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okay well thank thanks very much uh I I was a bit surprised when when when Richard invited me to to come to this conference I said I'd really like to come I'm going to learn a lot but I'm not sure what I can teach you uh so I'm not a I'm not a statistician and I I'm not a computer scientist I don't work in developing these methods but I actually I have been interested in following this literature for for a very long time uh uh in fact when I was just doing my when I was doing my uh postdoctoral research uh myself and my my uh supervisor we started looking at this very strange new and bizarre statistical analysis called structural equations modeling which for for biologists was just very very strange and when I started playing around these I started reallying these sorts of questions okay well if I get a model are there other models at work and just by chance I was wandering through the library one day and I came across a book called discovering causal structure and it was such a provocative title I had to stop and take a look at it so since that I'm interested in trying to follow up in this literature but obviously I'm not I'm not an expert in these sort of things so trying to think of what I might be able to uh show you that might be useful I just to give you a case in which one thing I found useful is in Sometimes using these methods forces me to conce ideas which I thought I understood very well the causal structure it forces me to think in a very different way so that's what I want to show you today um how uh these causal Discovery algorithms force me to reimagine the generating causes of something that uh plant ecologists call the worldwide Leaf economic spectrum so by training I'm an ecologist a physiological ecologist an evolutionary ecologist so I'm I'm going to be coming at it from from that that perspective so again I'm I knew preparing this talk that most of you uh are from other disciplines and so just to take a few slides just to give you a little bit of vocabulary of what uh uh evolutionary ecologist what we talk about so the two things important is the notion of evolutionary Fitness and adaptive value so of course the two important people are Charles Darwin here who gave the intuitive ideas and Sir Ronald Fisher who Quantified this in terms of the mathematics so the idea of evolutionary Fitness is actually very simple so if you take a a set of org organisms who express the same value of a trait or a set of traits uh so you can do a sort of a an age structure for each age what's the probability of surviving to that age and given that it's surviving what are the average number of offspring that an individual will produce and you some over the lifespan and this gives you something called the net reproductive output for this genotype so this trait value in this environment these are the average number of offspring that an organism would be produced and when you do that the the fitness is is simply defined as the net reproductive value compared to the average value over all trait values uh of that organism in that environment so that's when people when we talk about evolutionary Fitness this is this is really just all we're talking about and that means then if you look at different values of this trait in a particular environment we can compare the fitness of each of these we typically some sort of a a curve like this so those trait values which maximize Fitness are those trait values which you say are the best adapted to the environment those tradeit values which have very poor Fitness or producing many fewer Offspring than on average is poorly adapted tra values so the idea of course is that these this Fitness function varies with the environment so as we begin to change the environment uh these Fitness functions change so a uh a trade value which have high Fitness and one environment could be poorly adapted than the other and so we expect then that as we look across environmental conditions natural selection is then tracking this this Ridge here of Maximum Fitness so we expect that those individuals which we find are most dominant should be those which have highest Fitness so that's really all we're try interested in and uh the important question that people who study this is we're not when we look at not one trait but many different traits these traits are often very highly correlated with each other other and we want to know what is generating these patterns of correlation uh to what degree is it natural selection to what degree are these perhaps other causes independent or secondary to Natural Selection okay so when we look at lots of different traits we end up with something which you'll all recognize now so this is where you get into the idea of Discovery algorithms so you have a covariance matrix or some Matrix of dependencies and we want to know then what is generating these patterns of dependencies among these different genotypes as we change the environment we want to know well one way that it could changes that the averages are changing but the relationships between the the trait the variables are remaining constant so that whatever causal structure is the same causal structure occurring across environments and you're simply changing what the average values are or it could be that you're changing the underlying generating causes along the environment and we want to know which of these it might be occurring so really want to understand what are the generating causes here to what degree are these can be associated with common selection pressures to what degree can they be associated with different trade-offs between traits as you're trying to maximize Fitness and to what degree they could be other constraints I've put physical constraints here but we could also talk about what we call fenetic constraints as well so this is the question that's intered me over over many years and this is why I became interested in these methods of causal Discovery um we can't it's extremely difficult if not impossible often to do manipulative experiments you can't randomly assign size of an animal without changing all the other things as well you can do selection experiments but you're changing everything at the same time so you can't really do controll randomized experiments in this area but we still want to understand how these traits are related to each other uh in terms of a causal system so just to point out to you that when we talk about uh these different causes there are different types of causes so of course here we have different breeds of of dogs they as you all know these have evolved between 20 and 30,000 years ago from wolves and more importantly if you look at the different breeds that you see here most of this variation has occurred in the last couple of hundred years as people started selectively choosing different traits and if you notice here some traits are really easy to change so size here um I mean the the difference here in a couple of hundred years is larger than the difference in the evolution of horses from the very first organisms to large horses today and that's occurred within a couple hundred years shapes of shapes of the face are very easy to change as well so those are traits which are very easy to respond to natural selection on the other hand every breed of dog has only four legs we can't select breeds some dogs have eight legs others have seven and others have two and the reason is that this is very strong philogenetic constraints these are things which develop very early during the ontogeny of the embryo uh all vertebrates they start out with these four Limbs and if you want to change the genes generating this very early you basically kill the embryo so this never occurs so this is a very strong philogenetic constraint uh we're interested in knowing Sometimes some of the Cor ation we see may be due to these sorts of constraints not due to contemporary natural selection so that's the sort of introductory part the traits that I'm going to talk about today that I've been working on are traits related to leaves the physiology the morphology of leaves um so I just want to give you a little bit of an overview of how um evolutionary ecologists understand the how the different traits of Le interact and then we're going to see later on to what degree do we is this understanding actually agree with the the the patterns of co-variation so I think a good way to describe this this is for the for the economists in the field we actually think of this as a problem of Economics so you imagine a tree sorry a leaf as an individual Factory the the plant itself as a I know multinational corporation and of course the function of the tree is to generate profits for the function of the leaf is to generate profits for the for the tree so just as a if you think of an economic problem the company has to have a capital investment to construct its company construct its Factory once it's the factory is beginning to produce it produces a certain amount of gross income some of that gross income has to be repaid in order to so if there are if there are cars coming from this side you have to pay money in order to get the resources that steal everything coming in on the other side you also have to pay a certain amount of money just to keep the the pay the salaries for the for the workers to repair the machines and so on and whatever is let is left is the net profits and well we're assuming that the COO doesn't use that to buy Yachts or something so they're investing that to maximize their income so they're vaccin to produce new new factories and eventually the company has to decide that that this Factory has become so inefficient it's wor it's better to knock it down forget it transfer whatever resources are left then to keep it running okay so that's if you think about that we think of R between a leaf and a tree in the same way so the the plant has to invest capital in the form of carbon and nutrients to construct the leaf so there's a capital investment that has to be recovered once the leaf is producing the what is producing are sugars fixing carbon and converting it to sugars some of that those sugars have to be used then to allow input of resources carbon and mineral nutrients into the Leaf part of the car part of the sugars have to be burned off to produce energy to maintain the leaf to repair the the the various organel and the physiology and so on of the leaf and whatever is left over is exported from the leaf to the other growing Port parts of the plant the Marist stems to produce new leaves and eventually to to reproduce and As Leaves age they become less and less efficient at doing this and eventually then it's in the interest of the plant to allow that leaf to die to transfer whatever resource it can to new leaves and so the the economic problem then is how do we coordinate these different properties of the leaf how long do you allow the leaf to live before you killed off in order to maximize the fitness of the plant so that's how we see this this uh evolutionary problem so uh a few years ago 2004 this uh page this paper came out in nature a wonderful example of marketing and I chove a wonderful title the worldwide economic spectrum which just a wonderful title but basically what they did is uh they collected from the literature large numbers so over 200 20 2,500 species these are traits measured all over the world Arctic tropical rainforest deserts and everything and they measured a series of traits physiological and morphological traits of these leaves which relate to these different aspects of the economy of the leaf and they wanted to know to what degree are these traits correlated and especially to what degree do these patterns of correlation change as you change different habitats so if you go from a Arctic tunder to a tropical forest do you find the same patterns of correlation or do you find different patterns of correlation uh I think the the expectation of most people was that you would find different patterns of correlation as you change habitats because the evolutionary challenges the selection pressures are very different and so you should expect these patterns of co-variation should be different between different uh habitats in fact they found that wasn't the case at all so that's why they called this Universal Spectrum the same patterns of correlation occur whether you're looking in an Arctic Thunder or tropical rainforest or a desert the the correlations between the traits are the same the average values change but the correlations seem to be remarkably uh robust so they overall the modulation of Le traits and trait Relationships by climate is surprisingly modest and this was a surprise to people and it was a surprise uh was I think it was a surprise to myself as as well so what were the variables that they measured well first was a morphological variable called uh specific Leaf match Mass which is essentially the dry mass of a leaf divided by its surface area so the idea is the surface area is the amount of energy being captured by incoming photons the dry mass is in interpreted in some ways the amount of Machinery invested to convert this energy into sugars the second variable that was measured was the maximum photosynthetic rate so you put a in the field you put a leaf into a cuet you measure the change in the carbon dioxide concentration into and out of the cuet the difference is the amount of carbon that's being removed from the air and if you do that under optimal conditions highest light intensity that's the maximum net photosynthetic rate uh the third variable they measured was Leaf nitrogen content nitrogen is the the mineral element which is almost is most commonly limiting to plant growth and it's completely critical to to the leaf functioning because it is an essential element for all enzymes and almost all of the vast majority of the nitrogen Leaf is associated with photosynthetic enzymes so it's sort of approxim proxy the amount of photosynthetic enzymes that the leaf is able to to utilize uh the respiration rate of the leaf so of course the leaf is fixing carbon but at the same time it's burning carbon these are living tissues and so this is the leaf respiration rate so how rapidly the metabolism of the leaf is turning over and the final variable that they measured was Leaf lifespan the average number of days until the plant allows the leaf to die okay so just remember when I'm talking about Leaf span I'm not talking about the leaf is dying because it's being eaten most leaves die because the plant chooses to kill them off okay so all the the leaves are dying around here right now but it's because uh hormonal levels in the in the in the tree are causing the leaf to die question yes are you including deciduous absolutely and there's an incredible amount of variation so some species a two a life a lifespan is 2 weeks long uh there are some species where the the leaves are kept for 10 years or more in fact there's one species in South America that same Leaf is kept for the entire life of the plant which is over 100 years so these are incredible degrees of variation okay so just to resume what did they find so the very large number of species these were data called from the literature uh they chose them from as many different habitats as they could find the prior expectation was that the different the different environments would select for different patterns of co-variation because the selection pressures would be different and they found the contrary that in fact essentially they find the same patterns of co-variation wherever you go whether you split the data into different families of of trees or whether you look at angiosperms versus gimos sperms or whatever the same patterns basically keep coming back so this was kind of a surprise and the question was then why was that so so first of all just to give you some examples um these are all log log plots these you generally have to log transform these data to have vineer relationships and normality that's rather typical so the these are the sorts of patterns we're talking about I don't know if you can see that very well uh I can't even see it very well anyway this is uh photosynthetic rate nitrogen content Leaf mass per area Leaf lifespan Leaf mass per area respiration rate uh this is a phosphorus content nitrogen content Leaf Mass priority and so on so you see these patterns and these are all the data thrown in from as I said from tropical rainforest to AR Tundras and they all seem to follow the same general pattern so they didn't they didn't do any causal modeling they didn't apply these they simply did a principal component analysis so this sort of summarizes what they found so the first axis explains about 80% and basically you all these variables which are basically metabolically related variables nitrogen photosynthesis respiration phosphorus are all highly positively correlated with each other Leaf lifespan and this morphological variable of the leaf positively correlated and they go in opposite directions so they interpreted this is why they call it the economic spectrum basically a suite of traits which are designed to conserve resources which have already been obtained so slow metabolism long lived lives that don't lose their resources very rapidly on the other hand to Resource acquisition designed to very rapidly acquire resources acquire fixed photosynthesis uh but with very short lifespans and very that's very thin rapidly dying leaves so I sort of compare this that this is the this is the person who when they gain get a little bit of money they put it in a sock under their under their their bed and this is the guy who's playing the stock market all the time and so this is sort of the the range that you have between very conservative and very liberal I guess you maybe call type metabolisms anyway this is how they interpret it so why what was the the causal exp explanation that was proposed for this uh the first part the first part I'll explain it to you is basically one of based on ideas of natural selection so the the assumption is that the plant is natural selection is acting to maximize the cumulative net amount of carbon fixed by the leaf per unit time uh calculated over the lifespan of the leaf so basically we're saying up to a given time What's the total amount of accumulated photos carbon fixed you take off the the investment cost you divide by how long it takes for that to get that and we're assuming that uh natural selection is trying to maximize this value and this is these are this was a model produced by a person called kikuzawa many years ago and uh I just noticed that we've part of it was missing anyway um so part of part of that slide just was missing anyways the result is that when you calculate this you find out that the uh light leaf leaf life span of the leaf should be inversely proportional to its maximum photosynthetic rate and directly proportional to this construction cost and how slowly the photosynthesis decreases over time um so if you go back to the the discussion of the paper I just told you they didn't test these ideas but basically if you look at their paper and you translate that into a causal a directed ayc graph well it's not completely uh directed here this is the explanation they they they gave photosynthesis this is the part kika's model High photosynthesis would degrease Leaf lifespan uh the the cost of constructing the leaf so more if it cost were to construct the leaf the plant has to maintain it for a longer time to get its to get to maximize it its carbon output this is the morphological Leaf morphological trait which is basically how dense the tissues are um how thick the leaves are and leaf nitrogen and I modified it just slightly so these are the this line here comes because more nitrogen means more photosynthetic enzymes which increases the maximum photosynthetic rate this line here was negative because um this means T A leaf which is has denser tissues more carbon for per Leaf area which would therefore decrease this because you would have less nitrogen per dry weight this would increase the the amount that plant has to pay to construct the leaf which would then affect its Leaf lifespan so I modified it just slightly as you see it here and I I fully expected that this would be the explanation and this is sort of way people understood it and so just as a an exercise I decided to just test that as a structural equation model and it fails miserably and that was a surprise to me it was I think a surprise to other people it wasn't completely a surprise to me because although I agreed with most of this I had proposed a slightly different model based on the PC algorithm actually based on a very very small data set many years before and I thought that was going to work so I was decided smiling when I put my model in and it works even worse um so this was sort of an existential problem for me that now what do I do um so the folks said okay well we'll we'll just give these data to the PC algorithm and maybe they can suggest a structure which might make sense so basically I added to the PC algorithm I started out with a very low significance level basically it was a completely saturated undirected graph that it keep me back I started increasing the sance level you can go all the way up to 0 five and it's always a completely saturated undirected graph that it gives you back so now that's that was a real problem for me what I had no idea what was going on so an obvious explanation I wasn't quite sure this must mean there's some kind there's some latent variable hidden in there which is generating these covariant structures that we haven't measured and what might it be and I didn't really know so I decided okay well we'll we'll use tetrad Vanishing tetr to see if that can suggest so with only four variables or are three possible tetr equations we already it was already a saturated uh saturated uh skeleton so if these tetr equations vanish that means that it has to be a latent variable which is generating this process process so I tested that uh I don't I I I missed the Friday morning so did you talk about this and and okay you know this goes back a long this goes back to the 1973 even earlier I suppose uh method of detecting latent variables so that is basically a simple algorithm that given a set of four variables which no pair of variables are independent conditional on any other set including the empty set uh you test whether this tetrad equation is zero if it isn't you go to the next one if a tetrad equation is zero and if this part of it is true then that means that there is a there's a latent variable which is called a choke choke point which is generating all the paths between those two measured variables passed through this choke point so this gives you an idea of maybe where the latent variables are hiding and when we did and I did this one latent variable doesn't uh vanish and the other two do so that means if I just get over here uh all caal paths leaking every pair of variables except for this one that doesn't vanish so that's the leaf lifespan and photosynthesis pass through the same latent variable or at least some set of latent variables which are all going through the same choke point it didn't help me too much but uh at least it gives me gave me an ideas of what this might be so when you look at each of these pairs of variables and you say uh which pairs of variables do they are they being generated by some latent variable this is what this is what comes out of the analysis these are the only pair of variables which apparently you don't you can't um forced to be independent based on this latent variable so obviously what is this latent variable and that's where it started forcing me to try and rethink what I thought I knew so I'm going to explain to you how the how I think it's working now and I mean when we think about these cause causal processes at least myself I tend to forget that we're we when we talk about different scales we start talking about different processes the structure which was presented was basically based on what we know about the physiology of photosynthesis at the molecular level but we were actually looking at the level of a leaf not at the level of uh you know riscal molecules floating in the cytoplasm so I thought maybe the problem is it's a question more of scale so if we look at the structure of a leaf you've I'm sure you've all all seen these sorts of graphs these sort of pictures so this is this is a cell these little things here are called chloroplasts and this is where all the action is occurring in terms of of photosynthesis um basically a chloroplast is a chloroplast there's a chloroplast if you go from one species to the other there's really not much difference in their chloroplast what does differ is how many chloroplasts are being packed into these cells and so maybe the what was happening was that it's what the physiology is the same Within These chloroplasts what's really differing is how many how you're packing these things together into bigger bigger components so I I started to start to think about that so here's a photograph so this is one cell and all these little dots are these chloroplast are really packed in together there and the size of a chloroplast doesn't vary very much from one cell to another it's really how big the cell is and how many chloroplasts are in there so this is a basically if you think of a cell every every cell of every leaf no matter where you look at it every vascular Leaf you have a cell wall where most of the mass is occurring this is where all the the lignan and uh cellulose and so on in the middle of is basically a big balloon fold of water cyle plasm in which you have enzymes floating and these various organel mitochondria chloroplast and so on so basically if you think that a chloroplast are basically functioning at the same process then what's really important is how many of these chloroplasts in there and that should be basically a function of the volume of the cytoplasm bigger cells you have more chloroplasts and so the photosynthetic rate per chloroplast is as I said very much less very so mostly what's generating this is the numbers of chloroplasts in the leaf not really the molecular physiology occurring uh within one the same thing nitrogen will will should scale with the cytoplasm because nitrogen is primarily photosynthetic enzymes which are primarily Within These chloroplasts there are other components of the cell as well but all of these things scale basically with the volume of cytoplasm now remember these variables that were measured were all Express PR gram dry weight of the leaf that was the standard way well most of the dry weight of the leaf occurs in the in the cell walls on the outside so it's really playing off what is the volume of cytoplasm as opposed to what is the volume occupied by the cell walls I think which is really generating this whole process so photosynthesis per dry mass then should basically scale with the volume of Cy plasm within the leaf relative to the volume of dry of cell walls so the things that change mostly when you're looking at different leaves are the different tissue types so some types of cells have very thick cell walls are primarily there for structural components things like fibers and factors no cytoplasm at all inside the leaf inside the cell it's dead uh leaves such as this have very thin cell walls this primarily with the photosynthesis occurring and the choice that the plant has is do I produce more of one type of cell or more of another type of cell and once the leaf has made that decision it's basically decided how much of its volume it's allocated to one type or another and once it does that I think basically all the other decisions follow from that so this ratio of the volume of cytoplasm to cell walls should determine photosynthesis per Mass it should scale positively with the amount of nitrogen per Mass uh the carbon cost which is how much how much did it cost in terms of resour is basically should scale with dry Mass so this ratio should be inversely proportional to the the construction cost of the leaf and the more complicated one was this other various specific Leaf Mass but it is essentially a product of the tissue density divided by the thickness of the leaf and this tissue density is primarily determined by this and the density is the dry Mass so a leaf which has basically constructive lots of water filled balloons would have a very high ratio it would have a very low low tissue density and a leaf which is constructed lots of cells with very thick cell walls have a very high tissue density uh and a high slm so this then I think this was the model I I figured uh would be the most reasonable so this structure here if you look at it agrees with the tetr equations this Arrow here comes from that theoretical expectation I told you before maximizing a natural selection kika's model and these all these arrows here I think are just these are this is not a question of natural selection choosing these values this is the only variable which I think is sensitive to Natural Selection once the leaf has chosen basically how which types of tissues is going to construct this Leaf out of uh and and also the size of because obviously if you if a large cell would have a larger volume to surface area so that would be affecting this as well once you decide this all these other parts here are basically fixed so this is the part this part here can respond to Natural Selection and certainly you can do selection experiments and change these things without changing these other variables the rest of it I think is basically basically rather like uh you know you only have four legs on a dog and all all vertebrates have that because this is a basic Choice which you make all leaves are constructed in this way and once you choose this ratio you basically have to live with whatever comes out now if you fix if you to test this with a structural equation model it fits very well it's not surprising because the tetr equations told us it would in the first place um and by the way we've tested this two different independent data sets as well and this model continues to work is this now is this true is this really what this latent variable is I don't know that's the next step I guess we can measure these things in principle and if I can find enough of a research Grant to do it I I will but this is a lot of work so so so far I can tell you that the is this a model I I don't know if it's the right model it's agrees with the patterns of the Tet trade equations and so on makes sense to me but beyond that can't tell you any more than that but hopefully in sometime in the future we'll be able to actually say whether this really was allowing me to detect what this lat and variable might have been so I think yeah that's it so I can take some questions can you think of any other descendants of that lat variable if could think of one or two more is it two more you can find a a tetrad constrainted on the variance Matrix and that'll tell you if it's just one L in all those things I'm trying I was I'm trying to think of of sort of a an indirect proxy that I can measure quickly in principle if you just measure the total volume of water in the leaf relative to total volume of dry mass that should be more or less proportional to this ratio and that will allow you an another a descendant from this late which can then maybe help you to better identify that uh in the in the data set we set there the first one I did that and and the model contined to be working so I said I tested with two different independent data set the second one if we haven't published yet was is not based on different species they're all based on different genotypes of a single species and when we put the this ratio basically water to dry weight uh we take it out the model works but when I put this arrow in it doesn't work is that because it's not a it's not a a very good measure of the underlying latent or is it because I haven't interpreted the lat correctly I I don't know yet I I for myself what I would prefer to is to have a more direct so you can take leads you can slice them you can you know do image analysis and calculate these areas more directly which would be the proper way I think to do it but yeah I thought about that but that's as far as I've gotten so far how how uh I'm trying to understand the philogyny of this um isn't I'm not a biologist so deciduous versus Evergreen for example is does that figure here at all yeah sure both both deciduous and Evergreen were in here oh absolutely well in fact much wider than that so deciduous Evergreen angiosperms gimos sperms ferns uh quite but basically all vascular most of vascular I mean are some of those distinctions deeper down in the you know the four legs versus eight legs yeah than others of some of those variables yeah oh absolutely as I say things like photosynthetic rate and leaf uh lightspan those are things which you can select fairly rapidly um things like the ratio volume of cytoplasma to uh drrive weight I don't think so because these are all basically these are all constructed of same types of cells it's just different proportions of these things so any any vascular any Leaf of any vascular plant is constructed in the same way so this is this is a a philogenetic s signal which is coming from when did the when did the vascular pan evolve 4 400 million years ago or something like that I I have two questions one is you you use normal distribution tests um how normal to the DAT when you log transfer there pretty they're linear and they're pretty normal my second question is um how uh how stable the uh the linear coefficients were across your two two independent data sets i' have to go back they weren't identical the signs were that would that would be amazing the signs were the same I think in terms of orders of magnitude I think they were reasonable but I can't tell you off hand I suspect if you if you put it in in a multi-group model and tested you would probably find differences between them would suspect well I guess the question is um you probably didn't look at this but how how much they did differ compared to taking uh different uh uh subpopulations of this universe data set so you think splitting up the data set into different logical groupings you said the correlations are are pretty much stable across species so um I can't tell you directly but I know that for instance in the original paper if you add things like uh average temperature you can find little you can find significant differences in these in these slopes between these different groups there's small there there's a small signal but it's statistically significant is there so there's obviously more going on than I've I've talked out here have you published this what I what I said here what been published yet uh so this this is published yeah yeah you got a slide with the reference give it to you I yeah just I mean most most of my colleagues think this is all Voodoo but so I'm happy when I can come here and talk to people who feelings for these have some secret I take it somebody published it nonetheless Voodoo or um the do do biologists generally regard thises just bizar yes I was speaking I can't can't remember Jeff Jeff so Jeff is another person another biologist what what's your opinion my opinion is yes well on the one hand uh so I do the same thing but I work with with modeling animal Locomotion most of my colleagues we just stick with mechanistic models but on the other hand there's a large number of us that just are kind of at the opposite in the voodoo that are just used very naive Association and think of everything all you know aggression it's all causal so even though it's hammered in coration not causation corelation not causation C it's really hard for a human to look at these correlations and not think of them as anything other than causal so it's sort of that weird combination yeah but don't nobody in my field is this other questions okay