 All right, so let's go ahead and kick off our first talk for this week, and this is going to be delivered by Pietro Cicuda from Cambridge. And Pietro has been heavily involved in the hands-on schools for a number of years, co-directed. He's really been a driving force, and it's a pleasure to have him here this week. He is tag-teaming with Dominic Vella. Dominic returned to Oxford, and then we have someone from Cambridge. So it's a pleasure to have Pietro, and he's going to tell us about his work in biophysics. Thank you, Mike. Good morning, everyone. It's a pleasure to be here now, and I'm sorry I missed the previous week. As Mike said, I had a range with Dominic who is a collaborator on many projects to split half-half this year. Today, the title you saw is biophysics. I would actually talk to you about one of the projects we have in the lab that addresses a biological problem. But I would like to have that as an example of how physicists can address biological things, because you may or may not be interested in the specifics of what I'll tell you today. I would still like you to think, okay, as a physical scientist or an engineer or a computer scientist, at some point you may be challenged or you may become interested in addressing biological problems, or maybe you already are. And one has to think what are the good problems where you can actually contribute significantly, as opposed to problems where the science just isn't ripe for physical scientists to come in, or the other danger is just becoming kind of the technician for some biological or medical operation, which is okay. But if you do that, you have to be aware of what you're doing. Anyway, having said that, I'll return some of those things during the talk and at the end, and I welcome questions in that sense too, I will focus a little bit on one particular area that we're dealing with in my group. So I've been an API for about 10 years and I've managed to build up a group of people. These are PhD students and postdocs. Some of them, Nicola and Luigi, have taken part in previous hands-on schools as demonstrators, but this year it's just me from this photograph. Okay, so the subject that involves about a third of those people in the photograph is understanding how these little filaments called motile cilia work and particularly how they beat. So this is a slow down movie and how they synchronize and making such a nice wave, a traveling wave. So here, these filaments are about 10 microns long. They're microscopic. We have these things in our airways, so from the lungs to the throat. We have them in the brain. Females have them in the fallopian tubes and they basically cover the whole tissue covering many cells and they push fluid. In this case of the traveling wave, they're pushing fluids from the right to the left. They're also enhancing transports across the motion that they're creating. This is all low Reynolds number motion. I'll tell you what that means, but some of you probably know already. So there's no turbulence. And basically the question is, these things, they're not communicating electrically and probably neither are they communicating chemically through calcium. What we think they're doing is communicating mechanically. So they're like, each one of these has an inbuilt pendulum like periodic motion and something in the way that they're coupled makes them couple in a way that creates a nice traveling wave. If you want to make an aula or like a Mexican wave, if my talk is very good, you can try to kind of stand up and create a wave. These still are doing something similar by getting a trigger from their neighbor, a mechanical trigger in this case. Given you the plan already. So in general, I've moved from being a soft matter person during my PhD. I studied physics beforehand without doing very many experiments. I did an experimental PhD looking at liquid interfaces and never in my life I thought I would touch biology. I thought it was complicated that there was too much to learn. I couldn't understand anything. I didn't really think physics could do anything there. But gradually over 10 years, I now have totally changed this opinion. I think biology is creating really nice data, even without the help of physicists. Theologists do this anyway. And now this data needs explaining. And the data is of high enough quality that it looks like this type of measurements that a physicist or a physical scientist more generally would take of materials that are non-living. And therefore, as a consequence, also the kind of thinking and modeling that you can do on this data is now the type that we're used to in materials, for example, or in measuring other non-living things. And the beauty is that in physics particularly, many of you are physicists, we're used to both drilling down, being reductionists, but also kind of looking across scales and trying to understand how the rules that apply at one scale can give rise to some collective behavior at a higher scale that isn't really encoded in those rules. It's a completely unexpected, sometimes emergent behavior. And biology is really that. It's working out of equilibrium, close to thermodynamic equilibrium in some sense. It's working with many time scales and many length scales. And it's all about emergent behavior that arises from some simpler rules and some simpler interactions that the biologists have now deconstructed. So the beauty is that a lot of the detail, the nitty-gritty of what molecules do what has been worked out. And yes, that's the point where a traditional biologist, molecular-biological approach gets stuck because this link then from that level to understanding a process that involves many of these interactions together is something that hasn't really been part of biology up to now, whereas it has been part of physics. So that's kind of my pitch for why it's a good time for physicists to go into biology. And it's the physics of statistical mechanics, soft matter mechanics generally, but also from the theory point of view networks and the linear dynamics are the key ingredients in understanding this sort of biology that I've been alluding to, basically cell biology. But not just cell biology, that means cell biology is where a lot of people have gone into so far, but the same ideas could equally apply to simple ecological networks or to other bits of biology where, again, the data is good and the questions are really kind of have become physicsy questions. Okay, so back to my topic today. I told you we're going to deal with these microscopic cilia. They beat at a few tens of hertz and if you calculate the velocity and the time scale and given the viscosity of the liquids that they're beating in, this boils down to a low Reynolds number problem. So Reynolds number is the ratio of density, velocity, length scale over viscosity. So if viscosity is big Reynolds number is low and it means you have no turbulence. And this is a bacterium with its flagella. This is actually a different biological structure from the cilia that I've showed you which belong to eukaryotes, not to bacteria. But the length is similar and the speed at which they move is similar. It's just to say that E. coli moving in water, moving its flagella in water, creates a Reynolds number which is very small, 10 to the minus 4, whereas a good swimmer in water is 10 to the 4 for this ratio. So there's an 8 orders minus 2 difference. And if you want to reconcile that and understand how it would feel to swim at low Reynolds number, you would have to go and get kind of a chocolate spread like Nutella and try to swim in it. So that's the environment that something like the bacterium is feeling and also these cilia that are beating are feeling. Basically they're feeling a very viscous fluid and in the case of cilia in the airways it's even more complicated. They're feeling the mucus, the kind of disgusting stuff that we cough out when we have a cough. That's the stuff that the cilia are beating in. In fact that mucus is a beautiful biological screen to protect us from dust and from lots of airborne pathogens that get stuck in the mucus. And that mucus is carried up by the cilia that are my object of interest. And most of that mucus just gets thrown into our digestive tract. Okay so not all biological motions are low Reynolds number. If you look at large things swimming and also flight, those are all high Reynolds number problems. This is not my area of expertise. This is the area of turbulence. It has beautiful questions but they're very different questions. Whereas little things swimming in liquids are typically low Reynolds number. So hydrodynamics is kind of explained by a kind of momentum balance through this Navier-Stokes equation, which many of you have seen probably in courses and some mathematicians spend their whole lives addressing certain regimes of Navier-Stokes. In low Reynolds number you can simplify this a lot and you can get rid of the non-linearity parts because basically you linearize for high viscosity and low velocity. And what you get is an equation called the Stokes creep equation where you still have time dependence of the velocity if you want to. But the right hand side here has been linearized. And within this equation people have solved for example the flow around the sphere that is moving slowly in a fluid already almost 100 years ago. So the only thing which is relevant about that in the context of cilia beating is the fact that the drag on a cylinder moving this way, a cylinder of a certain length moving kind of perpendicular to itself is different and a little bit higher than the drag on a cylinder moving along its axis. This means that if you drag the same cylinder this way and then turn it go back this way and then turn it go back that way. You're actually pushing fluids to the left. And so you're doing a conformation change in the system and then a periodic motion and this has broken the symmetry and allows you to push fluid. What you can't do at low Reynolds number is break the symmetry just by say moving a little bit faster and coming back slower, faster and slower. Because this equation is linear if you just play with velocity and time in this way you just end up moving the same fluid to the left and to the right and you haven't pushed anything. Whereas this is a trick that we exploit when swimming because we're using higher Reynolds number properties of the flow. So at low Reynolds number if you want to generate a pumping motion or a swimming motion or any sort of net momentum transfer into the fluids you have to play a game of conformation switching and change your drag coefficient as you go through a periodic motion of the pieces of the system. That's going to be important in the cilia, the cilia that I showed you already in that video at the very beginning. They have a power stroke where they are extended and they push and then they can fold on themselves and go back and have a lower drag as they're going back and then a power stroke recovery stroke. Okay, the other thing which people, again, Ozin who just followed Stokes in the history of that branch of fluid dynamics worked out already a long, long time ago is that if you again at low Reynolds number where the equations are linear you can calculate for a point like object moving through a fluid. I already told you you can calculate the flow field around this object which is point like but with a sphere boundary condition. What you can also do is calculate the force that this moving object will put on another point like object somewhere else in the flow. So this is now hydrodynamic interaction. It's the force that a moving object puts onto another object in the fluid. So if you have objects labeled with a N, imagine two objects for example, if my object number one is moving, there'll be a force acting on it which depends on the drag of the fluid and that simply Stokes. So if I want to move my object through the fluid, there's a drag that resists me. But now if I have a second object in the system and it's also held by force, my object two manages to exert a force and object one. Okay, I made a long story but basically the fluid flow transfers a force between the two objects. So this force that the two objects feel mediated by the flow decays as one over the distance between the two objects. And again it depends on the viscosity of the flow. So this is a kind of a linear matrix that in principle if you have N objects couples each of the N to each other and this one over distance decay in physics we would call it a long range decay. It's a decay that basically means that your system of many objects really behaves as a collective many body system. So it's a tensor and the entries of this tensor are three by three objects, so it's a very, it's a busy thing to fill. But if you have a simplified problem where you just have two objects that you're dealing with and you just make a move on the axis then this equation here just becomes something much, much simpler which is down here. And now so the velocity of my object one depends on the force that I'm putting on object one resisted by the drag of the fluid. And then the force on the second object and the interaction depending on that depends on the distance between the two objects. So this is now fairly simple. So this would be if I just had two cilia and they were beating pushing fluid towards each other. Force one and force two I would imagine are the forces that are inside these cilia and that are coming from the biological molecular motors that consume ATP and turn this into force. And then the two objects would actually feel each other also through the fact that they're pushing liquid. So this was kind of had been discussed as a theory idea about 15 years ago now and as a possible mechanism for cilia synchronization. So the fact that these cilia existed has been known in physiology for a long time and people had started worrying about how these tissues managed to have those nice waves that travel for centimeters. So much, much bigger distances than the 10 microns of the single filament. So beautiful kind of collective wave. And then so one first set of experiments that we started to do about 10 years ago was to make a simplified model of the cilia to just see how far away hydrodynamic synchronization would work when you're in a fluid such as the water or slightly more viscous water and in the presence of thermal noise. And we did that by working with little spherical colloid particles and putting them in optical tweezers and using the optical tweezer to generate an oscillating motion of the beads and then putting the beads further and further away and checking at what distance we lost a short distances, whether they would synchronize and go together and then over longer distances whether they would lose synchronization. And if we work with spheres, then the hydrodynamics is really what ozine equations give us. And we can even write Langevin equations that probably some of you have seen for just the Brownian motion of single spheres. We can write that as coupled Langevin equations for the Brownian motion of more than one sphere, so two or more, also held by external forces like the optical trap. So the optical trap is also optical tweezer is a system that costs a little bit more than our budget for the typical hands-on experiments, but not that much more. So it requires a high magnification objective and if you want the tweezer to be really precise it's got to be a high quality one, but you can sacrifice a bit if you prefer to just kind of want to hold objects. You then need a laser. It doesn't have to be super powerful. If you're working with visible lasers, probably 30 milliwatts is enough. But if you're working with biological objects, it's better to work with infrared light and we work with 1064 nanometers. So this laser is basically passed through a series of telescopic lenses so that at the flexion of the beam that we generate here is transferred to a change of angle at the back focal plane of the objective, which then means it becomes a translation of the focus of the beam when we're in the front focal plane of the objective inside the sample here. So this is one way of how you achieve moving this point of focus of the laser beam. And the importance of the point of focus is that when you have this very tightly focused light, the momentum that the light transfers onto the objects in the liquid is in some conditions enough to hold that object at the point of the high focus of light, which is why this is called a tweezer. You can basically shine your light and have like a tractor beam from Star Wars holding your object at the point of high light. And this can be used to hold things like single cells or individual bacteria or in these experiments, these plastic spheres. The objects have to have a slightly higher effective index than the liquid around them in order for the refraction of light to add up in a way that gives you a dropping force. Okay, so you end up with quite small forces of the order of a few piconewtons, but these are enough to be slightly stronger than thermal forces that come from just Brownian motion. So you end up holding something and then the thermal fluctuations make this thing typically jiggle in the potential that you've created by having this dropping light. Okay, so we did we did a lot of experiments with this building up towards using the system as a as a mimic of the motile cilia. So this was one of the first experiments we in this experiment we just held a number of particles. So in this picture, there were three. These experiments, there were four, five, six, seven. And here the traps are not moving. So we just kind of created those three traps and put a particle in each one. And then we made videos. And unfortunately, this picture here is static. But in the videos, each of those particles just jiggles around basically exploring the minimum of this potential well, which is created by the laser light in each of the three positions. And then we could analyze the properties of the fluctuations. So, so there's fluctuations of each trap in the minimum and they, they kind of, if you plot the displacement from the minimum, it forms a Gaussian. And some of you must have done this problem as a physics problem if you just have one potential and and and a particle jiggling about in a liquid in that potential. You end up with a Gaussian of displacement. But what's interesting here is that we have, we have more than one and we can look at the cross correlations of the fluctuations of these objects. And if you do that, you, you back out that the cross correlations are explained by the Ozine tensor. Basically because if you have one particle suddenly moving, it's creating a flow field from that fluctuation and the other particles will feel that flow field and be pushed in the same direction. So that gives correlations across, across the fluctuations of the particles. This was particularly interesting in these kind of regular kind of polygon shapes because the Ozine tensor could be diagonalized and we knew exactly what, what should be happening analytically even in the, in the relaxation times of these particles in the trap and in the cross correlations between them. This is just the formula that says what I just said in words. We then moved to experiments where we started kind of moving the particles and the first thing we wanted to prove was could, can we have a set of particles that we move in such a way that they end up pumping fluid. Now I told you at the beginning that we, a Lorentz number, you need to change the drug. You need to, if you have just a cylinder, you will have to rotate that cylinder to, to end up pushing fluid that way because it's not enough to just move it this way. So how can we do that if we just have spheres? Well, Purcell, who was one of the leaders in this, this area about 50 years ago, I think, maybe a bit more, had shown theoretically what he called the, the scallop theorem. So the fact that if you have a, a scallop is this shell and maybe this is a movie that moves. Okay, this is a scallop. You may or may not, you probably have seen these shells as dead, but this is a real scallop moving. So it opens and closes this, this, this clam of two shells and, and it propels itself by, by jetting, jetting liquids out from the back. So this, okay, so Purcell called his, his concept the scallop theorem, because he said the scallop should not be able to swim if it were a Lorentz number. And the reason is that it only has one degree of freedom, this, this angle between the two, the two shells. And if you only have one degree of freedom, it's a bit like just my, my, what I was describing as moving left and right. You're going to end up with the fluid flow just reversing itself when you open as opposed to when you close, if you're a Lorentz number. This works because it's jetting and that's higher in a slumber. Okay, so, so what's the minimal amount of degrees of freedom that would enable you to, to pump or swim if you're a Lorentz number? Well, you need two degrees of freedom so that you can act on the first degree of freedom where, where the second one is in one condition. And then you act on it again when the other one is in a different condition. And then the fact that you have those, those two degrees of freedom to switch your configuration means you can make a set of moves that is non-recip- that is non-symmetric when you, when you turn time around. So, so Purcell sketched this. So here the degrees of freedom for him were the angles between these, these three rods. So first, if you flip the right hand rod, you go into this shape, you then flip the left and then you flip the right. But see this, this flipping of the right from down to up is occurring where this is in a different configuration from, from this one here. Here it was up, here it's down. So, so there's a different drag when, when this one goes back up. And then finally the, the left rod goes back up. So you return to this configuration. You can then cycle again. So this can happen again and again over time. And this object will swim or it will not stay in the same position because it's breaking the time, the time direction with this sequence of moves. And we can do something similar with beads. This was studied first by Jaffe and Goleztanian. I think, so Ramin Goleztanian is now in Oxford, but I think when he did this work he was in Iran. And certainly Jaffe is, Alim Jaffe is still in Iran and there's still a very interesting community working on these problems in Iran. Anyway, so what they showed is that this, this idea from Purcell could be realized with spheres by, by having three spheres and playing with the distances between sphere one, two and sphere two, three. And those are the two degrees of freedom that you need to play the same game that is happening in this sketch here. So I'll show you. So they studied it theoretically and, and we did it with the optical tweezers. So we held three beads. We moved the right hand beads a little bit to the left. Initially they are the same distances. We moved this one to the left and we moved this one to the right. Now all three are a short distance between each other. And then we move this one back here. So it goes back to this position, but it's doing this move in a condition where this one is close up whereas here this move when this one was far out. And then, then this beat goes back out and we return to this configuration. So this is time and it cycles and loops, loops over, over itself. So this, this sequence of just four moves done over and over again generates a fluid. So, so this is like a micro pump. And if, if this object. So, so this is the only thing we can do tweezers generate a pump. But if this, if these moves were created by a little piston holding the beats together, then this object will be a swimmer and you would have like a micro micro submarine able to move itself. So this is how it's quite boring experiments. These are the three trapped beads. This one we're not actively moving. It's just receiving the fluid flow from the other two. And we can analyze the position of that mean beads and work out that it's being subject to, to a net force, which is the fluid flow that this micro pump is generating. So I want to tell you the details of how we really analyze that data because it's all bit kind of too much. So, so that was how a Lorentz number we could play with optical chops and understand how fluid is generated. And then that led us to the real biological question, which is how do these things synchronize. So, so you now have to imagine that each of these cilia in our heads is represented by a bead and that bead is doing a periodic motion, say left and right. And we now understand a bit about how a set of those periodic motions can generate a flow. But now we have a separate question, which is how does this system actually generate that well defined set of moves that we know can generate a flow. Okay, so there's, there's a simpler biological system, which is an algae that lives in many ponds, freshwater ponds. And then it has various companion algae that lives in the sea that are much simpler and only have two, two motor cilia. So, so in these videos, the algae is here. It's barely seen because these are subtraction videos, but there's basically an egg like objects here. It's a few microns big. It's held by a glass pipette that again, you can barely see here. So it's not a, the cell here is not actually moving. What's moving are it's two, it's two 10 micron longer appendages. So for most of the time, if you, if you look at this algae swimming or if you hold it and watch it like this, the two filaments are moving in phase. So like it's like swimming breaststroke. But then sometimes the phase is lost and the two filaments start going in out of phase or, or random phase difference. So, so this was the work of a colleague of mine in the applied math department, Ray Goldstein, and they published a beautiful paper where they showed that this, this loss of synchrony in this algae is biologically relevant because it leads to the algae changing direction. So when it seems breaststroke, it goes straight and quite fast. And then if it wants to go towards food or towards or away from good lighting conditions, conditions, it needs to have a way to change direction. And having its Cecilia lose phase is, is, is the way it has to, to randomize its direction, choose another direction. So the strategy becomes a bit similar to how bacteria can go up gradients of food or away from things that they don't want by randomly changing direction and going into a tumble phase. This algae is doing something that is the very, very similar emergent behavior with a completely different set of, of kind of structures and, and molecular and biological circuit, circuitry behind the same phenomenon. But we can also go towards food and we're also, when we do that, we also have a different phenomenon for doing that. But the emergent behavior of going towards foods is shared between lots of different organisms. Anyway, so the algae is, is a, is a much simpler object to study compared to the human airways or the human brain. And so you can hold this in the, in the laboratory, do imaging. And so a lot of work has been done on understanding both what's going on inside. So it's going to be ATP consumption, but also the synchrony between the Cecilia on algae. This colored thing here is, is a section of a, of a mammalian airway tissue. There are cells that produce the mucus, which are labeled here. They're called goblet cells. And then cells that have Cecilia are, are black here with the Cecilia in green. So, so, so you see that the whole, the whole surface is really carpeted in, in, in these Cecilia. If we, if we just take a tissue of those cells and, and try to do microscopy, what we get is something which is gray and, and just fuzzy. And probably from far away, you can barely see that there's, there's any kind of dynamics in this movie. So here cells are, say the size from here to here. Each cell has tens or even hundreds of, of motile Cecilia. And it's very, very hard to kind of imagine zooming in and, and seeing, seeing nicely shapes of, of Cecilia or, or even individual states of, of phase locking to understand phase locking. So, so one first challenge we had when, when we wanted to understand airways and the brain tissue, which is very similar to this, was, okay, how do we actually quantify the, the synchrony between, between Cecilia and can we work out how that gets lost when, when Cecilia are far away from each other. So I'll come back to that. I'll, I'll show you. So, so we started doing those biological experiments together with continuing with the optical trap experiments. So the optical trap experiments are, are now an evolution of what I showed you with the three spheres, the, with the micro pump. But now we need a way not, not to be assigning what each sphere should be doing, but to lead, to make it into an oscillator and to leave the phase of this oscillator free. Because what we want to study is synchronization between, between oscillators. So in order to do that, you need some sort of feedback in the system. So, so the first, the first way in which we created a free oscillator was to, to have a, a bead in a trap, put the trap a little bit sideways, say to the left of the bead. The bead would then go towards the point of focus. And then we kept analyzing the position of the bead. And when the bead reached the position of the beam, we moved the beam back to the right. The bead went back to the right. And when, when it reached the position of this new laser beam here, we switched this beam off and switched the beam back on at this position here. So if you do that, and this requires constant imaging of where the bead is and the ability to put the laser beam left and right. If you do that, you create an oscillation that has a fixed amplitude, but the phase is free. Basically the phase is only set by, by, by where the bead is. So, so, so if there's no external forces and no noise, this will just be quite deterministic bead going left and right. But if you have thermal noise, this can randomize the phase. And if you have another object doing its own thing and putting a force onto this object, then that, then that can kind of make the phase of this oscillation anticipate or retard. So, so this, which we called geometric switch is, is one nice way to, to set up a system that has a, has a free phase oscillator and to study then synchronization of more than one of these objects. So again, in our minds, this now represents cilia. So each, each bead is creating a fluid flow, which is quite similar to how a filament generates fluid. The frequency and the amplitude that we make these workouts is, is fairly similar to the cilia operation. And kind of all the details of how a cilium can flex and, and have different power recovery stroke, etc. are, are then represented by the details of how we, how we, how we move the beads in the laser beam. And I'll just show you an example of what I mean by that in a second. So all of that, this I've already explained to you. Okay. So, so what we did was set up to, first of all, just a system of two of these oscillators. And we, we studied them as a function of the distance. So here's zero distance and here's 40 microns. And we created an order parameter just to tell us if these things are going in phase or an anti phase or a random phase. So in this notation minus one is, is in phase and one is anti phase. And this heat map just basically gives us the distribution of what's happening once we let the system go and analyze the, the, the, this, this order parameter for synchronization. Basically in, in our first experiments, which we published in I think 2009 or 10, the beads were always going in anti phase. And we could, we stayed in sync, sync up to about 35 microns away from each other. At that point, basically thermal noise becomes stronger than much stronger than the hydrodynamic forces that they feel. And, and the system just goes into, you see the histogram is spreading out. You can basically have any phase difference between the beads. So it's a random, there's no coupling anymore. This is just a histogram of that same experiment. What was more interesting and, and linked nicely with what the silly are doing was this experiment where again we had two beads, but instead of moving them in just harmonic traps. So I showed you the tweezer as being a harmonic well. But if you, if you scan your laser beam very fast, you can create out of, out of the light of the lasers, you can create a potential landscape that can have any shape. It doesn't have to have a harmonic shape. It can also be flat, flat lines of potential, which I've sketched here, or even these lines can be curved the other way. So you can, you can basically you spend more time here with your laser beam that you spend here. And you do this super fast, much faster than the beads move about. And so, so for the beads, this is an effective potential landscape that has this funny shape. And then in these experiments, we were doing the same game of geometric switch. So we pushed the beads in one direction. They reached the boundary that we analyzed and we just moved our harmonic system, our laser light trap to the left and then to the right again. Switching. And the synchronicity you see here is the anti-phase that I showed you from the previous heat map when, when the potential is curved up. But then if the potential is flat, there is no synchrony, even if the beads are closed to each other. So these spend, they spend some time in phase, some time in anti-phase. Basically, the average is a random phase. There's no phase locking. And when the potential is curved down in this active driven way, then the, the synchronicity is in phase. Okay. So, so this week we then had some hand waving way to, to understand this based on first passage times and, and also the hydrodynamic modes of the ozine tensor. But what's interesting biologically is that you can then, you can actually go and look at what, what the algae do. Okay. So we got this nice kind of transition as a function of the, of the curvature of the potential. But the, we could actually for the first time link with something biological based on those experiments. We, we took data, which came from an American group of Bailey about the, the confirmation. So, so, so the detailed shape of the, of the algae, the calamina monosalgae cilia as they beat. And we just segmented that, that, that filament at each position in this beating cycle worked out how much force it was putting on the, onto the fluid. And then represented as, turned that into a sphere that moves into, in a closed orbit, which is kind of this, this red, this would be in here. Basically the motion of this, of a sphere along this orbit is, is, has been defined as having this, as giving the same fluid flow as the sum of all the little cylinders along this thing here. So you can think of a sphere moving along here as being the center of drag of, of this filament. So in the, in the far field, the flow generated by the sphere will be the same as the flow generated by that filaments doing its complicated motion. And then that thing, we can actually think of, it's now become a sphere moving just left and right and end up and down in an orbit. We can just really represent it in terms of our simple models of how we're moving spheres with, with optical traps. And so we have, from the optical traps, we have a dimensionless way of, of understanding the role of thermal noise. So we have a number which is the thermal noise divided by the amplitude of our motion and the force that we, we move things by. This is dimensionless. And it's a way of understanding thermal noise versus the internal forces that the filament has. And then we can create this kind of curvature number that in the experiments we can, we can flip from, from positive to negative. And we can compare to, to biological, to, to the effective curvature of what the cinema is doing in a biological context. Okay. When, when we did that, then it turned out that we could explain breaststroke motion of, of the algae. Or at least, I mean, our, our simple model of thinking is consistent with the fact that the algae goes in breaststroke most of the time. Okay. In the very last minutes, I will just show you what we're doing now with the, with the real airway stuff. So these are now videos from our lab. Actually the very first video in the first slide, I think was, was not an airway, but it was a parameetium. So that's a single cell microorganism that swims in ponds again and has hundreds of filaments. These are actually human cells. You can see kind of, it's, it's complicated. And there's actually cell, cilia in the background are doing something slightly different from the cilia in the foreground. But this is what we have to work with. But what you can see is, is, is a cilia doing a very nice power recovery stroke in, in, in videos like this. Okay. Whether you, whether you're looking at an algae or a human, these, these cilia have exactly the same biological structure, which means you can really, depending on the question, you can go and choose to be investigating the simplest organism that exhibits the phenomenon you want to explain. There's molecular motors. Okay. I'm not going to talk too much about how things go on inside. I'll just show you how we're doing current experiments. So for, for airways, the cells need to be, need to live between kind of a medium that gives them nutrients that represents blood and air. So, so people have devised a semipermeable membrane that is represented schematically here as a line. The, the cell culture medium manages to get under and through this membrane cells live here and above here there is a gas, gas vapor that has to be humid and has to have the right percentage of CO2 for the cells to be happy. So, so basically you need some sort of gadgets to, to hold this nicely sterile and in place if you want to do experiments for, for hours of imaging. So, have to kind of, so, so the, the well, okay, these, these wells are called air liquid interface culture wells that the biologists already have these. There are commercial things because it's something medics do if people have asthma or cystic fibrosis or a whole set of more rare diseases that involve celia motion. The analysis of celia motion is one of the assays that doctors would do. So, so, so these wells to look at cells exist. But, but gadgets to actually do proper experiments over a long time don't exist commercially and you, you have to build something to hold temperature and, and gas and, and image at the same time. So, then you can do two experiments. You can, you can image the whole tissue and you get those fuzzy videos that I showed you before or you can, you can take the membrane, bend it over and look sideways on. And that's how you get the, the videos of, of the power recovery stroke. Now these videos are of, of, from the edge, but of, of cells that come from humans that have a genetic condition called primary celia dysclinesia. And you can see that these celia are just twitching, but they don't have the proper kind of beating pattern that leads to, to proper flow. So, so, so people with this genetic disorder are, are very ill. And one of the complications they have is they, they don't properly move their mucus, but they have a whole lot of other problems as well. Because this, the celia motion is important in the brain to, to, to move nutrients around the brain. Cystic fibrosis is a much more common disease. It's not very acidic, but it affects the way mucus is produced and mucus becomes too stiff. And patients with cystic fibrosis typically have lots of problems linked with infection because their mucus, mucus celia recurrence is not working properly. So, so, so these things can be looked at in the lab. But in order to, to investigate synchronization, and this is the last thing I want to tell you today, what we really had to do was address how to quantify videos. From, from taken from the top down, because, because the, okay, so, so videos like this one, these are, are what doctors take that they would, they scrape themselves from, from you or from the patients. They, they, they then look down and what they see is something like this. But it's, they can qualitatively tell you that this is twitching and it's not right. But it's not very good for, for understanding things like synchronization, which is then itself important in, in, in the mucus celia recurrence, which is actually the, the, the property that needs to be working fine in, in a person. For that, what you really want is, is large scale enough systems. But then if you're looking at the, at the whole tissue, you get a video which is complicated and looks like this. And it's almost impossible to think of segmenting and working out motion of individual things in, in, in an image like this. So what we did in the last couple of years has been to, it has been to develop a technique called DDM, which is an image video, video analysis method. And the nutshell is that you have your video made of many images and you first of all, you subtract every image with every other image. So you go from N images to N squared pairs of subtracted images. You then average together the pairs, all the pairs that have the same time interval. And that way you go back. If you had N images, you now go back to a stack of N subtracted images because you have N difference time delays in, in a video. So we're back to N. And then in those, but now these are kind of different images that they come from having subtracted two frames on that object. Okay. If you just add all the pixels in an image difference together, then if the time lag is zero, that means you were subtracting the image with itself. The addition of is zero because we just have subtracted the same frame. If you're, if you've got a little time lag, you subtract and you add up everything, you will have some signal. If you take a bigger time lag, you subtract and you add up, you'll have a much bigger signal because things have moved about more in those two objects. But the signal doesn't grow to infinity. If you, once you've given enough time for the system to rearrange, then from that time onwards, if you make your time lag bigger and you add the signal up, it's going to flatten out. So in general, if you take frame subtractions, you sum up and you plot, you're going to get a signal that grows and saturates. Okay. What's clever about this technique that was developed about 10 years ago by colleagues is that if you, if you don't simply sum up every pixel, but you first Fourier transform and then you work with the coefficients of the Fourier transform, you can basically see how the structure grows with a time scale for a signal, which is a structure to grow for each of the different Fourier modes. This basically tells you everything about kind of space and space scale and time scale in that movie. It tells you how quickly structure is rearranged for over each characteristic scale. And it's equivalent to doing light scattering, but it's much more direct and it doesn't need any equipment. You've basically just made a video. So it's been exploited now by us and by others in microscopy, but it's also a technique that could be used to analyze images from cameras, say imaging ant colonies or people moving about in crowds, etc. And that's all stuff that hasn't been done. So there's nothing special about microscopy is what I'm telling you. But in microscopy, it's very powerful because it gets away from trying to segment identifying objects and still it tells you the average of how things are moving about. So for the cilia, the cilia kind of beats so that they go back on themselves. So this is a quite special situation. And what we get is oscillating signals as a function of the lag time. This is because the cilia have a beating frequency and they come back. So by doing this analysis and plotting signal versus time lag, we get the beating frequency of the cilia. But we also get this decay here. And the decay time of the oscillation comes from two things. One is that the cilia are not perfect oscillators, so they lose coherence. And the second is that over a field of view such as something like this with many, many cells, a cell here is not going to have exactly the same frequency as a cell somewhere else. So when I image together everything and do Fourier transform, I'll get a decaying signal from the averaging of the frequencies. If I do the analysis on windows, so not just the whole frame of view but kind of tiles, I can actually work out the scale at which things synchronize with each other by looking at how that decay time scales with a tile area. So if I make my tiles small enough, I do go into a tile which is coherence with the cell. And if I make the window bigger, I'm averaging over things that are incoherent. So with no user input, so there was no segmentation, no thresholding, nothing of nothing, just Fourier transforms. I've looked at a very noisy video of things that are barely visible and I've calculated something complicated which is the scale over which these cilia are synchronized with each other. I think I've kind of run out of time. So I won't tell you very much. So there is very interesting information that you can get if you have the patience to segment cilia. You can actually work out how each cilium is beating. But that's almost another topic. Okay, so some of you may have been interested in this and you're very welcome to ask questions now or come and talk to me over the whole week about this. The one thing I did want to go back to was also some of you may just be wondering, okay, you've heard that biology poses interesting questions and just wondering how to attack a problem, how to find something tractable there. I think that's also another very interesting thing to talk about. And there could be questions now. And also we're going to have some sessions, an open mic session and various other occasions to talk collectively during this week. So you can also think about this sort of question for those moments when we will be discussing. Thank you very much.