Okay so good morning everyone. I can see there's a little bit of hindered diffusion in this room because everyone's sitting on this side they don't make it over to that side. Okay today I'm going to continue to talk about problems in biology from the perspective of control and today and tomorrow I'm going to talk about two very important classes of problems those having to do with pattern and those having to do with growth and size and in each case I'm going to appeal to some of these notions of control and reasons for control and strategies for control and so on. So today we're talking about pattern but I thought I'd begin by reminding you of one of the lists we went through yesterday which was some of the objectives of control processes in biology right some of the things you try to achieve using control stability speed robustness of parameters ability to reject disturbances ability to object changes in the structure of the model ability to adapt and homeostasis set point control versus homeoresis guiding something along your trajectory so today we'll see some examples of some of these and then tomorrow we'll see some examples of others. So the mechanism of pattern formation that you've already heard about multiple times in this course is pattern formation via positional information that's obtained through diffusable molecules or morphogens. So this is the basic morphogen theory right the idea that something localized produces a gradient from the gradient cells get information from that information about spatial location they're able to consult their internal program that enables them to differentiate and behave in positional ways. Okay so by construction and what I mean is by sort of the construction of this theory this is clearly not a system in which you expect set point control the idea being that set point control is something where a system knows where it's supposed to go and so it's trying to control in order to go there but of course if the cells already knew their positional information you wouldn't have to give them their positional information right so it'd be sort of pointless. So this is not that kind of a system where the cells know what they're supposed to do and are trying to you know avoid having say disturbances to that this is a situation in which you have to tell the cells what to do on the basis of some process. So here we're going to be looking not for too much say integral negative feedback we're going to be looking for situations to take a process and try to make it more stable more robust more fast and so on. Okay so to begin with as we did in a number of situations yesterday the best place to start asking questions about strategies for control is to take a system and say well how does it behave without control right we say the open loop system without any feedback how should it behave. Okay and so we'll talk first about things like stability and speed and then work our way down that list. So as you remember back in the early days of thinking about morphogen gradients folks like Wolpert and Crick envisaged these gradients as simple source sink gradients where you have a source of something that diffuses to a sink it's describable by fixed laws with boundary conditions that are sort of production and a zero sink and you can see dynamically over time these gradients should evolve in this manner they're clearly unconditionally stable this is not a problem and they evolve at a rate that goes with the square of the distance right and I mentioned that that was one of the arguments that Crick used to establish that there should be some maximum length over which you can pattern because it will take longer and longer in a quadratic fashion to pattern if you try to go too far but I also alluded to the fact that this model doesn't really agree with much biology because by the 90s or so it became possible to start actually visualizing morphogen gradients and when that became possible it became clear that they're not straight lines this is the DPP gradient I think you already saw pictures of this gradient from the wing disc from Frank Ulaker and this is the Bicoid gradient using I think in this case an antibody against Bicoid to visualize it but there are other ways it's been visualized as well the Bicoid gradient where the measurements are particularly good it's very easy to see by plotting these on a logarithmic axis that this is truly an exponential gradient right this follows the form e to the minus x over some characteristic constant called the decay length on the other hand in the case of DPP you can certainly fit an exponential into that whether this is truly exponential is much harder to say given the noise in these data but certainly it doesn't look like it's a straight line so how can we explain the exponential shape of a morphogen gradient well it turns out a very very simple modification to fix law simply superposing another term saying that morphogens are removed destroyed decay you can use whatever term you like but this refers to the molecule leaving the system in some way gives you this equation and when you solve that in the steady state and when you give boundary conditions that the far boundary is very far so then it doesn't really matter what boundary condition you put there then you can show this solves to this form right and this makes sense exponential is the only function where if you take a second derivative it's essentially a scaled version of itself okay and moreover this characteristic decay length constant here has a physical meaning right it's the square root of the diffusivity divided by this uptake or destruction constant over here and so that's where you can get these exponentials but the interesting thing is if you solve that equation dynamically rather than at steady state and as far as I know this wasn't really done until I think the early 2000s by Sven Bergman who who produced this lovely equation and you can see here now the approach to the steady state at different locations so x is one half of the decay length would be about here and then this would be about there and this would be out there and so at each location it's approaching the steady state at a slightly different time and if you plot all that you get a curve of time versus distance that's almost perfectly a straight line it's not exactly a line but it's very very close to being a straight line we're essentially the time to go a certain distance x scaled to this decay length constant is basically just related to this degradation or destruction or removal rate constant over here but you can also express it in terms of the diffusivity and that constant as well because of the definition of what the decay rate constant is okay so this is actually sorry yeah yeah so this represents the time it takes at any point to get one minus one over e or about two-thirds of the way to steady state okay so each each one of these reaches that time at a different I'm sorry reaches that percentage of the way to steady state at a different time and that's going up linearly with distance okay so remember simple diffusion it goes with the square of distance here it goes linearly with distance okay and so this is the other reason why Crick was wrong right because had he known that gradients were exponential right then he would have realized that the assumption that things go with the square of distance actually no longer holds things go linearly with distance and therefore you shouldn't have to wait an inordinate amount of time to make a gradient bigger right of course of course it's it's true that the time at which you reach steady state is going to get later and later farther away that's true both whether you have an exponential gradient or just a linear gradient it will still take longer farther away so there could be constraints on that as well yes it is counterintuitive isn't it but again it's counterintuitive for the same reason remember we discussed how negative feedback makes things faster and that seems counterintuitive it's the same reason is that you have to think scaled to the steady state so in other words by throwing in degradation certainly you're taking molecules away and therefore you should think it should take longer to get to the same value but here we're not asking the amount of time it takes to get to a particular value we're asking the amount of time it takes to get to the steady state for that location and that steady state is declining exponentially so you don't have to get to his higher value so that's a very good point right you have to scale these in order to plot them this way okay so and we'll come back to that because the values are definitely getting lower at a essentially at a faster rate when they're declining exponentially as opposed to linearly but the point is we can use this curve to make some very nice estimations about the time it takes to form morphogen gradients and for reasons we can talk about later you know most of the patterning that we see takes place within two or three length scales of the source of morphogens and within that region at least the time that it takes to reach steady state at those distances can be read you know right off this curve and it should be basically one to two times the half-life that's associated with that decay rate constant k okay but because of this relationship between k and lambda and d we can also express that time as just lambda squared times a constant over d so in other words if you know the time that it takes to form something and you know the decay length because you've watched you've observed the picture a static picture of the morphogen gradient then you should be able to extract the diffusion coefficient from that information okay so everybody gets that this is a should be a way to measure the diffusion coefficient that applies to a morphogen gradient by looking at the shape of the morphogen gradient and looking at how fast it forms or how fast it returns after you've perturbed it a certain amount that should work and in fact many people do that type of experiment using the following type of perturbation if you can make the morphogen fluorescent which as you've seen we can use fusion proteins to look at fluorescent morphogens you can bleach a very small spot of that fluorescence and then you can watch the recovery as molecules diffuse back in and reestablish the morphogen gradient inside your little spot okay and so you can do the math and calculate the amount of time it should take according to that equation to refill that spot you get a sort of relaxation curve from which you extract the time constant given that time constant which will give you K and given the decay length of the morphogen gradient lambda you should be able to pull out D so this was done for bicoid a number of years ago and the D that was pulled out was about point three micron squared per second from those experiments in this case the decay length is a hundred microns for the entire for the morphogen gradients it's a fairly large one and so if you put that together then for the whole embryo it should take about six point four hours for the gradient to set up in the embryo that of course is a bit of a problem because bicoid does its job in less than two hours would be fair and the grand in fact observations of the gradient show that it sets up in less than two hours so that's just one example in which frappe was used to measure diffusion coefficient another example I think frank may have referred to this in the dbp gradient in the drosophila wing disc frappe was used to sort of bleach these little rectangles and then watch the dbp fill in and again using exactly the same methods the diffusion coefficient of point one square micron per second was was obtained just for comparison diffusivities of proteins in water are typically about 100 square microns diffused per second diffusivities of proteins in normal fluids like intracellular you know cytoplasm or extracellular fluids tend to be more like 20 to 50 20 to 40 square microns per second so this is orders of magnitude lower than you might expect okay so is this somehow there's some kind of special control that's enabling these gradients to form faster despite having very slow diffusion coefficients or is it simply that the measurements are wrong and one set of observations suggest that the measurements are wrong and that's because there's an independent way to measure diffusivity I think some of you are aware of fluorescence correlation spectroscopy which is a technique in which you illuminate mostly lost the battery on this one in which you illuminate a little tiny spot and you look at the fluctuations and fluorescence in that spot those fluctuations are due to diffusivity of molecules in and out of the spot if you plot the autocorrelation in in those fluctuations you can calculate you can determine the mean time it takes for a molecule across the spot and from that you get a diffusion coefficient and in fact that's been done for DPP in the very same system you get a diffusion coefficient of about 20 micron squared per second consistent with free diffusion and in fluid and if you do the same thing for bicoid you get about 7 fairly consistent with free diffusion and at least more than an order of magnitude over what you get in frappe if you do it in for FGF in the zebrafish you get about 50 square microns per second so again you get very large numbers by FCS and you get these small numbers by frappe right yeah so you can from first principles try to calculate what the diffusion coefficients should be if you know the viscosity of the medium right and if you use a viscosity of water you get numbers about a hundred square microns per second now the viscosity of biological fluids is is you know higher than water and so that's why you typically get numbers up to an order of magnitude less than that okay so to get these very small numbers you have to have some type of very severe hindrance right and it would be unclear where that would come from yeah so that's I'm glad you asked that question so another thing that can influence the the apparent diffusivity of something is tortuosity right if it has to diffuse through a space in which there are many many obstacles tortuosity kind of is like a different sort of viscosity the nice thing about tortuosity is that while it can lower diffusion coefficient typically it can't lower it by more than about three-fold the reason be unless the tortuosity has been specifically ordered but if you have random tortuosity what happens is the longer paths that you have to take due to the tortuosity are to some extent mitigated by the fact that you also have much smaller volumes through which you have to diffuse which actually speeds you up right and so the two things eventually in the limit cancel each other out and you get to about a three-fold change so you can't get big orders of magnitude changes from tortuosity which which actually is a good point because a lot of biologists intuition it'd be to be really hard to diffuse in and around cells because it's such a tortuous environment but the reality is that's not such a big deal and a lot of years of experiments of people measuring diffusivity in the brain for example of injected drugs really validates that intercellular diffusion is not much slower than diffusion in free space okay so why these big discrepancies between fcs and frappe yes I'm sorry you can't you mean between the viscosity and the diffusion coefficient right but there's a relationship between the viscosity and what you measure right a thermal system and you mean as opposed to a system in which transport is by some other process I'm not sure that's a major correction to any of this though right I mean I think things are happening a very long time scales things really are pretty much just floating around in space here so we can talk about it later but I don't think there's a there's a there's a big issue here I think there are there are other more significant issues here so the thing to point out is that photo bleaching is an experiment in which you perturb and you recover according to this equation right so the idea is that you know given measurement of recovery which depends largely on K as you saw right the rate of approach is essentially dependent on K and given a knowledge of lambda you should be able to calculate D right but that's based on a very very simple model now that's under the hood of that simple model is something like this where you have cells and they have receptors and you have morphogens and the morphogens are floating around and it's these receptors that are responsible for this K the receptors are binding the morphogen they're taking in the morphogen they're destroying the morphogen okay in the case of bicoid it's slightly different right the receptors are presumably the nucleus and something else is destroying the morphogen so it's a slightly different model but for most of the morphogens the extracellular morphogens this is kind of a pretty good model and if you really want to write that model out explicitly it looks something like this right you have now a diffusion equation for the morphogen you have binding equations to the receptor and unbinding and then the receptor has equations for production and destruction and then the bound morphogen has equations for going in and coming out and the internal morphogen has equations for going out and being degraded and so on you get a whole bunch of equations even if you solve this in the steady state of course these equations all become algebra so you can isolate all the variables and basically get a single differential equation in the steady state which kind of looks like that original equation it's slightly different but if we make the assumption that morphogen concentrations are low then we can get it to look exactly like our original equation right just diffusion balanced by decay so the reason you have to make this morphogen is low assumption is because receptors are saturable right and so only in the regime of low levels of morphogens can we treat binding to receptors as linear right and so under those conditions right this this which includes the saturability of receptors now reduces to that and so to get that equation we've all been using right this is really what's under the hood of this k over here so the problem is that that means that the decay length of the morphogen gradient is determined by all of this together right and consider for example the possibility that if k off or k out either of these is close to zero meaning that things tend to go in but they tend not to come out then this k is going to be independent for example of this k deg term the day the actual degradation term that's why it's always bothersome when people refer to this as a degradation rate constant it's not it's a removal rate constant this is the rate constant at which the diffusing species is removed from the diffusing pool that's it and you can under the right conditions this can be almost completely unrelated to the actual rate at which the morphogen turns over absolutely all of those possibilities are in those rate constants depends on where you set them right so yes you could have a case in which the most of the time things just come back off the receptor but the point is if that's the path to being decayed right then that plays a role in k if that's just a dead end and then you come back off again it has no relationship to then that actually drops out of the steady state right well okay so if you just have something like this Bitcoin but plus something that's not a receptor it goes to that or Bitcoin gets destroyed by an enzyme right then this is going to affect the dynamics but it's not going to affect the decay length but on the other hand if you have something like that even as a possibility then this will affect both the dynamics and the decaying so it's sort of it depends anyway okay go ahead so if we yeah no in okay so two separate things this model here is for an extra cellular morphogen that's spreading between the cells Bitcoin is a little special right it's an intra cellular morphogen it starts in this biggest incisional cell and spreads within it okay so two different models but the same underlying math it's just the meaning of these constants maybe a little different in those cases fitting models okay I mean there's also experimental issues I won't get into them because there are experimental issues with how you get those constants out of frappe but I think the more dominant thing right is this issue that the decay length depends on this but frappe depends on some combination of these okay and particularly on whichever one is slowest right because essentially what you're doing is with each of these equations here is you're filling up different pools right a pool on the surface a pool inside another there may be another pool and so on each pool has its own turnover and each has a characteristic time of turning over and you're essentially taking a sum of all those times and that's what you're measuring by frappe and so for example if things go into the cell and spend a long time there that means k deg is low and frappe will essentially give you k deg remember I told you if k off is very small like zero then the lambda the decay length will not give you it'll be totally unrelated to k deg it'll be something entirely different and so that's the flaw in using that formula to get the diffusion coefficient out of frappe is you're assuming that the decay length of the gradient is actually related to the thing that's causing the relaxation time in frappe so just to give you an example right these are no it's not it's a it's an error it's not a defective so frappe isn't measuring movement frappe is measuring relaxation it's measuring a systems return to steady state and the system you're measuring is all of these compartments and some of these compartments are returning to steady state mainly because of these rate constants that's what you're measuring with frappe if you could only measure this compartment alone it should return on a time scale that reflects only diffusion but you have no way to measure that you can only measure the total morphogen no it's not a longer length scale so you get a shorter diffusion coefficient than you expect so for example if you fit assuming that the same k that determines lambda is the k that determines frappe you get the numbers reported in the literature if you take the same data but now you put in these pools and reasonable rate constants that fit the kind of concentrations that are observed at steady state within these pools you can easily get a diffusion coefficient that exactly matches the fcs so to do the exercise for bicoid right what you have to have again is some system in which bicoid delays in some state before its decayed that's all you need right so instead of having just bicoid goes to this it has to be bicoid goes to some state which hangs around for a while and then goes to that as long as that exists you can adjust that diffusion coefficient to whatever you like so the point is not that you can get the right number out of frappe the point is you can't get a number out of frappe the number you get it is is an illusion unless the model is exactly the original model right where k there's only one k and there's only one species okay so anyway that's I thought you would enjoy that part because it's physics all right so now does that mean yes question well you'll see that this is not a perfect so this one and that one don't have it perfectly the same shape of course you'd never tell that in the lab but so you don't get actually in neither cases it truly is strictly exponential approach okay because the formula remember the formula for the this time spread of a morphogen gradient was this ugly thing with error functions and so on so it's not perfectly exponential either but yes the shape is slightly different but that the point of that is to show you that it's very slight right for the total pool of molecule different yes very good point okay so so the point you're making and this is a point a lot of people say is fcs measures movement on very microscopic spatial and temporal scales right so for a period of time things might be moving fast but then they might stick to something and might be moving slow and actually you can sort of see that if you look at the fcs measurements for DPP you can see about half the molecules are actually in a slow pool so this is what you see in fcs is two pools is a fast pool in a slow pool presumably the real molecules may be going back and forth between those pools so you would have an effective diffusivity right due to trapping that would be lower now the important thing to remember about those kinds of effective diffusivities is they have a big effect on the transient behaviors of such systems right like the time it would take to form but they have no effect on the steady state they won't affect lambda okay and the reason is if you put in an extra equation here where something bound and then unbound from say a hindering site right that equation would completely drop out when you turned it to this right because in the steady state it just goes to local equilibrium right so you'll accumulate morphogen but as much will be going on as will be going off at any location in space so you can't really affect the steady state shape by hindered diffusion that's another misconception in the literature right but you can definitely affect the dynamics okay so it's think of like a capacitor right a capacitor affects the dynamics of an electric circuit but at the steady state right the capacitor does nothing right it just it just holds stuff I mean it takes some of the charge away it doesn't change the shape of anything okay all right so let's move on from that topic of speed so basically the idea is that because diffusion is fast and because things go linearly with distance really spreading morphogens is not a problem okay but it's very difficult to measure the real rate the morphogen spread at because of the fact that techniques like fcs are sensitive to things that are outside of what you really want to be measuring which is the speed of the morphogen itself okay let's talk about parametric robustness now and here I'm going to focus on the Drosophila wing and you've heard a lot about this so I'm going to just very quickly go through the basic setup remember the wing has these veins that have to be patterned along the anterior posterior axis it occurs in this epithelial tissue called the disc it starts off because the morphogen hedgehog is produced in one half the posterior half that corresponds to this part of the wing and one of the veins is patterned close to the hedgehog boundary over here which is the original anterior posterior boundary in the wing disc hedgehog is a morphogen at a distance from hedgehog it patterns the l3 veins so targets of hedgehog determine where this vein is going to occur hedgehog also determines where DPP is going to be expressed DPP diffuses out in both directions here's a picture of the DPP gradient using GFP DPP but very often what we track with the DPP gradient is phosphorylated mad protein because that's the direct target of the activated DPP receptors so this antibody basically picks up the receptors activity you can see there's two gradients going in the anterior and the posterior and these two gradients respectively set the positions of the L2 and the L5 veins okay so knowing that then one way we can try to ask how parametrically robust the patterning system is is by making heterozygous mutants that is what taking away one copy of the receptor genes or other genes so for example the morphogen genes for example here we take away one copy of DPP and you have to play some genetic tricks to get the embryos to live through that so these involve different genetic tricks for doing it but what you can see is in all cases the L2 3 vein spacing and the L4 5 vein spacing which represent the positioning of these DPP dependent veins show a small change in in width when you take one copy of DPP away when you take one copy of DPP away nothing happens to the L3 4 vein spacing that's the one that's set by hedgehog so that's kind of like a positive control it shows you or negative control it shows you that this doesn't change because it's not supposed to it's a different morphogen that sets that so there is an effect so it's not perfectly robust right there is an effect of taking away one copy of DPP here we take away one copy of the DPP receptor it's called thick veins or TKV and here it looks pretty good really can hardly tell any difference between these two wings and then finally here we take away one copy of the hedgehog gene and we're looking at one particular hedgehog target called Collier and again it looks pretty good so how good is pretty good right remember yesterday we talked about the fact that we need a metric for me for quantifying robustness the typical metric we use the sensitivity coefficient right which is the slope of a log log plot or the relative relative derivative you can look at a number of ways remember sensitivity of one means linearly related sensitive to quadratic sensitivity point five varies as the square root and so on and engineers like to keep sensitivity down below about point three that's usually the threshold when people say something's robust so now you can go back to these cases and say okay so this thing is about for a presumed two-fold change in the morphogen there's about a 14 percent change in pattern here there's less than a 10 percent change in pattern here there's less than a 5 percent change in pattern so from an engineering standpoint that's pretty good right that's we'd be happy with that kind of robustness very often now the question is what do we need to get that kind of robustness if we go back to the model we have of the exponential gradient we can ask the simple question how robust should position be all right so for example if we were to take that gradient and increase the amplitude of the gradient by a factor of two how much would pattern change so the idea is that if you're specifying a particular structure at this location then when the gradient goes up the location is going to move over by that much okay so really we want to know how much does the gradient shift to the right as a function of being shifted upward that's essentially the question we're asking but we want to ask that as a sensitivity so it's very easy to figure that out right because you can invert this and have X as a function of the concentration and then so therefore the change in X is just equal to that right which means that if this is a two-fold change in the morphogen that everything shifts by lambda the decay length times two times the logarithm of two okay but of course we want to do that as a sensitivity coefficient and so if we want to do it that way we take the derivative of that with respect to its concentration right the thing that's being changed multiplied by the initial concentration divided by the function itself rearrange that and then we can substitute X over lambda for that based on the definition here and so there you get that the sensitivity is simply the decay length over the position in other words as the thing gets farther and farther out it becomes less and less sensitive and that makes sense because even though it's moving by the same amount in fractional terms this is a much smaller movement than that right that's a big fractional movement and this is a very small fractional movement and so you can get a curve like this that essentially captures for this morphogen gradient here's what the sensitivity of positional information is to a change in the amount of morphogen at the start of the gradient okay so everybody follow that so one of the nice thing oh so just to put this in X needs to be at least three lambdas in order to get acceptable robustness right basically you're saying this falls and it hits this kind of good region somewhere around three times the decay length constant so that's saying for any morphogen gradient if you just pattern far enough away you can achieve good robustness you don't need anything special now just as an aside one of the nice things about doing sensitivity coefficients is something I alluded to yesterday which is that you can often calculate them even when you can't solve the initial equation that you're trying to calculate them for so for example if you take the equation that doesn't make the assumption of receptors being far from saturation I remember there was a form of the equation that looked a bit like this where I said you know assume this is small right because then receptors aren't getting saturated but let's take the full equation where we allow receptors to be saturated then here C stands for morphogen normalized in such a way that a value of one means receptors are 50% saturated anyway you can't solve this differential equation even in the steady state right I dare you to but anyway that there's there is no analytical solution that anybody's found for this and yet you can still find the sensitivity very very easily right the trick being that you can differentiate this once with respect to distance you just multiply it by dcdx on both sides and differentiate once with respect to distance and you can actually get an analytical albeit implicit solution for the first derivative and the first derivative is what you need to calculate the sensitivity coefficient so you can sort of go directly to the sensitivity coefficient one of the things you can see is because this number is always greater than one when you allow for receptor saturation the sensitivity is always greater than it would be without allowing for receptor saturation so in other words neglecting receptor saturation gives you a false impression that the sensitivity is better than it really should be okay in fact allowing for receptor saturation can have really bad effects on robustness if you take into account the fact that morphogens have a producing region and they have receptors too and they're turning over in that region and if they're saturated there then it's particularly bad because if you have say a two-fold change in morphogen there's no capacity to absorb it in the production region and you can get really big changes both in the starting point of the gradient and in how robust it is as a result of that so so we have to watch out about receptor saturation when thinking about sensitivities yes yes so when you have receptor saturation the apparent decay length will actually get bigger and eventually it'll get kind of more like that and underlying that if you look at the receptor bound morphogen it will go to start looking like that so yes and actually that brings up an important point which I have here when you have an exponential gradient of course you can define the decay length as the the constant that lies inside the exponential but if you have something that doesn't truly have an exponential shape you can still define something called a local decay length by analogy essentially it would be the decay length of the best fit exponential at that point but very simply it's just the ratio of the function to its own derivative because that's what this is as well so for any function we can always calculate in a local decay length and your point is if you have receptor saturation then you'll get an increase in the local decay length and then it'll go back to the original local decay length later on well no it's as you go farther out they become less and less saturated so saturation is always the biggest problem closest to the morphogen source so let's talk a little bit about receptors okay so remember starting with these equations this is what we've used as our equations for the morphogen right for in the very simplest term forgetting about receptor saturation how do we build in robustness to receptors when receptors don't appear in this equation right so they do appear they appear implicitly receptors appear here because this reflects the uptake by receptors and since k appears in the denominator you would imagine that if you increase the number of receptors what would you do to the decay length decrease it right so a gradient should shrink when you increase the number of receptors however the gradient you measure is the signal gradient the gradient of morphogen that's bound to receptors and if you have more receptors more morphogen will be bound to it so what you should have when you increase the number of receptors is greater amplitude okay and so in fact you have both if you do the equations we explicitly put receptors in there you see both effects occur as you increase receptors you increase the amplitude but decrease the decay length and you can see this here in these simulations where we're varying the it's not it's just a equation just solved but you're varying the concentration of receptor as it goes up right the amplitude goes up and then thing pulls in as it goes down the amplitude goes down the thing flattens out right this is in fact exactly what you get in experiments if you over express receptors pull in the gradient but the gradient gets a higher amplitude okay but since you can determine this analytically you can also turn that into us into a sensitivity coefficient curve and the reason why this has this inflection point here is because this is the absolute value the reality is as it goes down below here it switches from things before this point are actually pulled up by more receptors things beyond that point are pushed down it's the other way around think you know it's the things beyond that point are pushed down by more receptors and are pushed up by fewer receptors now my reason for showing this is to put this in and have you notice something very very important is there any position x in which you can be simultaneously robust to both morphogens and receptors there's not right so that's actually a really interesting result it's telling you that in the open-loop system characterized by the simple morphogen gradient with production and destruction it's categorically impossible to be robust to both types of perturbations at the same time at the same patterning position in space okay right there's no location where you can do it and so there's only one way out of that which is to say that the model has to be wrong okay so the model is right from the standpoint of fitting data but from the standpoint of achieving any type of reasonable robustness that you would expect in the real world that can't be the right model because that model can't be robust to two things that we know empirically you are robust to at the same time and so the point is that's an open-loop model right that's a model with no control in it it's just assuming hedgehog turns on dbp dbp turns on mad mad takes care of patterning and you're done you have the physics of diffusion which set the gradient and and so on right so how can we repair the model so if you go to the biology and you look at what's known in this system in terms of things that interact with dbp with the morphogen with pattern and so on you get something like this as of now it keeps getting worse as people do more and more experiments right but there's a hideous amount of feedback and feed forward and all kinds of stuff going on and let me take you through a little bit of it so for example the synthesis of morphogens and of receptors right for both hedgehog and for dbp are controlled by the morphogen gradients themselves the since there are these binding proteins that act as co-receptors for both hedgehog and dbp and their synthesis is controlled by the morphogen gradients themselves both hedgehog and dbp there's feedback regulation on the production of on the destruction of hedgehog and on the production of hedgehog by the morphogen gradient itself and then that feeds forward on to dbp right there's substances that feedback and control the spread of the morphogen which we'll talk about at the end I think Frank alluded to some of that and there are even extra morphogens in the system that people don't like to talk about things of the same protein family but expressed rather differently and work through the same receptors one of which is called glass-bottom boat and so the question is are these control strategies right as the reason these things are there to achieve the control that you clearly can't achieve at least from the standpoint of getting good robustness from the very very simple model okay so let's talk about some possible ways in which these extra loops could be control strategies so remember the simple model is just morphogens are produced transported and uniformly degraded as they're transported and now let's consider what happens if we add in the fact that there's something else that binds the morphogen that's being produced in this system and perhaps this is doing the uptake in the destruction rather than just that okay so that's easy we just add one more equation in the steady state it's just a little more algebra and now you can very easily get the sensitivity curves for for a variation in morphogen production and for variation of receptor production to overlie each other so bravo right we can be robust right out here but what's the problem right what's the trade-off so in order to do this trick you had to express another protein right you had to express this other non receptor binding site and have it do the destruction well wouldn't you want to be robust to that right and the problem is you're not right you're terribly non robust to that so you've traded robustness to one protein product for robustness to another protein product and that's not a big help right you're you're that's kind of even swap so that's not a particularly good strategy let's consider another strategy who's to say that the uptake by receptors is necessarily linear in other words is proportional to the amount of morphogen what if the morphogen controls its own uptake what if the morphogen's own signal controls how actively it's taken up so this is something that Nama Barkai postulated a number of years ago and noticed that in at least two well-known morphogen signaling pathways morphogens do in fact upregulate their own degradation or own uptake in hedgehog it's because hedgehog upregulates its expression of its own receptor in the case of wingless it's because wingless downregulates its own receptor but its own receptor also downregulates its own degradation the reasons that are still not terribly well understood through a numerical screen Nama noticed that these systems were more robust than the simple system where degradation was just constant and one of the interesting things about these systems is if you model this signaling dependent increase in uptake by simply putting an exponent here so now essentially degradation goes with the square of morphogen concentration these things solve to fairly simple power law equations which if you plot them generate curves that they look kind of exponential but they're not right this is actually a power law curve which means it doesn't have a single decay length associated with it right it has different decay lengths at different positions at the beginning it has a very steep decay at the end it has a much shallower decay than this and it's easy to show that in both of these systems the sensitivity to morphogen varies with the the same form lambda over x but here it's the lambda at position zero where it's very very steep and so is very short and so consequently this is more robust than that and that was the point that that Nama's paper was trying to make is simply using this shape means that if you shift up by a certain amount here you shift less over here and that's what what this is pointing out now you might imagine there'd be some price to pay for that strategy to sorry in the other direction mean if you shift position with it yes no it's still the sensitivity works both ways right so it's this is simply the constant that says how what the fold change in x will be for any fold change in y okay and that's whether you increase or decrease okay so what's the cost for this strategy every strategy has a cost so here it's a little trickier to see what it is and to figure it out I have to go on to the next performance objective and then it will become clear and that's disturbance rejection okay so let's talk a little bit about disturbance rejection so a lot of times when we talk about that we're talking about the effective noise on a system so noise can mean a lot of things right but what here in a patterning system when we think in a noise free way right we have this very french flag idea that there's some threshold at which you turn on a gene and so you would get this interpretation right but if you had a noisy morphogen gradient or a noisy interpretation of a morphogen gradient you get something like this right and so that in other words there's imprecision in figuring out where that boundary is and it's associated with something that we could quantify which we'll call a transition with and we know from empirical studies that having too big a trans transition with is a bad thing because the purpose of these boundaries that are set by DPP is to set up these pro or vein primordia and if your transition with is too big if the boundaries too fuzzy you get these very very fuzzy vein primordia and when it becomes time to form the pupil wing those fuzzy vein primordia give rise to multiple veins right so this is a very bad wing because you're not able to specify the position with enough precision there okay so we want to be able to quantify that transition with that amount of precision that we should expect in a morphogen gradient so typically we'll think about the noise as noise in the signal right so noise in this direction and we can approximate it as just a variance right we'll just use that moment to approximate the noise and we want to know how much does variance in that direction translate to variance in this direction right so we just need the scaling factor for going from one direction to the other and that scaling factor is simply the relative derivative right well the scaling factor is the derivative but if we're looking at relative noise that is coefficient of variation or relative standard deviation then it's the relative derivative that provides the conversion to the transition with okay so basically the transition with is just the coefficient of variation of the noise divided by the absolute value of the relative derivative but remember what the absolute value of the relative derivative is is one over the length scale okay the apparent local length scale so that then gives you right that the width is just the coefficient of variation times the local length scale alright and that makes sense right if something's very shallow and you have a lot of noise you're gonna have a very broad zone of imprecision if something's very steep and you have a lot of noise it's still only going to be a short zone where the noise confuses you before you climb out of it and so right away now you can see what the problem is with self-enhanced decay the price for making this curve steep here but having the same kind of average values as that there is it must be shallow there and the price for having a shallow there right is that the noise effects are going to be much worse here you're gonna have much more imprecision for the same amount of noise here as you are to have there okay so now let's talk about whether that really matters or not or this is just a theoretical problem so first of all where does the noise come from it can come from fluctuations in the morphogen concentration because diffusion is really a discrete process turns out those are very very minor because they happen on such a fast time scale it can come from cell-to-cell variation in gene expression that's probably a really big deal but we don't understand that much about it it can come from background or ligand independent signaling probably there's not too much of it but we don't really understand that very well the thing we do understand are fluctuations in receptor occupancy simply due to the fact that binding is a stochastic process right not really a mass action process right so for example if you if you simulate binding you know using the Gillespie algorithm and you have these average receptor occupancies this is what a cell is really going to see right in here we're simulating the whole system not including taking in and turning over but I'm only showing you this here right and so you would expect as a Poisson process the coefficient of variation is going to go with the square root of the mean so the lower the mean the bigger the noise that's not that surprising the question is are these physiologically relevant numbers to be talking about average receptor occupancies of 200 per cell 50 per cell people work with mammalian cells and culture like to have you know thousands of receptors occupied so you know they don't worry too much about stochastic variation but it turns out in a morphogen gradient you have to worry greatly about it and I won't go through all of this but if you do some basic calculations based on the sizes of the cells the extracellular volume fraction in fact none of these are very sensitively affecting the results so these are you know rough numbers you can calculate the receptor concentration in extracellular space and we know that the decay length is really the rate of diffusion divided by the rate of capture by receptors and that depends upon the number of receptors and the association rate constant and so you get a kind of limit on that and putting in the slowest possible association rate constant the slowest known for any tight binding event you get that the number of receptors per cell has to be below 700 okay otherwise morphogen gradients just can't spread far enough they're going to be captured too quickly so you got to keep the capture rate down and if you have 700 receptors per cell and they're only 20% saturated because remember at the beginning if they're very saturated that's terrible for robustness right so now you're left with receptor numbers in this ballpark at the positions in a gradient where you'd be doing patterning okay there are other tricks to try to get around this we could talk about later but on the face of it it looks like at least at the surface you have to have very very low receptor numbers and so these kinds of variations are likely you know are definitely expected at the cell surface question I'm sorry binding of a transcription factor to a promoter is all it's also a stochastic process yes but here we're just dealing with binding at the cell surface all right but all of these things are as long as the numbers of molecules are small these stochastic fluctuations can be large so what I've just done is gone through it so mass action is an illusion right I mean the reality is always stochastic and mass action is simply the approximation you can use when things are have large enough numbers that you know you can ignore the stochastics okay but it's always correct to treat things stochastically and then when you see that the stochastic effects are large enough then you know you have to take them into account okay yeah that's very interesting yeah what is make it yeah make that fast right right right so here's the problem yes yes yes we'll get to that there will be some averaging later but here's the problem in order to keep the morphogen from being captured and preventing the spread of the gradient the on the the association rate constant must be quite quite low in order for the thing to have any significant affinity that is specificity the dissociation rate constant has to be even lower than the association rate constant but the dissociation rate constant being slow means the dynamics of the whole thing are very slow so you can't average out the noise right the noise is very slow noises you look at the hours right so only averaging you could do is downstream which which I will get to hopefully okay all right so that's that we should expect noise but you know that's theory right now our is their noise so you can look with GFP DPP and you can see well first of all a lot of GFP is inside cells this should not surprise you because remember we said the model was the DPP gets taken up and probably spends a very long time inside the cell that's why frappe doesn't work well there it is you can see huge amounts of it inside cells you can see enormous differences from cell to cell okay but that's not a very good piece of data right because the stuff inside the cell maybe some of it's not even working anymore you really care about the morphogen that works you care about morphogen activity so one way to do that is to measure the rate of of transcription of some morphogen target and one way to measure that is to do what's called nuclear fluorescence in situ hybridization so you make a probe to the intron of a gene something that gets spliced out so it's only there transiently while the genes being transcribed and it's in the nucleus and while the genes being transcribed right if you do in situ hybridization to that cell the fixed at a certain point in time probes will to stick to that intron while it's on the gene being transcribed and produce a little fluorescent dot in the nucleus and depending upon where you put the probe in the intron whether it's early or late essentially you're integrating the transcriptional rate over a period of time that relates to how long it takes for the transcriptional machinery to move from here to here in this case or from here to here in that case so you can have different averaging times so you can do that with the DPP gradient because it has some nice targets one of which is optomotor blind which occurs about two to three length scales away from the start of the DPP and it positions the L5 vein and it has a nice big intron and so you can put a probe and you can actually see these dots each dot represents in a sense the rate of transcription averaged over about a 30 minute time period at each cell and the cells are outlined in red which is a stain for the edges of each nucleus and so you can do that and you do a lot of nice image processing to make sure you're actually measuring the total intensity of the dots and if you measure the average dot intensity of particular location in space from the center to the edge it looks just like what you'd get if you simply stained with an antibody to that target gene it goes up and it comes down OMB doesn't have that sharper border but if you look at the cell-to-cell variability this is what you see right at those locations and that gives you a coefficient of variation for the raw data that starts here and goes way above one now you have to correct for the fact that some of that variability is coming from measurement error so you actually have to put two different colors on the same thing and look at the cross-correlation between them and you correct for that and you get a corrected measurement error and you see that same thing that as you get further out the noise goes up which is what you expect right as the occupancy goes down the stochastics become more and more dominant and you get bigger noise as you go up but basically the noise is on the order here where l5 would be formed of about coefficient of variation of one so if you take that and you multiply that by the decay length the two is here because you have both sides of the of the intended original position you get the expected uncertainty in vain l5 should be about 22 microns and then you go and you measure vain l5 at the moment it's forming it can be spotted by a particular transcription factor and you measure the sort of steepness of its boundary and it's about seven microns so we have about a factor of three different right the noise should give you something that's fuzzy of about 22 microns what we actually see is about three times better okay so is there a way that the system can improve the precision right given that we now see that there's a lot of noise and the noise is having an impact on the system well one thing is since the imprecision depends upon the amplitude of the noise and the length scale the decay length you can just decrease the decay length right and then everything would be more precise but why won't that work as if you decrease the K link everything else be closer in right you just shrink everything so that doesn't really work relative to the size of the field you haven't haven't improved anything at all okay relative imprecision won't change what else could you do well you could try to improve the signal to noise ratio by increasing the the amplitude right remember the coefficient of variation is reflecting the fact that you get more and more noise the lower and lower the mean so let's kick up the mean right let's just put in more and more and more morphogen right and then at farther and farther away we'll have greater and greater receptor occupancy and less and less noise what's the problem with that receptor saturation remember once you start saturating receptors your robustness to parameters goes to hell right so you can fix the noise problem but now you get another problem and so you start putting this all together and you see there's this constellation of interacting trade offs such that you can push the you can push the gradient farther out by putting more morphogen in it but then you shrink the region that's robust the part of the gradient that's adequately robust until it eventually disappears and you get a hard limit somewhere and again there's a very approximate numbers but somewhere around 60 microns you just can't do it anymore now that's very interesting because we've seen that number before right we heard crick and watch I mean Crick and and Wolpert argue back in 69 and 70 about why morphogen gradients were short and crick appealed to a physics explanation but I've just given you a completely different explanation an engineering explanation that has to do with signal-to-noise versus robustness trade-offs okay so a completely different way of answering the same question okay so I could go on for hours because you can come up with all kinds of good strategies to fix one problem that are destroyed by you know the the impact on another problem and you know you can play those games but at some point eventually you want to ask well what strategy is the wing actually choosing right and that will give you some sense of what are the perturbations the development really cares about and which are the ones you can deal with okay so to do that I want to go back to this issue of response to variation and receptors okay and remember this curve that as receptors go up morphogen gradients get taller but pull in as they go down morphogen genes flatten and spread out right and I told you that even though receptors give acceptable performance only under a very small part of the of the morphogen gradient that the reality is that you get really great patterning in receptor heterozygotes and we know that this is essentially a 50 percent reduction in the level of receptors what I didn't tell you is if you take those same heterozygotes and you look at them earlier in development while the wing disc is doing its patterning that even though the pattern comes out normal in the adult it is definitely not normal in the wing disc at the time patterning is happening if you look at the phosphorylated mad which is the DPP signaling it is indeed spread out in the case of the receptor heterozygotes and it's spread out by an amount that's just about what you'd predict from the model so somehow the receptors are are non robust I mean the system is non robust when you look at the level of the DPP signal and yet the pattern is retrieved anyway at the end of the day now one way people I'm sorry what is that that's just a we're that's a ballpark that engineers like to use and we're just putting it there as an arbitrary yeah yeah sure sure but I mean it's it's just there is saying if you use this number those would be the conclusion if you used a more generous number you could sort of barely squeak by right but at the beginning I showed you a lot of the robustness we observe is much better than that number okay so that I think that's already a pretty generous number for biology okay so how is it right that this is pretty poor looks like an open loop model with no control and yet you come over the right pattern so it's very typical in developmental biology when things like that happen to say well it gets fixed later right something happens in the pupil stage and gets the pattern back and that's kind of a cop out right because if it was fixed later why did you even need to pattern it right in the first place right so I don't think that's a good explanation but turns out in this case it's not the explanation because even well before those later stages if you look during patterning at these transcription factor boundaries that mark where those veins are going to be they are also robust okay they're hardly changed at all in the presence of these heterozygous mutants in fact we can push it even farther rather than make a heterozygous we can use RNAI to knock down the receptors to very low levels in this green domain here so here's the DPP gradient here's the control part here's the part where we knock down receptors and you see the PMAT is radically expanded in those cases and in fact the shape of how it's expanded really beautifully fits the very very simple open loop model and yet the veins come out in exactly the right places so we've totally trashed the gradient and the positional information that specifying the pattern doesn't seem to have changed at all right and so this is true for all of the markers that we look at other than the phosphomad with Brinker which is the most direct target downstream of phosphomad there's a very small effect but by and large all of these are robust but the thing upstream of them is not robust right so that's very peculiar right so how do we explain that some again something's wrong with the model right everything seems to fit the model until you get to here and then it doesn't fit anymore when you get to the actual targets so let's go back and think about that model again where does that exponential gradient model come from it comes from this idea right morphogen produced uniformly transported taken up by receptors and that's why you get these shapes because receptors both play a role in how much signal but they also play a role in the decay length of the gradient so as you decrease receptors the gradient spreads out except very close to the source now just for comparison remember you know Crick and Wolpert's linear model the idea of just a source sink gradient how does that behave if you were to vary the number of receptors in that well that would be much simpler right more receptors more binding but the receptors are playing no role in the shape of the gradient and so in that model if you decrease the number of receptors position would move to the left okay so now you're beginning to think wait a minute what if we can build a model that was somewhere in between those two now you could have a model where at least in some places position wouldn't shift at all right now what's the reason for thinking that that might be feasible receptors are not uniformly expressed spatially receptors here stained this is production of receptors with thick veins lack of Z you can see here's the PMAG rating going out from the middle these receptors are almost all concentrated at the edges here's a scan of the receptor profile it's very very low where DPP is produced and it stays low and then starts climbing towards the edges and climbs about five-fold it's a little hard to estimate exactly and the reason for this is because the patterning genes are themselves feeding back on receptor expression so there's a big negative feedback appear on receptor expression now what's the consequence of that so again remember we have our simple model we can take an asymptotic approximation of having all the receptors over at one end as being like having a constant low level of receptors and then at the very end have a sink okay because that we can actually solve analytically that's the same as solving this equation with an added sink at a certain position and then that has a fairly simple form and if you plot it when the sink is relatively nearby compared to this decay length constant it makes a line and when the sink is really far away it makes an exponential okay does that make sense right the sink is very close the receptors that are present within the gradient really don't matter it may as will be free diffusion if the sink is far away everything's already taken up before you get to the sink right so essentially by varying the amount in position of receptors at a distance you can essentially get a gradient to lie somewhere between the linear and the exponential okay and in fact if you do that if you model this amount of receptors into that sort of gradient you get a shape that looks something like this green curve here and look what happens when you change the level of receptors down or up by a factor of two it's not very robust over here but of course here's not where DPP patterns here's where DPP patterns and it's very robust okay so you can get this robustness simply by being somewhere between exponential and linear and here this is just to show in terms of sensitivity curves the top shows what the linear would do the bottom shows what an exponential would do and the blue shows you that there's when these gradients are at least are not too large compared to their own intrinsic decay lengths there's this region of very very low sensitivity for the mixed curve okay so that gives you a way to explain how you can shape a gradient so that its sensitivity to receptors can be really low at least when the gradient is of an appropriate size but it still hasn't solved the problem of how it is that the robustness to to phosphorylated mad the thing at the top of the pathway here is so much worse than the robustness to the things downstream of it that that needs another explanation although it's related and for that we have to remember something that Frank referred to last week which is that discs are not static they're actually growing during patterning and as they grow the morphogen gradient is expanding with them we'll talk a minute about how it's expanding with it but remember that as morphogen gradients expand and have all the receptors at one end right they are going to gradually shift from being linear to exponential okay they're going to go through that transition over time and in fact if you look closely that's exactly what you see you can even see it in other people's data which is remarkable because in the very same papers where they tell you that the gradients are exponential and then they overlie them all on top of each other you can see they're almost a straight line rather than exponential early gradients tend to be quite straight and then as time goes on they tend to bend more and more and eventually be more and more exponential so you would predict that the phosphorylated mad gradient would show and that all these gradients which are different robustness properties depending upon the time at which you look if you look early when it's more linear late when it's more exponential or somewhere in the middle okay so timing matters not only because the gradient is changing but because all of these factors all these molecules have their own time constants yes not in this particular not in this particular graph but but I can yes if I did that then this would be more like that well so these are getting these ones that are quite late are getting to a point when the discs are actually no actually I think those are rescaled but those discs are stopping growth towards the end so things are getting exponential right about the time that the discs are slowing down as well but I think this is a funny axis you can see it's kind of messy here these are pooled based on which size categories and then they're scaled for the size category and I think these are all then overlaid so they start with more or less the same decline so I can show you other pictures but this is just the one I happen to pick out okay so the point here is that the timing at which you'll achieve different robustness properties depends a lot on the kind of intrinsic relaxation times associated with the different species in here and those are determined by the half-lives of those molecules so for example phosphorylated mad is a very very short half-life if I throw in a drug that blocks the signaling by the receptor phosphorylated mad is gone within 25 minutes on the other hand optomotor blind one of those transcription factors that sets the L5 vein if I induce RNA I in half the disc and watch the protein disappear over time I can calculate that the half-life is about five to six hours okay many times longer and furthermore to get from phosphomad to OMB you have to go through a brinker and again using a drug to block DPP signaling and seeing how the expression of OMB turns off we can estimate that there's at least two to three more hours in the brinker delay so you can put all this together into equations for DPP receptor phosphorylated mad brinker you can put in there OMB the point is you can follow the actual time evolution of the different gradients and what you see is for things that turn over quickly they are robust early that is they reach that stage of being partially exponential partially linear at about this point in time and then they lose that as they become truly exponential whereas things like OMB will start off being more linear here and will reach that stage of being partially linear partially exponential over here and so this will be robust and this will be non robust so it's not actually that the phosphorylated mad didn't control the OMB it's that in a sense you're looking at different eras of time when you look at different signals when you look at phosphorylated mad you're looking at present time and when you look at OMB you're looking at the shape of the gradient 12 hours earlier and if this is correct what we should predict is if you look earlier that non robustness of phospho mad should go away and in fact it does if you look earlier you can see that even phospho mad is robust so having a very long half-life for this transcription factor actually explains another puzzle because remember we used OMB as a measurement of the intrinsic noise in the gradient and we said oh it has this very very high level of noise right and the transition width of abrupt with seven microns not 22 microns but I was just looking at the noise in the transcription of OMB now that you know that OMB has a long half-life right it can by the you know what we were discussing before it can essentially average out a certain amount of that noise at the protein level even if the noise is there at the transcriptional level and indeed it can reduce the noise by as much as the square root of the ratio between the protein RNA half-life half-lives versus the transcriptional averaging time remember we transcriptionally averaged over about 30 minutes the proteins about six hours so you could get a three and a half fold improvement in the noise just by time averaging and so in fact if you go in and you stain for OMB and you look at the protein noise as opposed to the transcript noise it is about three fold lower and that is sufficient to explain why the imprecision is about seven microns so in a sense that all finally agrees once you recognize that the protein is so stable but the protein being so stable then is the thing that separates the time regimes over which it and the signal are robust okay I have almost no time I will very quickly talk about the last control objective which is adaptability because this was something that Frank introduced last week and I just sort of want to quickly pick up on this topic which is scaling so as he mentioned a lot of patterns will scale for different sizes a very natural thing through evolution through developmental perturbations the DPP gradient is a typical example of that one of the best examples of scaling I know of are these old experiments where Steve Cohen used a trick to cause one half of the disc the posterior half to either grow extra big or to be extra small compared to wild type and then he measured what happened to the DPP pattern when you made the domain extra big or extra small and the pattern got extra big extra small so as you stretch the disc the pattern stretches with it okay he did this in the posterior half of the disc and he used the anterior half as a control and it's also been pointed out by several groups that during normal growth not just in this perturbation scenario the disc is growing and the DPP gradients is expanding with it actually you can see how linear it is at early stages but nevertheless you can still force this to fit an exponential and you can get that the approximate X decay length is going up linearly with time okay so as Frank mentioned Barcai produced a model a few years ago called the expander repressor model to explain how this scaling can occur and in this model right you have the standard morphogen uptake model with its standard decay length the only way to scale the thing is to change the decay length right and so the idea is that the morphogen signal represses an expander which then diffuses back throughout the entire gradient and either changes the diffusion or changes the uptake right you got to change either D or K in order to change that decay length it's actually very hard to change D so we now know it's really K and then I also mentioned yesterday that in that model the decay of the expander is considered negligible giving this integral feedback properties you don't really need this to have it be a feedback control but to have set point control where you have perfect expansion then you need to have this as well okay and then Frank mentioned there's a candidate expander a molecule called pentagon or Magoo which is expressed at the very edges of the disc right where DPP the DPP gradient ends is where this thing is on its expression is repressed by DPP if you lose it you shrink the gradient you can see a tremendous shrinkage of the gradient if you over express it you expand the gradient and if you're mutant for it your gradients don't scale so this is scaling shown that's on a logarithmic axis which is why these don't look like curves but you can see that when you normalize your gradients to the size of the disc they all line up they're scaling and a pentagon means there isn't scaling notice by the way that this the absolute numbers are very different here because the pentagon means are so short so it's a wonderful wonderful model beautiful math unfortunately slain by several ugly facts okay the first fact which was already known at the time is that the expander oppressor model which assumes that the expander goes everywhere would predict that expander on one side and the other side mix with each other right because it has to go everywhere if that were the case it would not have been possible for Cohen to observe that when you expand or shrink one half of the gradient you have no effect on the other half of the gradient the two halves anterior and posterior are completely insulated from each other but there is no diffusion barrier between them in fact if you over express pentagon in the posterior it does its job in the anterior so it can diffuse across between the compartments there's no problem there nevertheless you can independently scale the two compartments and that's that's not consistent with the barcode model the second thing is if you just knock out pentagon on one side you only affect the length scale the decay length on that side that half of the gradient you don't affect the decay length on that half again the two sides are completely independent here so here we're taking out pentagon everywhere right here we're taking out pentagon just in the posterior and we only affect the decay length in the posterior the third thing is if you make a GFP fusion to pentagon which is completely functional and rescues wild type and so on you can actually measure how far it travels in a in a disk and it travels even less far than DPP it has a decay length of less than 10 microns so it clearly a is destroyed so forget about integral feedback but be it also doesn't travel very far right has a very short decay length and finally recent work shows that the way pentagon works is by causing the destruction of this thing called dally and dally like which are co-receptors for DPP so you can see when you express pentagon here you get rid of dally and dally like over there but if you make little clones where you over express pentagon you only get rid of dally and dally like in and just barely around those clones and hardly at any distance at all so pentagon is not a long-range acting factor okay so it doesn't have the characteristics of the long-range expander repressor model which then begs the question of how in the world does pentagon enable scaling because it's expressed way over there and you're scaling the gradient way over here right and we it is a diffusible molecule so if we go back to the expander repressor molecule model the reason why it has to be transported uniformly throughout the gradient is because the morphogen in this model is taken up uniformly throughout the gradient that is the decay length is set by this uniform K this uniform uptake but we already know that's not true right we already know that the receptors are concentrated over at the far end and what else is at the far end pentagon right so can we rescue this model to a model in which DPP in the periphery represses pentagon in the periphery which acts on receptor function in the periphery but receptor function in the periphery affects the decay length everywhere and consequently gives you closes the loop on scaling and so if you model that that actually works pretty well if the gradient is not too far away is not too close yet to being exponential as long as the gradient is in that middle zone between being a little bit linear and a little bit exponential then simply affecting the strength of the receptors so pink is measuring where pentagon would be expressed and it's simply acting by inhibiting receptor function and you can see that as the tissue grows out the gradient scales and the dotted line represents the best fit exponential so through most of this an experimentalist would never be able to tell that that wasn't an exponential if you do the same thing again but you don't have pentagon having that effect on the receptors you see the thing just goes out and becomes an exponential early on and just stops okay so indeed you can use pentagon to keep the gradient in that quasi linear state for longer but eventually you lose it and eventually you can't scale anymore and that's actually what's observed the scaling does not go on forever but it ends somewhere about half a day before the end of the last larval instar things stop scaling and then also things stop growing which gives you a nice way to sort of time the size of the disk you had a question yeah essentially it is which is a kind of cruel irony because if you go back to Wolpert's original 1969 paper the whole reason for the source-sync model was the fact that it perfectly scales and that was something that no one had noticed before in morphogen because the idea of morph defusible morphogen has been around before Wolpert but he pointed out if you use a source-sync model not only do you have positional information but it scales perfectly and as soon as gradients were appreciated not to be linear everybody forgot that aspect and now it's in a sense it's kind of come back because this resuscitates the sort of value of that but only temporarily so it only allows you to go so far right once you hit that magic one to one and a half length scales intrinsic length scales the system goes back to being exponential can't scale anymore and interestingly if you were to couple that with growth then the disk would always stop at a fixed scale and size last thing is is there any way to test this idea experimentally and there is one thing we can do which we could take away that non-uniform expression of the receptor so it turns out someone had already made a flystock in which thick veins the receptor is uniformly expressed everywhere and you can then put that over a no mutant for thick veins so there's no more endogenous receptor and those flies are pretty good it makes some little errors in actual formations of the veins but the patterning of the veins is perfectly normal and the flies grow a little bit slow but but other than that the discs are are pretty normal but if you look at their ability to scale under normal growth you can see it's terribly disrupted right all we've done here is flattened the receptor profile and now a lot of the scaling goes away okay so I've gone a little bit over so let me just end by by giving you a few take-home messages the first is it's very easy to make a morphogen gradient it's very easy to fit the profiles of morphogen gradients with very very simple models but making a morphogen gradient that's robust and noise tolerant and forms quickly and adjusts that's the hard stuff right and what makes it hard is not just doing any one of those things but doing all of those things together because of all the interference of one objective with another we currently know some of these loops here but I'm sure there's a lot of information we're still missing because the kinds of experiments or genetic screens as Eric talked about yesterday that would reveal components doing this kind of things those are difficult screens to design how would you find it unless you knew the perturbation that it was it was there to cancel so I think there's a lot we're still missing here but at least this gives us some source material we're thinking about how the feed back and feed forward circuitry might be giving you control it still doesn't tell us that's what the disc is really doing with that machinery that requires a lot more experimental work but you know in the end I think I think it's a lot better than just assuming we know everything I also just wanted to put up some of the names of people in my group who've been involved in producing some of the data I told you about I didn't have time to to properly credit them so okay thanks and I'll be happy to take your question


Hello and welcome to the session. In this session we will discuss a question which says that a coin is tossed three times and all possible outcomes are assumed to be equally likely. E is the event. Both head and tails have occurred and F is the event at most one thing has occurred. Show that events E and F are independent. Now before starting the solution of this question we should know a result. And that is two events A and B are independent of event A intersection B is equal to probability of an event A into probability of an event B. Now this result will work out as a key idea for solving out the given question. Now let us start with the solution of the given question. First of all let us write the sample space of the given experiment. Now here we have tossed a coin three times. Now remember that a coin is tossed M times where number of elements in sample space is 2 raised to power 3. Now here the coin is tossed 3 times. So number of elements in sample space is equal to 2 raised to power 3 which is equal to 8. Now when we toss a coin three times then we get these outcomes that is getting all heads. Now here H denote head and T denote then getting exactly two heads. Now here we have the outcome H H T it means we can get exactly two heads and one tail. Now here we also have head in the first toss in the second toss we can have tail. In the third toss we can have head again then the other outcome is T H H. The next outcome can be T T H then the next outcome T H T then the next outcome which is H P T then next outcome. Now this T T T means getting all tails it means here the total number of elements in sample space 1, 2, 3, 4, 5, 6, 7 and 8. So the sample space S can be written as a set containing the elements. Now we have event E that is both heads and tails have occurred. Now in the sample space the outcomes H H H and T T T have only heads or tails. Remaining elements have both heads so leaving these two outcomes rest belong to event E. So event E is a set containing elements H H T, H T H, T H H, E T H, T H T and H T T. So number of outcomes or we can say number of elements in event E is equal to 1, 2, 3, 4, 5 and 6. Now probability P of an event E is equal to number of elements favorable to event E upon total number of elements in sample space S. Total number of elements in sample space is equal to 8 and number of elements in event E is 6. So this is equal to 6 upon 8 which is equal to 3 upon 4. Thus probability P of an event E is equal to 3 upon 4. Now event F is at most one tail has occurred. It means maximum there should be one tail or less than one tail. It means they have outcomes having no tail outcomes having one tail only. Now H H H has no tail so it belongs to event F then outcomes H H T, T H H have one tail only. So they also belong to event F. So number of elements in event F is equal to 1, 2, 3 and 4. Thus probability P of event F is equal to number of elements favorable to event F that is 4 upon total number of elements in sample space S that is 8 and this is equal to 1 upon 2. Now we want to show that events E and F are independent. Now from the key idea we need that two events A and B are independent. A probability of event A intersection B is equal to probability of event A into probability of event B. Now here we have to prove that E and F are independent events. It means here we have to show that probability P of event E intersection F is equal to probability of event E into probability of event F. Now first of all let us find event E intersection F it will include both elements which are in event E as well as in event F. So from the event E and event F you see D H H H H H T are common to both events thus event E intersection F is a set containing element H H T H T H and T H H. So here we can see that number of elements in event E intersection F is equal to 3. So probability P of event E intersection F is equal to number of elements favorable to event E intersection F that is 3 upon total number of elements in sample space S that is 8. So probability of event E intersection F is 3 upon 8. Now let us find probability of event E into probability of event F which is equal to probability of event E is 3 by 4 and probability of event F is 1 by 2. So this is equal to 3 upon 4 into 1 upon 2 which is equal to 3 upon 4 into 2 that is 8. Thus probability of event E intersection F is equal to probability of event E into probability of event F. Therefore events E and F are independent events. So this is the solution of the given question and that's all for this session. Hope you all have enjoyed this session.


I have these two rings. When I was 18 years old, I went on my own for the first time to Pakistan, and I met some street kids that weren't that distant in age from me, and we became fast friends, and they gave me these rings. Over the course of the months that I was there, they started to tell me about their life, where they had come from, who they were. These rings have stayed in my bag that I've brought to almost every classroom since I've worked here at UVM, which has been a decade now, and that's sort of metaphor for the fact that it always, this thought about people at the very edge of society, the most vulnerable, always stays with me and my teaching and my research, and I want to remember that. It gives me satisfaction, it's captivating, and also very troubling. But I hope that it illuminates wider questions about justice, society in general, cities, identity, and our future.


I'm JJ Zacharias and with today's Fandall hurry up talking a little week nine waiver wire action I think it's obvious where we got to start we got to start in the Tennessee Titans backfield Derek Henry He's gonna be out an extended period of time Jeremy McNichols and Adrian Peterson Likely going to split that backfield now that they got Adrian Peterson on Monday Adrian Peterson's probably gonna see early down work and goal line work But he's not that much of a pass catcher. He's never been that much of a pass catcher throughout his NFL career Jeremy McNichols He has over a 10% target share this season I think he's gonna see some work in the passing game and then see more work on the ground as well So how do you approach this? Well, I think if you're more in a standard league You can go after Adrian Peterson because he's probably gonna see that goal line work He's probably gonna see more work on the ground the Jeremy McNichols But if you're in a PPR format even half PPR, I may lean McNichols But not neither of these guys are gonna take over and just be Derek Henry in this backfield There's also Taysum Hill James Winston. He tours ACL Taysum Hill He's coming back from his concussion hopefully this week and he gets Atlanta So he's a good streamer but on top of that we have four starts from Taysum Hill last year and in those four starts He averaged 52 rushing yards per game and he had four total rushing touchdowns. He was a QB one He was a high-end QB one when he was starter. So if you need quarterback help Taysum Hill can fix it Finally, there's Jamal Agnew for Jacksonville Jacksonville sitting there They're negative game scripts pretty often But the crazy thing is is that when DJ Chark, he's been out for three weeks now without DJ Chark Jamal Agnew not Marvin Jones, not LaVisca Chanel Jamal Agnew has led this Jacksonville team and targets volume alone should make him viable in fantasy football So you should add him off the waiver wire


Hi everyone, Joe for JazzPiesCaseBricks.com. We got a team random for Mosaic. So what happens is, in picker team number one, people were picking their teams in this case right here, and then what we did with the remaining teams, we pulled them out of the picker team and turned it into a random team. So it's a random team inside a picker team break. So this will help fill up picker team number one, a full case break coming up in a separate video. Also, one spot gets you two teams on this list, which will make you eligible for the promo in the full case break. So thanks to this group here for getting in on the action. It looks like the Red Sox with a solo shot, Raphael Deriz with a solo shot. So we'll double you up. Good luck. Let's roll it, and remember, we're just randomizing your names, not the teams. So the name on top, we'll get the angel, second name, we'll get the Yankees, so on and so forth. Let's roll it and randomize that list four and a five, nine times. One, two, three, four, five, six, seven, eight, and ninth and final time. After nine, Sean on top. All right, so Sean, congrats. You got the angel. We'll put a little rooftop next to your name so you know you won that in a filler. Josh with the Yankees, David Baker with the Cardinals, Joshua with the Mariners, Travis with the Blue Jays and White Sox, David with my Dodgers, Brian with the Rays and Phillies, Sean with the Giants, Carl with the Cubs and Royals, and Adam with the Mets and Rockies. All guaranteed teams, and you've got a chance at the free box as well. I'm Joe for jaspyscasebrakes.com. I'll see you in the next video for the break. Bye-bye.


Well, a good question needs a lot of answers here. First of all, let me say that, you know, from my experience in the emerging market, we think that now broadband is going to be the future of connecting the unconnected in those countries. And, you know, being a major player in the emerging market, I know the challenges facing the companies, and how can we bring the prosperity into those countries? Let me give you an example here. In Indonesia, more than 70 percent of the country has no bank account. So when it comes to mobile and broadband, it is the solution for mobile money. When it comes to areas and farmers in remote areas, they don't know what the prices of their products and crops, so what they do if they do not have access to mobile and information about the pricing of their products. So broadband is a solution for those people. Plus in education, people are getting limited access to education. Now with the new broadband and new tablets and digital education, you can, these students now can touch into the, you know, external education, which they don't have in their countries. And that's what we are trying now to do in Myanmar, by getting access or giving access to universities and school free of charge and, you know, a small learning lab where they can get a free internet and free access to information. And that is the broadband and the benefits of broadband in those areas. There are lots of areas we can also talk about in health and in other matters, but what we are trying to do, and with the support of governments and regulators, and, you know, we can do a lot and help and bridge that gap. Of course, also when it comes to gender, women's gender, and we've done so much in Indonesia and Iraq, and I would love to share those experiences with you.