We have seen this expression for how to calculate the entropy if what we know is the multiplicity, so that's useful enough. But in a chemistry context, that's often not actually what we want. If we're dealing with dice or cards or flipping coins or something, then we know the multiplicity. We know how many ways a coin can come up, either heads or tails, or how many ways our die can give us a 1 through 6, but in a chemistry problem we often don't know or don't want to take the time to calculate the multiplicity. But what we often do know is probabilities. We might know, for example, that a butane molecule might have a certain probability of being in the anti-state or the ghost state or a ghost minus state. We might know, for a given reaction, what the probabilities are that we form the product that we're interested in or some side products. We might know the probabilities that molecules are in certain energy levels. So it's common that we know the probabilities. Take some extra work to figure out the multiplicity. So what we'd really rather have rather than this equation is an expression that gives us the entropy as a function of probabilities rather than multiplicity. So that's our next task. Luckily, that's not terribly difficult to do. Nearly every chemistry problem we can think of as a multinomial problem. And as we've talked about before, we can calculate the multiplicity for a multinomial distribution. So what I mean by a multinomial problem is let's say we have 10 molecules or Avogadro's number of molecules. Then we might be interested in knowing how many of those molecules partition themselves into the ground state or an excited state or into the ghost confirmation or the anti-confirmation. Or if we have more than two options, then it's not just binomial with two choices, but multinomial. So in general, if we have n objects or n molecules and a bunch of different choices for what those molecules can do, then the multiplicity is just equal to the binomial coefficient. The number of ways of putting those big n molecules into states or confirmations or outcomes such that little n sub one of them or n state one, little n sub two of them or n state two, little n sub three of them or n state three, and so on is just this particular multinomial coefficient. So we can write the multiplicity in terms of this multinomial coefficient. If we'd like, we can write that instead of this dot, dot, dot, telling us how many factorials to calculate in the denominator. It's a little bit cleaner if we use product notation and just write the product of all the n sub i factorials in the denominator. That gives us an expression for the multiplicity. If we wanna calculate the entropy, we just need to plug this multiplicity into the expression for the entropy. So we can say that the entropy of any problem that we can write as a multinomial is k times the natural log of w, so k times the natural log either of n factorial over all these smaller factorials. Or if we'd prefer to use product notation, k times the natural log of big n factorial over the product of all the little n factorials. So the next step is to evaluate that logarithm. So we have the logarithm of a quotient, logarithm of numerator over denominator. According to the log rules is just, again, if we want to use the explicit formula involving all of the little n's, we can say this is k times the natural log of, let's put the parentheses here, k times the natural log of big n minus natural log of little n1 factorial minus natural log of little n2 factorial, and so on. Or using product notation, what that becomes is now what I have in the denominator is log of this product in the denominator becomes the sum of the logs of the small n factorials in the denominator with the negative sign because they're on the bottom of the fraction. So I could write that as k times natural log of big n factorial minus each of the little n factorials summed up overall the possible values of those little n's. So n sub 1, n sub 2, n sub 3. This is just the summation notation version of this expression, which is a little bit more compact. All right, so at this point, I'll stop writing out the explicit version with the dot, dot, dots and just continue with the version involving the summation notation. So what we want to write down is what we want to use for the entropy is k times this difference between the natural log of big n factorial and the sum of all the natural log of little n factorials. We know what to do with the log of an n factorial. We can use Sterling's approximation. So I can write k times log of big n factorial. Sterling tells us is big n log big n minus n.  So I've got minus n sub i log n sub i minus n sub i. That quantity appears once for each value of i. So far, so good. I can clean that up a little bit. The second quantity in parentheses here, the summation of this term in parentheses, is the sum of the n log n's and the sum of the n's. So I can break that up into two different summations, write that as still big n log big n minus big n. I have negative a sum of n i log n i. And I have negative, negative a sum of the n sub i's. That's making some progress because now we can see some cancellation. What will happen? What will happen here with the sum of the little n's? The sum of how many molecules are in state one, how many molecules are in state two, and so on. If I add those numbers up, that's just the total number of molecules. So this minus big n, negative total number of molecules, must cancel positive each of the small number of molecules, the number of molecules in each individual state. That's always going to happen when we have a binomial or a multinomial. The sum of all the n's and the denominator is always going to be the same number as the big n in the numerator. So those two terms cancel each other. And what we're left with is big n log big n minus the sum of the little n log the little n's. We can use the same trick, writing big n as the sum of the little n's. And I'll do that right here. When I rewrite this line, I'll rewrite this big n as the sum of little n's. So I've got k times the sum of the little n's. But I'm going to leave the second one of those alone. So I've got my big n has turned into a sum of little n's, each one of which gets multiplied by log of big n.  And the reason I've done that expansion in the first term is now I can combine these two sums into one term, each of which involves sum of a little n. In the first part, it's multiplied by log big n. In the second term, it's multiplied by log little n. So that becomes sum over i of n sub i times log of big n minus log of little n. But that last term in parentheses now is something that we can simplify with the log rules. Log of something minus log of something else is the log of the quotient. So I'll rewrite that now. So we're still calculating the entropy. Rewriting this equation, I've got the entropy is equal to k times the summation of little n times the log of big n over little n. Summed over all possible states that can have some occupation little n. All right, that's begun to look quite simple and compact, but we can make it a little bit more understandable still. This quantity, big n over little n, will make more sense if I think about it the other way around. I say, let's ask ourselves what little n divided by big n means. That's the number of molecules in state i divided by the total number of molecules. So for example, number of molecules in state 1 divided by the total number of molecules. That's just telling us what fraction of the molecules are in state 1. N sub i over big n, that's telling us what fraction of molecules are in state i. Or if I pick one of the molecules at random, what's the probability that it will have ended in state i? So this quantity is not so interesting, but the inverse of this quantity is the probability of being in state i. But if I want to turn this fraction upside down, if I'd really rather think of it as the summation of n i times log of little i over little n sub i over big n, then because I've turned the inside of the log upside down, I just have to stick a negative sign out front. And I can rewrite that quantity now as p sub i. So I've got n sub i log p sub i. And we're almost to the point that I told you we were trying to get to at the beginning, trying to understand what the entropy as a function of the probabilities rather than the multiplicity or the little n's. One last step is to transform this little n into a probability also. If I could just take that little n and divide it by n, then I could call it a probability as well. So if I divide on both sides of this equation by big n, then on the right, I've got minus k times the sum of little n over big n multiplying log probability. And that little n over big n is the thing we're calling probability. I've got minus k sum of p sub i log p sub i. And this quantity s divided by n, that's just the entropy per molecule. So rather than the extensive entropy, the amount of entropy for the total amount of molecules, this is just the entropy per molecule or the intensive entropy. The entropy per molecule is our intensive entropy. So we'll call that s bar for the intensive entropy. So that equation, I'll put a box around that one, because that's an important equation that we'll come back to. That tells us how to calculate the intensive entropy as a function of the probabilities. And we'll work an example in just a second. But if I know what the probabilities are for occupying state number 123, confirmation number 123, product number 123, then I just sum up probability times log probability for each of those states multiplied by negative k. And that tells me the intensive version of the entropy for that system. That's very convenient because I get to use the probabilities rather than having to think about multiplicity. Notice multiplicity is long gone. I don't have to know the multiplicity to use the expression. And as a bonus, I get the intensive entropy. I don't even have to know how many molecules I had or specifically how many of them are going to occupy each state. I just need to know what the probability is of occupying each state. So for example, if we return to the example of butane, where at room temperature as we've used before, the probability of being in the anti-state is 68%. The probability of being in the Gauche plus and the Gauche minus state are both 16%. That's enough information for us to calculate the entropy per butane molecule, the intensive entropy. According to our new formula, that's just minus k times the sum of p log p. So that's minus k times 0.68 log 0.68. And 0.16 log 0.16. And another 0.16 log 0.16. So if I do the calculation in brackets, that comes out to a total of negative 0.85 is the quantity in brackets. So that points out why this expression has a negative sign in it to begin with. Each of these probabilities is less than 1. It's less than or equal to 100% chance that we're in any individual state. So the log of those probabilities comes out negative. The negative sign here guarantees that the entropy will eventually be positive. So the quantity in brackets is negative 0.85. After canceling the negative sign, I get positive 0.85 times k is the entropy per molecule of butane. If we want to, we can plug in Boltzmann's constant. If I multiply 0.85 by 1.38 times 10 to the minus 23 joules per Kelvin, we can get a number with some units on it. But for now, we're not so much interested in that numerical value as the fact that we can calculate this entropy, let's say, as a multiple of k, because we haven't yet talked about where that value of Boltzmann's constant comes from. The important thing for right now is we know how to calculate the entropy as a function of the probabilities. What that will allow us to do now that we can calculate entropy, remember where we obtained the entropy was as an extensive version and now an intensive version of the multiplicity. The multiplicity itself was not an extensive or intensive property. So in the future, when we're interested in finding out what is the likelihood that we're going to be in different states or what is the probability that certain molecules will exist in different states, then what we want to find out is the combination of molecules that maximizes this entropy that's the most probable as the highest multiplicity. Another way of answering that question will be to say not which has the highest multiplicity, but which has the highest entropy. And so with this formula for the entropy, we can figure out what the probabilities are that maximize the entropy. So that's the next step is to be able to talk about how to find the values of probabilities that are going to maximize the entropy.


I'm Andy Oates and I'm going to talk to you today about the segmentation clock and that's a topic that my lab's been working on now for about 15 years, I think. And I had an interest in this topic when I was a postdoc too, so maybe I've been suffering from segmentation compulsive disorder for about 20 years now, which is not to say that I understand everything about it by any means. What I wanted to do for the next three lectures is to give you, to try and break the subject into three parts, characterised kind of by spatial scales, and talk about those three parts in order. Of course these parts relate to each other, and I hope the old tea set out has a go along. So I'm currently swapping backwards and forwards between London and Lausanne at the moment, so these are, I should acknowledge the people who have helped me work in the last little while, it's the Francis Crick Institute and University College London, supported by the Wellcome Trust, both in London and more recently now the EPFL in Lausanne. So right, so let's get started. The segmentation clock, the basic problem is kind of in one of embryology and it's how vertebrates form their body structure. And so you can see I've got a collection of vertebrates here on the slide and they've all got a segmentation, their body axis is segmented from the head to tail axis. And you see a pattern like this and you're in, sorry, let me just, so that's what we're going to talk about, how that pattern arises. So here's the overview of the talk. So it's going to introduce segmentation, then I'm going to talk about noisy autonomous oscillators, I'm going to talk about negative feedback, I'm going to talk about period mutants, which are mutants where the period of the process changes and what that's taught us. And then I'm going to try and discuss some open questions, which I think are relevant. So all right, so you, as a developmental biologist, you typically are interested in things like the shape and size and number of body parts or structures and what characterizes, I think, the questions that I'm interested in are the role of timing in creating the structures of the embryo. So we are interested then in, that was a human and it's going to go back, that was a human skeleton and you can see the segmented backbone of a human here, all the way down its backbone itself, but also the ribs are segmented and in fact, though you can't see it in this image, the muscle and skin is segmented as well. We use our segmented muscles between our ribs for breathing, called intercostal muscles. But other animals use them to move and one good example is a fish. The fish uses the muscles that it has between its body segments to swim and so there of enormous significance for a fish when it moves. So this is a zebrafish, which is the animal that we've studied and mostly a lot of what I'll talk about today and over the next couple of days is from zebrafish, partly because it's what I understand the best, but also because I think there's some aspects of that model organism that have led us to be able to see things that perhaps other have not been more difficult or challenging in other organisms and it could also be that the zebrafish is a bit simpler in how it works and that's also given us some insights. Okay, now here's a zebrafish skeleton prepared so you don't see the muscles, you can see the repeated structures of the centra and of the ribs of the neural arches which guard the spinal column, the spinal cord, the neurons along the back and the hemal arches which guard the major blood vessels that bring blood to the tail. That's the structures I'm going to be talking about and if you want to understand how they come about, oops, you have to go and look back in the embryo and this is a time lapse movie of a zebrafish embryo, it started when it's covering the yolk, you can see its head forming here, here's an eye, the ears forming back here and you can I hope you can see these blocks of cells forming, they're budding off from a tissue in the posterior one by one so you can, I hope you can see the boundaries forming, these form rhythmically and sequentially, that's the key concept here, the body segments forming rhythmically and sequentially, that should loop, nope, okay, okay, so all right, so it's sort of, yeah it's one of the mysteries of presentations, okay so now let's look at these things again, they're balls of several hundred cells that sort of roll up and each of these forming boundaries is caused because the cells on the outside of the ball epithelialize, their basal surfaces are the outside, so it's like a cyst and the apical surfaces point into the middle of the middle of the segment, so we're interested then in, these are called somites, we're interested in what's going on in that posterior tissue then that's producing the rhythmic morphogenesis of the segments, now each of these segments will then go and give rise to the bone in the muscle and then the skin that we just described, okay, so what you would have noticed from that movie and it's been noticed for a long time is that the addition of the segments is rhythmic and you can get an idea for that by time-lapsing and here's the sort of rig that we use, we make a small depression in agarose, in agarose, in a microscope dish, you make a little cone and that allows us to slot the yoke of the fish in so that the fish isn't pinned, it's not held firmly, it's just resting on top of the agarose and then we can film an array of these embryos growing at the same time and then from the movies we count the time point when each successive boundary is formed, we can record that then and for an individual embryo what you can see and now looking at the somites along the trunk forming with a lapsed time and for one embryo you can see the very regular formation of these boundaries and for, put a slope through that will give you the period with which the segment boundaries are forming and that seems to be constant in the trunk. What's really remarkable I think is when you look at a population of embryos, these are 15 brothers and sisters sitting on the same stage, this is the mean and the standard deviation of segment formation times in those animals, so this is an incredibly precise rhythmic process that's going on, okay, and this precision, I mean it wasn't I think until around this time that that we got a measure of the precision for the first time but the idea of the rhythmicity has led people to speculate for some time that there might be some sort of clock or oscillator that's behind behind this process and that idea goes back to the 1970s and it's an idea from Jonathan Cook and Chris Seaman and it's called the clock and wayfront mechanism so I want to tell you about that now it's a very generic proposal doesn't depend on the microscopic details, has two elements a clock and a wayfront so it's well-named and I'm going to illustrate them to you now so previously what I showed you was fish growing from a lateral side so they were sort of extending out this way and now we're flipping the animal over so you can see that when vertebrates form their body structures they're actually bilaterally symmetrical so the head would be out here this is the tailbud here are the two arms of the presemitic mesoderm and here are already formed segments and the clock is this population of cells in the posterior and in the simulation they're going to oscillate we'll keep track of the phase which is think about of a clock you can think of a the hand of a clock moving around the clock face so all we're interested in is in the angle don't care of the length of the length of the of the hand and it can be you can imagine that at 12 it's dark blue then it gets lighter and lighter to white at 6 and then comes back and gets darker and darker so that's how we could illustrate phase and that's what I'll illustrate phase today the granularity here that's that's a cell that's one of the oscillating units this little yellow unit and these will also these will all oscillate together okay that's the clock the other element is a wave front it's this black line and the wayfront is a rule that says when an oscillator hits the wave front it's phase angle is recorded now in this case it's going to be arrested it's going to be stopped and then whatever phase it was when the wayfront hit it will be color coded it will go from being blue to red but that's that's it being recorded and then it stopped left behind and so when we see these two when the interaction of these two things will be a is a way of recording permanently the temporal activity in the clock that's what the wayfront does okay and to get an animal to extend we're just going to add oscillators on this end at a certain rate and that rate at which the tissue then extends is exactly the same rate at which the the wayfront will move across the oscillators so in that sense the tissue itself won't change its length at all we call this the infinite snake we just keep keep growing our oscillators will be added at one end and they'll be they'll hit the wayfront at the other end so this is what happens the clock oscillates now elongates you can see that what's left behind then is a permanent periodic record of the temporal activity that was going on in here and you could also see I hope that that cell drifting in the it's at rest in the lab reference frame but it's drifting in the reference frame of the tissue we just do that one more time go the wayfront's moving and each of the oscillators that it hits goes from red goes from is stopped and goes from blue to red and so the pattern that's left behind then which will call the segment length so from two peak one peak to the next or from one trough to the next anywhere along that along that periodic pattern that's given by the velocity of the wayfront multiplied by the period of this clock in the posterior so that's it really that's that's the rule of course you can you can this is this is a there's two things to say about this kind of mechanism one is that the I just said the segment length is the velocity times by the period the other is that the total number of segments that an animal would make under this kind of rule is given by the duration so just how long it keeps this mechanism up divided by the period so very simple but the reason why I'm hammering on about this is because the the period here appears as causative for the eventual anatomy the initial anatomical structures that appear in the in the adult and so when this idea came up I think as far as I understand it was the first time that the idea of using a using timing to cause structure in the embryo had become explicit and I think that idea was then used in a number of other circumstances in embryology so I think it's a very powerful idea it's a very influential idea okay so for example in in in limb bud growth people have talked about having some sort of clock or timer in the outgrowing in outgrowing limb bud alright so let's see where are we so clock on my front and so that idea languished for a while people discussed it but it wasn't until 1997 that some evidence from an embryo was found about the molecular nature of a set of oscillators in the posterior tissue of vertebrate was found in a chick embryo sorry and by Olivier Porquier's lab and I'm just going to come to the dynamics of that but I what I realize I forgot to say is I wanted to show you because in these in these images you can't see the cells in here they're made up cells so I want you to see what real cells look like in that tissue as it's forming segments okay so this is now a confocal is a an optical slice with the confocal microscope we're using a transgenic line that marks the histones and you can see individual nuclei of cells sitting in the tissue so here's the developing central nervous system there's a layer of skin right around the outside and this tissue now down here is the mesoderm the tailbud here the pre-semitic mesoderm here and here I hope you can see the shapes of newly formed somites so let me let me loop this movie a couple of times two things to notice of this movie is that when a somite forms it does so actually by rather subtle rearrangements in the cells at the anterior of the tissue so if you look here that looked uniform but now the cells are reorganizing themselves and so cell cellular material is some set in some sense being ejected or segregated out of the the main body of the tissue okay at the same time material is being added at the other end so that comes into the tissue by a number of different means but the point being that there's a rate at which cells are removed from one end of the tissue and there's an and there's another rate could be the same could be different of cells that are added to that end so in that sense the pre-semitic mesoderm isn't a fixed when I talk about the pre-semitic mesoderm when we talk about it it's not a fixed set of cells it's actually a reference frame in which there's a continuous movement of cells they're continuously being added at this end and they're continuously being ejected at that end so so that means that we can also talk about a flow a slow flow of cells in the reference frame of that tissue to give you some idea and analogy then if you about how these cells move around can you see this tissue boundary look at the cells on the dorsal edge of the paraxial mesoderm on the tissue we're interested in and the cells there can you see that there's a there's a shear between those two that gives you some idea of the movement of the cells in the tissue in this case relative to the neighboring tissue okay so that I hope that gives you an idea of the sort of granularity of the events in a zebrafish and also of the displacement of cells in the system as it goes this is a slow displacement all right now now what's going on in those cells and this is brings to the segmentation clock so the first molecular evidence for this came as I was saying in the chick from Olivier Porquet's lab and that reference is down here so I've I've tried to put references through the talk and you can have the PDFs afterwards so hopefully you can chase up any of these things that you find interesting this is a zebrafish obviously and this is a transgenic animal that was made in in my lab which has two transgenes one is a gene called her one it's been fused to YFP so that the fusion protein becomes fluorescent has a short half-life it's about 12 or 13 minutes and so you can see what I hope you can see is that there are waves of green signal that are sweeping along this tissue so there's an elevated signal posterior and then the wave travels anteriorly through the tissue the wave shortens its distance as it moves anteriorly and then it arrests both the wave arrests and these the signal switches off right at the anterior end of the tissue and just where it finishes predicts the position of each newly forming segment there's a very strong coincidence temporal coincidence between the arrival of one of these waves and the formation of a new segment and the other trans gene in this animal called mesp which won't play a role again today I don't think marks quite a small cluster of cells on the anterior side side of each of those forming segments and there's a there's a moment in the development of each segment where the incoming wave is sitting in those cells and just before it dies out it co-expresses with this with this red gene so this might be some sort of illustration of a temporal pattern and then a permanent periodic pattern that's left behind right segmentation clock you again you can't really see the cells back here so let's zoom in again using again a confocal and this is now the same trans gene the the her 1YP I'm going to come back to her one in much more detail but for now I just want you to think of it as a marker of these of the of the dynamic expression that's going on in that tissue so here again the tail bud precemitic mesoderm the central nervous system sitting over here the yolk in the head would be down here and what I what you can see now our individual nuclei these aren't lit up by histone anymore they the signal you're seeing is the the transcription factor entering the nucleus and then becoming fluorescent it's actually degraded in the nucleus so it never leaves again so you see each of the nuclei becoming bright and dark and bright and dark so this is this is what those waves were if you looked at an individual cell it's clear that the cell isn't moving with the velocity phase velocity of the wave through the tissue it's switching on and off and it's the the coordination of the neighboring oscillations that gives rise to this wave that moves through the tissue okay so we have a population of genetic oscillators and so from what I've showed you yeah they are dividing not not so the dividing about I guess it's about ten times slower than the cycle that you're watching here but there are cell divisions that go on throughout that tissue yeah yes we can completely block cell division genetically or with drugs and the whole phenomenon occurs I'm not saying it's exactly the same but yeah it's not required at all to get this to happen in invertebrates there's some interesting cases in analytics where there's a one-to-one correspondence between cell division and segmentation but that's not true it's not true invertebrates the the half life of the trans gene is on the order of 12 to 13 minutes and the period the period of these oscillations we're talking about being in the tens of minutes and so for the temperature here and for a 12 minute half-life 28.5 degrees Celsius plus one plus or minus 0.1 degree and so this is quite a this is an interesting period for an oscillator it's not long like a circadian clock which one might typically associate with genetic oscillators and it's not short like a calcium oscillation or a neural oscillation which typically depend on ionic currents so it's somehow in the middle and actually took us five or six years to finally be able to see this by balancing the microscopy and the and the mitt and sort of I won't bore you with the details but I'm happy to share them with you of how to get the time scales right so we could could see this we're still learning actually obviously so okay there's a there's an interesting model for this which was actually the reason why I moved to London in the first place because they've installed segmentation clock models in all of the tubes in London and it's it's sort of a joke but it actually illustrates a bunch of the questions about the segmentation clock quite nicely so when you look at this you see three kinds of patterns or three kinds of elements you can see that an individual pixel or lamp or unit in the tissue is switching on and off and if you just watch that you'll see that it switches on and off with a rhythmic pattern it'll it'll it'll repeat over and over again so you can ask all sorts questions about okay why is that switching on and off why does it have the period that it does etc you can ask now you notice that if you pick your favorite pixel and you look at the neighboring pixels it's strongly correlated so if one of your pixels on there's a high chance that some or all of the neighboring pixels are on and conversely off when they're off and so you can ask questions about that you know does one pixel induce its neighbor to be on or is something else outside telling them all on to be at the same time or and finally the thing that's most obvious of course to us is the global organization of this oscillating pattern which is words telling you to get off at more gate at the next stop if that's what you want to do so now one so that this is really actually quite similar to the segmentation clock there's one key difference in the phenomenology which is that there are no new lamps being added at this end and there are no lamps being ejected at that end so this is kind of a steady-state segmentation clock there's no there's nothing moving no no no matter being added or removed from the from the from the tissue okay so and then maybe just to finish this analogy off you could imagine using this system to do something periodically even though that pattern is complicated every time the M in more gate arrives at one end you could use that to trigger a particular event because everything that's M leaves the tissue and the next time more gate comes up everything that's M leaves the tissue so okay that's I think gives you some sort of idea okay and actually these these units and I'm about to run out of batteries the topics of the three lectures broken down into biological units which I'm gonna list here so we can think of trying to understand what's the nature of the cellular oscillations we can think to ask what's the nature of this local correlation a synchronization between between the the oscillators and then we can think about trying to understand these tissue level wave patterns how they arise what they're good for if they're good for anything and although it's still not entirely clear the connections between the different levels will be important if you're an engineer you can imagine lots of different ways you could design that that notice in the train and many would work right there's potentially as many different ways to get it to work as there are people sitting in this room so we can't just look at the pattern and understand how it works we have to break it we have to try and build it and we have to make models of it to see whether our ideas are going in the right direction or not and I you know it's kind of an unusual question at this stage but does anyone have any triple-a batteries or a long stick or a laser pointer it's still changing but I guess the for a laser of this intensity we need a bit more sort of big cable or something all right so cellular oscillators that's the that's the subject now okay so the first question I wanted the first issue I wanted to discuss is whether or not these cells are triple yeah three days was to ask the question so I deliberately didn't say when I said cellular oscillators I deliberately didn't say anything about whether they could tick by themselves there are cells oscillating but we need to we want to to hey thank you I think a fundamental question to get started with is to understand whether these oscillators need to talk to each other to oscillate or whether they can do it by themselves and it will influence the way you try and write down models for the for the thing okay so this has been a topic of debate and interest for a while and what's been clear for quite a long time is that if you cut out this tissue the precemitic mesoderm this has been done in both chicken mouths and you separate it away from the central nervous system in the skin and the other neighboring tissues you put it in a dish it will continue to segment by itself so actually bud at one end it's not clear whether the size and the shape and the timing is correct but still qualitatively that tissue will will segment itself so it's been clear for some time that the tissue is autonomous so now we need to break the tissue open somehow and studies doing that from from this is one from chick looking at cells which change their this was done by growing cells and then fixing them and asking whether their patterns look different of course there's lots of there's another interpretation to seeing a sort of soul from pepper pattern here that so could be on because it's in the up phase and that cell there could be off because it's in the down phase or this cell could be stuck on and that so could be stuck off so it doesn't solve the problem what one needs is a live reporter which I showed you and that the first group to introduce a live reporter in segmentation clock was the with Rio Kageyama's group from the Institute of virus research in Kyoto and so this is the this is the entire data set from mouse yeah you just saw it and it consists of three cells which fluctuate in time I'll play them again and so this according to Rio it was difficult to find cells that would show any behavior like this and here's the trace of one of them and so the conclusion of the paper was that signals from other cells are likely to be essential for ongoing oscillations that was the that was the conclusion of that paper so we repeated those experiments quite a lot later using the new transgene that transgene that we built in the zebrafish and we did the following experiment we we noticed that the tailbud this is this black line is the trace of intensity measured from the from the tailbud in the movies notice that it had a quite precise rhythm which you can get from the order correlation function of the of the of the phase and then you can you can you can by the envelope of that you can see you can calculate a persistence time so it's quite it's quite precise and so what we did is we thought well maybe there are some clock like cells back there we chopped off the tip of the tailbud and incubated it in a in a medium that contained elevated levels of FGF which is a signaling protein and also serum so which contains almost every known signaling protein so this is a bit of a cocktail that's strongly stimulating and our idea was let's let's make the environment of a of a of a tailbud let's see if we can get keep these cells to oscillate I mean the mouse cells were also incubated with FGF and serum but we're going to try the zebrafish and this is what we saw the cells appeared to be quite happy to oscillate in vitro and if you look at that cell for example is not touching any other cell in this movie and it oscillates several times let me just show you that again look at that one switching on and then off and then on again and in fact doing these movies over sort of standard interval of 10 hours we saw many cells that would persistently oscillate through the time and what we also noticed was that if you take a whole bunch of these cells now separated from each other and record them they continue to oscillate but they're not organized anymore into local locally coordinated waves so the and so it looks as if the cells are quite noisy they'll keep oscillating but they're no longer either coordinated with each other and actually nor do they keep a consistent period an internal period they're not you wouldn't want to use them as a clock actually they wouldn't be very useful so then in the animal yeah if if we take the most precise of the of the of the individual cells and compare that to the precision of the tissue then the tissue is at least five times more precise that's the most precise of those cells even the best ones are five times worse than the tissue and some of those cells are so bad that the way we were measuring the quality factor it didn't work so so there's a and I'll come back to why the precision goes up when they're in the tissue tomorrow in fact so so in fact individual cells sitting all by themselves on the bottom of a dish they'll continue to oscillate and that that was a movie this cell is if you follow that that's the only cell in that well these experiments are done by serial dilution and this cell continues to oscillate so you can see it's it's trace over here so we we think that in the zebrafish segmentation clock cells are autonomous oscillators yeah yeah they can I'm not going to talk about that today but they can talk to each other yeah that's right so today what I wanted to do is to keep focusing on the individual cell as a unit what we can try and understand about that so looking at all so looking at these things the thing we noticed was how heterogeneous these rhythms were and we saw cells that sort of build their amplitude we saw cells that appear to stop some cells are not oscillating and they do two pulses and then they stop again is a little interval where the cells stopped oscillating so we want to try and understand the kind of noise that's here and a couple of things you can ask is whether the amplitude is correlated and the amplitude is quite well correlated so that means when a cell starts going up it keeps going up for a while and when it starts going down it keeps going down for a while but the period is is not correlated that means that if you know one interpeak interval it gives you very very little information about the next interpeak interval so this is a kind of unusual situation and the time scale of the correlations in amplitude is longer than the period much longer than the few not not quite sure exactly how much longer but it's and so without any internal knowledge of the oscillator we wanted to get a measure for these types of noise and we used a very generic model of an oscillator I'm just gonna sketch it for you now we keep track of the phase of the oscillator which is the angle of the hand that I was talking about before and that's given by the frequency of the oscillator so here we're assuming there's an oscillator right we're assuming the oscillator because we've given a frequency and this can this can be influenced by the amplitude the amplitude is r and that the time evolution of the amplitude is given by this parameter mu which you can see here which is some sort of distance that the system has to a bifurcation point so you can think of that as being equivalent to some sort of driving force that makes the amplitudes grow or shrink to nothing and die away and so you could this model really talks about the transition between a limit cycle and a fixed point yeah this should the amplitude have yes the effective omega goes down I think because the as your as your are goes up you take longer to get through your cycle so your your frequency goes down does it make sense that's that's the simple that's you know you you're producing something take a look you take you produce a lot of it and now takes a long time to get rid of it so now but but it's not clear that the two are coupled okay this is the generic form where you allow coupling between the frequency in and and the and the and the amplitude thank you but actually when you plot frequency versus amplitude it's very poorly it's actually very poorly correlated so we're going to simplify things by leaving that coupling out first approximation so now here's the here's the data from the from the from the cells and then we can use this model to so it's just be very clear this model is being used to try and understand the magnitude and the time scales of the noise what kind of noise it's not it's it's no surprise that this thing is stopping oscillating and then starting oscillating in because that's the behavior of the model and if we allow that the the parameter omega to sorry mu to to vary then of course we're going to stop oscillating and start oscillating the question is what values of the noise for the period and the noise for the for the amplitude match the match the data so that was a question and so I'm just going to say that there is a long time scale noise in the amplitude and that's actually very well fit by some sort of colored noise so an on-site Ullembach process whereas the the noise in the in the period I mentioned that was uncorrelated that's very well modeled by white noise so that's that's actually the conclusions at this point is that when we take these cells out of the embryo they look like they're really close to the edge of oscillating they'll oscillate and then I'll stop oscillating in and I'll oscillate and they'll stop oscillating again and so we think about this a long time scale in them in of them oscillating and then stopping again and then we think about a short time scale noise in exactly what the frequency the period will be if they are in the oscillatory phase yeah we're missing these these high these high events yep and that is that is definitely a shortcoming in the model yeah I don't understand where they are there no there they are correlated with some multiple of the period so so two three maybe more yeah that's right so maybe if I put that in in molecular terms I might say you know maybe what's causing the drift between the oscillatory and the non-oscillatory state is something like ribosome queuing or or blocky or or competing for entry into the nuclear pause cell cycle state remember I mentioned that cell cycle was much much longer than the period but it's not the microscopic events in the birth death processes of the oscillator okay so that's I think that's a lesson that we learned just from looking at the noise and I haven't even talked about the genes oh sure sure sorry I didn't explain that at all I beg your pardon so AI is any given peak the amplitude of any given peak and I plus one is the next one so we compare the correlation between any peak and its successor and the same is true here we compare the period the interpeak interval and the next one we run along all the time traces comparing what what the so now let's talk about the genes so this is the next topic which is negative feedback so there'll be no surprise if we've got an oscillator that we need some negative feedback and we oscillators can work with negative and positive feedback and all sorts of other bells and whistles but but it's but it might be the case that in that here there's quite a simple negative feedback loop simple is always relative and that's what I want to talk about now what the model in the field for for that is so the model in field depends on the has come from knowing what the genes that are oscillating are so the green gene I showed you in the movies at the beginning encodes a family of encodes it comes from a family of a trend of transcription factors called basic helix loop helix transcription factors and that's not important except to say that they dimerize and that is important they can homodimerize and heterodimerize with other protein family members they bind to DNA in a sequence specific way so when you when we've been talking about enhancers and promoters previously that's exactly the kind of thing we'd be talking about here and we and others have measured those proteins binding to their own enhancers so that lets you sketch out a very simple scheme where the the expression transcription of one of these genes translation into protein dimerization and binding to its own nuclear own gene could complete a negative feedback circuit and that could in principle give rise to oscillations so for for that to happen of course it's not the only dynamical outcome of a of a negative feedback loop it could go to some sort of steady state but these systems will oscillate if the time scales of the of the if the the half lives of the mRNA and the protein are short so they turn over quickly in comparison to how long it takes you to get around the cycle one so maybe so first return time coming around the cycle so otherwise those products will just accumulate and you won't see any strong effect of switching the gene back off again and under those circumstances the other thing that you would want to say is that the period is given to first approximation by by the delays to go around the cycle because I can't go any faster than the cycle can't go any faster than it can deliver the off signal back to itself very sort of simple yep they dimerize very fast they've got a very fast on rate also very fast off rate I would be surprised if they weren't dimerizing in the cytoplasm but I don't actually have any direct evidence for that like they can't bind DNA unless they dimerize they can dimerize without DNA being there that's for sure in vitro but whether in these cells they are dimerizing first in the cytoplasm and then entering the nucleus I don't know actually don't know the answer to that okay so this this I wanted now to talk about a model that we've used in the field which is really been extremely influential dominant in fact and it's got some strength and it's got a couple of weaknesses and I'm going to I'm going to talk about it briefly now and then I'm gonna use it to think about three period mutants that have been reported and basically form our knowledge about how to think about this how this feedback loop is constructed and how it works so basically you keep track of the protein dynamics which is being made and those are given by the by the amount of mRNA you've got so m stands for mRNA and there's a so that's being produced and then this is being degraded so there's a degradation rate here and the amount of protein is read off the amount of mRNA at some time in the past so there's this is where the delay comes in this is the delay of producing the protein got this right and then the change in the mRNA is given by some production term which depends on the amount of mRNA in the past and some degradation term and so I'm going to go into these in a bit more detail so where mRNAs being produced this is a regulatory function here and the proteins being produced up here and produced and degraded so putting in a fixed delay means that we've got a delay differential equation and these are these are very difficult to treat analytically I just got frightened and didn't even bother to try treating them analytically but some informed colleagues of mine say that they're they're very difficult if not impossible to solve analytically but they are good for simulating they they work very well for numerical simulations and they're also in some way producer they give you an intuitive way to think about the problem because the delay turns up explicitly like that so I'm so and I'll come to some other models in a sec so the degradation rates here B and C are reciprocal of the half lives of the molecules which is the half life here and and here I'm just trying to that so that the half life of the protein which is which is B is given by the reciprocal of the the half life so the degradation rate is called the half life and conversely for the for the mRNA and the repression function is basically some production rate divided by a hill function which basically gives you a switch going from being all on as a protein rises there's some non-linearity which converts that into an off switch so you can you can say two things about this model in order for the oscillations to happen there's got to be a balance between the production rate of the protein and the degradation rate protein mRNA and their and their and their and their degradation and that's got to be larger than the amount of protein that will feed back on to the feedback is required to get an off and and then if this holds then the period can be approximated basically by the total delay and then modified by the by the half lives by two yeah I would have to go and look at the derivation I can't tell you off the top of my head yeah so this these this is work done by Julian Lewis and I waited that to come from I'd be making it up let's let's have a look afterwards see if we can find where the two come from yeah it would be it would be interesting if it was seven because seven doesn't turn up very often right have you noticed that not nearly often enough because it's actually my favorite number right good so so let's so let's say that so you can parameterize this this model and here's one of the weaknesses one of the one of the weaknesses is that the delays are hard you you insert the delay and so the model behaves according to the number you insert so the the total time of the oscillator isn't doesn't emerge so naturally from the dynamics it it's dominated by the delay okay and but you can make guesses for these various parameters and if you're happy to allow a bit of fiddling in certain and tuning you can produce simulations which match very nicely your favorite oscillating animal okay and so yeah absolutely I mean I think that so this is this is clearly massively simplified and what I'm trying to argue is that that might be a price worth paying it if you're you're somehow always searching for the right granularity of your model which will match the things you can measure you could put lots of stuff in the model that you can't measure and it's not clear it's not always clear that that's going to help you understand the situation if you think you might be able to measure something and you should make it maybe make that as explicit as you can and then maybe the art is working out what the coarse-graining or the abstraction is for the other the other parameters so I shouldn't I I think this model is already very complicated actually and so I'm trying to introduce it because it's being used this is the model in the field everyone talks about this model you can argue whether it's the best one to choose and I'll show you some I'll show you an adaptation of it but there are others you could pick so maybe I've got yes love your work so there's a bunch of other ways yeah there's a bunch of other ways you could choose to write write down even even a very simple and so I've got this I've got this more detailed diagram here where I've got dimerization I've got I've got monomers decaying I've got diamonds decaying I've got transport you could include things like transport out of the nucleus splicing yeah there's a bunch of things you could you could do so and then you could be much more explicit about what's happening on the DNA you could choose to model the binding explicitly you could put in multiple binding sites and it's a bunch of work being that's been done about that the big problem is that none of those models up to now have got any experimental counterpart that lets us say oh okay so this model has seven sites you this this model tests the difference between having one to ten sites which I feel the different occupancies but no one's ever built a fish or a mouse over there where there are one to seven sites so that's why I'm leaving those I think this should be interesting but that's why I'm not I'm not going to discuss so that's that's what that's one thing about consider other events but you might still be in a sort of ODE framework and the other thing would be to say well maybe there aren't very with this oscillators going quite fast maybe there aren't very many molecules involved and so maybe the approximation of an ODE is not is not the right one if we want to understand the the noisiness in the system and so there are some papers here which I'm missing it which have used Gillespie algorithms and other techniques to to get it sort of a more maybe more realistic chemical situation okay so now what I want to do is I want to talk about three period mutants in the remaining half an hour right and two mouse mutants and a zebrafish mutant all from the same family I'm going to start off introducing period mutant which we've borrowed the term from the circadian field and it means any mutation or treatment alteration to the genome which changes the period of the main observable which in this case is so much is segment formation and so going from the S6 mutant which which we isolated by a retrovirus landing into the S6 gene we noticed that the the fish made its segments more slowly these are the the orange segments and the blue segments are a wild type sibling it's about six percent six seven percent slow it's incredibly reliable it's not very much but it actually changes the size of the segments that are formed by six to seven percent and in the final animal it reduces the number of backbones by seven or eight percent so what that means is quantitatively the animal that was made from a clock that appears to be ticking seven percent slower has an altered segment size and it has an altered segment number exactly predicted by the change in the period of the of the of segment formation so that's a good that's a that's a good period mutant and it was the first sort of I think general test that these two relations might be true and that we might be thinking in the right direction that you you can reach into this putative clock in the posterior can change its timing and what comes out of that are changes to segment length and number okay I'm going to leave that there I'm going to come back to it what I want to do now is to tackle some of the claims that some of the ideas that come up in considering the delay differential model these these are sort of viewed in in the light of the delay differential model but things like half life of the protein will appear in almost any model that you make where you've got that that protein in there so I think it's still generally maybe less than to be learned okay so this is a wonderful piece of work from the Kageyama lab and they reason that maybe we could make this that they could make the cycle tick slower if they could change the half life of the protein it's a clear prediction from the model so they scanned through the seven protein they looked for all of the lysine residues which has the abbreviation K because lysines are the target for a modification that destabilizes the protein and causes its degradation it's ubiquitination that detail doesn't matter but what they found was they mutated each of these lysines in turn and they're listed down here and then they expressed the protein in a in a cell a cell culture and they assayed the proteins ability to repress so here's is an unrepresed promoter at one and now this is the promoter they're going to try and repress the inbox will bind that protein and then you ask the question can I switch off the lyciferase okay and the wild type the normal has seven protein switches it off the k14R does these variants can't switch the can't switch the reporter gene off so they were no longer considered because they're not repressors anymore and these these two also worked effectively well talk they only made one mouse and it was the k14R I think no it was a hundred and eighty in G sorry let me go back and now in the same cell line they measured the half-life of these of the various proteins so they wanted to say we want to make a mouse we want to put this altered gene we want to know that it at least in cell culture it can still repress a target and we want to know that we made a different half-life so they chose one particular version and here they've altered their numbering so they chose they chose a protein with a longer half-life let me just k14R and here we see what happens when they made a transgenic mouse with the animal so what i'm showing you now is an early mouse embryo with its head here and its tail here and here it's formed just three segments i don't know if you can see that it's it's a bit small and here are two two different mouse embryos both wild type and you can see these striped patterns of the segmentation clock genes in the posterior if the seven gene is removed altogether so now the whole gene is is deleted then you see some sort of patchy expression of these genes but no real stripes and now you don't have any stripes in that tissue anymore and this is interpreted as having completely wrecked the mouse clock there's no more oscillations you can't make segments and in fact the skeleton of this animal is badly deterred um when oops and so here's one of the mutant versions with a longer half-life and you can see that it's still forming anterior segments and it still has some oscillatory pattern in the back so it doesn't appear to have an obvious change but when they look just a few hours later here's the wild type and the full mutant and now they look at their mutant variant you can see that the that over time the system has come undone so just a few hours later there's no remaining striped patterns and you can see early segments have been made and then the posterior segments are defective so this clock has undergone a transition from from having rhythmic behavior at the beginning and then losing the rhythmic behavior so they've come they've come returned to this model and they've said okay well um they've parameterized that they've changed the half-life of the protein to match what they got in the cell culture it's an open question whether that's the half-life of the protein in the animal but that's a good start I think and they've run the model strangely enough they've kept all the other values the same as the original zebrafish model doesn't matter I don't know and here's the behavior that they get with the the oscillator having a longer period and then damping and so so is this convincing I think is the question and I think it's a it's a an amazing piece of engineering of genetic engineering to think about trying to engineer the period of the clock to make a variant protein to go through the testing to make it but um what you really want to see is an oscillator that's persistently ticking with a different period because there's lots of ways to break a clock and to play the devil's advocate I could fiddle these numbers and get almost any dynamic behavior I liked out of the system so this is consistent with the change in the phenotype that we saw being because of the longer half-life but I I would argue and I have argued many times to Richiro that this is not this is not conclusive okay this is consistent but not conclusive okay cool so that's that's the Hirata paper so this is this paper is actually cited quite often as proving that a longer half-life will slow down the oscillations but but it doesn't okay so that's important tonight okay what about speeding up the clock so again this is now this is now nearly 10 years later again from Richiro Kageyama's lab and this is a an astonishing piece of work because Harima what she did was she made she took the S7 gene and she said what if we can speed up the production of the mRNA by deleting the introns so when a gene is transcribed you have to polymerize the RNA up across all the introns then you have to splice them and both those events take time it's not clear exactly how much time they do take but they take some time and so the reason is if I remove those events then I can finish making the cell can finish making the mRNA quicker and so the total delay to go around the loop should be less I think it's a perfectly reasonable proposal so they're about twice the length of the exons yeah the introns are about twice the length of the exons yeah so so there are there are some so polymerase has moved pretty fast maybe two kilobases three kilobases a minute and some genes have massive introns kilobases megabases even in some cases and in this and in those cases you can take it can take hours for the polymerase to clear it to move along the to move along the gene and this is this has been recorded this is not a gene that's this short where you've got as you point out that's not so different and for a high speed for a polymerase is it going to make any difference splicing introns seems to take about five minutes per intron and it's not clear what sets that time that's a very loose estimate coming from a number of different systems so so that's so they said well let's take them out and see what happens and so again testing this in a cell culture line these constructs are expressed and you can and these these plots here are the heat shock they serum shock the cell and then that that causes the the gene in the cell to be expressed and now what they're measuring is if it's got introns or none or just one intron how long does it take to reach maximal value so this is in a cell culture again but it's it's a reasonable test to try and see whether it does make any difference and in fact in fact it does if you if you look at the if you look at the both the rates and then the time to maximum you can see that deleting the introns systematically speeds up the production of the protein so okay so qualitatively that's working so now again they made transgenic lines and I'm I'm you might say well how come I'm only showing you one version actually Harima made all possible versions and only one worked so here's the one that worked goes into the mouse and this is this is true this is truly astonishing phenotype if you I guess it does help to be obsessed by segmentation for this to be a truly such phenotype but so wild type mouse has seven cervical vertebrae always have seven this is the first thoracic which is the first rib bearing vertebrae and actually all mammals do all mammals have seven so giraffes have very long necks but they have exactly the same seven vertebrae that that we have or that all that mice have they just in fact in in in embryogenesis they actually don't form bigger than the other segments they grow by they grow in utero differentially so differential growth makes those segments big not early patterning differences so in contrast birds have different numbers of segments in their neck so the reason why a goose can do that is because it actually has more segments in its neck than a duck for example so there you go okay now what we've got here is we've got the duck goose transition in a mouse because what they saw was that depending on how many copies of the transgene were present sorry no the ratio of the of the of the wild type to the transgene they could make nine or eight neck segments that doesn't sound like a big difference but actually it's a it's a really striking difference for a mouse embryologist and this suggests that because you produced more segments and they're a little bit smaller uh that the clock's running faster by 2013 Harina could take advantage of the transgenic the heseven transgenic uh reporter line that they've made in the meantime and what she did was she placed a region of interest over the tissue and measured the the period of the of the gene activity that was coming off the transgene so this is using not a gfp but a lyciferase uh but it's the same general idea the the reporter gene produces a signal and you measure the frequency so here's the result uh for one animal that they've plotted and when they they're measuring uh interpeak intervals here and they see that when in the transgenic animal that has the single three intron that it's systematically shorter period so this is um this is this is now an animal that's been engineered to have a faster rhythm there's one interesting note so now so this is again how this was discussed in the field you come back to the model this time you leave the half-life of the protein the same but you change the the delay corresponding to uh different lengths in the 29 or the 24 is the value of minutes that corresponds to the estimated difference in production of the protein that they measured in half-life and so here they go here's the here's the shorter period here's the wild type here's the shorter period and it so here's the reduced delay and the model gives you back a shorter period okay so so this is then used in the field to say well okay so the the delay coming from the introns it's real this delay is playing the role that we thought it was there's a couple of a couple of the two caveats to interpreting this though which i think are worth they're worth trying to understand and i certainly don't understand at the moment one of them is what i what i the the dirty secret here is that um this animal doesn't make normal segments in its posterior they're they're strongly disrupted so the clock ran faster but then it crashed the same problem that happened with the other animal when the half-life was adjusted you seem to be able to alter the period for a while but then then then it breaks actually in that other there was no evidence in the other animal a big part that the period had been altered there's no evidence in this case there's good evidence that the period has been altered but it's not stable for some reason the clock can't sustain an elevated period and that you can see a hint of perhaps in this plot here with perhaps an amplitude damping here so it's not clear so we're still we're still missing something where the other caveat of course is that all of the other combinations of introns uh didn't all of them caused a complete failure of the clock so just one variation of those introns was was the right so now i know that that the karyabas group is now trying to understand what it is about that intron that gives the desired timing but i still think that uh we don't quite understand that it's a really good suggestion so the idea is um what if this were an excitable system driven by noise and if you drive with strong enough noise uh you're not going to be able to recover to your reset point and you're just not going to see i i think it's i think it's an open question i mean um when one considers the this uh that feedback loop then it looks like it doesn't look like an excitable system but you know maybe we got that wrong right and maybe it's it is behaving as an excitable system so um i'm trying to think now whether i know any experiments that conclusively rule out an excitable system let me think about it yeah what a good idea okay that is an extremely good idea it's a really good idea so what i want to show you now is a mutant that does that exactly what you said uh or something very similar yeah yes is there any way there's a there they are the the problem with the zebra fish clock up to now is has been that it's sticking quite fast and the kinds of changes that in the past we've been able to do take several times longer than the cycle to be introduced that's not an excuse but it is an excuse it's but it's a call for better technology and actually i know that rio's group has been using a blue light inducible transcription factor so you don't so it dimerizes so you don't have to wait for a production delay it's present already and the time the the timescale is the dimerization and there's an amazing set of work from uh someone in his lab using pulse or light pulses to try and entrain so i won't i won't steal that thunderbolt but keep it keep an eye out for it still i don't uh i was discussing with him whether there are any signatures of the bifurcation i don't know if it's if still our measurements are too crude to uh to be able to see them yeah i don't know uh so i'm i'm where i'm aware that i've got uh 15 minutes left okay so what i want to do then uh i'm going to skip over some of the data in the next section to talk about what i think are the the principles it's from as you've gathered most of this stuff is published and all the references are here i'm trying to i'm actually trying to work out what we know and uh as usual the the field of ignorance is much lighter than the field of knowledge um so let me let me give you a sketch of this idea of having more than one clock there um so up to now we've discussed this simplest feedback loop but there could be other feedback loops or operating at the same time and now i'm going to come back to the the notion of hess hessix and the first thing you have to know is that in the zebrafish there's more than one oscillating her gene there are two her one this was the trans gene i showed you before there's another one on the same chromosome it's only 12 kilobases apart it's called her seven and this is its pattern so these are sort of snapshots of the stripe pattern moving up up the tissue there in marina there's another gene that's also expressed in the tailbone and preseminic mesoterm called hessix and that's the gene we hit with a with a retrovirus uh by accident um and it forms a a gradient of expression across that tissue and so uh and so one is now has the possibility of forming various kind of feedbacks with all of these different proteins so this is this is just um this is just a sort of all to all feedbacks sort of a mean field um coupling of all those we took some time and we in vitro we measured all of the dimerization and all of the dna binding capacity in that network and this and this is what we found we found that all of those three proteins can form homo and heterodimers with almost the same affinity so these things are continuously exchanging uh dimers um here at at equilibrium and that's fast on and off um however the and and there are six possible heterodimers there's six possible dimers in the system only two of them bind dna so this limits the number of points of feedback onto the express onto the expression of these two genes to just two her one can form a homodimer and repress itself and her seven actually requires hessix to form a dna dna binding heterodimer and come back on to the onto that so this is the biochemistry i'm not going to show you the data for that um but this all to all gets severely reduced to just having two two loops so okay here we go so remember i said that hessix was a dimer hessix was a period mutant can i show you okay sorry folks um right what i want to do is give you a very brief snapshot of what what each of those mutants looks like and then the doubles so i'm going to accumulate all the hits into this circuit okay uh hessix um it segments almost normally so this is what the circuit looks like no hessix her one can still complete the feedback loop and we see uh essentially normal segmentation we see the clock with its uh strike patterns and that's the one where we had the period seven percent change in the period okay her one mutant segment normally almost um as well here's a her one mutant and actually have to look pretty hard to see that the defects sit here and again you can see the strike patterns in the her one mutant so clock looks to first approximation intact and that's explained in the topology for the biochemistry because this feedback loop is still intact so both those guys should be fine the her seven mutant is interesting it should still have the her one feedback loop running so it should work but actually here's where we don't quite where there's a mismatch to the biochemistry in some way because the posterior doesn't segment properly so somehow in the presence of extra hessix this isn't stable any longer something's going on up here probably um and the clocks um still ticking early on and it seems to be badly damaged in the posterior now I'll make the doubles if you take out her one and her seven you would expect that nothing should happen right it should be badly damaged here's her seven and the her one individual and when you combine those two together you see segmentation defects along the whole axis okay so that's in that's in line with what we'd expect I will point out that there's a couple of really good segments in the middle of this animal and every now and then a good segment turns up in these in these creatures so I that's a fact I can't explain it um but when we look and when we look at the gene expression patterns we never see any stripes in those guys so it looks like we've got some sort of severe problem in our clock uh if we take out if we leave her seven in but take six and and one out so now we don't have we see an almost identical phenotype I find it difficult both in the segments and in the clock genes I find it personally very difficult to tell the difference between those two combinations one and seven out or one and six out but if we take six and seven out we we rescue the phenotype so this double if you've already lost seven or already lost six it doesn't get that much you you actually rescue the phenotype you actually you you no longer have this defect in the posterior can you see that so taking her set taking her six out in addition gets rid of this defect so so hess six so hess six when it was present was somehow poisoning the system causing these defects and when you pull it out now you're running purely on her one and and it's fine so that that's that's a clue and um and that that uh mutation oscillates as well okay so now right so now we can measure the period and what we'd expect if her seven and hess six are equivalent all they're doing is forming this feedback heterodimer we should expect the period difference to be the same but it's not so her seven actually has the same period as wild type within the error of the measurement her one has no change in the period so this is now you can now see this redundancy in the system we're not changing the period by pulling things out but hess six uh you saw that data already but now we're combining it with the hess six and her seven that um that the her seven double homozygous has the same period as the hess six so the hess six period change is dominant it doesn't get any worse when we pull out her seven in addition so how do we make sense of this okay so what i'm going to do is uh i'll tell you the conclusions um and just say that we used a variant of the Lewis model with delays can you see the delays is the product this is the um the delay for her one uh this is the delay for her seven and we explicitly included this time dimerization because the topology that I just showed you that can't explain the differences in the phenotype her seven and hess six should be identical the differences in the phenotypes have to come from something off the dna and so that's why we included explicit modeling of the dimer so let me tell you um what we what this model predicts initially we set all of the um parameters to be one or equal to each other you didn't put any asymmetry into the model apart from the asymmetry in the topology we couldn't simulate the data properly uh so we introduced two asymmetries uh the first one was um came from looking at the hess six mutant so you look at a hess six mutant by simulating that system and setting production of hess six to zero and now you increase the production of hess six until the offset matches the observed difference between mutant and wild type and that gives you a production rate for hess six that's nine times bigger than that for uh one and seven okay that's the first asymmetry that we had to put in and then um the next asymmetry was that when we simulated her seven mutant we actually get uh a damping um but we get the period wrong and to get the period to match the wild type we have to say that the her one gene takes a little bit longer to make than the her seven gene so that's consistent with the intron-based delays that we saw in the previous study um and then it then it matches quite well so this we're not trying to parameterize this with seconds or miniterremies this is all this some sort of uh internal time scale and of course the seven six uh matches okay so the point here is that the dynamics uh matching the periods getting the periods right all of all of that comes from these interactions up here so the dimerization with each other uh changes the um so let me just get this right um in the hess six mutant you in the hess six mutant her seven and her one no longer degrade at the same rate because their their degradation uh depends on a non-linear effect with forming all the dimers so by by pulling her six out of system one and seven can't degrade at the same rate so they become their effective stability goes up and the period slows down so the presence of hess six is changing the way the other the other proteins degrade uh in the model and um the her seven phenotype comes about because um in the absence of her seven hess six is freed up and now titrates her one out and without her one enough her one this feedback loop won't complete properly and the oscillations damp so that in the model the explanations for these two phenotypes come from uh changing degradation and sequestration all in this dimer cloud they're not coming from the activity down here on the dna okay so i'm i know i went over that too fast uh in the interest of time the message is that in the model at least what's predicted is that it's uh the dynamics in the dimer cloud are playing an important role in setting the dynamics of of the oscillatory circuit not just the guys binding dna but also their interactions in non-dna binding partners so here's an example where we have two coupled circuits they're coupled by dimerizing so in the sense each of these could be a clock but they're coupled through the ability to dimerize into this into this dimer cloud up here okay there's one prediction that the model makes which is that the hess six protein level should oscillate you can't have this working unless hess six protein levels oscillate and that's because the uh just as the one and seven are being degraded along with hess six well hess six is being degraded along with one and seven and so it follows an oscillatory path this is the plot from the from the model so we raised an antibody to hess six and we had a look at its pattern and we found stripes of hess six now hess six transcription doesn't oscillate this is entirely the protein levels oscillating driven we would argue the claim is by the interactions between these dimers so this was a this was a complete surprise it's a non-trivial prediction of the model of course it doesn't mean the model's right just means it's useful for now okay so um there's a conclusion there there's a two loop parallel negative feedback the short period um maybe maybe if you have a short period if you have more genes in parallel you're more robust you're less susceptible to fluctuations in the molecule number from cycle to cycle um and the presence of hess six enables you the system to tune from using only her one to increasing amounts of her seven being able to come back onto the DNA and you can think of situations where that might be useful and to get the basic phenotype the basic phenotypes of oscillating or not are explained by the topology but the dynamics to get the dynamics right you need to include these effects in the dimer cloud good so that's it first tier um there are noisy autonomous oscillators in the segmentation clock there's a negative feedback mechanism which we think depends on the hess her family of genes by auto repression um and we have a set of mutants some engineered some naturally occurring which do change the period but i think what you'll gather is we don't really understand how they're doing it we're still not in a position where we can design a change to that circuit and get a stable oscillator back and predict the change in the period um so i think that's these are still open questions these are things we need to be able to understand just to quickly list them now it's not clear to me that oscillators in amniotes mouse and chicken are autonomous so that i think is is we need to understand what the real effect of having multiple proteins is still not well understood so we need to understand this idea of having the dimer cloud um in more detail what are the sources of noise that we don't understand can can we go from a genetic regulatory network type description or understanding and explain the single cell data the way the noise in that single cell data work that's only that's very recent data we haven't actually done the experiments and i'm not sure that that we'll be able to do that yet um and then there may be other oscillators and i sort of cheated by giving you other oscillators from the same family but it could be that there's other genes oscillating it could even be that the idea of a simple negative feedback uh is not right and that some sort of excitable system is in instead at the heart of of the system okay that's our limit today two big open questions are communication between oscillators which will be tomorrow and then the tissue level control of these oscillators which will be on saturday evening so thank you very much