 Good morning. I hope everyone had a good weekend. My name is Henry Lester. TAs, do we have any announcements? If there will by any chance be a quiz this week, and the average is one quiz per week, it would also cover the material in the second half of the first lecture about deriving the ions in and near a cell from first principles. I think we have to stop deriving things from first principles today, though, and actually start some tough work. I do want to remind you about the Buy Neurobiology CNS 150 homepage. Please note my office hours. They will also be on the final slide of this lecture, and please read the book. Now, we exist here at Caltech in two worlds. We have the world of research and education, and we have the world of compliance. Many people who work in the world of compliance have a tenuous arrangement with the rest of the Institute, and the registrar in particular is usually a couple of weeks late with regard to what goes on in a course. And because of this, if you drop the course or if you want to register late, email the head TA, Geron, don't just talk with the registrar, who is an extremely nice person but has lots of other things to do. Now, this morning, ion channels in the news. The Nobel Prize in Medicine or Physiology was announced this morning, and it went to William Campbell, Satoshi Omura, and UU2 for therapies that have revolutionized the treatment of some of the most devastating parasitic diseases. The first two winners in alphabetical order, Campbell and Omura, discovered the ivermectins, including the drug ivermectin, and I will tell you about that drug in a moment. UU2 discovered an anti-parasitic drug that works on malaria. It's always fun to teach by 150 in the fall because you always need to step up and announce the neuroscientific connection with the Nobel Prize that gets announced at the beginning of October. So the ivermectins are the world's largest selling anti-parasitic drugs, the world's largest selling drugs for agriculture. Ivermectin and its analogs irreversibly activate an ion channel. It's a chloride channel found among many invertebrates, but not among vertebrates. However, its closest vertebrate homologue is the glycine and GABA receptors, which we will discuss at great length in this course. Right, I'm going to show you ivermectin in its channel. Many, as you know, there are various courses at Caltech about protein structure and about biochemistry. This is a molecular viewer called Paimo, which is one of the most common channel, which is the target for ivermectin, is called the Glucel channel. I will tell you a little bit about that in a moment. Here is an x-ray crystallographic structure of ivermectin down to the channel. The channel actually consists of five identical subunits, each with a different color here, and we are now looking down the axis of the channel. So that's where ions flow when the channel is open. Ivermectin would actually bind to all five subunits. I've only shown one of them here in this position, and it is an irreversible activator of the chloride channel. So we would see only chloride ions going through here, and this has the physiological consequences that I will describe in a moment. Does anybody have any questions about this structure, what it represents, or any other aspect? I'm going to shut down Paimo because having too many things at once does tend to crash my computer. So today's news very much exemplifies how contributions to neuroscience come from many other fields. In this case, from parasitology, but some of you are math majors, cns majors, chemistry, biology, bioengineering, other fields that have contributed to neuroscience are that, choreography, public health, biochemistry, molecular biology, statistics, physics. I was actually majored in physics and chemistry in college. So it is not surprising, it is not surprising, typical that parasitology should contribute to neuroscience and vice versa. So when Ivermectin binds to the invertebrate glutamated chloride, gated chloride channel called Glucel, this forces the cell, clamps the cell, if you would like, to go to the nernst potential for chloride. So the dominant ion whose permeability is in the plasma membrane becomes chloride. This is around minus 70 millivolts, and as we will learn in today's lecture and mostly in the next lecture, this prevents the cell from firing action potentials. And so the parasite, the invertebrate, cannot feed and it dies. And here's an experiment that we published a couple of years ago, taking the Glucel channel and making it into a tool for neuroscience. But you would see the same sort of records in a native invertebrate expressing the Glucel channel. Here we have transfected a modified version of the Glucel channel into vertebrate neurons in culture. Am I using any words that you would like explained, such as transfect? So this is an empty DNA, it's the control, and you inject current into the neuron as though you were a synapse, and the neuron starts firing without ivermectin present. Now, we first tried this technique a few years ago with an older version of the Glucel channel, and it did inhibit firing when we added ivermectin, but not enough. And so the optimized version of the Glucel channel, which we mutated and put GFP on so we could find it and all that, inhibits nearly all of the firing of the neuron. And so if you think you have a neuron whose function you want to understand, then you can use various techniques to engineer silencing into that neuron, or as is done in other labs around Caltech to activate that neuron. So they allow the experiment to silence the neuron by applying ivermectin. So very important drug, world's largest selling agricultural drug, ivermectin, and it has also become a tool for neuroscience. Same action, activating your chloride channel. Today's Nobel Prize. All right, moving on now to the usual substance of the lecture. You may remember that we said at lecture one that ion channels have a selectivity filter and that potassium ions lose their waters of hydration. Well actually, ion channels have two major functional characteristics. They have an ion selectivity filter, which determines which kinds of ions will permeate through them, and they have a gate, which determines what opens and closes the ion channel. Very few ion channels are constitutively open, although some are, and that's what causes the resting potential. For instance, nature has tricked today's Nobel Prize winning channel into being opened by ivermectin. Ivermectin gates the channel open. It's not usually open. Usually glutamate gates it open. So we are going to be talking then this week about two major types of ion channels. We're going to be talking about electrical transmission in axons, and here the gating function is accomplished by an electric field, actually by a change in the electric field. And in future lectures we're going to be talking about chemical synapses, chemical transmission at synapses, and there the ions are gated by neurotransmitters. Or in the case of today's ion channel by an external drug. Could anybody else name an external drug that activates an ion channel in the vertebrate central nervous system? This is an experiment that has done around the world 150 billion times a day. Yes. Okay. Very good. Thank you. Nicotine. Okay. I've told you that there's that electrically gated channels are gated by the electric field across the biological membrane. That is surprising because we don't usually think of cells have a resting potential of only the nudged potential minus 70 millivolts. Let's calculate from first principles the electric field across a biological membrane. So if we look that way toward Mount Wilson and toward the St. Gabriel Hills, and incidentally today is a beautiful day finally, we see high voltage transmission lines that carry current at about a megavolt, 10 to the sixth volts. The ceramic insulators which prevent the lines from shorting through ground to ground are about a meter in length. And so the field across those ceramic insulators is about one volt per meter. Engineers like to play it safe. And so that's presumably much less than the dielectric breakdown voltage of that insulator. And in fact, if we do look at dielectric breakdown fields in the real world, ceramic has a dielectric breakdown voltage of about eight times 10 to the seventh volts per meter, silicone rubber a little less, PVC a lot less. Right now, let's calculate the field across a biological membrane. The resting potential here, we're saying it's about minus 60 millivolts. The membrane thickness, however, is really small. It's about 30 angstroms. And so we do the division and we come up with an electric field of two times 10 to the seventh volts per meter across the membrane of nearly every cell in our body. Now, this is pretty close to dielectric breakdown for many materials. Well, what does dielectric breakdown mean? It means that dipoles in molecules get extended and sometimes they get extended so much that they break. Just so, in a biological membrane, the dipoles in a channel protein get extended and distorted by changes in the membrane field until those extensions and distortions make a conformational change that's great enough to open the channel. And I can tell you from experience that if we accidentally apply a membrane field only twice as much as usual, those ion channels will undergo dielectric breakdown. The membrane will break. We have to stop the experiment and try again on another cell. So, biological membranes exist very close to dielectric breakdown and instead the proteins are moving around under the influence of an electric field. Any questions? So, here, another simple electrical fact that I believe we talked about in lecture one is that we can think of an open channel with its complex alpha helices and beta sheets and fluid conducting pathway. We can treat it most of the time in neurobiology as a conductor, a little resistor with a finite resistance. Now, I prefer to call ion channels conductors rather than resistors, although they mean the same thing. And that's because treating them as conductors will allow us, as you'll see in the next lecture, to add very nicely in parallel. But they are little conductors. Any questions about that? Okay, let's go back to 1973 when I first came to Caltech. This lecture is full of Nobel laureates. We had one already. Now we're going to get another couple. I went to see Max Delbrook, who was a biophysicist and had indented molecular biology and molecular genetics. He died in around 1980 and had won the Nobel Prize for essentially indenting part of molecular biology. Max said to me, well, Henry, what are you going to do? And I said, well, gee, Max, I don't know. But, you know, I like ion channels and maybe I could measure individual ion channels. He said, well, I don't know how to do that. But it's getting to be noon. I have lunch with Feynman every day over at Chandler. Let's go there and ask him how to do it. So Delbrook and I went over to Chandler and Delbrook said, Feynman, this is Lester. Lester, this is Feynman. Dick, Henry wants to measure single channels. Tell him how to do it. Now, also present that day was Carver Mead, who is now an emeritus professor of electrical engineering at Caltech. Poor Carver was the only person in that group who did not eventually have a Nobel Prize. But he does have the Lemley Award from MIT and the National Medal of Technology and a couple of other distinguished recognitions. And probably among his more famous accomplishments is that he invented the touchpad on your computer that his company called Synaptics did that. And just recently, the New York Times, sorry, the New York Times had an article about the future of Moore's law and actually Carver is recognized in the Times and other places as the person who actually told Gordon Moore about Moore's law. Okay, and then there was me in 1973. So Feynman asked me, well, these ion channels that you're trying to measure, how are you trying to measure them? I said, well, we have these sharp glass electrodes and we stick them inside cells and then the potassium channels are open, sodium channels might open. And so currents obey Kirchhoff's law and flow in a complete circuit and that changes the membrane potential across the membrane. And if we can get low enough noise, we measure those ion channels individually. He said to me, no, I don't think you're going to do it that way. He said, the problem is that the membrane has a resistance and a capacitance. And he asked me what the resistance and capacitance were. I said, well, the capacitance is about one microfarad per square centimeter and the resistance is about 10 to the fourth ohm centimeter squared. And then there's the Nernst potentials in the membrane. He said, yeah, I was afraid of that because this capacitance, this RC circuit is going to act like a filter, which you all learned about in PHIS 1B practical. How many of you took PHIS 1 analytic? Oh, good. Oh, just one. You'll have a little trouble in this course, but the rest of you should do fine. So Feynman said now the RC circuit is going to filter out the noise, filter out the signal due to the ion channel and you won't see it. You need to think of a better way. The time constant is going to be about 10 milliseconds if you multiply this times that. Think of a better way. And he suggests, why don't you come up to the cell with a device that allows you to measure the currents directly rather than the voltage and just put in an electronic ammeter here in the circuit. And so the currents that flow will be forced to flow through the electronic ammeter. And the whole key though, said Feynman, is to make sure that the currents don't get shorted underneath the pipet and go back on their own. So he said to me, how close can you get? I said, well, I don't know. Never thought about it, but maybe a tenth of a micron. And he thought for a minute and he said, no, that's not close enough. Come back when you can get closer. So, of course, the field never stops and other people figured out how to get much closer. It's called the patch clamp or the gig ohm seal. And so now with the patch clamp and the gig ohm seal, all of the current that flows through a channel has to flow through the electronic ammeter. And here are the results, a single voltage gated sodium channel. So this is five picohamperes, which equals 10 to the fourth ions per millisecond, roughly, in 20 milliseconds. Now, the patch clamp is a wonderful device. For one thing, it has a tremendous dynamic range. One can record a single channel, even if it opens for only 10 microseconds. And even if it does this only once every 20 minutes, which is a dynamic range of 10 to the eighth, almost as good as the human genome. And the patch clamp can record currents ranging in size from two picolamps, which is about the size of this sodium channel to 100 nanoamps. And there are roughly 50,000 channels per cell. So we will come back to stories about Feynman and Mead next time. But meanwhile, the people who actually discovered the patch clamp, oh, let me show you a better version of that. So this was, my goodness, 25, almost 25 years ago, 1991 Nobel Prize. So it was sort of 24 years ago today. Erwin Nair and Bert Sackmann, function of single ion channels in cells. So the story that I just told you had got around. The morning that the Nobel Prize was announced, the editor of Nature called me and he said, listen, there's a rumor that Richard Feynman told you that you could never record ion channels. I actually said, no, it's not the way it went. He told me how to do it. I just didn't pay attention to him. So the Nobel Prize announcement said, this new knowledge and this new analytical tool has, during the past 10 years, revolutionized molecular biology, facilitated research and contributed to the understanding of the cellular mechanisms underlying several diseases, including diabetes and cystic fibrosis. So here we're back to this annoying concept today that actually electrophysiology and ion channels is more than simply neuroscience. Neither diabetes nor cystic fibrosis are neural diseases. And here's a simple diagram of a patch clamp as it existed in those days. Nair and Sackmann's scientific achievements have radically changed our views on the function of the cell and the contents of textbooks of cell biology. When they say textbooks of cell biology, they actually mean the early editions of the alberts, which you used in by nine or by eight. Both, okay. And finally, toward the end, here's a little bit about neuroscience. Drugs against anxiety act on certain inhibitory ion channels in the brain, alcohol, nicotine, and other poisons act on yet other sets of ion channels. So we have brought matters up to near the present. We're going to talk today about shaker, which is a very well studied, voltage gated potassium channel. Shaker, as Ralph explained, shaker is the name of a channel, but it also is the name of a gene that encodes the channel. It's a Drosophila gene. It was first studied in the late Seymour Benzer's lab by graduate students Lily and you knowing Jen, who are now very famous professors at UCSF. And it was isolated simultaneously by their lab and by Mark Tenoway, who was both a Benzer postdoc and a Caltech professor now at Berkeley, still doing wonderful science. And if I have my wits about me, we are going to simulate the gating of shaker, and we're going to use Francisco Betsenia's simulation programs at the University of Chicago. So here goes. This is Java, and you have to convince Java that it's not going to rob you or steal your identity. Yes. Okay. So we are going to talk about the so-called Hodgkin-Huxley formulation of a neuron membrane. Hodgkin and Huxley made this formulation in the early 50s, and they emphasized the gating of channels. But actually, they never mentioned channels or they ever measured a single channel. If you've been following the lecture, you know that single channels were not measured routinely until the 80s. And so Hodgkin and Huxley talked about macroscopic properties, permeability of membrane, which we now know is the sum of all of these little conductors in parallel. And Hodgkin and Huxley measured these macroscopic properties as functions of voltage, not of time. And so the conformational changes that I'm going to show you today in individual proteins, microscopic conformational changes that underlie the macroscopic properties are mark of processes. Would anyone like me to define what I mean by macroscopic? Okay, macroscopic means much larger than an individual molecule or an individual event, but characteristics of the population itself, macroeconomics as opposed to microeconomics. And so these macroscopic processes are typically sums of individual microscopic processes. And a Markov process is a process that involves a transition between two states of a molecule or a state or a system that has a defined simple probability of occurring with time. That probability does not depend on the history of the system. So ion channel gating depends instantaneously on the electric field across the membrane on the voltage, not on the history of the voltage. But it's easier to show you this in a simulation. Okay, now here is the simulation of a membrane. And we're going to show you the following events. The channel is actually shaker that's been simplified. It's been mutated to remove inactivation. And I will tell you about a little about activation later. The model is based on several types of evidence. It's based on biochemistry, electrophysiology, lots of site-directed mutagenesis, x-ray crystallography more recently in fluorescence. The model shows only two of the four subunits. So there are four subunits. They are not color-coded by subunit. They are color-coded by transmembrane helix within the subunit. And so in front of the screen and behind the screen, there are two other subunits. There are several states in this model which would be associated with conformational changes of the protein. So you can put the protein in one state or another. Here is closed number one. Here is closed number two. You see there is a rather small change, but that's enough of the change to open the pore when it occurs to open the channel, when it occurs simultaneously in all four subunits. And if we now make the change further, that's the one that actually opens the channel in all four subunits. So let's open this one and open that one. And now the green arrows go all the way through. Those are potassium ions leaving the cell. So we don't need to think about current, we just need to think about the flux of ions here. Is anybody lost or requiring a further explanation? Tell the Java app change the confirmation of the protein into states that do not allow the channel to be open. The question is, when you press the buttons, what happens? And the answer is we're modifying the confirmation of the protein to go into states' conformational descriptions that are not associated with the open channel. So here we've opened one. The channel doesn't open with only one of the four subunits in the active state. It needs all four of them, but only two are pictured here. So here we are activating the channel manually. Now we're going to do what, yes, why do the white and yellow, the question is why do the white and yellow not move at all? These are, the white and yellow are actually helices within a subunit. They are not by themselves subunits, show me. And so you may remember that the pictures we showed of ion channels have big complex proteins, and in the middle of that complex protein is the conducting pathway. You could think in this case of the white and yellow helices as insulators that hold of the rest of the protein in place to respond to the voltage, and also interface with the lipids in the membrane. So they are the, if you like, if we made the analogy to a wheel on an axle, that would be what holds the bearing in place. Any other questions? You know, I've been explaining this for so long that I think everybody knows what I know, but now is the time to say, well, Professor Lester, actually you need to step back. Right? So now we are going to let the program do a voltage step. If all, oh dear, start. Okay, now that all this is very complicated, what the program is doing here is repeating a voltage step. Now you remember the Markov process, the Hodgkin-Huxley formulation says that the rate constants for opening and closing channels are not functions of the history, they are only functions of the voltage. And so the essential trick of the voltage clamp experiment is that the voltage is always constant, then it's jumped to another voltage, and then it's constant there and jumped to another voltage. That's what clamping means. Now cells do not do this, their voltages vary all the time, but in order to understand how ion channels work, experimenters have developed the voltage clamp paradigm. The patch clamp is one example of a voltage clamp. The old-fashioned intracellular electrodes that Feynman told me to throw away are another example of a voltage clamp. So we are repeating the voltage pulse every trial for many times, and that's because each trial which concentrates on only one channel is a microscopic event. Now by doing the experiment many times on that one channel with its statistical stochastic behavior, we sum that behavior until we get a macroscopic signal, but actually this cell is doing it all at once during a single trial by summing simultaneously the microscopic behavior of a lot of channels, in some cases 50,000. Okay, so far? All right, so there's a lot of stuff in these traces, and you can see that the traces differ from one to the next. That's because the Markov process is inherently a stochastic process that is governed by statistical transitions from one state to another. So the left shot and the right shot really mean the states that we previously controlled manually and now that voltage is controlling them, going from C1 to C2 to A. That's simply what the shots mean. Eye gait is something that we won't discuss at the moment, although it's really cute. The main current that we want to talk about, the main phenomenon we want to talk about is eye single. Now, if you look monomaniacally, you'll note that when the green arrow goes all the way through, the current trace is up, and when the green arrow stops in the inside the cell and can't get through the trace is down. So this is simply the ionic flux through the channel. When the channel opens, current flows. When it closes, it doesn't. And you'll note that there are fluctuations in the opening. That's because of the stochastic processes that keep the channel more or less open, but sometimes closed, and more or less closed, but sometimes open. At the resting potential, by definition, the cell is at rest, and the channels do not have a high probability of being open. They are gated by voltage, actually by changes in voltage, actually by changes in the electric field across the membrane, which pushes and pulls on the dipole moment in the protein until these conformational changes occur and open the channel. Now, down here at the bottom is what we are used to seeing. At now after 105 summed trials, we actually have a fairly smooth macroscopic action average of all of these individual guys here. So this channel takes a couple of milliseconds to open all the way, but then it stays open. Any questions about the individual stuff here? You can run this yourself, and it's a pretty good simulation of the actions of a particular type of channel. Remember, the channel is the Shaker potassium channel, and you will come to understand this code. The code is who or where, and there's only one correct answer usually. So where was this channel discovered? Which institution hosted the discovery of this channel? Caltech. All right. Now, there are a couple of interesting little electrical characteristics here. Remember, I told you not to worry about this current called iGate. Actually, this current called iGate is a capacitive current, and it has to do with the fact that the charges here in the membrane that move in response to the voltage actually move in or out of the membrane. And if you think about physics one, the charge moving in a dielectric is a little bit of a capacitive current. And so sure enough, since the charges don't leave the membrane, the time integral of the charge movements is equal at the beginning and the end of the pulse, because those charges don't leave the membrane. And another interesting point is you see this little blip down here, pause, this little blip right here, and this little blip right here. When we step back to the resting potential, the channels don't like to be open, but they don't close instantaneously. They close according to stochastic processes. And so therefore, the currents stay on for a fraction of a millisecond. So, electrophysiologists have talked about all of these characteristics, and they give great insights into the way that ion channels actually make information in a cell and conduct information in a cell. Now, the charges here are not surprisingly positive charged side chains on the protein. They are lysines and arginines, and typically they occur every third residue in a helix. In the helix, there are great arguments about whether the helix twists or moves up and down still. Green stuff in the middle here, without those insulating helices that Shaman asked about. Roughly corresponds to the ion channels I've shown you before, the very simple ones, that have only two transmembrane helices per channel. What is the counter? Now, because where is the voltage, actually? Well, we're in the membrane here, and the voltage is somewhere around, across somewhere around here, probably toward the top of the channel here. This is the extracellular solution, this is the intracellular solution. Now, note that active state, a closed state, the gate is near and prevents ions from flowing. And so, this helix actually consists of a break and then another helix, and this is not surprisingly to those who've studied protein chemistry. It's a glycine, which does not have a side chain and is often associated with breaks in proteins. And so, if there's a hinge, it's right here in this glycine. All right, I think we have done enough, any questions before we turn this simulation off? Question is whether modeling ion channels as a marker process is relevant to processes that occur in the nervous system? Absolutely. It is a very good approximation, but sometimes unexpected environmental influences can change the transition probabilities, and sometimes those transition probabilities do change with time and with other events such as phosphorylations, proteolysis, etc. I'll get to it on the next slide. There are many voltage dedicated potassium channels, and each has its own voltage sensitivity in kinetics, and so one needs to understand the details of the voltage sensitivity in the kinetics for every potassium channel. Now, we're going to move on from potassium channels to sodium channels, and you remember that sodium channels, there it is, sodium channels are selective to sodium, and their side chains make a sodium ion feel more comfortable than a potassium ion. Sodium channels have a water-like pathway that trick the ion into being comfortable in the channel and make it a little conductor for sodium, but returning to the question that was just asked, how good is a Markov process? Well, the way you make a better Markov process, a more realistic Markov process, is that you keep adding states to the model, and so in order to understand the gating of sodium channels, you need to add another state, and that is the inactivation state. Ion channels open, and most of the time they don't stay open, they close again, even though the stimulus may still be there, even though the voltage may still be there, they close again, and so typically, back in the form of the years, people added another state called the inactivated state, and the more we know about sodium channels and other channels, the more we understand that typically the inactivated state looks like a flap that can cover the pathway and prevent ions from flowing in a different way. Now, not only are there flaps in ion channels that can inactivate those ion channels, and as you know, sodium action potentials are rather brief. They last only a millisecond or so. That's because sodium channels inactivate after a millisecond. We'll talk about this in the next talk. Not only are there parts of the channel protein that can come in as a flap and inactivate the sodium channel, but there are lots of external chemicals that can come in and inactivate channels or activate them. We talked about ivermectin. This one is tetrodotoxin, the puffer fish toxin. So in order to do a fair analysis of the sodium channels, which are after all the basis for the action potential, we need to look at another simulation again from Betsania's website. This model is a bit more venerable than the potassium channel, and you'll see the graphics are not so great. The potassium channels have four identical subunits. Sodium channels have four nearly identical subunits, but they're all actually hooked together in one polypeptide chain. So you could call it a pseudo tetramer. The individual repeats resemble an individual subunit. The part that actually selects for ions, the so-called P or poor region, differs. We are going to see now that the single channel events in this simulation, sodium channel simulation, there we are. So the sodium channels, the graphics are a little cruder, consist of these four homologous domains. Here's the inactivation flap down at the bottom. We're looking from outside the cell. So now we go to the simulation. We're going to start the simulation, and we're going to start it. So this green stuff around here is the membrane. The orange balls are sodium ions, which want to enter the cell, and as in the previous simulation, they only enter when the gate is open, but also they can be prevented from entering when the inactivation flap is closed. And so we have an additional so-called Markov state. I don't know why it should be running all the time. Here the channel is open, ions are flowing. You see little plus signs on the helices. Those are responding to the electric field. They're being driven one way or the other to produce conformational changes that open and close the gate. And then after the channel has been open for a few milliseconds, there's a statistical probability that the inactivation flap will come in, stop conduction, and terminate the action potential. Did I answer your question? Any other questions? The question was both the activation gate needs to be open and the inactivation flap needs to be out of the way. Those are two separate conditions, and they need to be met. Yes. In the Hodgkin-Hurksley equations, which are macroscopic, there are two parameters called brilliantly enough, M and N. M describes the activation gates, and since there are three of them, you have to cube them. N and M runs from zero to one. N describes the inactivation gate. It runs from one to zero. Since there's only one inactivation gate, it is there in the first power. So Hodgkin and Huxley, 40 years before the gene for sodium channels, was known. Guest from the kinetics of how they had to model the channel, that there were three subunits that could open it, and one subunit that closed it, even before they knew what subunits were. Okay, our next lecture does employ electrical circuits. Please review your material from This 1b Practical, Huxley-Ohm's Law, and you can also see Appendix A in Candel. And my office hours are Mondays, not Wednesdays and Fridays, entering near the red door from 1.15 to 2. See you Wednesday.