Phys498BIO; Prof. Paul Selvin Hw #9 Assigned Wed. 4/18/12: Due 4/25/08

Transcription

1 1. Ionic Movements Across a Permeable Membrane: The Nernst Potential. In class we showed that if a non-permeable membrane separates a solution with high [KCl] from a solution with low [KCl], the net charge on either side of the membrane is zero and therefore no voltage difference exists across the membrane. However, if we allow the membrane to be selectively permeable to K + ions, we find that K + ions will flow from the side of high concentration to the side of low concentration such that the entropy of the system will increase. The result of this ionic flow is a buildup of positive charge on the side of increasing K + concentration and a net negative charge on the side of decreasing K + concentration. This separation of charge produces a voltage difference across the membrane, which resists further K + flow. Equilibrium is reached where the gain in entropy from K + flow towards lower concentration exactly balances the increase in electrostatic energy from moving K + ions to a region of higher potential. (Examine the pictures below and review the lecture notes.) The equilibrium voltage difference across the membrane (the Nernst potential) is given by the Nernst equation: V = -(kt/q) ln([c inside ]/[C out ]), where C is the concentration. Keep in mind that this calculates the potential of the inside relative to the outside. The outside is taken to be 0mV. a) For the concentrations given in the picture, calculate the Nernst potential, i.e. the equilibrium voltage across the membrane. b) We would like to know if making the membrane permeable to K+ and forming the Nernst equilibrium has significantly altered the concentrations of K+. Let s calculate what percentage of potassium ions flow out of a nerve axon in order to produce V nernst. We can treat the membrane as a capacitor with capacitance, C, which stores a charge, Q, producing the Nernst potential, V nernst. A nerve axon is cylindrical in shape, but if the diameter of the cylinder (diameter of the axon) is much greater than the thickness of the dielectric (the membrane), the membrane can be viewed as a parallel plate capacitor rolled up into a cylinder. (See picture below.) Treat the axon membrane as a parallel plate capacitor of thickness d and dielectric constant ε. The axon length is 1

2 L and the radius is a. At equilibrium, write an expression for the total charge that is separated across the axon membrane. (Hint: remember or look up the equation for the capacitance of a parallel plate capacitor.) c) For a realistic cell, L = 10 μm, a=2 μm, ε = 9ε ο, and the membrane thickness is 5nm. Calculate the total number of K+ ions that moved out of the cell to produce equilibrium. Be careful with units! Calculate the total number of K+ ions that were originally in the cell at 145 mm. What percentage of K+ flowed out of the axon to produce the Nernst equilibrium? Would you say that the Nernst equilibrium perturbs ionic concentrations significantly? When an action potential fires and Na + rushes in, and then K+ rushes out, has this significantly changed the Na + or K+ concentrations? 2. Energetics of the Na + K + ATPase Transporter The Na + and K + gradient across cell membranes is partly maintained by the Na + K + ATPase transporter protein. The protein uses the energy from ATP hydrolysis to translocate 3 Na+ ions out of the cell and 2 K + ions into the cell. In this problem we will examine how much energy is required to transport the sodium compared to the potassium under physiological conditions. Let s consider a mammalian skeletal 2

3 muscle cell, which has a resting potential of 90mV. (Remember that the extracellular solution is defined to be at 0mV and the intracellular solution is at 90mV.) Ionic concentrations are as follows: external sodium=145mm, internal sodium = 12mM, external potassium=4mm, internal potassium=155mm. a) Given the above concentrations, use the Nernst equation to calculate the equilibrium potentials for Na + and K +. Which of these equilibrium potentials is closest to the actual resting potential of 90mV? Will it require more free energy to move 1 Na + ion out of the cell or 1 K + ion into the cell? Explain your answer briefly. b) The cost of free energy for moving an ion up its electrochemical gradient is given by the entropic loss plus the increase in potential energy of the ion. ΔG 2 1 = kt ln([c2]/[c1]) + q Δφ 2 1 In order to transport a single Na + from inside to outside the cell, how much of the free energy required is entropic and how much is electrostatic? What is the total ΔG for a single Na + in units of kt? c) In order to transport a single K + from outside to inside the cell, how much of the free energy required is entropic and how much is electrostatic? What is the total ΔG for a single K + in units of kt? In a few words, how does this compare to your answer in part b)? d) The Na + K + transporter ATPase moves 3 sodium ions out and 2 potassium ions in, how much total energy is required for these ionic translocations? If the protein gets 25kT of free energy from ATP, what is the transporter s efficiency of free energy use? 3. Understanding the Action Potential Below is a picture of the voltages during an action potential. Describe what determines the voltage at each part of the action potential in terms of sodium and potassium ion channels and Nernst potentials. For example, why is the resting potential about 60mV? What causes the sudden rise in potential, and why is the peak about +30 mv? Why does the potential then fall? Why does it go below the resting potential? Why does the potential then return to the resting potential? Additionally, does the concentration of sodium and potassium inside and outside the cell change appreciably during the action potential? If so, why? If not, how can the potential then change so dramatically? What initiates the first action potential (i.e. at one end of the nerve) in terms of type of ion and where these ions are injected? How do ligand-gated ion channels play a role in forming an action potential? And list one or more types of ligands involved. 3

4 4. The Voltage Gated Ion Channel Open Probability. The opening and closing of voltage-gated ion channels can be understood in terms of a gate the highly charged S4 segment -- that opens or closes the pore. Hence we can think of two states: the open and closed states. These states have different energies and therefore we can use Boltzmann statistics to extract the probability of being open versus closed. Furthermore, the difference in energy between closed and open is a function of membrane voltage. This is what makes the open probability voltage dependent! The voltage dependence arises because the gate mechanism of the protein is coupled to a positive charged protein element (S4, containing a charge q) which is moved across the membrane (from the inside potential of -V to the outside potential defined to be 0 mv. See cartoon picture, noting that in the closed state, the charge shown is supposed to be in contact with the inside potential of -V.) a) Assuming the only contribution to the energies of the open state is the charge q in an electrical potential, what is the Boltzmann factor for the positive charged gate element 4

5 being inside the membrane (when the channel is closed)? Phys498BIO; Prof. Paul Selvin b) With the same assumption as part (a), what is the Boltzmann factor for the positive charged gate element being outside the membrane (when the channel is open)? c) Now write out the open probability. (Don t forget to normalize the probability using the partition function.) d) Examine the open probability you found in c) in terms of limits. What is the open probability when V gets really large? What about when V gets very negative? What value for V gives an open probability of ½ (the so- called midpoint potential, V o )? Make a sketch of the open probability as a function of voltage. e) In reality, voltage gated channels have a midpoint potential which differs from the one you found in part d). The reason is that the charged protein element controlling the gate has voltage independent interactions within the protein, which also effects the open versus closed energy. This can be modeled as a spring (of stiffness k) that is stretched a distance δx when the charged segment moves to the open state (see new picture below). Rewrite the Boltzmann factor for the closed and open state taking into account the energy from stretching the spring. Now write down the modified open probability. f) What is the midpoint potential, V o, in terms of k, δx, and q? Rewrite your expression for open probability using V-V o, conveniently eliminating k and δx. Is V o positive or negative? Make a new sketch of the open probability including V o. If we change the picture so that the spring element is relaxed a distance δx when the charge segment moves to the open state, is V o positive or negative? 5

6 6 Phys498BIO; Prof. Paul Selvin