 So I want to talk about how the brain becomes wired up during development, because of course for the brain to function correctly, that depends on a very intricate and precise pattern of wiring during development. And I'm going to tell you three different stories related to how the tips of developing axons find their targets in the brain. So if we zoom in on the tip of a developing axon, we find that it's guided by a large number of molecular cues in the local environment. So at early stages of brain development, activity does not appear to play a role in establishing the initial pattern of wiring, rather there's a number of molecular cues in the environment. And one of the most important type of molecular cue are molecular gradients. So here we have an in vitro experiment done by Andrew Thompson in my lab, where we're releasing here a nerve growth factor from the tip of this pipette here. We're looking down from the top at a culture dish here. And this is a rat neuron, which is being guided towards that pipette, because there's a gradient being set up here. And the structure at the tip of the developing axon, the growth cone, is detecting that gradient and changing the direction of movement of the axon in response to that gradient. So there's three things I want to talk about today. The first is how a small sensing device, such as a growth cone, which is only about 10 microns across, might be able to detect a concentration gradient. So that's a difficult computational problem. The second thing I want to talk about is how once a growth cone decided which gradient points, how it decides whether to be attracted or repelled by that gradient. So that's a different kind of computational problem to do with understanding the signaling transduction pathways with growth cones. In the last part of the talk, I'll discuss some unpublished data where we tackle the problem of trying to understand the shape of the growth cone. So let me just play that movie one more time. As you saw in the movie, the growth cone shape is constantly changing. So the frame rate here is one per minute, and you can see that there's a very complicated dynamic morphology, which is constantly evolving in time. And how that relates to external cues and what kind of computational role it might be playing in axon guidance is still a bit unclear. So we'll discuss that in the last part of the talk. So how could a small sensing device such as a growth cone detect molecular gradient? Well, the way a cell knows about the external environment is by the binding of ligand molecules to receptors on its surface. And here we have a simple model of one-dimensional array of receptors, which are feeding into some complicated signal transduction network. And the output of that network is a decision about whether the gradient points left or right. Now, what makes this a difficult computational problem is that there are many sources of noise in this system. And the one I want to focus on is receptor binding noise. So the binding of ligand molecules to receptors is a fundamentally stochastic process. And it's only the probability of binding, which is determined by the external concentration. So using standard Michaelis-McMenton kinetics, we can say that the probability of a receptor being bound is the concentration divided by the concentration plus the dissociation constant. And so each time we look at this array of receptors, there'll be a different set of receptors bound. So given these continual fluctuations, this sensory uncertainty in the environment, how can we decide what's the best way to decide which way the gradient points? So we hypothesize that the growth cone may be thinking about the Reverend Thomas Bayes and in particular thinking about the optimal way to detect a gradient, which is simply to compare the probability that gradient points right given the pattern of binding with the probability that gradient points left given the pattern of binding. So using Bayes' theorem, we can calculate the probability of the binding given that the gradient points right and the probability of the binding given the gradient points left, which are easy to calculate based on what we know about receptor binding kinetics. And then using Bayes, we can invert that to calculate this ratio here and thus the optimal decision for deciding which way the gradient points. So we're taking an ideal observer approach here. So the result, oh, I should say that this was the PhD thesis of a student, Duncan Mortimer in the lab, helped by Peter Diane in London and Kevin Burridge at UQ and now at Oxford University. So I'm not going to show you all the maths, but essentially Duncan was able to derive this formula, which is a prediction from this ideal observer model, of the proportion of correct decisions this sensing device should make. So the proportion of correct decisions is just a half, which is the random value, modulated by a scaling constant, times the gradient steepness mu, so this is fractional change across the width of the sensing device, times a function of the overall concentration. So this is the dimensionless concentration divided by the dissociation constant. So this is a prediction for how chemotactic performance should depend on the gradient parameters. So this is a strongly hypothesis-driven modeling and this model has, I mean, this prediction has basically only two parameters that you fit to the data, which are the scaling constant and the dissociation constant, or you can look this dissociation constant up in which case you just have one free parameter. So it's a very tightly constrained model. So the predictions of the model look like this, I'm sorry this is going off the edge here, but this is just a measure of chemotactic performance. This is the concentration and the different colors represent the different gradient steepnesses. So you can see that this is complicated nonlinear dependence on concentration. At high concentrations and at low concentrations, you can't detect the gradient because there's either too many receptor bound or not enough receptors bound. Around about KD, you get the optimal chemotactic performance. So this is a prediction both for growth cones and in fact any chemotacting device. So we're interested in growth cones and interested in determining their chemotactic performance experimentally. And it turned out this was a difficult problem because the assays that have been proposed for studying axon guides have mostly been looking at the identity of the molecules involved and not very well set up for asking very quantitative questions about how a response varies as you make subtle changes in gradient parameters. So we developed a technology for creating very stable gradients in collagen gels where you print onto the surface of the collagen the chemotactic factor you're interested in. This diffuses into the collagen and creates a gradient which is actually stable for a couple of days. And this is the kind of time scale which corresponds to the time scale of which axons are being guided by gradients in vivo. And so using that technology we're able to map out this sensitivity surface how the chemotactic response of growth cones depends on both concentration and gradient steepness. And so this is the experimental data here. So the experimental data was generated by several people in the lab and this is collaboration with the lab of Linda Richards. And so you can see at least qualitatively the shape of these curves matches quite well with the predictions of the Bayesian ideal observer model. And I don't have a slide here but you can make more quantitative comparisons to show it's a pretty good fit. And so essentially apart from this scaling constant which when you plot the chemotactic, when you plot the experimental versus the theoretical data that goes away. So essentially all we're doing is fitting the dissociation constant and the dissociation constant turn out to be the same as what's being measured experimentally using more standard techniques. So this formula essentially applies to any chemotactic system and can be used to predict chemotactic performance in any system. And what we've shown here is that consistent with the prediction growth cones are only sensitive to gradients over a relatively narrow range of concentrations back to waters of magnitude. So this has consequences for understanding, for instance, therapies for regeneration. It's no good just to dump in a lot of some chemotropic factor. It needs to be at the right concentration in order to be effective in guiding axons. Okay, so that's the first story I wanted to tell you. The second one. Yes, so these are the gradient sleepinesses experimentally. And so this is over 10 microns. Yes, so these were very, very shallow, very shallow gradients. There's a lot more I could say about that but we'll come back to that at the end. So the second story is about how a growth cone interprets a gradient once it's decided which direction it points in. So here is exactly the same kind of neuron, exactly the same kind of gradient of no growth factor but now the growth cone is being repelled rather than attracted. And Mooming Poo and colleagues showed that you can get this behavior when you reduce cyclic AMP levels in the medium. So reducing cyclic AMP converts attraction to repulsion. That calcium also plays an important role. And so this is schematic summarizing many experiments of people like Mooming and James Jeng. So green here represents a chemotropic factor which is normally attractive. So when you have a normally attractive factor that produces a steep internal gradient of calcium, perhaps I'll use the point there. Steeping internal gradient of calcium and that causes the growth cone to be attracted. When you have a normally repulsive factor shown in red, that produces a shallow internal gradient of calcium. And that, even though the calcium gradient is still pointing in this direction, that causes the growth cone to be repelled. This is also dependent on the absolute level of calcium. So if you lower background calcium levels, then a normally attractive factor produces, okay perhaps I won't do that. Normally attractive factor produces repulsion. If you raise background calcium levels, a normally repulsive factor produces attraction. So you get this very complicated set of behavior in terms of how calcium and particle KMP levels determine the response of growth cones, how the growth cones interpret gradients. So in order to make sense of that, we proposed a very simple model of a growth cone consisting of two compartments. So this is the very simplest way of introducing a spatial dimension across the growth cone. And based on work of people like James Jeng, we imagined that there was a simple signal transduction network operating in each side of the growth cone. So the input is calcium and the output is the ratio of camkinase 2 to calcineurin. And so we hypothesized that it's the side of the growth cone with the largest ratio of camkinase 2 to calcineurin, which is the side to which the growth cone turns. And this is interesting because the output of this network is non-monotonic. And that's how you get lots of interesting behavior to do with differences in input calcium levels. So the assumption of the model is that you have the gradient produces a difference in input calcium level. And then we read out how that affects the output level. So this was work of Libby Forbes in the lab who did the modeling. And Andrew Thompson and Jaja Uanna did the experiments. So for the signal transduction model, we borrowed a model from Michael Grampner and Nick Brunel, proposed in the context of the switch between long-term pretentiation and long-term depression, which actually uses, they hypothesize it uses exactly the same networks that are able to borrow their model but have kind of two copies next to each other. And the results of the model look like this. So this shows the ratio of the outputs from the two sides. So if this ratio is above one, we hypothesize that leads to attraction. If it's below one, that leads to repulsion. And the three curves show different cyclic AMP levels in the model. And so black corresponds to normal cyclic AMP levels. So you can see there's a critical range of calcium for which you get attraction for low calcium or high calcium when you get repulsion. And then this blue curve shows the effect of reducing cyclic AMP levels which shifts this curve to the right on the calcium axis. Now one of the most interesting things about this is the predictions it makes. So these points here correspond to experimental data, which was already available, which we use to help validate the model. But then the model makes a number of novel predictions. And perhaps the most interesting is that at high calcium levels, so if you raise calcium and normal cyclic AMP, you don't get attraction anymore. And the way to recover attraction in that case is to lower cyclic AMP levels to put you onto this blue curve here. So contrary to textbook wisdom, this suggests that there's a complicated interaction between calcium and cyclic AMP. And sometimes to promote attraction you actually need to lower cyclic AMP levels rather than raise them depending on background calcium levels. And again, this has implications for therapies for regeneration. People have been very interested in cyclic AMP levels, for instance, in the regenerating spinal cord. And it's not just as simple as raising those levels. In some circumstances, you may need to lower those levels to induce attraction again. So we recently reviewed the role of calcium in axon guidance. This is in the current issue of trends in neuroscience. And I apologize for the cheesy picture. OK, so in the second half of the talk, the last part of the talk, I want to discuss some unpublished work we've done on growth cone morphology. So this is rather different. The first two parts were sort of hypothesis driven. Here we're taking a more neuroinformatics approach and generating a large data set of movies of growth cones and trying to extract information from those movies in a more sort of bottom up way. So the basic questions we were trying to address are what are the basic shape primitives of growth cones? How do these change over time? And how do these relate to growth cone movement? So for that, we had many people in the lab. Richard and Daniel here did the computational analysis and the others did the experimental work and this is a collaboration with Ethan Scott's lab. And so here is the basic set of data we were working with. I won't go through all these different conditions, but the main point is we had several hundred movies, time-lapse movies of growth cones growing in vitro, and all together about 50,000 individual frames of growth cones. So there's been some beautiful prior work trying to understand growth cone morphology based on looking at just a very, very small number of growth cones and using the intuition of the observer to try and classify what kind of shape patterns those growth cones are going through. Here we used a more sort of big data approach where you just take a whole bunch of data and you try and extract in an unbiased way what are the key sort of statistical principles underlying this data set, in this case, growth cone morphology. So the way we did that, so first we had to automatically extract the outlines, of course. We can't have a student sitting there tracing 50,000 growth cones and so Richard wrote some image processing software to automatically capture the outlines of growth cones from these movies. And then we performed what's called an eigen-shape analysis. So if you haven't heard of eigen-shape analysis, don't worry because it's just principal components analysis in a space of shapes. So essentially we take each of these 50,000 frames, we extract the outline, and then we parameterize it. So you need some way of parameterizing the shape. So we parameterized the shape by putting a lot of dots around the outside of the growth cone, so 250 dots. And each of those has an X and a Y coordinate. So in other words, we have 500 numbers which describe the shape of the growth cone there. So you can think of that as being a point in a 500 dimensional space. This is at this point, in my experience, biologists head start to explode. But here are two dimensions and you just have to imagine the other 498. And so that's a single shape in that space. And when we take our 50,000 frames, you get a whole cloud of points in that space. And then we perform principal components analysis in that space to extract the dimensions along which there's most variance. So this is trying to ask the dimensions of the shape space which have the most variance. So here's the answer to that question. And the way you normally represent eigen-shapes is that you show the mean shape in green. So each of these is one of the principal components. And the mean shape is the same in each case, that's in green. And what we're showing is the shape, one standard deviation in each direction along that shape axis. Because you have to try and represent how shape changes as you move through that axis in the space. So here are, so here's the first principal component captures 37% of variance in the data set. And very neatly that represents bending right versus bending left. So this is saying if you had to characterize growth cone morphology by one number, the one number which preserves the most variance is just telling you whether the growth cone is bending to the right or bending to the left. The second principal component, 20% of the variance, is basically measuring thinness versus fatness of the growth cone. So moving one way, you have a thin growth cone. Moving the other way, you have a fat growth cone. Then moving further down, you have these more subtle variations. So here you get a bend to the left, then a bend into the right. Here you get another version of thinness versus fatness. And here you get bending to the right, then bending to the left, then bending to the right again. And so you get this kind of hierarchy of these kind of shape primitives which capture the most important sources of variance in growth cone shape. OK. So the next interesting question. So this is a property. These shapes are property of the entire data set. The next question is how the projections onto these shapes vary through time for an individual growth cone. So here is a growth cone doing its thing. Here's the outline. And this will start again in a moment. Here are the projections, measured by a z-score, onto the top four modes. And so this is saying that this growth cone is bending one way, then bending the other way, and so on and so on. Similarly, go through variations in thinness versus fatness. And you can hopefully start to see here there's a hint of some periodicity in this behavior. So to analyze this periodicity, we measured auto-correlations and Fourier power spectra. So here is a set of auto-correlations and Fourier power spectra for the top few modes for one growth cone. These movies are a few hours long. So here we have, you can see, oscillations in the first mode, the bending right versus bending left. You can see very strong oscillations in the second mode, which is thinness versus fatness. And so on, you see oscillations going down in all the modes and seeing our sharp peaks in the Fourier power spectra. So that's one growth cone. That's nice oscillations. Here's another growth cone. It also has nice oscillations. But these are a different set of oscillations from the first growth cone. And basically, each growth cone seems to have its own unique set of shape oscillations, so this periodic behavior. Now these don't actually predict which way the growth cone is going to go. So these growth cones are generally going straight, but they're kind of wiggling back and forth as they go straight. They're not doing that. They're just kind of doing that. And I should say the time scale of these is about, there's a variety of time scales, but the basic time scale is about 20 minutes of this oscillation. So we come back to try and understand where that time scale comes from in a moment. So turns out these mode scores and oscillations actually predict the movement. So if you look at how far a growth cone has traveled over an entire movie and ask, are there any properties of the oscillations which predict the amount of movement, the strongest correlations are with the score here, which is a measure of the strength of the oscillation. So the strength of the oscillation in this mode, which is thinness versus fatness, there's a correlation with the frequency of the R1 mode, which is bending left versus bending right. And if you put the top few modes into a regression prediction, trying to use the strengths and periods of those oscillations to predict the distance moved, you'll find that there's actually gives quite a strong prediction. So in other words, growth cones with strong, fast oscillations are moving more quickly. And I challenged the lab to come up with a catchy way of remembering that, and Daniel came up with this. Okay, so I'm almost done. So we found that similar modes and oscillations were present in vivo. So far, all the data I've shown you is in vitro. And of course, one might say all this behavior, this dynamic behavior we observed is just an artifact of the in vivo environment. In vitro environment. So in collaboration with Ethan Scott, we use some of his zebrafish, where there's a very small proportion of retinal ganglion cells, which are labeled with green fluorescent protein. And we did time lapse imaging of the movement of those growth cones across the tectum as those axons try to path find. So this is just a static picture, but you can do time lapse imaging of these growth cones. And then you can do the same analysis, right? And what we found from that analysis was that we get exactly the same kinds of shapes, the same kinds of oscillations. So it's not just a property of the in vitro environment. So just in the last couple of minutes, I want to discuss a possible explanation for this. So I don't know if anybody's had any thoughts about what kind of processes inside the growth cone might lead to periodic behavior on a time scale of 20 minutes. It stumped us for a very long time until we had a revelation that it might be something to do with dynamic microtubule instability. So he's a schematic showing microtubules invading the peripheral region of a growth cone. And you see sometimes they extend quite a long way into the growth cone. And microtubules undergo these characteristic phases of growth and retraction. And so maybe this has something to do with the oscillations. So to investigate that computationally, we discovered that there was this fantastic model from right here in the Netherlands from General Vesius, my apologies for the pronunciation, Van Pelt and Van Oyen, where they considered microtubules growing together in a limited volume where they're competing for a supply of tubulin monomers. So the basic equations of the model are like this. So you have a growth phase. So this is the length of the microtubule and it's related to the concentration of free tubulin. And then you have a shrinkage phase where they shrink and then they switch between these growth and shrinkage phases with a frequency which also depends on the concentration of free tubulin. And these authors showed that this model produced a very nice fit to the behavior of microtubules measured from real data. So we looked at a very simple version of this model where we just have two microtubules competing in a small volume in a growth cone for tubulin. And this shows the length of those two microtubules as a function of time. So blue is one and red is the other. You can see as one starts to grow, the other one doesn't tend to grow at the same time because they're competing for the same supply of tubulin. And so if you plot, if you look at the length, the ratio of the lengths as a function of time and then plot the autocorrelation, you get an autocorrelation function that looks a bit like this is a very preliminary data. But essentially the period as measured from this is about 20 minutes. So it exactly matches the period we've measured in vitro. At the moment we're pursuing the idea that at the root, these oscillations may be driven by the dynamic instability of microtubules. Okay, so that's all I have to show you. So to conclude, I discussed how a Bayesian model predicts the response of axons to molecular gradients, how calcium and cyclic MP interact to determine attraction versus repulsion in axon guidance, and then how eigen shape analysis can help reveal new features of the dynamic morphology of growth cones. So thank you very much for your attention. So the first model, it seemed to me, the Bayesian model sort of assumes a kind of static position of the head and then you show that the oscillations. So how do you, you've got good data for the first one. Right, so the first one sort of simplifies things into you have a series of snapshots of the gradient. Obviously you'll improve your performance if you average over time. But essentially it's just looking at the most reduced kind of idea about gradient sensing possible. But you're getting at a very important point which is maybe these oscillations, especially waving back and forth, are in some way helping to optimize gradient detection. And that's a very interesting idea which we are kind of thinking about how to pursue at the moment. The shape changes in the growth cone and the redirection are strongly mediated by the actin mesh work in the Lamelli podium. So the dynamic behavior of depolymerization and polymerization, can you make a link between your chem kinase two and cyclic MP model with respect to the behavior of the actin mesh work? Right, thank you. So let me just answer the question which I thought you were asking first to start with which was aren't we ignoring actin in this model of microtubule dynamics? And well at the moment we are but we're not saying that actin is not important in driving these things as well. But in terms of putting actin in the picture in the calcium and cyclic MP model. So we haven't really thought about how to do that. So we're basically looking at the signaling at a particular level. So we're assuming the receptor signaling has been converted into calcium concentration and then we're not considering what happens downstream of those chem kinase two to calcium urine levels in terms of rack and rower and CDC 40 and so on. I mean that would be an interesting thing to do but we haven't attempted to do that. Thank you.