 So, what I will do for today and the next day is this topic of pattern formation, will be sort of the last topic that I covered in this course. So, what I will try to do today is to talk a little bit about how patterns form in during development. In particular what I will be talking about this is the fruit fly the drosophila and some generic reaction diffusion systems and then I will do the formal mathematical theory the next to next class on Tuesday. So, one of the generic things that you often see in biology is that regardless of this complexity of constituents and stochasticity, this reaction kinetics and so on you will find very reliably that nature produces these very beautiful patterns and across it produces patterns across the length across length scales. So, for example, this is cells in the inside of a ear think of a mouse which has the patterns on the order of some 5 microns. These are cells on in the eye the rod and the cone cells of the fruit fly drosophila which is of the order of again some around 10 microns. Then if you look at flowers you will often see very beautiful spots and patterns basically that are appearing and these are you know the scales of a flower centimeters and so on. Or you could go to even larger scales and for example, it is the stripes of a tiger or a zebra on the order of a meters where you again you have this reliable. So, here you could think of patterning as patterning of these pigments that produce the C. L. O versus this blackbird right. So, this was a problem that originally Turing thought about a long time back in the 1940s and 50s and he said that given that I see that generically nature forms these sort of patterns across across scales across multiple length scales can one come up with a mechanism which will explain the process by which different different constituents chemical constituents be it pigments be it proteins and so on. They can interact with each other and give rise to these sort of patterns. So, that is the this is sort of background with which he started thinking as it turned out the theory. So, the generic theory is due to Turing himself pattern formation of reaction diffusion systems. It was not very applicable in the context that he was thinking of, but it has taught us a lot of how to think of systems like this. So, I will expand a little more as I go along. So, I will talk mostly in the first a with regard to development. So, for example, when I have a embryo. So, all life starts with a single celled embryo. So, some embryo like this it is homogeneous it is a single cell. Then slowly this embryo will start to divide and you will form multiple cells right you will form multiple cells. And then what started off as this homogeneous ball of cells will develops spontaneously some sort of an order right. You will have a head side and a tail side let us say you will have a head and you will have a tail right. So, there will be an axis that is going to be formed then different cells in different parts of this axis or along this axis they are going to develop differentially and then ultimately you will get a full-fledged organism right. So, the question often is that how do I start off with this homogeneous mixture of cells and how do I spontaneously then generate order. So, one way that one could think of is that there are some spontaneous events within an uniform field within an uniform homogeneous field and this somehow breaks the symmetry that is there in this original embryo and that symmetry breaking then cascades downstream and creates some sort of a large scale order ok. So, there is some spontaneous symmetry breaking with respect to this embryo that sets your let us say the head tail axis or whatever axis and then things it is a complicated process. So, I am just waving hands and saying, but that is a rough idea that you have some sort of spontaneous symmetry breaking that helps you at least get started on this development process. Or another way you could think of is that there is it is not as homogeneous as it looks there is some sort of external signal at some point let us say on the head side that introduces and built in asymmetry and that asymmetry then propagates and tells the organism how to develop. So, that is another possible mechanism yes. So, what it means is that you have some homogeneous ball. Now, let us say the concentration of one species the some species mixed uniformly and homogeneously inside this let us say the concentration of one. So, there are fluctuations in these concentrations because you know whatever stochastic processes are going on. So, this fluctuation let us say beyond the certain strength can sort of generate such that you have more of this molecule on this side less on this side that happens spontaneously. So, in some cells let us say this half gets more. So, this becomes your head that becomes your tail in other cell maybe there is a high concentration fluctuation here and high low concentration there. So, that becomes your head tail axis in that sense. So, another way to frame this question is that how in this in this system of different cells, how do cells interact with other cells to detect information about their spatial location in order to form these large scale patterns that is how does the cell over here know that it has to form the head how does the cell over here know it has to form the tail how does something over here knows it has to form the hands and so on. So, it will talk to other cells in order to sort of determine where its relative position is in the embryo and then have an appropriate sort of developmental cascade. So, Turing was this has been this sort of reaction diffusion systems has been used to sort of study various patterns. For example, patterns on butterfly wings is a relatively recent paper where through the use of these systems that we will look at the reaction the equations that we will look at today you can sort of explain these sort of macroscopic patterns and I use explain in a very loose way as I expand the question. So, let me start before I go into the genetic theory. Let me start off by taking this specific example of development of Drosophila ok the video I will play the video separately maybe. So, here is the idea that Drosophila is a common fruit fly. So, here is my Drosophila embryo it looks something like this like an oval. The Drosophila embryo is part of a class of organisms which are called syncytial which are called syncytial in that when it is. So, initially it is a uniform mixture then slowly what happens is that you get cells as you have nuclei started throughout the bulk of this embryo and then at some point this nuclei all move to the surface. So, they all move over to the surface and you have these nuclei coming over here. So, these are all the nuclei they all move to the surface the interior the yolk sort of face separates out and it goes to the inside. So, this is the nutrient from which these embryos will sort of take sustenance. Then as time progresses these embryos will divide. So, you will get more and more embryos, but it is slightly peculiar in the sense that you do not have proper cells until some time later. So, these nuclei are not divided up by cell walls in that sense. So, that is what is meant by syncytial cell walls form after certain number of divisions are taken place. Initially it is like this continuous domain where you have these nuclei sort of multiplying. So, this is what this movie shows we see if I can play this can you see yeah. So, this is the Drosophila embryo. These white spots the fluorescent spots over here are these nuclei and you will see this spots will get as this divide these spots will get denser. So, this is when the division is happening. So, let me here I lose some fluorescent intensity. So, let me display from the front. So, there are very few then as it divides there are more nuclei it divides once more there are more nuclei and so on. It has already established a sort of axis in that there are more nuclei on this side than on this side right. This side will ultimately be it is called it is called the anterior side it will ultimately become the head of the fruit fly this side is the tail of the fruit fly. So, it is called the anterior posterior axis. What is marked is a particular. So, what is fluorescently labeled in these movies is a particular protein called bicoid which I will talk about a little bit. It is one of these precursor proteins that will set this AP axis basically and on the inside what you see over here this dark region is olio. There are no nuclei over here in this bulk and this process will continue. So, here right now there are no cell walls after some time the cell walls will form and you will get proper cellularization and then the level of material content on this side there are actually very few nuclei. So, well there are nuclei, but yes the density is less ultimately it will get of course, filled out there is another thing over here. So, what is labeled like I said is this bicoid protein ok. So, it is correlated with the nuclei because these proteins get sort of localized inside this nucleus to a higher degree inside the nucleus which is why you can see this spherical sort of objects as nuclei. The fact that you cannot see them on this posterior side means two things. One is that initially the nuclear density is low here and secondly the bicoid also actually as I will tell the bicoid is actually produced over here. So, it needs to go over there it has not managed to do that yet. So, that is why you do not see any density. If you allow if you let this process sort of play on here is what the fly looks like at some later time. So, those were within 1 hour or so, about 1 to 2 hours these are much later I think this is an hour. So, 5 hours and so on. So, this is sped up and you will see that various sort of patterns have started forming the thorax and so on of the fly have started appearing. So, this is the dorsal view from the top, this is the ventral view from the bottom and these nuclei sort of separate segregate out into different regions and then you can see the sort of abdominal the marks sort of coming clearly over here. So, it. So, what starts over in this simple from the single single celled embryo, simple single celled embryo it goes through these nuclear divisions and then ultimately as time goes on you have this whole organism emerging. Also if you look at there is another interesting thing let since I am showing let me show. So, this is that initial phase when the cell division cell compartments have not formed the nuclei are just dividing. You will see that the way that the nuclei divide is actually very nice. It forms a the way that it is colored is that when the cells are resting they are not dividing they are colored in this cyan and then when the cells are dividing it is colored in this magenta or purple ok. So, the cells sort of divide in a wave it starts off at this anterior pole then this these divide then these divide then this divide and this wave sort of propagates from the anterior pole to the posterior pole and then again the next division cycle the wave propagates from here to here. So, not only there is sort of pattern that emerges there is also sort of traveling waves that emerges. The question is if I think of this simple sort of a system how do I get this sort of an emergence of order how do I get these different divisions that form starting with a very simple system like this. So, here is one of the proposals. So, the idea is this that let us say there is some signaling molecule which is called let us say a morphogen and that has some sort of a great concentration gradient along this AP axis. So, mostly I will talk about this this is the anterior pole this is the posterior pole. So, if I go like this this line constitutes my AP axis. So, let us say somehow I have formed a morphogen gradient that goes from the anterior pole to the posterior pole like this. So, there is a high concentration at the anterior pole there is a low concentration at the posterior pole ok. Then the cells or the nuclei in this case they might determine their final state in response to the concentration gradient of this morphogen ok. So, if this morphogen is let us say there is some threshold. So, cells which see this concentration above this threshold will become differentiate to become the head. Let us say cells which see concentrations below this threshold maybe will differentiate to become something else cells in the middle will become something else and so on. So, there is some sort of a controlling concentration morphogen which sets up a gradient and other cells. So, the further division or differentiation of this cells follows as a response to this morphogen gradient. But you could then ask that well in the case of Drosophila this is a sort of picture I have in my mind then what is this morphogen gradient what is this basic morphogen gradient and how is that established. So, the basic gradient as it turns out in Drosophila there is a unique answer to that, but it is this bechoid protein it is this bechoid protein that I talked about. So, what happens is that when this embryo is formed the maternal the mother sort of deposits some mRNA in this part near the anterior pole and this mRNA. So, there is some mRNA over here and this mRNA codes for the bechoid protein ok. So, what that means is that there is a reservoir of bechoid proteins or continuous source of bechoid proteins because this mRNA continuously gets read and bechoid proteins are produced at the anterior pole. And you can see that by so this is experimental plots again of this bechoid intensity as a function of this x x being the length along this k p axis as a fraction of the embryo length. So, it goes from 0 to 1 just to give a sense of the numbers this Drosophila embryos are roughly let us say around 500 600 microns. So, 0.2 means 0.2 of 500 microns. So, so this is the bechoid intensity that has been plotted at different time points ok. So, at very early times it is this blue line then as time goes on it rises it rises it rises at very late time. So, this is in minutes. So, this is after 2 and a half hours or so you get this sort of a red line ok. So, you form a concentration gradient initially there is nothing. So, here is my whatever plot of bechoid let me say bechoid concentration as a function of x initially there is nothing. Then as this mRNA gets transcribed and the protein gets produced the protein sort of diffuses slowly ok. It covers out this whole cell and ultimately you get a profile like this. And this you have seen in your I think Midsim thing. So, I gave this Drosophila problem this bechoid. So, this is the context for that basically. So, the so this is a very simple example of a reaction diffusion system a very very simple example in that sense. So, in general a reaction diffusion system is something where you have some sort of a chemical reaction. So, you have reactants and products reacting interacting to specific reactions, but they are also diffusing and that can give rise to patterns in general. So, but before I go to this sort of generic thing you can think of this bechoid as a very simple case of this reaction diffusion system in some sense. The reaction being that this bechoid protein sort of diffuse this sort of degrades. So, it degrades with some rate let us say I do not know some rate kappa ok. It is produced at the anterior pole. So, it has a source and it diffuses ok. So, it is produced over here it diffuses and it degrades which is why it is called the synthesis diffusion degradation model. So, the source is at this anterior pole which I say is x equal to 0 and let us say it is some concentration at this anterior pole. So, I scale everything by the concentration at the anterior pole which I call some maximum bechoid. So, then what this tells me is how this concentration of bechoid evolves in time. So, the rate of change of this bechoid concentration comes because of the diffusion of the bechoid molecules along this AP axis and the degradation of the bechoid. So, I have written tau in terms of tau. So, kappa is basically 1 by tau. So, tau being the lifetime ok. So, it is a first order decay process and a diffusion with the constant with the source at this anterior pole. And then as you have solved this the steady state concentration of this is an exponential profile. So, it is like e to the power of minus x by lambda right where this lambda the characteristic length scale is square root d into tau or d by kappa whichever way you write. So, you get a profile that looks basically like this right you get a profile that looks like this. So, that is the basic idea that we have that the mother deposits some mRNA which acts as a source of bechoid. So, there is already something that sort of breaks the symmetry. Then as this bechoid mRNA gets expressed into bechoid proteins that protein diffuses out and it degrades and because of this diffusion and degradation it forms an exponential profile ok. So, that is the first thing that happens in this whole cascade of in this developmental cascade. And you can do this you can check this for different sort of Drosophila species and you will see that. So, these have different embryo lengths for example, this the common fruit fly which is melanogaster has something around this 500 600 microns. This one is smaller that one is larger, but you get exponential profiles in all of these cases. Yes yes until this point it is still being continuously produced ok. So, another way to write this equation is to say that what was I writing del del bechoid del t is d del 2 bechoid del x 2 minus kappa bechoid plus a source term. So, let me write the source term as something like a source strength which happens only at the origin. So, delta x and which happens for all time t greater than 0. So, I put a heavy side theta function. So, for any t greater than 0 t equal to 0 being fertilization, you are continuously producing this bechoid protein at x equal to 0. In reality it happens for a long enough time that this exponential profile has had a chance to set in. So, as long as the mRNA is there the mRNA itself does not get degraded. It will keep producing bechoid proteins and the mRNA lifetime is large enough that you have enough time to form this exponential time scale ok. So, as far as this graph goes even at these times I think at least till 3, 4 hours definitely it is known that the bechoid protein is being produced continuously. The mRNA was to refuse. Yes. Yes. Yes somewhat different. So, you will have. So, you have to write some equation for the mRNA protein right which will have some diffusion coefficient of its own it will have some degradation of its own and so on. And then this source instead of being a delta x will respond to this mRNA concentration. So, it is a valid model people have done this and they have seen how this how the length scale changes once you take into account that the mRNA itself can diffuse. In the 0th order model what has been found experimentally is that the diffusion constant of the mRNA is relatively much smaller compared to the diffusion constant of the bechoid which is why to first approximation people treat it as a strictly localized source, but people have relaxed that approximation and seen what it gives. It gives something that is slightly better than this model. So, it is good in that sense right. So, you can if you look at this if you look at this exponential profile, you can find out what is the length scale of these profiles and it turns out for example, in Rossoffila melanogaster the common fruit fly. The length scale comes out to be 100 microns and if you put in measurements of these diffusion constant of the bechoid and the lifetime of the bechoid, you get a number which is very close to this experimentally observed value. So, these particular numbers give you 120 microns. This is some sort of cheating because over the last 15, 20 years there have been a bunch of measurements of this diffusion constant and people have given values from 0.1 to 10. So, it is like a huge variation depending on which number you put in you would get a different length scale, you put in the number that gives you the best length scale in some sense. These measurements are very difficult to do. So, people do frappe like we discussed if you remember in the earlier is not very good people do some fluorescence correlation spectroscopy FCS and so on, but these numbers experimentally. So, inside the cell it is somewhat difficult to sort of do it in a controlled sense because if you do it against the background of all these other nuclei and all other cytoskeletal filaments proteins everything. So, there is some debate about what is the right diffusion constant used, but general agreement is of the order of micron square per second. Nevertheless, the point I want to make here is that what this sort of a theory tells you. So, you can immediately see what are the shortcomings of a theory like this. So, what it says is that this characteristic length scale that I get is D by kappa or D into tau right. It does not matter to me what the length of the embryo is what is this L ok. So, regardless of what L I put I would get a lamp if my diffusion coefficient and the degradation rate were the same I would get the same lambda irrespective of the embryo length ok. And as you saw in these earlier pictures that is definitely not true. If you do different species for example, which have roughly different average length you will see that the lambda scales more or less with the embryo length. So, for an embryo which is larger you will have a larger lambda for an embryo which is smaller you will have smaller lambda. So, simple this 0th order model like this the synthesis degradation diffusion model it explains one thing which is that you get an exponential profile which is observed. So, that is good. On the other hand it cannot explain a few things. So, for example, it cannot explain how this lambda scales with the length of the embryo that is something it fails to do. And just in that context that various of these other models have been proposed that what happens if it is not steady state for example, if I take the time dependent solution of this equation then I will get some lambda as a function of t does that match or if I take this mRNA that the fact that the mRNA itself diffuses. So, the source is no longer a point source, but an extended source then does the length scale of that source for example, come into this lambda and how does that affect and so on. So, people have tried a bunch of models which are variations of these this sort of an SDD model. As far as I know no model has satisfactorily answered this question of how do how do these flies regulate the length scale so that this length scale is lambda is sort of proportional to the length of the embryo itself. Various models are there, but no model completely answers this question.