 Where's it gone? Yes, OK. All right, thank you, Ali, for introduction. And I proceed to my talk in which I describe a quantitative approach to the diffusive dynamics through the first passage statistics, as has recently been explored by a few authors, including us. I believe that everyone is familiar with the phenomenon of diffusion in its chemically pure form. We can fancy what happens with a droplet of ink immersed in a glass of water. The substance with time will spread over the whole volume available to the liquid. How fast this process goes is determined by the diffusion coefficient, which in this case we would call the diffusion constant. In a more general setting, however, we may need to consider inhomogeneous diffusion with a space-dependent diffusion coefficient d and chemical potential u. This situation applies to many problems of biophysical interest in which the diffusion takes place in an aqua solution, solutions near organic surfaces such as proteins. To name just a couple of such systems, I can mention the permission of ions through a cell membrane or the absorption of carbon dioxide by plants and bacteria during the process of photosynthesis. To model the inhomogeneous diffusion, we use the Smoluchowski equation. Or sometimes this equation is also called the Fokker-Planck equation. This equation, this diffusion equation, relates the evolution of the probability in the N-CTP of finding a molecule at the given space point in space or in macroscopic terms, the concentration of the substance to the divergence of the metaflow caused by gradients of the potential u and by the gradient of the concentration itself, p. A problem that often emerges in studies of biophysical systems is how can we measure d and u simultaneously? In principle, many of you know that u could be inferred by taking the logarithm of the concentration p, thus the determination of d usually poses the main issue. In the rest of my talk, I will focus on the transverse diffusion that is in the direction perpendicular to the surface. This problem is both more difficult and more relevant for the studies of the permission and absorption processes that I have just mentioned. Let me choose the coordinate set along the direction perpendicular to the surface as in this sketch. Here, I use the over dot to simplify the diffusion the over dot denotes the time derivative and the prime denotes derivative with respect to the coordinate set. To quantify our observations of a diffusion process, we adopt the first passage statistics as follows. Take a population of molecules in a narrow bin and observe how each molecule diffuses until it passes through a given distance capital L in any direction for the first time. Hence comes the name, the first passage. Then a fraction of the whole population of molecules will travel to the positive direction of z to the right here, whereas the other fraction in the opposite direction in the negative. Thus, we can determine the probability of the positive and negative first passage events, p plus and minus, respectively. That must sum up to 1. In addition, we can measure the time of each first passage event and calculate the mean first passage time, the average of all these observations, tau. First passage, yes, and the probabilities of the positive and negative events characterizes the preferred direction of diffusion, whereas the mean first passage time tau characterizes the speed of the diffusion. To describe the spatial variation of the first passage statistics, we apply the same procedure by varying the position of the molecules' initial populations as we do when constructing histograms of the probability density. Now, to relate the first passage statistics to the space-dependent potential and diffusion coefficient, we need a little bit of theory. We transform this Molokhovsky equation by introducing what we like to call a dual potential Upsilon. The green equation relates Upsilon to the external potential U and the logarithm of the diffusion coefficient. So by substituting Upsilon instead of U, the dual potential instead of the external potential U, we transform the original equation in an entirely equivalent form. In this form, we observe that we pick up a third metaphor term, which is proportional to the gradient of the diffusion. I want really to emphasize that equations 1 and 2 are entirely equivalent because the second equation has been featured several times in the literature as an alternative model of the diffusion equation. Now, it is not as established through this simple substitution rule highlighted in the green color. The dual potential is very convenient when solving the first passage problem for this Molokhovsky equation. Formal solution of this problem is well known and using a few approximations in the limit that we observe the first passage events with a very small first passage distance L, we can derive the following formulas to relate the slope of the effect of the dual potential Upsilon to the log ratio of the first passage probabilities P plus and P minus. And then using just the mean first passage time tau, we can also find the diffusion coefficient. In addition, using numerical integration techniques, we in principle can infer the chemical potential U from the dual potential and the diffusion coefficient that's found. Today, I will not have a report examples of this approach. All right, now from the theory to practice, let me give you an example of the potential and the space-dependent diffusivity curves reconstructed from our molecular dynamic simulations. In this example, I report our results for the diffusion of water molecules near a surface of a glutamine crystal. As the coordinates said, we use the distance to the Gibbs dividing interface that delimits the protein weighting layer from the rest of the liquid water in this instance. You can see that at nanometer lengths away from the Gibbs dividing interface in there, we recover the diffusivity of the bulk water, the purple curve here with diamonds. And the diffusivity of the bulk water, the simulation conditions is about 4 nanometers squared per nanosecond. Close to the protein surface, the diffusion of water molecules slows down. And near the Gibbs dividing interface, it is by an order of magnitude slower than in the bulk liquid. The chemical potential reconstructed here from the log by using the logarithm of the steady state density shows that the hydrophobic surface of the protein expels water as manifested by the growth of the chemical potential, the external potential in this case near and beyond the Gibbs dividing interface. Our estimates can be validated independently through stochastic simulations of the Langevin dynamics. Using the inferred potential and the diffusion coefficient, we can construct a model of a Brownian particle whose phase-space probability evolves according to the same diffusion equation that we have been discussing so far. In the slide, I report the mean first passage time of two Langevin dynamics simulations along with our original molecular dynamics data. The orange curve, which uses the external potential and the diffusion coefficient inferred from the molecular dynamics data, matches very well the original data that we tried to reproduce. However, in this second case of Langevin dynamics, we set the diffusion coefficient to the bulk value. So in this simulation, the diffusion coefficient was constant, peaks to the bulk value of diffusivity of water. And you can see that close to the surface of the protein, this model cannot reproduce the slow down of the dynamics of the diffusive dynamics. This result actually gives through the credence to the incomaginous diffusion framework to describe the dynamics of molecules in these complex environments. And finally, before concluding, let me say thanks to all the members of our team working on these problems. And I would like to advertise a recent paper published in SoftMatter, from which I took the examples reported in my talk. Here, I finish. Thank you. Well, thank you very much for the very interesting talk. We have one more talk left, and time for questions, I believe. Yeah, we have time for one question. So there is enough for me. Right up to me, please. Yeah. Roman, how did you measure the diffusivity by molecular dynamics? I mean, the profile you showed. Yes, so we measure statistics for a population of molecules studying at a given coordinate. That way, we obtain diffusivity at a given point. Then we repeat the same procedure, but taking population of the water molecules at a different coordinate. So we obtain values of diffusivity at various initial conditions z. Then we use, in our case, we use spectral methods, numerical techniques, to reconstruct the interpolating curve. OK, thank you. Great, thank you very much. There is one question more, if I have a question. Ali, OK, sure, we can handle that. Roman, so regarding your way to infer the U of z, the numerical estimate, does it work well for the glutamine surfaces? We have not tested this for the glutamine surfaces. It would be interesting to see how good or bad it is. Well, yeah, we could test it, of course, not on the glutamine surfaces. I don't know, but that's the work that's there. You have the data that's published. So I mean, we know what the real U of z is in some sense there. Yeah, anyway, we'll argue later. OK, well, I'm going to go back to the slide.