 For recent work, the title is the universal thermodynamic bounds on symmetry breaking in living systems. Before we talk about the bounds, we need to understand what we talk about, about what we mean by symmetry breaking. So in living systems, there are different forms of symmetry breaking. One example is the kinetic freedom. So in biochemical systems, in many times, it always need to discriminate different molecules. For example, in immune systems, the living system need to recognize the antigen in the copy of DNA RNA. The enzymes need to discriminate the right and wrong monomers to add to the DNA RNA. However, if we only discriminate these molecules by their energy difference, the error rate is too high. So Hopfield introduced this kind of kinetic freedom model, which used some energy to maintain stationary state, so that the error rate is dramatically reduced. So this is the kind of symmetry breaking in the chemical space that the energetic symmetry is broken. So another example of symmetry breaking is the pattern formation, which is also quite common in bio systems. We can see different patterns on animal skin. Also it's important for the cell division. So this kind of pattern formation is also auto-equilibrium and highly londinear. And due to the londinear nature and auto-equilibrium drawing force, we can show some spatial pattern emerges from uniform distribution. So these two different symmetry breaking are still have some similarities. The common thing is that both of them need energy injection. They need to consume some energy to break this kind of symmetry. So what we want to answer is that can we say something more general to unify all these different symmetry breaking mechanisms in a unified scope? So before we talk about the unified bounds or the cost to break symmetries, we first need to write down everything in a thermodynamically consistent way. So what helps us to write thermodynamic consistent time evolution is the local database relation. The local database relation tells us for transitions in biochemical systems. For example, here we have an H. So the forward transition divided by the backward transition rates should satisfy the local database relation, which is determined by the energy difference and the drawing force along that H. And when the system auto-equilibrium is auto-equilibrium, when we can find a cycle and the thermodynamic affinity to find a cycle is no 0. So in this case, it was just an auto-equilibrium. So but a master equation, which is linear, is not enough to describe a nonlinear symmetry breaking. So here we want to introduce a nonlinear rate equation, which can characterize the nonlinear nature of symmetry breaking, for example, the pattern formation. So the form of this kind of nonlinear rate equation is still similar to master equation. But the transition rates here are nonlinear. And there are some pre-factors, omega ijp. So this nonlinear factor should be understood as the catalytic mechanisms. So these catalytic terms, for example, the autocatalytic reaction, we should have this kind of term, which can make system nonlinear, but still have this kind of master equation form. This catalytic term is very good because the local database balance relation still holds. And if we take a ratio of these two, of the forward and the backward rates, we will see that this nonlinear term cancels. And we only have a linear term, which still satisfies the detailed balance relation. So as I just mentioned, we can understand the non-equilibrium nature from the cycle affinity. But here I want to go back to the pathway interpretation of non-equilibrium system. And we can see how can we construct cycle from the reaction pathways. So in a chemical reaction network, for example, here we have state i and state j. So if a network is very complex, we can define multiple reaction pathways in the network. We can find many pathways. And along each pathway, we can find the equilibrium constant of that pathway. So how to understand the equilibrium constant? This equilibrium constant is that if we think the pathway is very fast, and this pathway dominates the reaction, and all pathways are very slow, then we can get the ratio of these two states, the population of these two states, reach the corresponding equilibrium states determined by the chosen pathway. And the equilibrium constant is defined as the product of all forward rates divided by the backward rates along the pathway. And it is related to the thermodynamic properties in the form of the energy difference between these two states and the drawing force along that pathway. So for a network, if it's complex, we can always define many pathways. And each pathway can give us an equilibrium constant, which only determined by the thermodynamic property of that pathway, not related to the kinetic details of that pathway. And so that different pathways may have different equilibrium constants. So that if we find the two pathways with different equilibrium constants, then we can combine them together to form a cycle, then the cycle affinity is non-zero, so the system out of equilibrium. But apart from the cycle affinity, the pathway itself tells us a lot of information. So we can find, we can maximize and minimize over all the equilibrium constants identified in all reaction pathways. And intuitively, we can understand, we can think that if we find, finally, if we system reach a equilibrium state, a stationary state, this stationary state should be bounded by the maximum equilibrium constant from all pathways and the minimum equilibrium constant also minimized over all pathways. So which these two bounds define a non-equilibrium phase space in the space expanded by the probabilities. So yes, we can understand this bound by intuition, but also we show in our preprint we can have a rigorous proof based on the matrix three theorems. So while the benefit of matrix three theorem is that we can decompose stationary state in terms of all spanning trees. And because a spanning tree don't have any cycles, so it represents the equilibrium properties of a system. And this is slightly different from the master equation here, the system in non-linear. So the directly spanning tree also has the part which depends on the system itself, but still finally we can cancel all the non-linear term and found this found. Now I want to talk about two applications. The first one is the kinetic pruning. So in kinetic pruning, so if the system is driven, we have some cycle and system is driven out of equilibrium. If a system don't have this driving force, we cut these two red ages. So we, the discrimination will go back to the energetic discrimination. The error rate is too high for real biological systems and Hopfield introduced this kinetic pruning scheme. So we have this cycle which use energy to improve the error rate, to lower the error rate. In Hopfield scheme, so the drawing for this kind of external drawing transitions is unidirectional. So which means that the energy consumption is infinitely large. However, the error rate still finite because due to some kinetic constraints in the proofreading network, there are some symmetries in the kinetics. If we lose all the kinetic constraints, we should able to find the thermodynamic bound of error rate. So here using our reaction pathway argument, we can see that there is a lower bound and upper bound of error rate. Lower bound of error rate is found when we found the reaction pathway which use the drawing force to push the wrong state to the right state. And there's the upper bound which means that we use the energy in the wrong way which push the right state to the wrong state. So here we get the upper and lower bounds directly from the thermodynamic property of a system and we don't have any kinetic constraints. And here we can fix the data mu and randomly choose the kinetic parameters and we can find that the error rate is well bounded by this upper and lower bound. So these two bounds are not entirely new. So that have been found in these two papers. But here our reaction pathway interpretation, I think it's very intuitive and can help people to understand what really happened here. And also in a more complex property network, a system is very complex and we don't know all the kinetic details but we can determine the thermodynamic bounds solely from the thermodynamic properties. You can find all reaction pathways and we know the thermodynamics of all reaction pathways. And then we know the upper and lower bound of the error rate or the accuracy of a system. Another more interesting example is the bound on pattern contrast of the rectifying pattern. So when we see a rectifying pattern, we can see a lot of interesting properties. And the one observable is the contrast of pattern or the visibility. For example, here we can see two different patterns and one with low contrast and one with high contrast. How do we define this kind of contrast? So we have a very conventional way in optics and also we can borrow that definition here. The contrast is defined as the maximal concentration minus the minimal concentration which is the range of a concentration and divided by a sum of the maximum and the minimal concentration. This contrast using our approach we can show that this contrast is smaller or equal to the tension of the injection energy. So with a finite amount of energy injected to a system we can only achieve, we have an upper bound of the contrast. And we can do simulations of this kind of 2D pattern and see the pattern visibility is well bounded by this upper thermodynamics upper bound. I want to show how we get this bound. So thanks to the face-based geometry method developed by in this paper, it's from a reinforced group. This approach is very nice because this approach allows us to study the properties of reaction-tiffing pattern using the geometry of the face-space, the space expanded by the two concentrations. So for a reaction-tiffing system, we can always write it in forms of diffusion terms and the plus of reaction terms. And we have- Just mentioning that we are already in the time for questions. Oh, sorry. I will go through the rest of the question. We have, we can, the reactive look lines is the reaction zero line and also we can have another conserved quantity which is the flux-balance subspace. The interception of the flux-balance subspace and the reactive look lines tells us the minimum and the maximum concentration of the stationary pattern. But here, what's not answered is where is the non-current nature of the system? So what we tell is that we can define the non-current face-space and all the reactive look lines must lie inside the non-current face-space. And we can have an example. We have non-linear systems and we can have the two boundaries contributed from these two reaction pathways and the interception of flux-balance subspace and the two boundaries defines a thermodynamic bound. And we can see for a 1D pattern and we get a thermodynamic bound of the concentration and then immediately we can get a bound on the contrast of pattern and we get, we can do assimilation to a numerical verification. And finally, as summer to the signal message that we can understand the bound in two ways. The first way is that if we know the driving force inside the system then we know accessible climate face-space, the accessible non-current face-space. The other interpretation is that if we want to design some certain biological functional system, for example, we want to have multiple stationary state. We want to have pattern with certain contrast. We want to have a best stable state. And then all the stationary state will occupy a finance space in the face-space and the non-current face-space must be larger than the space occupied by the non-current stationary states. So that we will know the minimal energetic cost to design the desired symmetry briefing system. So finally, it's up to my supervisors and our work is, our program is already on archive. So we can check all the details. So that's all. Yeah, thanks. Thank you for the interesting talk. We have time for one question. I'm afraid. So anyone of the students or postdocs want to ask a question? If not, then I've seen that Tom has something to say. Okay, Tom, maybe you'll ask a question. Can you go back to the kinetic proofreading picture? Yes. Thanks for the talk, brother. Yeah. Yes. Yeah, so here. So...