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Master class with David Nelson

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so why don't we get started on it's a great pleasure and indeed an honor for me to go to visitors from those of on and on I actually 1st came to the Netherlands and has a graduate student myself there was a series of summer schools on fundamental problems In statistical mechanics and so on I went to a heartening and was the rainiest summers in 40 years but otherwise it was very enjoyable and and a lot of fun but it be grateful before I get started just to have a sense of my audience I understand that you almost all of you are working for the Ph.D. and can have a show of hands people that think they might go into some sort of an aspect of series 1 experiments and people that might have held both of having been trained in physics perhaps have an interest in biology In presumed some of you OK good and I hope to get to know more of you maybe during the break and then as we go along with what I often say in In when when I give talks to enter any group is that there is it's a collaborative effort and so I welcome and encourage them interruptions in and questions as we go along and even tho this is a distinguished group of of students and and so forth life is feeling that if you have a question there are at least 5 other people around this table had the same question and if you ask it they will be secretly grateful and I will certainly be grateful to to have your your interruptions and won the reasons I enjoy coming to the Netherlands is it's a country where the people don't hesitate to ask questions so relying on you to to stop me from behind going using .period PowerPoint I think of as a kind of form of asymmetrical warfare I can just throw up all the slides and drowned you in a sea of information and your job is to slow me down but I will go to the whiteboard induced few simple calculations which will also I hope slow me down but I'm going to need your help OK so we live in an ad in a remarkable time in human history because we for the 1st time can rapidly and inexpensively come sequence of DNA not merely of humans but of all sorts of organisms and there was a milestone that allegedly was passed recently called the thousand-dollar Gino and this is just the human gene on which can be at least approximately sequence but maybe not a thousand dollars but relatively cheaply compared to the billions that were I spent a sequence that the 1st genomes of James Watson and and other people backed only 13 or 14 years ago so this map comes out of that Div technological development I won't go into the details but is the result of sequencing the minor Condrieu albeit of a present-day indigenous populations populations that have been around a long time various parts of China also in the New World's not shown here in India and so forth and by looking at those minor kind real owns you can actually infer a migration history of human beings a kind of a family tree but laid out on geography let laid out on a map of the world and so 1 of the themes of this set of what I'll tell you about is In a hopefully our context helpful for physicists what happens when we have spatial population genetics spatial migrations and and what's going on if you look here this is a day of something will talk about more detail it's a population wave coming out of Africa How many of you heard about the out of Africa hypothesis for human origins but so it probably happened several times it might be that the 1st Aboriginal Australians came out in an earlier wave and there were subsequent waves we now can sequence meanders all DNA and see if humans had some sort of interactions with Indiana souls and that the demand 1 of them prominent migrations probably happened around 130 thousand years ago these are years before present but the 2 things that are a little worrisome thing about this this beautiful picture of 1st of all in the still life presented here in cartoon form there no error bars associated with the arrival time Nike issue there are bars and they're quite the 2nd thing regarded as an experiment you we think of the context of the physics or chemistry or or biology whatever is experiment that's not easy to reaping you probably wouldn't want to repeat and for this reason that we got interested this is my my my colleague Oscar lexicon I ate in trying to examined the kind of questions that lead to inferences in a petri dish in a dish that maybe 8 to 10 centimeters across and the idea is that whereas in 500 generations of say humans or you could also apply to the a cane toads invading invasive species coming into Australia but for humans at least and 500 generations of try to convince you that there is a wave-like behavior but that also will invite the generations would enable this this wave of humans to expand maybe 10 thousand kilometers of any repeat that experimental explain where this comes from on a much smaller scale this is 100 microns here in a bacterium called Pseudomonas aeruginosa in the same 500 generations that produced this picture but that produced this this idea of human migrations you go a centimeter and so the arrange expansion stay in the country and you can do it 40 times and and and try to understand that these patterns and try to understand where they come from Sofia don't mind a change in length scale of 10 to the 9 we can perhaps begin to understand some aspects of spatial reveled in spatial evolution in a simple reproducible context so without a
deduction let me go back this is the the abstract of what I was was hoping to cover and is how I won't have time to cover everything in detail and no I I'm not planning to so there so there's plenty of time for questions but but if you want to know more about some of the after some of the ideas others a popular article that appeared in Physics Today a couple years ago and then there's a more detailed review article Reviews of Modern Physics where we discuss these ideas and genetically mixing and evolution in something called the one-dimensional stepping which is a famous smile and population genetics so any questions so far about where we're going to have a made myself intelligible OK so In this context if I look at that have a look at bacteria growing across a petri dish and if I have time in the 2nd half and make sure you will be lower than what what's going on so what we have here is there is a picture of a model later of bacteria pushing their way upward on and I want you to imagine that the left half is a species which is very similar to the to the right half but it has a marker on it will talk more about this later it's a fluorescent marker which it under the right excitation shows up bread on the left side is a market that shows that green so the idea is that there's only maybe the 1st 20 layers of bacteria these these Rod shaped organisms are cola and anybody hazard a guess on the number of recall lies in a typical adult intestines wild guesses are allowed 10 billion OK thank you there are more call in your intestines and mine as well then there are people on the surface of the earth and you get that by just to calculating the Volume 1 lowered the length of this ride is 2 0 2 2 microns micron across think of it as a cylinder divided and the volume of typical prisons large intestine where they live and and you'll get 10 billion 12 billion so there are a lot of them are not so many unexpected dish and the 1st 20 layers or so are the ones who actively divide so there's a kind of range expansion that's happening to sell divisions of relatively thin layer at the frontier of the population leaving behind perhaps a mixture of red and green and that will be 1 of the questions that were tried it addresses what what is that mixture was italics about the range expansion self and so there's also lots of interesting scenario some of which will will try to understand today for example the red and green strains could be neutral they they could be genetically identical almost every respect except for the color is typically grown in a dark refrigerator and so that they can't tell the color of but what I want ultimately convince you of is that there is a boundary kind of legally boundary between writing Green has illustrated here which behaves like a random walk OK how many of you studied random walks in your various courses on statistical mechanics a polymer physics or whatever so why is it around what let's imagine I have 1 generation of growth so I go out another 10 bacterial thicknesses and I if the green accidentally pretty reproduces the red during that generation this walk will bend a little bit to the left but red awarded to do the same would bend a little bit to the right and if they're selectively neutral the discrete number fluctuations at this interface but at this frontier will cause stink to wiggle back-and-forth and the statistics of this wiggling boundary this genetic boundary between red and green behaves like a very much like a random walk where this direction is like time because as outright show you these populations is a typically advance at a constant velocity so distances little bit like time if you multiply by velocity and the transfers wiggling is like a one-dimensional around walk in the perpendicular directions might call that yes please the great question but typically what happens is they're sucking up nutrients they're sitting on park had got a petri dish so this is like a Jell-O image Jell-O are lots of nutrients and by that by the time this population goes by the bacteria back here have sucked up virtually all the nutrients that's available they don't die but they they they they reprogram themselves to stop self thank you the other question the please that you 2 what you yes and no good question will we humans don't go into status as soon as the population goes by that's a good point but on the other hand what happens for humans and you can do this with bacteria that swim in soft AG are is that you swim around aimlessly in a two-dimensional Random Walk and they this boundary will slowly diffuse away but because the frontier is advancing linearly and time you develop long-lived sectors sectors which can persist to the pleasant present-day which only has blurred out but by a much slower process which is this two-dimensional diffusion that you bring up you can do experiments that mimic that but we think that even for humans which are diffusing the sector structure that I'll show you and I showed you earlier for example here is remarkably long-winded may blur out back in the wake and eventually all the signal will be lost and it's a very slow process the questionnaire the wealth yes please we you OK yes I'll tell you more detail but that like I can tell you briefly now they would defy go back here this central region here we we like to call the homeland and so what we did was what this group of experiments did this is Kevin Foster and colleagues was to grow up the things the red and green strains overnight in a liquid culture shaking them vigorously mixing them up as the 4th and then in the morning and nite than this to you you you you take a small of portion of the mixture say a 50 50 mixture could be some other proportion in a carrier fluid and you pipette down maybe 5 Michael leaders the center of a petri dish to carrier fluid dries out in about 3 minutes and then you have an initial condition that that you asked about of fairly well spaced bacteria there's an excess of them on the rim of the untimely can talk about that and then the expansion starts to take off dividing time of these bacteria is about once every 36 minutes so this may be took 4 days to go up like this under question OK so we have this random walk boundary perhaps but if further the Green is out reproducing the red we wouldn't say there's a selective advantage the green breeds faster produces daughters faster bacteria simple because they have a happily asexual reproduction mothers divide in the daughter's divide so it's kind of like the the eternal process of simple cell division but and if that's the case the vendors Random Walk which systematically bend to the left of the green is not reproduced in the red so would be a biased Randall Monica request go the other way but they can also be kind of nasty these bacterial strains 1 or both could create toxins that impede the other than some bacteria do do that to foster their their own growth and finally in hope will get to this but you can have mutually the lists are bacteria yeast or some other microorganisms that are there the forced them to play together because 1 or both could secrete amino acids useful to the other and so then that he don't like to be segregated I'll give you an example later for yeast where the doomsayers the patch of organs of on the right that can make loosing but not tripped a fan with a reverse on the left they can make at the fan but not losing so the the 2 strains that they need to get together and be well mixed and yet if there is a tendency and I hope to convince you that there is such a tendency for this genetic the mixing that serve fights against cooperation but not inevitably the torpedoing cooperation
but it's another interesting issue and and you can create strains in the lab that that do all these other question OK so and this is what I'm going to tell you about when the focus on frontiers wife frontiers range expansions are very common in biology not just humans and at the frontier where the population is very spots and thin those number fluctuations are gonna play extremely important role and you'll see that it affects population genetics and you can buy a survival of the fittest which we all learned about from Doralin into survival of lucky and both both things are going on both the important and it's very much like trying to understand random walk with a little bit of drift a little bit of bias and so will the be biased random walks will be going on a great deal in this in this lecture and with as I said try to see if we can test theories of frontier evolution in cooperation with colored bracts bacterial strains or yeast strains with variable mutualism and hopefully I'm pretty sure we'll do this I'll get to out a famous model sort of like an Ising model on Anderson model Hubbard model condensed matter physics which played a prominent role in opposing an and challenging questions with a stepping model of cooperation and competition that I will tell you about as part of his lecture which underpins much of spatial population genetics and we can actually do that kind of spatial version of game theory associated with these organisms as you might guess when I talked about poisoning and dualism and you can even get some phase transitions speaking of the Ising model but in in these range expansions that might even be important biologically which is not work that was done by myself it started with Oscar check and I've had enormous good fortune and being able to work with a whole bunch of in very talented people whose whose work will come into play as soon as we come along so here's and again 1 of Iran's expansions to focus you this is a different colors it's blue and yellow I think this is a movie of arrange expansion of Pseudomonas aeruginosa and it almost looks like oil and water face separated but it's not In fact these 2 strains of their their their their colored versions of this picture here over the Pseudomonas aeruginosa organisms are as identical as we could make them they differ only In the color that is expressed on and that only under fluorescent excitation and the role in the dark and yet even tho they're quite similar they're not chemically different the way oil and water chemical can be different are there that they face seperate but it's not like chemical phase separation it's like if you leave salad dressing and refrigerator for a year I occasionally done you find a lot a lot lighter fluid at the top and a heavier fluid at the bottom a separated what's happening here is the descendants of this inoculation around the homeland is what's face separate Pseudomonas aeruginosa is an interesting organism it's a it's a pathogen it's you don't have to worry too much about it working in a lab if you have a good immune system but it goes after people with cystic fibrosis and so there's a lot of interest in trying to impede its range expansion so here's a simulation which could be a range expansion of 2 different colors along a catheter to In a hospital that unfortunately wasn't taking adequate precautions to keep these bacteria outside the hospital room and then and so it it's all worth understanding how they migrate what about what's going on with such genetically mixing and 1 of the things will see there's 2 kinds of waves that play an important role was a is a population waves of I just look at this in white light you couldn't tell that there were different genetic variants segregating out genetically mixing there would be an outward velocity of their the this wave I'll show you where that comes from and about 1 slide on so that's a what's called a fissure population where this is like the way the out of Africa but here along the frontier of the circular wave we can see genetic variation and there's a boundary between yellow and blue and so I right angles there is the potential to to this population where there's a potential for a fissure genetic wave wear yellow systematically squeezes out blue and at this time in the 2nd half I'll show you why that might be relevant but 2 simple models of the tumors that give rise to cancer I am hopeful that the 2nd port in any event there that there's a there's a genetic wave in principle at right angles going along the rim here will come will come to that I hope and I hear it because the neutral that genetic the sort of stalled out this is just some sort of random walk it's not really going anywhere except for random fluctuations those fluctuations noticed can cause this the yellow genetic domain to pinch off as he gets in golf by the Blue but otherwise is no systematic motion but in the yellow could out reproduce the blow you might imagine that there would be a perpendicular and genetic wave going on right angles to this population ways that was after tell you what waves are you know it's not like sea waves it's it's a wave of kind of a tight but it was a very unusual kind that is unfamiliar to most visitors so these
waves of rise in the following way this is the 1 slide summary or to slide summary evolve population dynamics no genetics yet just all population dynamics and so the ingredients in the initial ingredients this is a hazardous illustrious history if I have to see is the number of people were organisms in attest to the little Beazer bacteria and if you put them in a rich nutrient broth may grow up overnight will have many more of them the next morning and I just focus on births and deaths but there is a famous calculation the goes back to my office and 1789 mn officers book the population sometimes called The Population Bomb is erroneously it's not really a bomb but what he said was that the crime rate of change of the number of individuals could be this test to could be humans on the planet or whatever is going to increase proportional to the number there already there and they're having lost proportional harmony that already there so this if births exceeded deaths this constant is positive and this would lead to exponential growth and an exponential is not really an explosion prosaic but it's a pretty fast growth and he was extremely worried about that .period Darwin red Thyssen was quite aware of this exponential growth and it's certainly figured in his his ideas about evolution I don't know why it took so long but never Holst who I believe is that with the Belgian scientist and recognized that said this could go on forever and that competition for nutrients between organisms even in a well next test too would have to limit the growth and so he said that the that there's going be a negative contribution proportional to the density square basically of these organisms and so all although the terrible things associated with that I'd like to say death famine destruction before cost of the apocalypse are summarized in the coefficient b so that's a nonlinear differential equation very simple 1 out and you can solve it and compare it against the biomass and well stirred test tube of yeast and what is it saves as you rise up exponentially but then begins to feel this coefficient b bends over and then saturates out at this fixed point values the star that's given by over so that's that's never holds picture of a well mixed population now humans as we saw that involve proves spread often by spatial migrations and they're not in a well mixed test to except during the present time when everybody can hop on an airplane and I could jump on a plane in Boston and get over here fairly easily and and vise versa so no diffusion is sort of really really strong and it's almost like the world is becoming this Arab globalization a well mixed test tube and if you fit the exponential growth of human populations to this small office for a whole series of you and we are here so about the year 2000 we passed the inflection point you can decide whether that a good or a bad thing I say yes so did this is for a yeast and indeed the hours and so that a typical test to the east would grow up to this inflection point in about 8 hours they're doubling time is about 90 minutes humans different scale and you know that I know 100 hundreds of thousands of years but we we we we are approximately here but before we became well mixed by virtue of efficient transportation and how did organisms get around well the simplest model was written down at 200 years or so after 4 holes by R. Fisher famous British population genetics system he said well there's a population density say two-dimensional density in space could be on a petri dish and so forth but and so in addition to the mall office and a terms he just said Well the spatial diffusion so this could be who humans aimlessly wandering around or bacteria with gel or anything that is slowing it could be ordinarily Stokes Einstein diffusion due to Brownian motion for small enough organisms in the test tube so this diffusion constant and what Fisher and independently ,comma got off and collaborators were able to show in in the decades the former Soviet Union but was very interesting characteristic of this this equation it's a parabolic equation it's not 1 linear rides the wave equation at all just has grossed saturation diffusion but it turns out this nonlinear partial differential equations admits of wave-like solutions of the form function X minus Vt and you can put this on and differentiated show what what what the function what equation this function f satisfies encouraging do that but and the hard part is determining the velocity but it's sort of plausible in hindsight what happens in the early stages of I insane 1 dimension just as the one-dimensional petri dish let's say a little strip on a petri dish is the thing locally for the of inoculated at the origin grows up exponentially do this term and then due to this linear diffusion RWS spread and so you probably know that the definitive of processes spread "quotation mark controlled by this diffusion constants like the square root of time but that only happens in Telsey gets large enough to reach this ceiling in the allowed value and eventually this wave the this growth hits the fixed-point value and so here that the population is saturated and what's called the carrying capacity and then sure enough there is a constant velocity outward emotion on the right hand and a similar motion on the left here without velocity is given by the geometric mean of the diffusion constant and the growth rate and up to the factor of 2 which is the hard part you can estimate that by just looking down here in the foot of the wave when the linear right approximation works and say OK I grow up at a rate 89 I spell out words to the diffusion at this rate you put those 2 things together you can convince yourself that the least plausible there's a constant velocity and is also meant a facial with hi-fi beat this term here against this linear terms was a characteristic with which is given by the of the square root of this ratio of diffusion constants which has units of cm square per 2nd and this growth rate I think yes please it was the of patients I don't know that they're all in all the details of the
mathematicians have indeed tried to look carefully into this summer ,comma Gore often and his his colleagues were great mathematicians I don't know if they'd in the 9th 1937 however was that they they proved it rigorously but we we we certainly know that this this equation and if you can change the nominee arity a little bit for example still has wave-like solutions and certainly on a computer simulation robustly show that that that that's what happens eventually notice this is to waves 1 after the right 1 of left not an isolated wave as he said the subtle thing is that the so-called velocity selection problem which which says that it when it's when you saw that that the actual coefficients also on well in general yes because if I if I had the walls of the petri dish over here of the of the wave would that would eventually collide with the wall and so forth but but in large enough petri dish before you get near walls of this seems to be the stable a form that this nonlinear parabolic differential equations achieved but the question OK so that's the that's a wave and I now want to give you approximately 1 slide but then I'll go overboard summary of all of population genetics justly population dynamics malice talk about genetics and let's talk about it in a very simple context where this circle here is what would happen if you could look down on attest to that had bacterial organisms of 2 different types of little a and bigger so presumably all of you have view of Henderson has studied in some stage in school primary school whatever High School of the population genetics of blue eyes and brown eyes the little BOK OK so these are happy Lloyd so there's not too chromosomes there's just 1 and so it's a big game the way that it really simple is simpler than blue eyes and brown and and what this is intended to do in this cartoonists and illustrators that you could imagine an experiment where you start with only 8 of these organisms and let's suppose there 4 of them are capital late and 4 of them a little and I suppose they have labels on them so you can keep track of the descendants but otherwise for the moment there is no particular bias a selective advantage they would you let them grow up in a test tube overnight and another doubling every 36 hours so you know after I a full day they they may have doubled their 40 times to 40th big number maybe they're hitting that Malta's were Holst ceiling had a very full test to men here attest to has elastic walls in itself will expand from Khartoum so the experiment which is actually mimics in many respects a real the evolution of real a bacterial populations is to pipette the same amount as close as you can get that you had here out of this stub potential set of next-generation bacteria and put it back in the test same fresh medium that you had here and the same number just to keep it simple and doesn't have to cost end up being for capital lasers and for little ladies if this number and which is 8 in this example is reasonably small W. binomial fluctuations random chance by the next generation produce of 383 capitalized and 5 delays and if I follow the going all you know for a long time but and I start here with the let's say population was 20 and I let these things sort of random walk around and I start off with a with 1 capital a and and I follow this this this procedural-over over again which I can usually do on a computer what you get looks very much like a random walk jumps around however it's a very special kind of random walk not often encountered in most areas of physics are 1 reason why special but this isn't the most important is that is absorbing boundary conditions so it is if if the case if there is a fight the neglects the infrequent mutations and I just ask What is the fate of 1 capital Hill but it will often go extinct especially if you start off with only 1 copy of itself might have to the next generation drop down to 2 1 and 2 0 0 and so forth but occasionally we'll get lucky and go all the way across in this this is allele frequency space I should save for you you all know something about biology that I'll have to probably tell you that frequency does not measured in hertz frequency is the fraction of the population and that's what I mean the frequency and so we can go from 0 per cent to 100 % and they'll be this this perhaps 1 lucky in a strainer lineage which manages to random walk its way across this chasm and it's a random walk with not absorbing boundary conditions so if it if it ends up down here it never gets off this axis and within this approximation if the aid takes over a department from 1 out of 20 the 20 out of 20 but it also will love will in fact stay forever on this upper boundary or the slowdown so it's up to random walk with absorbing boundary conditions and what they have a great population analysis of the 1950 s and early 60 showed Kumar normal to Camara was that just like in physics if this were a conventional Random Walk with Brownian motion we would describe things in terms of a Fokker Planck equation there an equation for the probability distribution here as well so you is the probability that allele a has a certain frequencies certain percentage of the population at time t and some fixed number that the number is maintained at the same value just to make life simple and you end up with a strange-looking diffusion equation but that that describes what's going on and this diffusion constant depends on the the the allele percentage or frequencies appeals like percentage divided by the population and so on but it's just binomial statistics so there's a genetic diffusion constant comes inside these derivatives that controls so what's going on in this might be a good time to let me show you if I can a little computer program which you can download it free on the Web called populists and if you played with popular in the Arab so I'll show you how it works here it's going to go down to Mendelian genetics when I go to genetic drift
and run 1 of these
things so what what what is this program doing it's just what I described in fact that the picture was taken from this populist programs so this this runs and so that there are population sizes 8 instead of 20 but it runs 3 and generations that turns out to be about how long you need to have most of the random what's annihilate get stuck at 100 per cent or 0 per cent and it says here a number of low that's population genetics speech for the number of US sites on a gene on each each each side energy Duncan independently be going on these around the what each nucleotide choice could be doing this for example are not always but sometimes hear they're starting off at 50 per cent and in analogy to how many of you have studied Brownian motion How many Tuesday the launch of an equation fuck apply so those this will be somewhat similar but starting off a 50 percent you could end up appear down there and you can play around with these numbers for example I could that make it said 8 different realizations if you want this this random walk and I could stored all of the had an initial frequency it's a many go up to 20 here just as in the case that we we're talking about and that sample dental initial frequency of 1 individual cells with 1 out of 20 watts the initial frequency With the smallest initial frequency that could have it's not the 1 out of 20 otherwise known as five-person consumer that such see what happens I when do that put in a serial there and I will run this thing OK will good they'll fix whatever that means OK so look here here we have the the danger of the absorbing boundary conditions here's here's our advances on neutral model started off down here and sadly all of these realizations of by Generation 21 the observing boundary condition and in every case the allele capital which is only 1 out of 20 to begin with 1 extinct it's a very sad outcome this is this after this is this is a little more fortunate because this lucky black allele managed to random walk its way to fixation so this is and again it's important that the small so but I slammed on the Icelandic population started off with a very few number of founding members and the number was relatively small not 8 or whatever at 20 but could there there's often no there's a genetic level Jack purity because many possible genes Sarah alleles have fixed allele is fancy population-genetic talks for very variants variations like red and blue like Brown license for 2 years when it actually fixed although the other 1 sadly went extinct and you know you can play this for hours in hand amuse yourself in various ways peso so that's that's 1 study and
we'd like to study that in with the name to determining how a probability distribution evolves starting with some initial conditions as a function of time yes please well that's a great question so 101 simple example so that it is certainly you get very similar kinds of manner by some sort of random mating model so if you have an organism with 2 chromosomes like humans and and very very they make promiscuously but there that then get some random pairings and a very simple but similar kind of analysis because of the Fisher and and sole right showed will give you a kind of a random walk see origins but in the oceans spew out their eggs and sperm and it literally is random and I know for these bacteria at the frontier what's happening it is not survival of the fittest that's that's what I wrote at the top but it it is indeed like this this this random situation described in the survival of the luckiest and the reason that that happens is if I fly had 1 of its frontiers let's suppose I have a solid and they open circles representing my 2 types of bacteria there really rod shaped and so forth but but what happens is is in a very simple model where I just have 1 layer is lots of stuff down here as well the way the Fisher wave is coming up here what can happen in the next generation is he there I could have a mother with an offspring of a daughter she's quick enough that goes here thus blocking this 1 all of this could do the same to it's it's it's a neighbor and so we even in this very very tiny population size like to but effective but they say is about to you can have the use of these fluctuations but what happens in no more realistic model would be that everything going back maybe maybe 10 layers of bacteria will be dividing but no further and then it's like a crowd going forth jostling forward another 10 layers at all it is the small size of the 2 small sizes 1 is in the 3rd dimension it's just 1 typically oneself thick and bacteria so that's a small size and then only 10 this way and so roughly speaking you could say that the the affected population size is maybe 10 to 20 that are jostling forward in the next generation that's where it comes from and in the microorganisms and their similar effects elsewhere yes please and so it is in the middle of absolutely and so on White right now just to make it simple I'd like to focus on a situation where they have exactly the same fitness that is actually the same cell division time both 36 minutes but but then we of course have to look at what happens when I have a selective advantage superimposed on these number fluctuations and a jump ahead a little bit what will see is that instead of having a kind of neutral random the walk with the same probability of going up is down it'll become a biased ran more and so you'll be helped on your way across this this is dangerous for region between 0 per cent and 100 per cent by systems the survival of the fittest still be survival of the fittest interacting with survival the lucky other questions OK so I'll just mention as a further than technical aside this is like a Fokker Planck equation there is an analog for Population Biology of a large of an equation and for this simple problem of neutral is called neutral genetic drift and drift is used by the population genetics to mean unbiased motion it it's the random walk aspect of it even those in solid-state physics we think of the drift velocity of an electron or something metal that that's not what I mean by graft but this this this random process which sometimes is called drift With this genetic diffusion constant ends up in a lot of an equation you just get the PTT and is somewhat of an oyster sauce with delta function correlations and but now this nonlinear prefect comes out front and the subtle thing which you can read about in those references I gave you use that to interpret this to a differential equation correctly in other words to get this differential equation which up I'll show you how derived in a minute but you have to use not the fashionable physics interpretation of the Kazakh differential equations after all to nonlinear was going to be subtle even on a computer how you stepped forward in time you that is something called the talks and arises because in population genetics and evolution of the previous generations with determines the statistics of the next 1 and the continuous times to Tynebridge interpretation for if any of you care and know about this and the audience is not the right 1 to use here but if you if you're uncomfortable This is this has a rock-solid derivation and this is just a way of getting out shorthand in stochastic differential equations that will give rise to I'm now going to the sketch the was that the derivation of this equations so you have an idea of what I am going to be talking about it and just remind myself I guess I got a quarter of 9 we should stop and have a break for coffee so if you notice media and rambling on words of might wanna just raise your hand and say its core of mine and I will stop and we will come back so anyway what about what's going on with whole how would you drive this this kind of thing wanted Dodgers sketch a derivation that is kind of an intuitive it actually would work bacteria that are being jostled around in what's called a came a step so this is a test tube I haven't shown the extra connections that make it a chemist who has played with him is that anybody around the table Everybody knows 1 so it's too it's just as like an incubator
and all its missing a bunch of cells right red and green jostling around and wondered what it does is it new fresh nutrients are constantly fed in while ministerial by Lucian that I showed you love fresh nutrients come in from the left and waste materials taken out on the right as well as dead cells and this test to was always at that Malta's overhauls equilibrium fix .period up to fluctuations and to make it simple with assume there's always a fixed number and inside the cell the model that has got very briefly sketched for you is call them around model its it's just makes a calculation very very straightforward and easy and intuitive and then this Miran model turns out to give the same results as more sophisticated models as a kind of universe Saladin In the longtime large number of large but finite number limit Kate anyone of the scenes of a lot of what I'll say Is there there's an order of limits question in population genetics will typically be working in a limit where the number of particles is vague but not infinite 20 is pretty big for some purposes and in the limit where there is a selective advantage and will will define as in a minute but it like advantage a dimension was number of that describes how much faster but the reproduction is so if I had a bacterium that would normally be reproduced in 100 minutes but that it has a more vigorous neighbor only takes 98 per cent of the 9 98 minutes to reproduce then he would say there's a selective advantage of 2 % so this is typically much much less than 1 and so were going have eventually see that is the product of 10 times as much determines and determines the importance of June of genetic number fluctuations versus the bias and survival the luckiest of the luckiest is measured by the product of a large number times a small number and of course that could be anything that's that's sort of where we're headed OK so this model the the the the Miran model came 50 years after the original insights into population genetics is just a lot easier to calculate with the sort of set up for the King estate that in mind what happened you do you you randomly select 2 cells from this test to the scheme is that 1 of which is allowed to reproduce that that will typically happen and the other How will will die because there's only enough nutrients for certain fixed number and if you pick them as if they're neutral and there's a fraction after that green and then this event where the green reproduces and effectively 1 Green replaces a red the curdle probability of times 1 Of course we could go of the other way you could you could pick out the right 1 1st and say OK you reproduce but the the green the hostility picked out agreements and that's the 1 the dies and so now instead of of increasing the number of red cells by 1 viewing you decrease the the type that increasing number of green cells by 1 you decrease the number of brain cells in favor of a red and of course you could also pick out accidentally agreeing agreeing in that case nothing happens backers with probability of square this occurred with probability 1 minor staff to get the 1st red and improbability F to get the next to green and of course you could pick 2 reds which occurred the probability 1 minus of course OK so that's what we're going to be I was studying and then thinking about this very very simple model and basically you can think about it fly by drawing a kind of ladder state OK so it's a little bit with this would be time and 1 of these Miranda what steps 52 to really have a step of real-time only do only acting on oneself or what 1 pair of cells and of course they're all really competing and dividing in parallel so in fact if Delta to use the time step that I showed up on the slide I have to do it and times not to let the whole population have the option of reproducing itself were gone and so the generation time the 36 minutes or whatever want this round model will be capital and without the population size and the stilted to focus over a year I have time measured it's a generation times and you know it sort of you know Jim it's not it's not over its overlapping generations in the sense that you sweep through I could do it on a computer and all the whole population and 1 by 1 taken at random you advance that uh unit Calgene then there's also a discretization the vertical axis which will be the fraction and never go from 0 2 1 right so what's the what's the spacing Of these discrete quantity of on the left Axis Francesca you might know 1 over and because what happens thank you Is that a particular value have I will either go out when this time step delta T 2 you get 1 more of these guys so f goes from what was staff plus 1 over and that happens with probability times one-liners you could go could stay the same that happens with probability that the EC squared possibility that a show the bottom and then there's the 1 minus squared ,comma possibilities and then it could go down as Francesco said To have minus 1 over and when that happens with probability 1 minus after pick the year the red 1 and after the agreement so what we have is is basically transitions are going up and down this latter and this is a job for the master equation this is going to define how the probability of our foreign so what really is happening in this little picture Is that I might be starting here so I start
with some value of they have not and then to get to the next generation of possible this were dealt a team when repeated over and over again but that 1st step I heeded all I can happen within this Miran model is either status saying Are go up 1 run and the latter are down 1 run the latter With these these probabilities shown over here and when I say this is a job for the master equation I simply mean that but all of this this jiggling that is is going to happen and can be described by so let's say the probability is you as we wanna know what what what is the probabilities the firefighter I repeat this process you can see that I am basically it's going to be getting some kind of you to just basically to figure out what the probability distribution out here they have to some overpass basically doing a path integral it's it's it's it's it's like a fireman patentable but not far away function it for a probability distribution and so if I ask myself What is the probability of having frequent given that I started at frequency F 0 at time t was 0 of the war war how hard I figure out what happens the next generation well I wanna know what happens Delta the later OK and this I can write in terms of the these transition rates the transition probabilities so what I'm going to do is called "quotation mark these transition probabilities and we give them names of the name will be lost some rate far and these these rates Pfizer and tell me what happens to have been 1 of these Delta Tema and tough time steps and 1 of the 3 forms of the function and that meant mainly after his one-liners staff squared therefore forms in general but there's there's just to that have the same outcome here I'll summarize that by Sigma equals 0 plus or minus 1 those are the 3 states and so now I can write down you a kind of a master equation for what's going to happen and are the next generation will be given by summer Of all the places that the particle or in the bacterial concentration could come from sigh some over Sigma equals 0 0 plus or minus 1 hears a transition rates and this is a transition right into the state F the into the particular rung of the ladder here could be any of these runs latter noticed the transition rates depend on where I am on the last year that Appendix F has 1 miner said he might be biggest when it's a half at least for the for this term here and so forth so I had a transition rates and energy coming from half equals so F minus signal overhand receiving a 0 plus or minus 1 the things change I think we can have either signed or could be 0 2 here's the transition rate and then this gets multiplied by the probability that I was at a particular place in the previous generation so here we are stepping forward in time and the and will I have I have you comments -minus signal over and that's the previous generation I started with if 0 way back at equals 0 . 0 what's happening at at a certain time interval the previous-generation what's called a T I O a step forward in time here so that's just that's really all there is to be a master equation and if you stare at this and say sale population is maybe 20 or reasonably sighs then but a natural thing is to expand this equation this matter is always true it's combining probabilities says the product of of events that have to occur in sequence summed over all possibilities and so forth so expand in much the small parameter year Sigma can be bigger than 1 so that if 1 over an so basically expand in signal ran on the right-hand side and we will expand In delta T In the left hand side and you have to acquire show that this is consistent with 2nd order on the right-hand side only first-order left-hand side and the subtle issues whether that's justified or not the same kind of derivation you do for a Fokker Planck equation but because the rates are frequency dependent on you get something it doesn't look very much like Stokes Einstein diffusion course it isn't it has this subtle differences and and this this is the equation that comes out allergy you work through the math and but you see that there's this frequency dependent or a percentage dependent genetic diffusion constant because like 1 overran and again and has to be large for this to work so here is out of the Wiggles that we should start with that populist program and you can run many realizations and what happens not too surprisingly few star offered a 50 percent probability that the origin they all start like that there is a probability cloud is like diffusion near the center of this low frequency range from 0 per cent to 100 per cent but and then it spreads out sort of as a Gallician but there are these absorbing boundary conditions on the left and on the right and noticed what happens to the diffusion constant about I'm at the bottom at a time you kept a few goes away your stock so it's it's a diffusion constant that varies dramatically and you get near the boundaries of diffusion constant goes away when people 0 or people's 1 and so once you wonder why random walk away Tuesday's 0 percentage of capital a 400 percent capital later you get stuck now there is this is not a simple analytic solution of this there there is for Fokker Planck equation but for this 1 there will be 1 in in a minute I'll show you later but but but but but you can solve it numerically and that's
what Camara did back in 1955 starting with the sort of a delta function here this is this is the allele frequency space with representatives X and you start with a delta function here and you get these sort of eventually you get delta function distribution house at at concentrations 0 per cent for 100 per cent and this is a population of fruit flies that have been bred for the 16 of them and they go out some poor graduate student had to do 19 20 generations of these fruit flies and I they they chose some neutral mutation of some W 75 variation it was easy to read off maybe they had black eyes were black white coloring or something I forget now in and had had the wild type at the start up of the 50 50 % and they can all go into read from a freely and not only did they do it for a further banking generations or so with populations of 16 it 107 of those populations thus enabling good statistics and you know it's not a perfect match but it doesn't shouldn't be offensively noted that this is only really true for a but did not roughly follows what Camorra predicted the diffusion out to the wings fixation on all Little a moral capital any questions OK so what about the so that this is what we just talked about here is a summary of finite populations and finite go fixation all the a all capital way for long time what is the probability of fixation I'm starting off at a certain frequency well energy basically any 1 of these so far here I have 1 out of 20 of these and the initial conditions being capital a and 19 out of 20 being a little late everything's neutral and so on there is an argument that says Well that did democratically speaking senses no selective advantage any 1 of the 20 could win and take over the population and so there will be a 5 per cent chance that the Capitol label fix and and its related to and that's the idea and then as I said in the vicinity of these expanding populations like on a petri dish or even humans coming out of Africa the affected population sizes are small and so these these fluctuations are big yes you this this this thing here OK thank you of this delta T is 1 of those Miran steps which of them is simply allowing 1 Celtic potentially reproduced another 1 to die 2 the time the test to is playing out Pearla every sale is not looking around saying turn my turns they're all doing massively parallel in the actual laboratory time and so the Miran model tries to simulate that by doing this and times in a test tube to represent 1 generation so that's the idea and other ways of doing it you can know of allow them to have lots and lots of gametes like the sea urchin and then just pick them out yet all year yes thank you so you yes and no I mean I'm crime including both of them if you and what I should have said thank you for the question is you do this expansion and then you repeat 10 times together 1 generation so what I showed originally and you have corrected me With what happens for forward for delta T will what happens Delta tee time steps and this is 1 over and I do that over and over again to build up an entire generation that's on the left and then I I do this over and over again adding up various variances and that is the 2nd order In the past as in front of it 1 over and square than what comes in is the expectation of squared did you can check if you want calculation but then and do it times that's what knocks down this 1 over and squared related the 2nd order yet about the 2nd order to get something trivial and that's what gives you at 1 over and has opposed to the rule of law this is a yes that is what that the rest of the world you don't to thank you and then I would build up I would have 10 of these To give to get them to get a generation and then this differential equation is for the generations regarded as a short the question is that you can OK you know this is not listed in the that we're still we assume that these bacteria are so dire loot that there is no problem of self-avoidance crowding in that sense there may be crowding because of nutrients but they're not jostling each other saying was simply a label for these 3 possible
transition rates so the 1 over here would be think this is this is just a shorthand for the formula the good thing labels 1 is a shorthand for this process we got the letter sigma equals 0 this is shorthand for these 2 processes which occur with this probability we stay the same rung on the ladder In time delta T Consignia calls minus 1 is going down and you OK sorry they didn't understand the question so so so we hear all this year over here Excuse me so this is the initial time thank you so this is a function of 4 variables but it's kind of just being a little mad overly precise mathematically but I should do it so that it makes sense I timekeepers 0 bytes could be 0 and I have this fraction and then at time t plus delta T I have this fraction thank you for the on clarifications of questions OK so we were almost there the this is a summary of what was happening and I'll just just mention that it is not hard to put in the bias that someone mentioned that are supposed the red the greener out with reducing each other and here is a very simple scheme for the bias would save it there OK fly health and faces the signal over and basically new we lists these 3 possibilities said OK excursion go up the ladder it would normally if it was neutral B at half time 1 minus but I'll I'll say others those vigorously reproducing organisms that are green they actually have an enhanced probability of being picked to reproduce which is not enough but times 1 plus I to assess over to and then the death also was independent of color and so this is with a selective advantage selected and then but that that would be signal equals plus 1 go up 1 latter if I stayed the same Well let's assume that stays the same the Getty multiplied by 0 anyway most of these averages and then down here I would say OK I take the red instead of probability 1 7 slightly inferior so I'm going to give a little lower probability would normally and I have enough so that was 2nd was once repeat the algebra 1st delta T repeated on time you get a complete moron time step and I'll let you know what that Allah God look at that reference if you wanna to look at the details analogous something a little more interesting here I've tried putting the question that you asked about where where are these all where these variables F F 0 and T T 0 so I had fraction f 0 at time t 0 infraction F at time t and now we get the term I had before which is genetic drift random-number fluctuations an alligator bites OK and I did it in and it turns out that this this bias is sort of just what you might have expected but it is if you if you neglect this term here and he just sort of take a look at what what's going on and what you'll find Is that up to noise this fraction DFT tea is equal to s the selective advantage that I just introduced here and here times one-liners but I just ask you to accept them it's plausible because now superimposed on noise so they'll be nice here as well OK to symbolize the noise but putting in half times 1 might served over N time some stochastic functions they'd have to there has to be interviewed with the calculus there was some noise made not just leave it this way as noise VI leave out the noise what's going to happen what's a longtime solution of this equation DFT TNS is positive there's been a saturated 1 I invite you the check at the solution of this differential equation is half not the 2 the S T Over 1 plus that's not EDS team minus 1 and that's a basically says that if I don't have any Jennica reference I don't have any noise efforts to you would just drift higher the well-to-do the Woody we would be would be biased toward growing exponentially in saturating just like our introduction to the fish but now it's not the actual population density that's going it's and saturating it's the fraction of say green if Green has a selective advantage and it again it's big some sort of intuition and what this more general Fokker Planck equation means it really helps to go back to that program populist and make a few runs let's do
that don't come up here and I'll
go to the I mean the union genetics with drift and selection 2 of them on an incentive to selection this when get rid of
this 1 I will run and I'm going to set this up in such a way then we have this was originally this program was written for a
diploid organisms like ourselves with ,comma zones and adjusted so that we have a reasonable model so 1500 and let's go to a population of 50 the initial frequently or frequency of 5 % and let's win this thing see what happens right so here we have exponential growth and saturation with case noise images to perturbation theory and forth but this noise because of the absorbing boundary conditions can have drastic consequence because if before you escape this region near this perilous absorbing boundary conditions you hit then it's all over even those who might have a selective advantage and so survival of the fittest is having a very delicate interplay with survival the luckiest you see as I cycle through the trees sometimes it looks like it's going to be successful it's it's it's escaped the sobbing bound and this is gonna be absorbed appear allele capillary is going to try trial but doesn't have to be that way and population geneticists Leichter Imagen that may be over human evolution there all these great favorable mutations like X-ray vision and wings and ESP and and how many of them died and we'd like to know the fraction that will make it hard to to this this final stage any questions from the 1st half of the lingering question just leave yes in the way yes thank you so In the Malta's overhauls to picture we we had a population that was doing this and the why would there be a term like like sees great there's couple answers to that the other 1 is that you you you just rewrite this summer In terms of an effective it it's almost Justice symmetrical thing you would say that could rewrite it as an effective growth rate that depends on the concentration of vectors we have specified what this function is but there is some growth rate when the concentration it's 0 which will call a and it's plausible that if I make the concentration bigger and bigger the effective linear growth rate is going to dry and if it's anything that goes to 0 I can only arise and get an approximation that basically is what I which looks like this but it could be some polynomial and see as long as it drives the population down when the density is you could also say that these bacteria afterward so that that's 1 argument the other arguments which actually has a number of fluctuations underlying it is that these bacteria wandering around In the end the test too this is this is the world line of bacteria there's a process like this where the bacteria have wondered at the same point point in space knowledge imagine there's some spatial variations that are possible and when they get to the same point in space In a given reference fine of nutrients is only enough food for 1 organism and so coming out of this box will be at a single better we can't tell which 1 and this this death process will be proportional To the concentration squared so that's another source Of this term and in the same Fineman world picture you can say that cell division is represented bye a bacterium that splits into 2 that's or if the bacterium dies of old age that's death and if we take into account the finite size of the bacteria and we try to drive in two-dimensional space time this becomes the pair of pants diagrams of strength and other questions right and so we can now start this go back a space right unless that's what's going to happen the loss of population genetics and so forth when space is important I cannot tell you about genes Thai media surfed off the coast of the Netherlands California California Hawaii anyone OK so here's a here's a population where I want to show you how genomes conserve from population with so this is just a population lived the fisher way saying that we were we were talking about and here in the foot let's say on the foot of the fish away a mutation and it could be a favorable mutation unfavorable Baluchis for now say it's neutral well this is where the mutation and its offspring would have the greatest chance of taking over because the population density is low In the foot of the way if it occurred back here it would be stopped competing with 100 other organisms in its own little reference the region and wouldn't have much chance because of genetic drift a video so absorbing boundary conditions of taking over but here it has a chance and so on I noticed this nice smooth function it can't be smooth forever because down here we get to less than older ones organism you can have smaller than 1 and that Fisher wave stuff that's so beautiful that is actually slightly inaccurate down here because of number fluctuations the same thing that we talk about the genetics of he supposed to go existentially 2 0 well you can't because when it drops below 1 it's going to be Rick rigorously 0 so but anyway down here the numbers of small and so now it's moment let's work on a cold morning frame there that way with with this is moving forward velocity v right so we stay at framing move along with we see what happens to to the descendant of this music and it could be that it fixes locally and takes over the population thus providing an example of successful the wave comes by the Gino mutates locally just the right time and place and the skiff it's lucky enough to escape genetic drift who will take over but could be that well it has a chance but the wave goes by a little too fast it's now step back here competing along the x-axis with some vast numbers of the original strain it might not die out but it's no longer find the wave this is an example of unsuccessful surfing heaven you experienced and successful serving in England OK so you can go both ways and although we want to find a way to study the surfing phenomenon in a petri dishes and here's here's just to remind you how to get started but this is a simulation somebody pointed out the brakes easy call I arrived shape and this is the model growing up and a lot of interesting features the as the tangential boundary conditions around the edge of these rods a defect here with color discrimination and I kind of liked his that to think that it looks like a mirror image of Africa and if you just put in numbers for equalize that don't have jealous their dividing in this fashion large mammals like humans that tracking the mighty Condrieu DNA by the way my country are the descendants of what but bacteria so it's not crazy to use bacteria as a surrogate for the mighty kind real that are used to track these range expansions and the bottom line is I said earlier is that in 500 generations you go 10 thousand kilometers year-ago happenstance so
they go back and experiment that I started this discussion with them and here too we collided schematics of 2 ecology announced this is the main chromosome promise on those ,comma zones where the identical as their clonal and then there was a plasmid a 2nd chromosome and these plasmids were also nearly identical there are quite small compared the main chromosome and on the plasmid which were identical almost all respects there was 1 change and it was the change in a constitutive Li expressed means genes on all the time in the color of the fluorescent protein so this was a yellow this was science and the the days when in fact it only took 1 amino acid changes to shift the cold so there is a Damocles we could make every mix them up and somebody asked what was the protocol and the protocol was was inoculating down a few who leaders of the pipette the carrier fluid evaporated away and then under high magnification after the evaporation is what looked like 4 days later this scales about a centimeter we got this rather boring looking no white coin shake colony growing on the petri dish How many of you have done this experiment where you are shockingly discover what's inside your mouth by the little wire putting it on the package looking at a couple days later and sulfur horrible should brush achieve consistency same experiment is but but we had these 2 colors and fifty-fifty proportions and under fluorescent excitation it looks kind of Inter here's a more globally so this region inside the white circle we call the homeland there's lots of stuff going on just outside the homeland is this great migration proceeds outward in radio fashion but eventually genetic drift at each point along the perimeter it's going to cause fixation of green or red and when it sort of yellowish in here it's because they're still fighting out there still in the middle of this 0 per cent to 100 per cent gap that we've been talking about and starting off a 50 percent half of them end up red and half of them end up green and then your stock but it has different outcomes have a different histories as you as you go around here and so I my by my colleague Stephen Jay Gould looked like to speculate what could happen if you replayed the tape of life you can do it With these organisms at least in the context of simple homeland inoculations added to question the details are different here we have the genetic boundaries but there are universal aspects we think of what happens with his range expansions as I and
it doesn't have to be a 50 50 mixture in this illustration of survival the luckiest you can see that a 95 5 % mixture most of the red around the perimeter of the homeland you can see there the fraction is is estimated by looking at the area in the interior which is red but other 5 % some of them started off but then got squeezed out here got started off and then it may be survived this definitely survive and this is an example of Jesus and if you can't you blow this up you can see you could almost traced back and find the bacterial Eve that could determine the time and place that even gave rise to this progeny that came out yesterday but by contrast you mean you see red and green and the book Oh yesterday so that might just have to do with the way we set up the microscope and so forth what we think happens is that like in this in his previous experiments we end up your half of these are red happened green this will create a little red colony this 1 might create a little green colony than they collide and is probably weren't good enough in the microscope to get them equally balanced died in in the sense
of Europe interesting questions so that I think that's my that's my
best guess as to what happened and notice that because these stop dividing we
have a record of the entire history of the range if this is the present day we could do a measure making measurements around the rim and maybe further size the homeland we can see what we can do that accurately by trying our theory on a repeatable experiment in that that that's what our goal but
that here's another experiment where I stayed 1 evening but with a razor blade that I sterilized over a Bunsen burner and I touched the razor blades to own a liquid pan head of the 2 different bacterial species in touched it to the petri dish then and so we have a range expansion up and arrange expansion down OK and so we can think of I'd look at any Antioch look along with black lines what's happening it's exactly like that that's serial dilution experiment or that he missed out experiment because if I go from 1 generation to to the next what's happening is a variation due to number fluctuations of the fraction of red and green sitting inside each generation to actively dividing patches in the front moving along the black so rather than in the West it is as if the range expansion does serial dilution for you and if you did it grab it portions will be changing in some places rebel win office linear inoculation some places Green will win if you a dust motes sitting on the petri dish looking at this anomaly of bacteria coming at you at a sort of your eyes were down there and in petri dish you'd see this mixture of things coming initially but then it would segregate into red sectors in green sectors as you OK and so said these
generations map onto this population genetics thing and if there were a selective advantage that that equation that that that that took launch event type equation thing that I've said earlier for selection with draft beer selection his draft would apply and as we saw with populist program there on average maybe you might go up and saturate but there readers noise mn noise could sometimes be deadly causing even favorable mutations to die that's an issue for tumors we may not get there today but if you have a tumor but I I I don't actually have a tumor fortunately but I do have this ball bearing the red and this is taking this as a growing mass of cells in the human body that has gotten a little out of control but it's not really cancerous it's a benign tumors growing slowly but what people think about cancer and so that not an expert is that requires 4 or 5 hours 7 more mutations and for the the cells and is slowly growing tumor to become malignant they they would have to acquire the ability to throw off all restraint and and divide them like they were bacteria again or yeast or something rather than the control cell growth to carefully controlled sellable growth that characterizes a healthy human but never would be 1 mutation another mutation might recruit them blood vessels to supply nutrients to the interior of the growing was of cells because they'll be nutrient shielding of the budget just like you can grow in the middle of these two-dimensional colonies and others green represents 1 of the that those mutations you might have to be 5 or 6 of them and the question we really care about is whether this green mutation on the surface of a slowly growing tumor will and as it grows take over eventually would have all agreed on the surface and then well laughter that happens there be another 1 and 7 other mutations later if you're unlucky you'd have a full-blown of malignant rather than the 9 so it is analogous problems in female law passed this around people it's not going to have to worry about so that's the kind of question is that you can can that little green splotches overcome the genetic drift 2 spread all around the surface of that fear could be a circle with OK so how do we deal with the spatial problems I just tell you briefly cause they wanna get to some of the other stuff here and this is a stepping stone model and you can imagine that each point on this surface like the justices and white light after it 10 hours bundles razor blade inoculations each black arrows doing this but the black arrows talk to each other because the jostling cells at the front are around is pushing each other left and right and so they can be effectively some kind of spatial communication or spatial diffusion not genetic division with spatial diffusion between neighboring Arab states and the steppingstone why was nothing more than a bunch of of little while population genetics reactors that allow also some exchange of cells for 1 to the next and if I sit in an hour and
a local reference frame moving with his outgoing population we have a one-dimensional steppingstone model going along this frontier and so then we can have a fraction of of red-eyed attacks it's registered agreements time and the ice stepping stones was be itals 1 2 and 3 will have a certain fraction there's a genetic diffusion constant that we've talked about some some conventions have effective to there's some don't don't worry about that and this is another diffusion phenomena it's a spatial diffusion constant and if I just put in mutations basically all of population genetics is summarized in this master equation in this 1 from this one-dimensional context not subdued 2 dimensions this the genetic draft we talked about the 1st half of this isn't just a discrete version of the spatial diffusion operator and if you wanted to put in something that will involving mutations are just right down the equation and you can you get a sense of what's happening out there the spatial diffusion stadium in higher dimensions is the fraction of green what these models on the surface of that sphere but there would be asked times times 1 minus that's a selective advantage that we've talked about if I want mutations then there would be the abuse mutation rate how Of the state have safe from green to red the goes like that and maybe another mutation rate knew goes like 1 minus and then finally there be this genetic drift term which would be the square root of St times 1 one-liners In the end the 2 Tuen time some noise that would depend on on space and time OK so this is this is a kind of master equation for all of spatial population genetics which yes so Oh thank you so so for 1 1 is negative it was positive .period so the prevent your question if Green mutates to red at a rate New that would be summarized by this negative contribution added red mutates into green at rate proportional to the amount of red that's already there than anyone mind-set that would be this time this is mutation this is as we've talked about selection this is dialing Dow and didn't know about genetic drift you lot with another and this is the future and this is noise but population genetics is called genetic drift random drifters not it's not that there's no bias here is just going back and forth as we discussed and so I just extend this the additional plots in 1 dimension discrete model this is equivalent to the term here turns out and everything in it as a linear range expansion is going to be described we think by this model and I knew it turns out amazingly because it's this is nonlinear noise it's exactly soluble and you can compute something called The Herald's I Garcetti's correlation function which is the probability of if along with razor blade you measure 1 color year you go a certain distance away probably the color has changed depends on distance moralizing model and Green were up spins and red were down spins with looking at some non equilibrium kinetic Ising model has been spin correlation function at a fixed time and this can be calculated and at that time t equals 0 it's just flat it is random initial conditions along the frontier with equal probability of getting red or green in the sense of being have and you can do
simulations this is an experiment operator and you get pretty good agreement but there is a
Selby that I haven't the focused on but I'm not going to tell you about this is the razor blade inoculation and this is that there is a gossipy and so forth most of these range expansions on circles in and terrestrial environment it even a of foreign organisms lands on a coastline would be semicircle as it invaded until the virgin territory and these radio growth growth growth problems are slightly different because what happens is you get random walks but the radial random walk up here the random walk served are going around the goal like the square root of time these these genetic boundaries they collide into each other and pinching off and so forth and that happened for a while over here and over here from the coal iron Pseudomonas aeruginosa but eventually the parameters just like the inflating universe are getting further and further apart because you're going out linearly in time in radial direction from my arms are going getting further and further apart tips of my fingers a further apart and my elbows and battled a linear inflationary separation which eventually wins out over the square root of time annihilation of these random walk like appear and if you work it out you can show that eventually the number of sectors to develop up here we think either green or red woodwind and longtime it's going up or going but on the road with Israel expanded eventually inflation sets in and the number of sectors stops changes and you can show that the number of sectors in this simple random walk model goes like the square root of the initial radius of the homeland times the velocity times this abide by the the square root twice this diffusion constant at the base of the is spatial diffusion constant describes the wiggling of these boundaries the hard thing was getting private the rest of his sort back-of-the-envelope estimate but notice what you can do you can go around and count the sectors once you reach this inflationary epoch that you can get experimentally the velocity you could measure out at the rally in the present day and you can measure the wiggling of these lines out of the room in the present day inverters formula and determine the radius of a home and you can see by repeating this is from 40 times with theory is any good that's that's that's the idea and the error bars the theory says Arab hours which you can also check so that's the idea showing that we can do replicates we can calculate Hatteras either correlation functions in a radio context you go this universal shape larger larger farms I
want to spell out time and because they want a flippant and tell you about some very recent work with Macs event of edge on the inspired a whole fleet may be a little bit of insight into some small corner of issues involving cancerous tissue but this is to show that we can do experiments on this is yeast so that the strange shape is just as a budding yeast it's spherical of organism it has a nuclear you carry it and buds off smaller daughter cells the blue 1 can make losing but over produces tryptophan which is great news for the other 1 because although it over producers losing it can make its own kept a fan and so they have to play together and so the question is will this the world will these this pair of Exhibit genetically mixing as neutral strains might and I will go through the man you can work out what was going to happen with generalized version of this model known the selective advantage terms changed a little bit but but I will tell you
that's the if you work out this phase diagram and only for a limited regions of growth parameters Alphand data do you get this mixing things will go back
here many of the next year ago so
that the way you the way you do this is to say I now has a frequency dependent selective advantage the comes in front of Afghans 1 minus F which we've seen before and that comes from these different growth rates of red and green and so G is a background growth rate for both of them but state is a coefficient that tells you how helpful it is for the red they have a green nearby because this is the concentration of green and a growth rate goes up various positive this Alpha because you How fortunately it is but had to be for agreeing to be a red with concentration of and that the coefficients Alpha these you work it out there is this kind of inverted W shaped potential that this crisis going on and you can work out
in spaces Carol Korolyov did in the paper we work together and there's a phase diagram is a function of alpha and beta in here there's mutualism In here say blue or green wins already here there's the genetic the mixing like that razor blade inoculation took place right here at the origin and that that behavior continues along which seems to be a kind of line of first-floor face transition here the other color winds and this is
an experiment experiment is really quite beautiful done by Melanie Mueller and so I think I can say of unit and with the help of worry and what what was done here on the left was 2 2 2 makes these guys up in the homeland and they have a plate and Hagar plate which had some nutrients but absolutely no Lucien absolutely no tryptophan To survive at all they had to stay mix we managed to suppress very powerful force of genetic mixing Demick Meet genetically mixing however if you add tryptophan losing so they don't need to cooperate they can get much of a they want from the plight of they stopped genetically D mixing over here somewhere between here is things come and that that's what we're starting now any questions on this point before I switched gears a little bit and talk about mutations OK so let me say that I go over to another
talk actually similar to a
talk I gave in Paris last week and it comes at a conference on them physicists meets cancer tissue growth through summer slightly strange name so I passed around this this this tour a few the cells of migrated actually in 1 room into that of the fatal diffusion is happening on the surface of Sphere and so on he hears although not I'm not an expert is what I think is useful to know about a vascular tumor growth a vascular means that haven't got rid of you haven't gotten capillaries and and arteries in there to supply nutrients there is a thin growing region on the outside the sphere just like we hadn't seen growing region on bacterial colonies appeared there is a region of acquiescence they stop growing like the bacteria never got this severe in the interior of these two-dimensional colleagues probably because they could get nutrients from the 3rd dimension but here UK conditions from 4th Dimension audience freight and so there isn't a chronic or whether there can sometimes be volume loss and and this is a really cool experiment that the Max event its and John touch 1 assisted from where they look at hemispherical colonies not actually a tumor cells although the real people will do this experiment but but but he sells and basically when they get big enough Green which indicates active cell division only occurs on the rim of the hemisphere not not just the rim but that what all the whole hemispherical surface red beans this this region here of clear sense non non division still proteins produced the red for the onto the green ones on the outside a nutrient screening length showing dramatically in this case that you only get growth a certain distance and toward the center so the very much like a three-dimensional version of what we've been talking about with these microbial colonies and as I
said cancer often results from genetic changes in somatic means that the not the germ line but that the cells in the body they they stopped cooperating it's amazing what the cells on the core body doing and they commit suicide when they need to he stopped growing they're very polite well-mannered and and occasionally get mutations that cause growth there contend deleted this abnormal forms and so forth and as I mentioned you might need a number of driver mutations to take over at the surface and then eventually you get something at the surface where they might resume we have room to grow old with which would have enough of these mutations 5 to 7 could could lead to serious problems malignant Neil pleasant I think is a fancy name for it and we want odds when asked what is the probability of surfing on the surface of the sea some of spherical cell cluster and could we use some of these ideas and this is the inning in existence proof that you can study these things in vitro In a laboratory there's their cell lines that came from breast cancer and most the time they're reasonably well-behaved and they just slowly grows vertically within you can activate them and enhance their growth rate and so that's it that's an example this is and just finishing up his thesis and most of work and we have lots of help as usual on wonderful collaborators so that's what I wanted to spend the last 20 minutes questions .period OK so here is the more pedestrian but still quite beautiful experiments in a petri dishes this is not a cancer cell line this is penicillin standard for which means the praiseworthy bacteria that conform dendrites this doesn't formed and rights are under these conditions but it does have mutations but this mutation this has a slightly different coloration to conceal right away this is the basic experiment of microbiology people been doing this for 150 years or so and you know he's you see something like that you you might the state's school about this little piece here go sequence and see what's different from the host this is a mutation is as wild-type missed mutation is taking over some interesting way and we just we went through a little bit about this razor blades stuff I just wanna tell you briefly that I if you this is used but if you do a razor blade inoculation of yeast you can use the growth velocity Of the red versus the green strain as a kind of proxy for the selective advantage Survival of the Fastest and so there the week we could put in the formula for the the Fisher wave growth velocity of the background green strength and then compared to that velocity will just say that the red strained the mutant strain for that's what I means is veto 10 times quantity one-plus tests so as just like the selective advantage talking about it turns out we just measure this angle at the end of the day you get a selective advantage and that that calculation is a high school trigonometry just say I know well I calling out this little sector here that had ultimately has a slightly Bode frontier and I go along this distance and basically this point this distance had better match distance of the better arrived there at the same time these 2 population one-to-one along the inside of his status line and the 1 that went on a straight line shot the 101 straight line shot is going to distance teatime V but that has the equal this distance the times the velocity of the mutant strain middle and the coastline of firewood trigonometry and if you saw for the good of the equity angle means in terms of selective advantage this is the form of that angle is twice the Arco scion of 1 over one-plus S 4 for small so often the case goes like this square root of so if you're good enough to measure angle the 1 per cent and everything else worked beautifully this has not been done but might be able to the future 1 presented in FY 5 give you 1 part tender for selected him so it might be a useful way to read off the selective advantages and you know the way status as a perpendicular velocity here of the front at the 4 at the at the frontier and you can get that from trigonometry also and now what is the fixation time the fixation time of this domain on the razor blade has size L it's going to be Al divided by this perpendicular velocity so it's Al divided by the square of the best Turns out pesos so there could be you wanna know In the case of a spherical tumor this is a linear problem in 2 dimensions but in the case of a spherical term we wanna know how long will it take for these green cells to sweep over the surface and if it were flat then at that time would be proportional to the linear dimension that might be the perimeter of the worst circle divided by square us but this is flat it's not a circle but you can get a selective advantage it's about 13 per cent OK what's but I
happen for Circle and will be with fear circle easier to do mathematically so let's take a little colony here this is a simulation nodding experiment but also its minutes and at 12 o'clock will put a very small piece of news the grows faster and what Carroll Korolev did part of his thesis was to solve some reaction diffusion equations mutualism as described by these epsilon times if you want to put it and using different diffusion spatial diffusion and stuff like that and you can figure out what's going to happen anybody have any suggestions about what might happen run the simulations forward in time experiment that's about a favorable mutations so as his possible take a look so here it is this great that's that's that's the news he grounds it starts to envelop it hopes eventually it's a beautiful
heart discovered on Valentine's Day OK what's the shape of the heart well this is not high school John F. trigonometry This is a little bit of differential geometry so you basically say curve about a real coordinates this is the angle this is the radius of the function of 5 and this distance here had better map of the mutant had better match this radial distance of the wild type and if you work that through I invite you to do it you discover that in terms of the mutant radial velocity envy while tiebreaker velocity the angle is given by the logarithm Of this radius and if you put it in the end equals 1 possessed and discovery goes like this where again as multiplying lottery and so now we can ask What is the fixation time how long is it going to take to make this hot but it's curve now so it so that the more complicated calculations and you can get a budget setting fight equals please otherwise known as 180 degrees OK the fixation time which related to a fixation radius is the heart goes around Is the radius of the homeland plans the to the pipe over square root to us so now for small this is exponentially sensitive to the selective advantage is 1 over square reverse the 1 over square and that's probably worth knowing why would you wanna know that well but you like to know if this new grain is somehow expanding in spherical analog of what I've just described it was going to take a really long time to envelop maybe another mutation will come up with that and at the same time to get in the way or change might it might even now be deleterious mutations and so that's called clonal interference and decide these issues of common offense is important to know the fixation time and here we have a simple formula for it in 2 dimensions now it turns out this same system but let let me summarize hears the argument this is a famous mathematics called a logarithmic spiral it was discovered by Jacob renewal Is it occurs occasionally in physics but a lot of times in biology it's it's the the describes the shape of a trajectory of an insect that attracted to the flame of a candle he goes around and around and around and it comes in added to a fiery deaths on a logarithmic spiral related to the fact that has eyes a little bit off-center and NY would do such a stupid thing as to kill itself for our you know mosquito trap at a party where guests and UV light something that could be a bug zapper probably because that algorithm if the source of light where the moment or the sun and was effectively if only for a way that would enable to fly in a straight line but it evolved to fry in a straight line but now we nasty humans have .period light sources which closed the spiral deaths of these bugs that being insects so that 1 application is also the shape for other reasons of the nautilus shell an fossil hammer go back 70 million years also have to shape and I hope I've convince you that sectors of microorganisms also described the logarithmic spirals Jacob regularly was so pleased with his discovery he said I want a logarithmic spiral scrapping inscribed on my tour 2 on my migraine so this is the rest inscription says something about what a great guy he was but sadly the engraver inspire inscribed there are comedians spiral with equal spacing instead of elaborate so he got stuck with 1 curve that he did not invent the Archimedes invented I'm not nearly as cool as Lebanese paralyzing OK in
fact that this is very much of our excellent theorist when in the lab and and tested these things and Pseudomonas aeruginosa here's here's occurred the plotters curve log barbers selective advantage it Percent and this is melody Miller's data slightly curved were less dramatic but still it takes a while for the sectors get established but eventually they to form what spirals and significantly perhaps for these 2 words this
argument works in 3 dimensions but the so this is work by Martin Nowak and collaborators and they just went through the same people who simple man in an honest sphere and so this is this is like a continuum model of news of the background slowly going to work this is a driver mutation is like those screen of ball bearings and the driver imitation is taking over an go to the yard when the fixation time again that's likely the Pirates grew to now this it leaves out clonal interferences they point out but this as this blue starts to take over from the gray you could have yellow red and green different mutations all getting in the way of interfering and that's important study that I'm going to neglect but the there there is this interesting what would expire on 3 D as well what's missing from this picture can anybody in this room tell me what's left out With the smooth curves noise genetic drift the thrift is left out the this and that and that is the crucial thing that's missing is that always like this and these circular colonies of microorganisms are bumping the 2 Ficken post of discrete cells and some of the year in an effect that show here this is at 8 1 of these is not a razor blade inoculation of my colleague Will from Mobius and works in a group but it basically soak some filter paper In bacteria of 3 different and let the range expansion go on and these wiggly lines there's a reflection as we've seen of the discrete all the cells at the frontier Einstein used a similar argument when he was betting Brownian motion and said you know that's why we know that the waters discrete before people ever measured it because the the pollen grains jerking around the way the water molecules were worth 3 orders of magnitude smaller they would be willing because all the different impulses on the pollen grain would average out here too weak and we know that this is made of discrete entities because of his wiggles and fact that wall diffusion constant of its Amana layer and ended only goes right at the surface is given by the sell-side squared divided by a generation time just what we talked about the 1st half and if the cell size were 0 there wouldn't be any whaling so you can read off on a centimeter mm size scale of this filter paper the conversation of this cells which are occurring 10 thousand times smaller than the migrants and if left out of the out of this that 6 of their calculation the fixation time is right when she was escape genetic drift so we have a spatial problem of genetic drift here and we need to see what's going to
happen I don't have time to tell you the new exactly with what goes on there are just give you flavor but what we did was we wouldn't we we had come ball bearings are a ping-pong ball to something on a computer we could grow them in a plane away analogous to razor-blade inoculation or they could grow them sort of disorderly mass but it's a lot like dance ran the packing but now it stands random packing with population-genetic superimposed and we would put say 1 new himself at the frontier and ask you know what what will happen and we'll give it a little bit of selective advantage and will ask what's going to happen so it still sorting out what happens on the surface of a sphere we think we know what's going to happen but but only roughly we don't know what's in and out of simulation that shows so this is the logarithmic spiral that and and this is 1 of Max's lovely simulations the 1st successful gene surfing on the surface of a trauma by agreeing driver mutations but it didn't have to survive no no it happening purity and so what we did it was what we asked the both simple question you could ask which which I alluded to briefly in a discussion of conventional population genetics in Israel dimensional test to we could ask What is the survival probability of a single mutant a frontier and this is a problem that we can at least solve with Bakassi in 2 dimensions we think we know how to do it and 3 would mature but we think we can do it so here's what's happening in a two-dimensional world but I have a little opening angle here at the service of the homeland in radio inoculation and and here I have some sort of the driver potential gravitation might have a selective advantage might not about those those random walks was real random walks have pitched off and sadly for the 2 but fortunately for the human it gets encased and no longer can reproduce now this survival probability which is what we're trying to say something about depends on a lot of things it could depend on a selective advantage could depend on the radius of the homeland will depend on the size of the cell a depend on the generation time on the video this low angle here of the initial state on the velocity on time fortunately in in a long time limit is not a function of 7 variables those of you who do experiments might be glad to hear that I and in fact we can show that it collapses down all the long time limit only to function of 2 variables and for tumors you can make a reasonable approximation which is that the velocity of range expansion plans generation time is a cell size but the grows 1 cell with like an onion at a time and then another useful thing to say is well we really care about is when happens if the mutation rates wealth is a single new at the premiere which case this angle is up to geometrical factors is just a over a so if you put that in this becomes a irreducibly irreducibly complex but only to variable function of the square root of a over a 0 and then this dimensional if selective advantage which we call cap it's like advantage that comes from the velocity of the cell growth rate which is small number multiplied by the square root of the radius of the toll road when it 1st gets the mutation divided by the cell sites this is a large number maybe it's 20 or 30 and so small times larger key parameters you know also could be bigger small and in terms of this parameter it's lodges a pearly big selective advantage and that what we have here we have a noisy while spiral here we we have was trying to build a wanna-be logarithmic Spiro again squeezed out and here we have had for cap equals . 0 5 a mutation whose fate is unknown at this time indeed it is .period assimilation and at least until dimensions this noisy logarithmic spiral this is it can be described by a large that equation doesn't even have these weird to and 1 minor steps in just look at this the cast the behavior of this 550 This is the selective advantage this is the deterministic port and this is noise in the usual ones about sense this Delta has a weak temperature the time-dependent and it's exactly soluble and
you can compute this survival probability can get functional both variables should finish up I notice there's a finite survival probability unit as equals 0 why because of inflation if the the the the the mutant on the surface and the service is growing it in its descendants are inflating away from the dangerous encapsulation by the wild-type neighbor and that gives it a little enhanced survival probability and like math this is this is a formula that but it but actually seems to work pretty well and I'll close by saying that we we we we can do simulations on the surface of of of a three-dimensional sphere like this 1 it's sort of like the surface of those ball bearings and sadly for this tumor but not for the human it got engulfed in died out that happens some fraction of the time the other fraction is this 1 that survived the Tories getting bigger it's just being really scale that's whether spheres are getting smaller so you can also see it the same theme of dome volume that area domain on the flight but there but the tourist getting bigger and it's has this logarithmic spiral cap and in the 3 dimensions we can calculate the survival probabilities functions the nationalist time and here too there is a finite survival probability it as equals 0 and so neutral mutations can sometimes called Hitchhiker's mutations that will hitchhiked onto the passenger mutations and sometimes mess it up mess up it's selective advantages by giving some surgical overhead so that's the the story :colon by just saying if the mutation rate gets higher you could have lots of competing logarithmic spirals that are born at various times this isn't mutation to simply ones the driver mutation 1 type but really there should be many we can solve
added 2 dimensions were now trying to solve 3 and
with that I think I will close and thank you for your
attention but the and I'm happy to answer additional questions people have maybe tired want to go to the bar yes this week was in use at the time in the history of the planet of he was the trouble but well you you you knew you could but I think we will only look at the simplest possibility of of 2 different alleles 2 different variations what actually happens in humans since we have a gene known that you know the actions billion to face pairs long is that there's lots of things going on in parallel we think of them as different colors and to to really do the correct inference for humans it's much more subtle than you might have thought but in fact there is there is there is at least you can get a sense of how to avoid some some potential traps but I don't think we're I'm at the point where I can reliably say anything about humans but here's a potential trap is James River so if this for humans and there actually is nature of science papers that makes such claims buffer humans by looking at special population genetics the the may have been retracted by now you saw this as all what's going on with this red but that maybe that's for humans that would be like in the gene for opposable thumbs 1st speech because look the red the Reds taking over there must have been some amazing thing that happened right In fact In this case that's not true the red it hasn't changed from his ancestral 4 minutes it virtually identical to green you can sequencing various tests and it simply viable luckiest there as to where the warts 1 on the left 1 on the right and they just didn't happen to me on this timescale and now because of inflation they almost never meet but it's not because of some great favorable mutation that you can attribute that to humans that that can happen till then you get a noisy logarithmic spiral and if he can show that slaughter in Mcdaniel knows something I'm so I I I will but I will certainly say that we we can't yet say anything directly about humans but at least we can confront simpler questions laughter like a hydrogen atom in physics isn't the same as really complicated thing like uranium but understanding how urgent is the 1st step toward understanding the rest of the periodic table and we hope we we've made a small contribution let's will be 1 answer if there are no further questions let me wish you all the best of luck in here scientific careers and the thank you for all your questions good luck to you
Atomphysiker
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Summer
Intervall
Bergmann
Konfektionsgröße
Tonfrequenz
Wolke
Leisten
Dienstag
Satz <Drucktechnik>
Minute
Grau
Prozessleittechnik
Kopfstütze
Biegen
Gasturbine
Kotflügel
Langwelle
Speise <Technik>
Familie <Elementarteilchenphysik>
Theodolit
Verpackung
Kaltumformen
Konzentrator <Nachrichtentechnik>
Großkampfschiff
Front <Meteorologie>
SIGMA <Radioteleskop>
Drift
Schwarz
Niederspannungsnetz
Juni
Diffusion
Nivellierlatte
Angeregtes Atom
Weiß
Jahr
Buntheit
Modellbauer
Lineal
Zentralstern
Quantenfluktuation
Takelage
Verpackung
Elektrisches Signal
Pulsationsveränderlicher
SIGMA <Radioteleskop>
Grün
Halo <Atmosphärische Optik>
Bikristall
Rauschsignal
Ruhestrom
Computeranimation
Feldemissionsmikroskopie
Drift
Prozessleittechnik
Jahr
Buntheit
Quantenfluktuation
Analogsignal
Theodolit
Summer
Magnetisches Dipolmoment
Konfektionsgröße
Linearmotor
Störgröße
Leitungstheorie
Computeranimation
Schlauchkupplung
Prozessleittechnik
Bildfrequenz
Fahrgeschwindigkeit
Vorlesung/Konferenz
Randspannung
Speise <Technik>
Familie <Elementarteilchenphysik>
Array
Wellenreiter <Aerodynamik>
Konzentrator <Nachrichtentechnik>
Kaltumformen
Feldstärke
Optische Dichte
Rauschsignal
Elektronische Medien
Übungsmunition
Drift
Wellenreiter <Aerodynamik>
Brillouin-Zone
Quantenfluktuation
Gesenkschmieden
Tonfrequenz
Regelstab
Postkutsche
Rückspiegel
Fuß <Maßeinheit>
Kotflügel
Speckle-Interferometrie
Klangeffekt
Morgen
Gasdichte
Röntgenstrahlung
Strukturelle Fehlordnung
Band <Textilien>
Bremswirkung
Mitlauffilter
Videotechnik
Mikrowelle
Buntheit
Modellbauer
Kapillarität
Münztechnik
Pelz
Schaft <Waffe>
Mikroskop
Maßstab <Messtechnik>
Stoffvereinigen
Kohlebürste
Flugzeugträger
Regelstab
Schlauchkupplung
Videokassette
Fertigpackung
Speise <Technik>
Energielücke
Kontrast
Tagesanbruch
Gelb
Tag
Schindelmacher
Gedeckter Güterwagen
Abend
Angeregtes Atom
Drift
Elektromigration
Weiß
Satzspiegel
Eisendraht
Buntheit
Sprechfunkgerät
Brillouin-Zone
Messung
Anrufbeantworter
Konfektionsgröße
Linearmotor
Erdefunkstelle
Staubmessung
Leitungstheorie
Monsun
Computeranimation
Klinge
Patch-Antenne
Vorlesung/Konferenz
Speise <Technik>
Familie <Elementarteilchenphysik>
Chirale Anomalie
Grün
Tag
Schwarz
Tonbandgerät
Drehen
Zylinderkopf
Läppen
Fußmatte
Klinge
Quantenfluktuation
Wärmeaustauscher
Myon
Kugelblitz
Linearmotor
Geokorona
Satz <Drucktechnik>
Kernreaktor
Schwache Lokalisation
Kabelader
Zelle <Mikroelektronik>
Bildfrequenz
Vorlesung/Konferenz
Familie <Elementarteilchenphysik>
Längenmessung
Grün
Optischer Verstärker
Rauschsignal
Diffusion
Rootsgebläse
Spin
Angeregtes Atom
Drift
Stapellauf
Weiß
Edelsteinindustrie
Jahr
Klinge
Analogsignal
Synthesizer
Treibboje
Regler
Masse <Physik>
Klangeffekt
Kotflügel
Blechdose
Tiefgang
Ruhestrom
Feldeffekttransistor
Avro Arrow
Umlaufzeit
Eis
Kommunikationssatellit
Urkilogramm
Front <Meteorologie>
Stunde
Umreifen
Front <Meteorologie>
Eisenbahnbetrieb
Mitlauffilter
Thermalisierung
Grün
Zeitdiskretes Signal
Modellbauer
Buntheit
Frequenzsprungverfahren
Myon
Greiffinger
Zelle <Mikroelektronik>
Nahfeldkommunikation
Fehlprägung
Satz <Drucktechnik>
Computeranimation
Klinge
Bügeleisen
Elektron-Positron-Vernichtung
Sternmotor
Trenntechnik
Kopfstütze
Erdähnlicher Planet
Fahrgeschwindigkeit
Stunde
Waffentechnik
Eisenbahnbetrieb
Tag
Weltall
Wechselrichter
Diffusion
Rootsgebläse
Passung
Gleichstrom
Sprechfunkgerät
Modellbauer
Flugsimulator
Windpark
Klinge
Feinkohle
Kaltumformen
Front <Meteorologie>
Grün
Tissue
Gesenkschmieden
Tonfrequenz
Wechselrichter
Begrenzerschaltung
Computeranimation
MAC
Angeregtes Atom
Mitlauffilter
Spheric
Hintergrundstrahlung
Jahr
Modellbauer
Vorlesung/Konferenz
H-alpha-Linie
Randspannung
Trägheitsnavigation
Randspannung
Modellbauer
Atomphysiker
Drehen
Zelle <Mikroelektronik>
Summer
Blechdose
Leisten
Stoffvereinigen
Geokorona
Leistungssteuerung
Eisenkern
Leitungstheorie
Computeranimation
Amplitudenumtastung
Woche
Vorlesung/Konferenz
Neutronenaktivierung
Stirnrad
Luftströmung
Theodolit
Grün
Tissue
Diffusion
Übungsmunition
Druckkraft
Plattieren
Buntheit
Betazerfall
H-alpha-Linie
Klinge
Papier
Neutronenquelle
Schwingungsphase
Kapillarität
Erder
Zelle <Mikroelektronik>
Konfektionsgröße
Treiberschaltung
Rasenmäher
Reaktionsprinzip
Minute
Leitungstheorie
Feldeffekttransistor
Woche
Domäne <Kristallographie>
Brennholz
Zelle <Mikroelektronik>
Fahrgeschwindigkeit
Randspannung
Speise <Technik>
Tonnenleger
Wellenreiter <Aerodynamik>
Eisenkern
Kaltumformen
Feldstärke
Front <Meteorologie>
Beschusszeichen
Längenmessung
Grün
Tag
Optischer Verstärker
Standardzelle
Diffusion
Rootsgebläse
Übungsmunition
Angeregtes Atom
Mitlauffilter
Spheric
Kernbrennstoff
Hintergrundstrahlung
Jahr
Mikrowelle
Buntheit
Flugsimulator
Ersatzteil
Fußmatte
Klinge
Leistungsanpassung
Anstellwinkel
Klassische Elektronentheorie
Ultraviolett B
Spiralgalaxie
Mikroskopobjektiv
Magnetisches Dipolmoment
Comte AC-4 Gentleman
Rastersondenmikroskop
Energiequelle
Sonnenstrahlung
Lötlampe
Hammer
Radialgeschwindigkeit
Steckkarte
Computeranimation
Discovery <Raumtransporter>
Kopfstütze
Fahrgeschwindigkeit
Schneckenpumpe
Magnetische Resonanz
Eisendraht
Längenmessung
Licht
Hydraulikleitung
Tag
Proof <Graphische Technik>
Teilchenfalle
Rootsgebläse
Trajektorie <Meteorologie>
Tinte
Mitlauffilter
Atomhülle
Sternmotor
Jahr
Atomhülle
Abhörgerät
Analogsignal
Anstellwinkel
Leistungsanpassung
Unterwasserfahrzeug
Spiralgalaxie
Zelle <Mikroelektronik>
Kugelblitz
Konfektionsgröße
Siebdruck
Treiberschaltung
Spannungsabhängigkeit
Rotierender Radiotransient
Geokorona
Leitungstheorie
Computeranimation
Zelle <Mikroelektronik>
Fahrgeschwindigkeit
Familie <Elementarteilchenphysik>
Wellenreiter <Aerodynamik>
Kaltumformen
Pulsationsveränderlicher
Rauschsignal
Diffusion
Rootsgebläse
Übungsmunition
Drift
Angeregtes Atom
Konfektionsgröße
Kontinuumsmechanik
Jahr
Flugsimulator
Spiralgalaxie
Irrlicht
Klinge
Natürliche Radioaktivität
Anstellwinkel
Temperatur
Treiberschaltung
Konverter <Kerntechnik>
Masse <Physik>
Atmosphärische Störung
Maßstab <Messtechnik>
Interferenz <Physik>
Hobel
Begrenzerschaltung
Schraubverschluss
Flavour <Elementarteilchen>
Edelsteinindustrie
Klangeffekt
Sonnenstrahlung
Gruppenlaufzeit
Minute
Raumfahrtzentrum
Mitlauffilter
Videotechnik
Nassdampfturbine
Zeitdiskretes Signal
Ziegelherstellung
Hintergrundstrahlung
Sternmotor
Modellbauer
Rucksack
Sprechfunkgerät
Eis
Pulsationsveränderlicher
Direkte Messung
Kugelblitz
Maßstab <Messtechnik>
Treiberschaltung
Geokorona
Satz <Drucktechnik>
Schraubverschluss
Computeranimation
Mitlauffilter
Domäne <Kristallographie>
Flugsimulator
Experiment innen
Straßenbahn
Luftkabel
Verkapseln
Exosphäre
Faltenbildung
Schwingungsphase
Fliegen
Parallelschaltung
Cocktailparty-Effekt
Teilchenfalle
Minute
Übungsmunition
Computeranimation
Woche
Kopfstütze
Uran-238
Buntheit
Schalter
Wasserstoffatom
Interferenzerscheinung
Planet
Papier

Metadaten

Formale Metadaten

Titel Master class with David Nelson
Untertitel An introduction to population genetics and evolution for physicists
Serientitel Physics@FOM Veldhoven 2014
Autor Nelson, David
Lizenz CC-Namensnennung - keine kommerzielle Nutzung - keine Bearbeitung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt in unveränderter Form zu jedem legalen und nicht-kommerziellen Zweck nutzen, vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/18029
Herausgeber Foundation for Fundamental Research on Matter (FOM)
Erscheinungsjahr 2014
Sprache Englisch
Produzent OpenWebcast.nl

Technische Metadaten

Dauer 2:15:19

Inhaltliche Metadaten

Fachgebiet Physik
Abstract Important ideas about mutations, genetic drift (survival of the luckiest) and natural selection (survival of the fittest), originally developed in population genetics, will be reviewed in a form suitable for physicists, with the aim of understanding the growth of bacterial or yeast colonies in a laboratory environment. When migrations of one- and two-dimensional populations are considered, results for mutation, selection and genetic drift are closely related to 'voter models' of interest in nonequilibrium statistical mechanics, suitably extended to allow for inflation of a thin layer of actively growing pioneers at the frontier of a colony of microorganisms undergoing a radial range expansions on a Petri dish.
Schlagwörter genetics
evolution

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