 Can I ask the final speaker in today's morning session, which is Andrea Puglisi, on the thermodynamic limits of sperm swimming precision? Yes, okay. So can you see the screen now? Yes, just a minute. I'm just going to mute this person. Right, there we go. Yes, we can see your screen. We can hear you. Go ahead. Yes. Okay, so thanks for the opportunity to speak, and I wish also to thank the collaborators, which are Roberto Di Leonardo, Claudio Maggi, Benetana, Viriliana Carmona-Solza, and Filippio Sallimbeni. So I'm going to discuss and show you some results, some experimental results about sperm beating precision and its thermodynamic limit. For the reason I'm going to introduce to you the two main actors of this small talk, which is the sperm cell and the thermodynamic uncertainty relation. Okay, so the sperm cell is a very fascinating biological system. Here you can see a nice animation created by Ex vivo and the group of Daniel and Castro in Dallas. And inside the tail, there are nine doublets of microtubules, and there are a very packed arrangement of molecular motors, which are called D-names. And there are roughly four, a tail of 50 microns, a number of the order of 10 to the 5 motor domains, which actuate the flagellum. And here you can see freeze fracture electron microscopes, pictures which show you the very regular and packed arrangement of these molecular motors, these proteins, which actually are so close that they really need to interact with each other. And of course there are several proposed mechanisms to understand how the way that these molecular motors, some kind of spontaneous coordination creates the way beating the, so the traveling wave that make the sperm swim. So the other actor of this talk is the thermodynamic uncertainty relation. You basically measure in your molecular system any kind of current, you measure for instance in the steady state which is the simplest case, you measure its average and its diffusivity, and you define a precision rate which is the ratio between the square of the J divided by D, and this precision rate is bound as an upper bound, which is the content of the thermodynamic uncertainty relation this upper bound is the entropy production rate, which you can compute in the simplest case of isothermal motors as the ratio between the power injection divided by KBT, and there are of course several derivation of this result in several different conditions and generalization etc etc so the literature is very wide in this direction. And of course the molecular motor like the DNA is one candidate for checking how good is this estimate by this bound. And so for instance there is this recent paper on John of physical chemistry letters. When the authors use the Markovian models for several multiple models such as the DNA in using empirical transition rates to compute the precision and to check how the thermodynamic uncertainty relation is satisfied how well is satisfied I mean how close the bound is. So they found for instance that the DNA is semi optimized in the sense that the precision with respect to its bound is 20 to 50% in cellular conditions. Okay. So how to go from the molecular motors to a mesoscopic system and this is in the in a recent wave of studies where several different mesoscopic systems have been studied like flagella or Celia. So basically the first step is to reduce this the complex space space of shapes into a few degrees of freedom like the main coefficients of the principal components that describe the shape. In this way you can reduce and measure some kind of current current in a in a limit cycle. So for instance the sperm shape cycle, which is roughly a traveling wave, not exactly the one I'm showing here, but not so far from this one particularly close to detail that is not not a lot of difference between idealized traveling wave and the sperm traveling wave and you can basically represent the shape cycle into the cycle of these two coefficient A and B, which basically move in this plane, and there is so a rotation of a face. This is basically the macroscopic counterpart of the motor limit cycles. Okay, the DNA is chemically doing transitions in a network of states, which of course is not exactly a limit cycle is more complex but still it's dominated most more or less by a main cycle. And of course if you want to then check and verify that the thermodynamic uncertainty relation you need also an estimate for the power injection rate in the so the consumption of energy for the sperm. So for instance you can get estimate considering that each hydrolysis of itp you get 1020 kbt and then you have an order of magnitude of the given by the frequency of beating and the number of motors you have. And then you get estimates which are very close to what is experimentally measured in experiments where for instance atp molecules are fluorescent died and and so it is possible to to see the real depletion of atp near sperm cells. And of course this is also close to the estimate of energy that you need to swim which is this is the state or formula for swimming with a given frequency F amplitude a length L and the viscosity of the fluid. So going to our experiment. Basically we micro fabricated the cages small cages that you can probably barely see in this photographs of the microscopes by two photo polymerization setup in the laboratory of Roberto Leonardo, and the sperm cells easily and spontaneously go and remain trapped into the cages. With this trick it is much easier to track the motion of this detail it's it's reduced to a planar movement, and there is no motion of the almost no motion of the, or the center of mass you can basically track just the shape of the, of the tail, and you can reconstruct. Basically the, the, the limit cycle, based upon the two main coefficients of the two main modes of beating. Okay, and you can really see this is experimental data you can really see the cycling of the face representing the shape that beats. And then you can do this for several sperm cells we analyze more than 50, and we get the data in time so that an average of the current and the diffusivity can be computed in the end you can measure the precision the precision rate that I told you before and the maximum that you can find for the very the most precise cells is of the order of 10 to the two hairs. And this number is as two important comments. First, it is much smaller than the bound given by time that I'm going to say that the relation of the order of 10 to the five smaller, and it's quite close to the single DNA in multiple precision. We're interested to see if the thermodynamic aspect of the relation has any meaning, even if the first observation is not very encouraging but we decided to seal the box where the sperm cells are in order to make them suffocate in order to to make an oxygen the oxygen is depleted and the cells are going to die. And this is seen in a few hours that beating frequency the case quite strongly, and the current and the diffusivity change and the precision, it is seen to decay. And if we put the maximum precision we're observing time over the energy consumption rate which is estimated by the frequency through the Taylor formula, we get quite a good agreement with a constant factor so the, the, it is sort of the precision fulfills a kind of thermodynamic uncertainty relation with a constant. This constant coincidentally is of the order of the number of motors inside the detail. So, we took a bit seriously this coincidence, and we, we decided to propose a possible conjecture where basically there is this this number is a kind of correlation of the number of motors which are strongly correlated. And if you think of a chain of motors, of n motors, which are coupled and correlated for some length, you can easily, I mean, argue that the, the, the correlation length enters as the ratio between the bound and the precision of the, of the system. So we try to see if in the literature there were, there were results in the, in this direction, and we reconsidered experiments with sperms in other conditions, as well as with the flagella of the chlamydum monosalga which are made of the same structure there are axon names exactly as the sperm tail. And this, this, this is the result of this, this attempt of a collapse of very different data with different lengths. Okay, and we found, I mean, I would say decent agreement. Then what we needed, this is the last slide, is a model which would support this, I mean, quite bold conjecture, I think. And so we can consider the model by Julica and Prost like 25 years ago for a minimal motor model. In this model, you have a potential with it fixed, which represents the bending potential in the, in the, so elastic potential for instance, in the tail of the sperms and there is, there are n motors which can attach or detach to the, to the backbone of the, of the flagellum and there are also standard forces so the, the model is, is described by the states of the n motors zero detached and one attached and on the relative position between the chain of motors and the backbone, which is X and which fulfills a simple dynamic equation, following a gradient motion with respect to this potential, but where only the attached motors are important. And then there are transition rates to the, to decide if the motors attached to the touch based also upon the, the, the local potential. And this, in this way you can break the balance and you can really create a limit cycle here for instance you can see the limit cycle in the position and the total force, and you can measure how the face correlation the case in order to obtain an estimate of the sensitivity and you can measure the precision. And then we decided to modify this model, adding a binding potential between the motor so the motor are the probability to attach the touches enhance. Depending on the states of the motor nearest neighbors okay with this factor K with this parameter K, and then you can get again and you recycle and you realize that you get a much stronger noise the face the correlation is much faster. So you can measure the precision of this model, the thermodynamic precision and how it skates with them. And you can see that if there is no coupling between the nearest neighbor motors, if K is equal to zero, the precision increase with them. Okay, this happens because the fluctuations of the motors are basically independent. Okay, so the diffusivity scales as one over M, and the precision increase with them. What happens when the motors are covered when case equal to 10 for instance, you discovered that the precision is over the one always as apparently how we observe for the precision of the sperm beating. Okay, so this is all thanks for your attention. Okay. Thank you. Do we have any questions. We're going to move directly into the discussion so questions here and then we'll bleed into questions for other people. Peter right. Yes, I was wondering whether the fact that these motors share a load, could that not couple them. It is a bit tricky this fact because this is exactly what is already in the model in the original model they the transition rates so that the fact that they attach and the touch depends on the local potential so in a sense that it is so it is already in the original model this dependence on the local for example local bending. Okay, so the importance of the load. So, but apparently this new ingredient is something different I mean it is a sort of direct coupling, the direct coupling make them more coupled and the precision get much more lower for this for this kind of it's like, in the first case you have a coupling an indirect coupling by the average. So the average is modulated in space and the transition rates depend on space so there is a kind of coupling. Okay, but then you are also trying to couple the fluctuations rather than the average. Yeah, thanks. Okay. Yeah, hi, Andrea. This is really fascinating. I mean, it's a very important question to see how the uncertainty relation behaves for for larger and larger systems. And with Timor Quio, we recently looked at a German lattice gas also as a function of the number of particles. And the upshot was that the tour remains of order one if the current scales as the number of particles. So my question here is, I mean, how does how does your current which you observe and whose fluctuations you're looking at, how does that scale with N. In the in the model where we have a full control of N the actually the current is very weakly depending right right and then it's and then it's no surprise that the tour ratio is really bad right. Yes, probably yes this is a this is a difference. So I think I mean that's my understanding of of of this large and limit you need to look at a current which is proportional to the number of degrees of freedom which are driven. Yeah, do you think this can be related to the way they are coupled I mean if they are coupled conservatively maybe the current is less dependent on them. Well, I mean in our lattice gas the couple the coupling among the I mean this was just hardcore and some next nearest neighbor interaction and the driving was just an external field. Okay. So, yeah, it certainly may be related to to the coupling I mean your last graph here on the right bottom, somehow seems to indicate that you could change the behavior as a function of and depending on the coupling right. I'm just looking through this paper. Thanks. Thank you. Cool.