 Everyone for having me here, I'm Gabrielle. I'm a PhD student at the group of Jordan Horowitz at University of Michigan. And today I want to talk about how non-eclipid response is fundamentally limited by topology and thermodynamics. And in order to motivate our results, I will use a receptor binding model where we have a receptor here in green. In this case, we assume it has two binding sites. And ligands here in yellow can bind and inbind. So we can have one at the left, one binding at the right, or two of the sites being bound. And the binding rates are proportional to the concentration of ligands in the environment. So if you're interested in characterizing the system, we can look, for example, at binding curves, which gives us what's the average concentration of ligands of bound ligands in the receptor as a function of the concentration of ligands in the environment. So what we see is a switch-like behavior where for low concentrations, in average, there is no ligands in the receptor. And for high concentrations, all the binding sites are occupied. So if we look at the steepness of the scourge, which is given by the response of the average of the ligand occupation number in respect to changes, logarithm changes in concentration, we can study how sensitive the system is to changes in concentration. And an interesting question that we can ask ourselves is a system that's out of equilibrium more sensitive, the one that's in equilibrium. So if we are interested in studying a non-equilibrium system, we always start by asking ourselves what happens in equilibrium, right? And then now we're talking about a general observable F and we have an equilibrium, the fluctuation dissipation theorem tells us that this sensitivity is equal to the covariance of F with N, which is the ligand occupation number. And if we further assume that the observable F is bounded between zero and one, which is natural if we are thinking, for example, as the average occupation number normalized by the number of binding sites, this covariance can be bounded by the average of the observable one minus its average. And we have this proportionality constant in the front, which is equal to two for this case, which is exactly the number of binding sites in the system. So we also can ask ourselves which curve saturates this inequality at all values of concentration. And this is what is called a hill curve, which has this functional form over here where age is known as the hill coefficient, which in equilibrium, it's always equal to the number of binding sites. So now we can ask ourselves what happens out of equilibrium and in order to have non-equilibrium driving in our receptor binding model, we couple it to ATP hydrolysis, right? So now we have a chemical potential difference that is driving the system out of equilibrium. So now if we randomly sample rates for the system and we plot the sensitivity divided by this F one minus F and we look at this ratio as a function of the unknown equilibrium driving, we do see that it's not bounded by two anymore, which is what you would expect in equilibrium. And there seems to be a bound over here that even depends on the non-equilibrium driving. So a question that we want, we're interested and we will be able to answer at the end of the talk is what is this term over here that's bounding this sensitivity when we are out of equilibrium? So of course we need to describe what we're talking about and the overall problem that we're interested in looking at what is the, we're looking at steady state averages of observables that we call Q and there is an external parameter lambda which can be externally perturbed. So what we're interested in looking is the response to the response of changes of this parameter lambda. And of course there are many ways to describe non-equilibrium systems and non-equilibrium dynamics and what we are talking about is Markov jump processes which have this very good way to visualize them using graphs, using graphs where each node represents a state of the system and the edges represent the possible transitions between states. And we always assume by thermodynamic consistency that there is always the forward and the backward rates and all of them are weighted by a transition rate. A very important characteristic of such systems are cycles which are trajectories that connect a state with itself without self-intersection and for each cycle we can associate a cycle force which is the log ratio of forward and backward rates in the cycle. And the cycle force tells us how far away from equilibrium our system is. So, and in the way that we think of response and perturbation in the setup is to have the transition rates being dependent on the external parameter. So, our approach to be able to de-tangle and understand all the possible perturbations that the system can possibly have is to decompose the perturbations in different categories. One specific category is called an equilibrium like which accounts for changing uniformly all of the outgoing rates from a specific state in this case N. So, if I wanna give you a little bit of intuition we can assume that if the system comes equipped in an energy landscape this type of perturbation accounts for changing the energy of the state. And as the name says this is an equilibrium like perturbation which satisfies the fluctuation dissipation theorem. But now we also have what we call a symmetric perturbation which accounts for changing the forward and the backward rates between two states. So, perturbing this edge M and M here which accounts for changing the energy barrier between states M and M. This perturbation is zero in equilibrium but we might ask ourselves what is, how is bounded when it's out in equilibrium? And this is exactly what our main result tells us which is a fluctuation response inequality that's telling us that the response of the observable to changes in the energy barrier depends on two different terms. One is a topological term another one is a thermodynamic, it's related to thermodynamic driving. So for the thermodynamic driving part what counts is the maximum cycle force which is the maximum cycle force over all cycles that contain the perturbed edge in this case M and M. And over here for the topology term what we're interested in is structure called the topological consistent splitting which is the splitting between the states of the system in two different subsets. One that contains states M and the other that contains states N. And the splittings they need to be consistent with the topology of the system. So what we have over here is the maximum over all the possible splittings of the covariance of the observable with a D indicator function over all states that are part of the subset that contains states M. And of course we can numerically verify this result. So if we plot the response now normalized by this topological term over here for randomly sampled rates and look at as a function of the maximum cycle force of the system, we do see that indeed it's bounded by the tangent of the maximum cycle force over four. So now we are actually ready to answer the question that we had in the very beginning and go back to our known equilibrium receptor binding model. So now if we use our approach and we look at the sensitivity, we see that this response can be decomposing five different response while three of them are equilibrium like. So they're equilibrium like in states one, two and four. And the other two are symmetric like. So they correspond to perturbing energy in barriers between states one and two and between states one and four. So now using the bounds that we found we have the sensitivities bounded by an equilibrium like term, which is the covariance between F and the ligand occupation number plus now the maximum between topological splittings between states two and four as the covariance of the observable F over indicator functions over subsets that contain state two times the hyperbolic change of the non-equilibrium driving over four. We can also ask ourselves what happens when we don't have access? We only have access for example to the average of the observable. And when we do that we have access to we obtain a coarser version of the bound which is where we have again this functional form average of F one minus average of F. And here the proportionality term is now two which is exactly what we would expect in equilibrium plus the non-equilibrium driving. So we answered the question that we had at the very beginning of the talk which is this term over here it's exactly an equilibrium like term plus a non-equilibrium term. And we can also ask the same question we asked before it's which curve saturates this inequality for every value of concentration. And when we do that we see that this curve has exactly a heel curve like functional form but now the heel coefficient is equal to the equilibrium term plus this non-equilibrium enhanced term. So here in red we have this optimal non-equilibrium heel curve. And now here in purple we do see this curve over here for an equilibrium system and in dashed for a system. Here we lost your sound. Yeah, I was just trying to ask you if you still hear it or not. Yeah. We lost your last 30 seconds. Sorry, we lost your voice. We don't hear you. Maybe the headphone charge so it's the batteries empty or something. I don't know. Yeah, if you take out the headphones maybe it's better. Can you hear it now? Yes, yes. Okay. So I was almost finishing and what I was gonna say is that the steepness of the curve, of the purple curve and the dashed curve are always upper bounded by this heel-like curve out of equilibrium. So if, thank you for listening to the talk and I would like you to take us hateful messages that utopology and thermodynamics constrain non-equilibrium response. And if we apply those results to a receptor binding model, we do have that the heel coefficient, it's equal to a big little heel coefficient plus a non-equilibrium driving. Thank you very much for the talk. There's a question by David Sivak. Maybe you want to unmute, David. Sure, can you hear me? Yes, I can. Great, really nice talk. I was just curious, this one over four factor appears everywhere in the TANGE argument. Do you have any intuition for sort of where is that coming from? The, well, I guess the intuition comes, you need to go through all of the math and how, where everything comes from. I'm not sure if I have a good intuition but it comes naturally if you're using the matrix three theorem and solving these terms over here. And then, so basically this term that always depends on the maximal cycle force over four, it's a ratio between rates of cycles, unnormalized rates of cycles. So I'm not sure if I actually really answered the question but I'm not sure if there is a good intuition of why the over four. And there's a comment by Tom. Maybe you can comment briefly or KVT. Yeah, it just seems to come up quite a lot. Like factors of roughly, well, factors of four exactly I've seen in a number of papers. I'm not sure if I've seen anything in a TUR paper which has a four rather than a two but I've seen a four in Peter Ryan Ten Walder's paper on, and Chris Govind's paper on energy resource costs of sensing which is obviously a closely related system. And I've seen it in a paper on thermodynamic error of time. Yeah, so I don't know if there's anything deep there but this isn't unique that a four it's kind of fallen out of the analysis. I need to think about it. Interesting. Presumably it's not specific to the network's apology here. No, no, it's not. Yeah. And nothing that I showed is specific to this, I use this example but nothing is specific to this network apology at all. And it's time for a quick question by Jonathan who's raised hand, please. Hi, thank you Gabrielle for the great talk. I was wondering if there would be any if you thought there'd be any merit in using this it's sort of an inverse problem if you had some sort of like hill like graph.