, the flux will be due to both diffusion and drift. Flux is due to 2 phenomena Diffusion Drift ac(x, 1) -H, 2, FC (x, 1) ay(x, n) Because the movement of charged transport is easier to think about in terms of current we convert flux to current densit z F Of course there is still the Continuity equation a. (x, t) ac (, t) If the membrane is permeable to more than one ionic species, you will end up with a Nernst-Planck Equation and a Continuity Equation for each ion. So, if there are k permeable ionic species that makes 2k equations. (In this example, there is only 1 permeable ionic species and therefore there are 2 equations)

Week 6 Review What was covered: - Ion transport - Active transport - Steady state, rest, quasi-equilibrium, and equilibrium - Indirect and direct effect of active transport - Circuit review Ion transport: inside outside f The membrane is only permeable to ion n that has a valence, zn. (Valence is the amount of charge on one ion.) So if there is a difference in concentration between the inside and the outside, the ions are going to move down the concentration gradient because of diffusion. But this will begin to make charge build up on the membrane, that effect, drift, will cause the ions to flow in the opposite direction. Therefore, the flux will be due to both diffusion and drift. Flux is due to 2 phenomena: Diffusion Drift f = -Dn ¶c(x, t) - mn znFcn ( x, t) ¶y (x, t) ¶x ¶x Because the movement of charged transport is easier to think about in terms of current, we convert flux to current density. J z F x n ( , ) n n n ( , ) n n 2 ( , ) ( , Of course there is still the Continuity Equation: ¶J n ( x, t) = -z F ¶cn (x, t) n ¶x ¶t If the membrane is permeable to more than one ionic species, you will end up with a Nernst-Planck Equation and a Continuity Equation for each ion. So, if there are k permeable ionic species that makes 2k equations. (In this example, there is only 1 permeable ionic species and therefore there are 2 equations)

However, we have 2k+l unknowns (in this example, there are 3: Jn, Cn, and v). So we need one more equation. Remember, 8.02? Well you can use it here! We can use Gauss's Law and the definition of potential and get (total charge) Poissons equation ay-∑:Fe(x) Electroneutrality approximation To simplify the equations On the length scale of cells and on the time scale of biological processes, we can make the approximation that the total charge is 0. That ∑=nFcn(x,1)=0 Thats because all the charge build up on the membrane occurs very fast(charge relaxation time is on the order of nanoseconds ), is on the order of nanometers( Debye length"is-lnm for physiological conditions, and the amount is very small compared to the amount of ions in the bath Steady State Electrodiffusion through membranes Steady State still has the same definition(->0). This means that a (x, 1) ac, (x, n) 0 from the continuity equation and so it implies that Jn Then you can solve for Jn through the membrane from the Nernst-Planck equation( see the lecture notes) G,(m-vm where Vm is the total membrane voltage(Vm=yo-y(d) Gn is the conductivity(units of s=1/@2)of the ion species n through the membrane. It's a function of the ion mobility, valence and concentration in the membrane Finally, Vn is defined as the Nernst Equilibrium Potential y RTC,(outside) From this we define a circuit model for the conductance of one ionic species through the membrane Vn o>M

However, we have 2k+1 unknowns (in this example, there are 3: Jn, cn, and y). So we need one more equation. Remember, 8.02? Well you can use it here! We can use Gauss’s Law and the definition of potential and get: ¶2 y 2 = - 1 (total charge) ¶x e Poisson’s Equation: ¶2 y 2 = - 1 � znFcn (x, t) ¶x e n Electroneutrality approximation To simplify the equations: On the length scale of cells and on the time scale of biological processes, we can make the approximation that the total charge is 0. That is: � znFcn ( x, t) = 0 n That’s because all the charge build up on the membrane occurs very fast (charge relaxation time is on the order of nanoseconds), is on the order of nanometers (“Debye length” is ~1nm for physiological conditions, and the amount is very small compared to the amount of ions in the bath. Steady State Electrodiffusion through membranes Steady State still has the same definition ( ¶ fi 0 ). This means that: ¶t ¶J n ( x, t) = -znF ¶cn (x, t) = 0 from the continuity equation and so it implies that Jn is ¶x ¶t constant. Then you can solve for Jn through the membrane from the Nernst-Planck equation (see the lecture notes): J n = Gn (Vm -Vn ) where Vm is the total membrane voltage ( Vm=y(0)- y(d) ). Gn is the conductivity (units of S=1/W) of the ion species n through the membrane. It’s a function of the ion mobility, valence and concentration in the membrane. Finally, Vn is defined as the Nernst Equilibrium Potential: V = RT ln( cn (outside) ) n znF cn (inside) From this we define a circuit model for the conductance of one ionic species through the membrane: + _ Vm Jm _ + Vm Jn G _ n + Vn

So, if we have more that one permeable ionic species you will get one branch in the circuit model G k What is the Nernst equilibrium potential? RTc(outside) v. c,(inside))=o log(n))mv at room temperature(T=250C) This means that if Vm=Vn then ionic species n is not moving across the membrane Whats the problem with this model? Well, since all the transport is passive, you will eventually make the cellular concentrations change but this doesnt happen The cell uses active transport to maintain its internal concentration of ions constant. So to account for this we add active transport as current sources in the model Jmm GK Steady State, Rest, Equilibrium, and Quasi-Equilibrium: What's the difference? (Steady State and Equilibrium still have the same definitions) Steady State: nothing changes with time (i.e.->0) at Rest: the net flux of charge particles is zero(i.e. Jm=2P+Ja=0.This is only one equation.)This implies that: V ∑c(2Gl-

So, if we have more that one permeable ionic species you will get one branch in the circuit model: What is the Nernst equilibrium potential? log( ) at room temperature (T 25 C) 60 ) ( ) ( ) ln( = � � � ł � � � Ł � = » mV c c c inside z c outside z F RT V i n o n n n n n n This means that if Vm=Vn then ionic species n is not moving across the membrane. … G1 G2 G3 Gk + V1 � + V2 � + V3 � + Vk � + Vm � Jm What’s the problem with this model? Well, since all the transport is passive, you will eventually make the cellular concentrations change but this doesn’t happen… The cell uses active transport to maintain its internal concentration of ions constant. So to account for this we add active transport as current sources in the model: … G1 G2 G3 Gk V1 + V2 � + V3 � + Vk � Jm … J p Ja a J1 a J 2 a k J + Vm � + Steady State, Rest, Equilibrium, and Quasi-Equilibrium: What’s the difference? (Steady State and Equilibrium still have the same definitions) Steady State: nothing changes with time (i.e. ¶ fi 0 ) ¶t Rest: the net flux of charge particles is zero (i.e. J m =� J n p + J n a = 0 . This is only one n equation…) This implies that: Vm = 1 � ��GnVn - J n a � � �Gn Ł n ł n

Quasi-Equilibrium: The net flux of each species is 0(i.e. JP+J4=0 for all n. This is a set of k equations: one equation for each permeable ionic species.) Equilibrium: There is no flux of anything(i.e all the J=o). This can only happen if the active transport is blocked Indirect vs. Direct Effect of Active Transport Indirect effect: If you turn off the active transport, then if you wait long enough the concentration in the cell will begin to change. (over a long time at least several hours) Direct effect. If the active transport is electrogenic, then turning it off will cause an immediate change in Vm(see equation for Vm at rest) Electrogenic active transport means that∑/n≠0 Example: The Na/K ATPase covered in lecture pumps more Nat than K+ every iteration 3Na outside Review circuits Remember our friends KVL and KCL. If you dont, come talk to a TA for a quick refresher course. (Also, Gn is conductance so now the constitutive relationship is Jn=Gn/n)

p a Quasi-Equilibrium: The net flux of each species is 0 (i.e. J + J = 0 for all n . This is a n n set of k equations: one equation for each permeable ionic species…) Equilibrium: There is no flux of anything (i.e. all the J = 0). This can only happen if the active transport is blocked. Indirect vs. Direct Effect of Active Transport Indirect Effect: If you turn off the active transport, then if you wait long enough the concentration in the cell will begin to change. (over a long time at least several hours) Direct Effect: If the active transport is electrogenic, then turning it off will cause an immediate change in Vm (see equation for Vm at rest). a Electrogenic active transport means that � J n „ 0 . n Example: The Na/K ATPase covered in lecture pumps more Na+ than K+ every iteration: 3Na+ inside outside 2K+ Review Circuits Remember our friends KVL and KCL. If you don’t, come talk to a TA for a quick refresher course. (Also, Gn is conductance so now the constitutive relationship is Jn=GnVn)