 So, this is sort of the source strength. So, that is the flux. So, it it is like del c current at at at x equal to 0 yes. So, you are saying that you want to vary this current . So, in one slide I had written. So, the thing is that this is the correct formulation this is the current which is why at x equal to 0 also the concentration will rise with time. But if you are just interested in the exponential profile you could just say that I will normalize everything by the value at 0 that could be different at different times, but whatever. So, if I normalize everything at 1 at x equal to 0 I would get things like you know I would get sort of profiles which would evolve keeping this at 1. It is a different way of saying, but this is if you think about the physical process this is the right equation to write down there is a constant source at x equal to 0. You could say that well I could vary the source strength for example, that may be more mRNA is being transcribed at some time and less at some other time. So, the source it is strength itself varies that would be a different that would be an improvement on this model. But yeah it is as far as I know there is an experimental data on this mRNA sort of transcription rates as a function of time. So, people just take it to be constant. So, here so, that was across different species three different species. This is the same species, but two embryos one which has a largest embryo length simply because of random variations and one which is a smaller embryo length and again you will see that the lengths the lambda sort of scale with the embryo length. So, a larger so, if you just plot x. So, one embryo is around 500 the other is around 700 the lengths the lambdas are different which you can see. So, if you scale the x axis by the length of the embryo itself. So, it goes from 0 to 1 then the two curves fall on each other the collapse on top of each other which basically tells you that this lambda again even within the single species this lambda is proportional to L. So, this bechoid gradient forms it is an exponential gradient which we sort of understand in this context it scales with the embryo length which we do not understand how it does that. But this is the first step to this sort of to the developmental cascade that you get this meta this bechoid gradient the constant gradient profile of this bechoid protein. What does that do next? This is very bad maybe I can show here. So, here is my bechoid over here the top curve top figure is bechoid it is the highest concentration at the anterior pole the lowest concentration at the posterior pole. These are different these are these three are different downstream proteins that are expressed once the bechoid concentration has sort of set in. So, this is one protein called hunchback this is one protein called giant this is one protein called kruple. And if you plot the intensities or the concentrations of this protein again along the AP axis you will see that clearly there is a variation. So, this blue curve over here is the bechoid right it has an exponential profile. The hunchback which is the red one over here is roughly constant until a certain level and then it drops down and becomes 0. The kruple has a peak somewhere in in the middle of the embryo this giant has sort of two peaks one here and one here. So, the idea is this that once you have set in this bechoid gradient all of these other proteins respond to the bechoid gradient and express in a particular way. So, as to give these sort of spatial patterns. And then in response to these spatial patterns of these other downstream genes hunchback giant kruple whatever you will get different parts of this fly body developing into different organs organs. Some will be abdomen some will be wings some will be head and so on. There is this patterning cascade. So, this I am just showing a few there are of course, many other proteins and so on. It is a extremely complicated process, but this is the basic idea that once the bechoid has set in that is the thing that controls all subsequent developmental processes and all other proteins respond to this bechoid gradient level ok. So, for example, if you look at this. So, this is just the bechoid gradient profile the two curves being from two sides of the embryo just as a consistently check. If you draw a line along this the top half or you will draw a line along the bottom half you get roughly similar sort of the red and the blue are that thing. So, that is my bechoid concentration profile as a function of the ap axis along the ap axis. So, I will just talk about one of these proteins in particular which is this hunchback protein ok. So, if you plot the hunchback intensity against the bechoid intensity what it shows is that there is some sort of a sigmoidal response and a very sharp sigmoidal response. When the bechoid intensity is low the hunchback is low once the bechoid intensity crosses some sort of a critical threshold the hunchback level sort of jumps up ok. So, you can imagine that when I have a bechoid gradient like this and let us say the critical threshold is somewhere over here the hunchback can respond to this. So, on this side the bechoid intensity is greater than the critical density. So, I will have a high hunchback on this side the bechoid intensity is lower than this critical intensity. So, I will have a low hunchback profile right and I will get a concentration which looks something like this it is high on this anterior half then it is sort of false sharply and it is low again in the posterior half. How does this happen? How does this sort of control happen? So, that people have understood to a certain extent what bechoid does is that so, you have the fly DNA right. So, you have the fly genome which has all these base pairs. Let us say here is the gene that codes for hunchback starting from here ok. What bechoid can do is that it can come and bind to these sites upstream of the hunchback gene and it regulates the transcription of this hunchback gene itself. So, in particular it is known that bechoid has 6 binding sites upstream of this hunchback gene. So, 6 bechoid proteins can come and bind and once it binds the hunchback starts expressing ok. So, you can think of this if you go back you can think of this in terms of the MWC remember the Monod Weiman-Shango sort of a model. So, in the states and weights of a sort of a thing that when this DNA is in the off state the bechoid cannot bind and it has some particular weight let us say it has some energy E naught epsilon naught. In the on state the bechoids can come and bind and in particular there are 6 sort of binding sites. So, you can get any of these confirmations no bechoid bind bound 1 bechoid bound 3 2 bound 3 bound till all 6 bound ok. And let us say this on state where this DNA is accessible to the bechoid there is some energy let us say epsilon on and then you can write down those weights corresponding to this state the off state or these on states. So, for example, if this is an energy epsilon off this is a weight e to the power of minus beta epsilon off. If this has an energy epsilon on this on state it has as e to the power of minus beta epsilon on and then because there are 6 sites remember this was the concentration went as if you remember k d sort of hill function of first order. So, k d plus c by sort of k d. So, 1 plus c by k d the concentration in this case being the concentration of bechoid and k d being the dissociate equilibrium constant. And because there are 6 sites and if I assume that these 6 sites are sort of independent of each other. So, I do a z to the power of n sort of the thing. So, there is a power of 6. So, given this sort of a thing you can say that what is then the probability to find this DNA in the on state what is the probability and therefore, the hunchback to be expressed and then the probability of being in the on state is just this divided by the whole partition function this on state plus the off state ok. So, this is just the kinetic spot fit that given a bechoid concentration if I know that there is this sort of a 6 fold interaction of the bechoid on this DNA maybe I can write an probability of being in the on state which depends on the bechoid concentration. If on top of that you say that well the bechoid concentration I know from this SdD model is a function of position. So, the bechoid has this exponential sort of a profile and I put that in over here what I will get is that I will get a profile for this hunchback protein itself assuming that the hunchback concentration is proportional to this on rate ok. So, I have this from the kinetics I have this bechoid profile gradient profile from this SdD sort of a model it is an exponential profile. If I put them both together I will get a profile for this hunchback protein itself and how does that look. So, this is if I just put back lock back everything I put in the bechoid exponential profile. So, this is how my hunchback profile looks like as a function of x x being the distance along the V axis. How does this curve look like? So, for example, if I take an exponential profile for the bechoid this is what my hunchback as a function of bechoid concentration looks like and if I. So, if I take this exponential profile here is how my hunchback profile looks like ok. So, it looks exactly like this sigmoidal function as we expect. The sharpness of the sigmoidal function depends on the fact that you have this to the power of 6 sitting over here right. So, the higher this number the sharper that sort of sigmoidal curve will appear. So, this sort of a model which sort of takes in this SdD model then this sort of a Mwc model for the bechoid interaction with the DNA then tells you how this hunchback profile will look like in response to this bechoid concentration profile. And this is precisely well not precisely, but this is roughly what you see in experiments ok. It gets even the transition region sort of correct in that if you look at these experimental graphs this hunchback actually falls hunchback level falls very precisely at around the midpoint of the embryo. So, it falls exactly around 50 percent of the embryo. It does not care what the embryo length is what if you take a large embryo versus a small embryo the hunchback this transition will always happen very precisely at around 50 percent 49 percent. In fact, even that is a puzzle. So, for example, if you look at different embryos and you go to plot this bechoid bechoid concentration as a function of x you get an exponential profile let us say with some lambda equal to 100 microns on an average. So, different embryos some will be 110 some will be 90 and so on. So, you will get a spread with a mean of around 100 let us say, but in all of these somehow this regulatory cycle is such that if you look at the hunchback proof if you look at the hunchback concentration the hunchback will still precisely happen at 50 percent of the embryo length. So, there is some sort of an error correction built in the fly can very precisely regulate this domain boundary between the high hunchback and the low hunchback midchains. And there is a lot of work in trying to explain how this high level of precision of the hunchback actually comes about. So, this is just for one of course, you can do this for other proteins as well. In particular all of these proteins talk to one another. So, all this hunchback giant crouple they all have those. So, they all giant crouple bechoid. So, they all talk to each other and bechoid talks to every one of them. So, you have this very complicated network of chemical kinetics. So, these are chemical kinetics networks and on top of these these things can diffuse to form some sort of the gradients that you see. And there have been models like this which have actually done very well in explaining the observed gradients of all of these or many of these proteins simultaneously. So, this was for Drosophila. Let me now switch gears to a slightly more general model which is to say that how do I if I have a complex network like this of chemical reactions and then I imagine that these species themselves are diffusing. How do I generally think about, generically think about patterns that form? So, this was very simple case of a pattern of high, low sort of a thing, but you can have more complicated patterns that form and then how do I think about cases such as that.