Okay, so we're going to do one last thing. So remember we're talking about evolution. Okay, I can start with showing you pretty pictures just to warm you up. We were talking about, we're talking about evolution of bacteria yesterday. Remember the long-term evolution experiment. So you can do the same thing on yeast. This is just a summary table of what people do in lab evolution. So we were mainly talking about this and climbing up the landscape starting from some initial individual who tries to mutate and do that. So you can do something similar with just DNA, mRNA, or just protein, some molecule. You generate a random library of different mutants, and so they fall in different places on this rugged landscape. And then you select them for, say, binding to something, like a protein, whether it's a good enzyme. And the ones that are left behind are left behind. And as time goes on, you find the winner. You find the one that lies here on this landscape. You can also do these kinds of in-lab experiments with breeding, so with mating. Yeast do mate in general. In the lab, they don't like to mate. Okay, it's kind of like animals in the zoo. So what they do is they mate them beforehand in happy conditions, and then they usually, what happens is it turns into more of evolution like that. And flies, on the other hand, if they want to reproduce, they have to mate. So people are now starting to do these kinds of experiments also with flies. Flies are difficult because they fly, okay? And so, you know, this is actually a problem. So already with E. coli and yeast, you have a problem with contamination. How do you make sure that something from this dish didn't somehow come into the other? Now imagine that this thing flies and tries to get from one test tube where it's supposed to be breeding with your controlled population to another one. So these are very hard experiments to do. And of course flies also reproduce much more slowly. So it takes more time.  And this is what you get as a result of these experiments. So for the in vitro ones, as I said, you really should select for a winner. This is like picking the best protein out of a randomly generated library. So at the end, so you start off with a lot of diversity. And at the end of the day, you get one winner. In asexual organisms, you get these patterns of what we talked about yesterday, clonal interference, where somebody wins for a while. Then somebody else that's better comes along and so on and so on. And this goes on and you're generally getting more diversity at the end of the day than you started off with. In yeast, the time scale here is shorter. So it's going up until here, there. You start off with large diversity because you outbred them. You made them made in the beginning. And then on short time scales, there's one that will win before the process starts looking like that. And this is just flies where, of course, they mate. So chromosomes get mixed up. That's what mating is. So you start off with specific chromosomes. And then you get these mosaic patterns, which is what happens in real life. So this is just to tell you that these kinds of things happen. But let us now think, go back to what we were talking about. And as you say, see, in a lot of these cases, you get this fixation phenomenon. So somebody wins. So yesterday, we derived this equation that I will just write back. This is the equation with selection and mutations. So we have the mutations and then genetic drift. So I think it looks like this. So now we're going to ask a question about fixation. And we're going to say, given that our population is initially at time zero, that there's some initial condition that we have, well, one individual of the new mutant, what's the probability that this mutant will fix? OK, so at time zero, you have one individual out of n. And so we want to solve this equation. So of course, we could go ahead and solve this equation. So we're after the question of what's the fixation probability. But that's rather difficult. There's an easier way of doing it. So physically, what is fixation? Grow. So remember, we were drawing these. This is the steady state probability. And fixation is when you get to one. And the extinction is when you get to zero. That's symmetric. So maybe extinction is easier. In terms of boundary conditions, does that look from, if I say boundary conditions, does that look like anything? Both of these are absorbing boundary conditions. Once you get to one of these, you don't leave. So how do we solve these kinds of problems? If I ask, if I'm telling you I start off with this frequency, and what's the probability of me getting here, what kind of problem is this in physics? First passage time, yes. So we're going to solve it as a first passage time problem. And how do we solve first passage time problems? What kind of equation? This is a forward equation. And we solve them by the backward equation. So we're going to write down the backward equation. Everybody's seen a backward equation in their life, right? OK. If you didn't, you will see one now. But so OK. So the general idea of backward equations, I'll write it down and then I'll explain it. I'm sure you've seen it. You may not recognize the word backward. If you've been overly tainted by mathematicians, you'd want to call it Kolmogorov something. So this is an equation for p of x, y given t, where x is the frequency at time t, which is the end of the interval given, what is it, that y is the frequency at time t0, at the initial time, OK? So we know, it's called the backward equation because we know what we want x to be. We want x, we want it to fixate, and we want it to be one. But we're going to ask, it's an equation for y, and we're going to solve it for depending on what y is, what is the probability that you get to x equals y, OK? That's why it's backward. We're asking, we're writing an equation for the initial condition instead of the final condition. In a normal equation, you're given an initial condition, and you're asked, what's the frequency at the end? Now I'm going to tell you, I know what my frequency at the end is, what is the initial, what's the probability of getting there given my initial condition. This is the adjoint equation to the forward equation, and this is how you derive it. And we're not going to, I'm not going to prove that, so, OK, if I tell you it's the adjoint, this form should be obvious to you. You write this in matrix form, and you take the adjoint of the matrix. So that's why these things come out of the derivative, and that's why the sign changes here. This is like a momentum operator in quantum mechanics, right? It's the first derivative, the adjoint of the first derivative gives you a minus sign. I'm giving you analogies, right? I'm waving my hands, because in principle you should have seen it. But if you haven't, that's where this comes from, OK?  So specifically we are going to solve it, so we're actually going to solve for the extinction and then say that the fixation is one minus, the probability of fixation is one minus extinction in the approximation that you're in this limit where you're either go extinct or you don't. So we're going to say we want x equals 0, so extinction. If you want a title for, yeah, OK, fixation probability is what we're after. And so that means that we want p of x equals 0 y of t, and I'm going to shortcut my notation and just call it p y 0 t, OK? So we're writing this equation, maybe I don't need to rewrite this equation, OK? Let me just rewrite it for the sake of being general. So I'm just rewriting what you had, OK? I'm even going to drop this 0, because I'm going to drop this 0 I feel from. We're going to look at the long time limit. We're going to look at t goes to infinity. So we're going to ask what's the probability of extinction at long times, which is essentially we're asking what's the probability of it going extinct at all, OK? So we're solving the steady state form of this equation. And now this becomes easy. I can define an auxiliary function, which is just the derivative of d pi ui dy, OK? For those of you that are bothered by, I'll try and keep my capitals and non-capitals the same. So if I define p y dy, well, first of all, if I have a 0 here, I can get rid of d's. So then I have, from the second one, I just have dy, whoops, I'm sorry, this is dy, of course, yeah, OK? So I have dy equals ns with a minus sign r of y, which is just an exponential equation, right? Everybody agrees? This is solved by an exponential, good, r of y equals 0, OK? So now we have to unsolve that. So p of y dy equals, so this is r at y equals 0. I'm just going to call it r0 e to the minus sy. So I can do this by direct integration of the two sides, get minus r0 ns e to the minus ns minus 1. And on this side, I get p of y minus p of y equals 0 equals to that. Nothing interesting happened. But now we know that p of y equals 0 is equal to 1. Because if p of y is the probability of going, basically, I'm asking now the question, what's the probability that I end up at 0 given that I started at 0, right? So if I'm already there, and it's an absorbing boundary, I won't move. So this is 1. So that means that p of y is 1 plus r0 ns 1. Sorry, I missed the sign here. Negative. All my signs are OK. OK. And then I happen to know another thing, that p of 0 of y equals 1 is 0. Because if I start off fixed, so if I start off in 1, again, another boundary condition, I'll never get to 0, right? This is the probability of getting to 0. If I start off in the other sticky side, I'll never get to 1. So that's 0. So that gives me a way of calculating r0, which is 1 plus r0 ns ns, right? So this means that r0 over ns minus 1. So if I put all of this together, then the probability is 1 minus 1 minus e ns y 1 minus en. And so since this was the probability of extinction, and I'm interested in the probability of fixation. And as I said, I'm working in this limit of weak mutation, essentially of weak mutation, that I just look at the case. So I should say that mutation is much less than 1. I'm considering a case where I just had one mutation, and there'll be no other mutation before this process manages to go through, right? So the idea is that basically all of the probability is either here or here. So I can approximate the probability of fixation as 1 minus the probability of extinction. So that means that the probability of fixation is this nu mu much less than 1. Because I can write it t fix is up to 1 minus 1. OK. Sorry? Where does the time, if you know the time, the system will go through one of the points. But what is the difficult time? That's a very good question, and that's what we'll do in the second part of today.  But in order to do that, we need the answer from this. So this is, you know, just be clear about the approximations. There's clearly this approximation, and it's, yeah, nothing interesting happened in the time that it takes for this to happen. You can calculate the time for fixation, too, but we won't, we found this formalism, we won't do that. OK, so we'll start off by analyzing what this gives us. OK, so the first limit, let's look at the limit that selection is strong. So in this limit, p fixation, which is given on that board over there. So I'm going to expand in the limit of ns large. So basically, I can delete whatever, you know, in the denominator, the one will dominate, and I'm left with e to the minus ns y. And if we say now that y is 1 over n, so initially we have one individual at the time we're looking at, then that means that this p fixation becomes 1 minus e to the minus s, which again, for strong s, I can expand this 1 minus 1 in the s, and so this is equal to s. OK, so this has a name. It's called the Haldane result, that the probability of fixation is proportional to the selection for a beneficial mutation starting from one individual. OK, if we start from k individuals, which essentially means we start looking at it when this k individuals, which is often what happens in experiments because before that, we may not see them. We may not be able to detect them. Well, then we have the probability of fixation to be 1 minus sk, and there's a threshold here because it equals 1 if k is 1 over s, and it equals roughly ks if k is less than 1 over s. So that tells you that there's a threshold at 1 over s, above which if you see an individual, so if you see them in k individuals where k is larger than the selection, 1 over the selection coefficient, that's almost certainly going to fix. Well, certainly going to fix. However, if you see less individuals than 1 over s, well, they still may fix, but their faith is much less likely. OK, so that's case 1. But this is a result to remember that generally, if selection is strong, then the probability of fixation of a beneficial mutation is proportional to the selection strength. However, if selection is weak, and this can be either beneficial. So this is beneficial because it assumes it's a beneficial mutation. So it's a mutation that's good for you because it assumes that this is positive. OK, we're going to do the negative one in a second. OK, so this is either a deleterious or a beneficial one, but with a weak selection coefficient. So this is the general result. And then we expand in the limits, and we get that it's proportional to y. So now, if selection is weak, the probability of fixation doesn't depend on the selection strength. It actually just depends on how many individuals you had in the beginning. So really, so what's interesting here is that whether you're beneficial or a deleterious mutation, so your mutation that's good or bad for you, your probability of fixation is completely the same. Right? It just depends on how many of you are there at the beginning. So if I now ask you this question in a different way, can a deleterious mutation fix? What's the answer? Yes? Yes. With a certain probability, but this tells you yes. It tells you it's not impossible. So just because you see a mutation somewhere, and like all of us, we can all have the same deleterious mutation. And all the yeast in the Petri dish can get a deleterious mutation which can spread and just by chance. So it's not impossible for bad things to happen and survive. And it's even stronger than that because you can imagine having strong deleterious mutations. Okay, so this is deleterious, which means bad. Okay? Beneficial means good. Deleterious means bad. Okay, so in this limit, we do the same game and we expand and we find a certain probability, which is arguably less pretty, but it's an exponentially decaying probability with the selection coefficient. Now remember that this is this, you know, so okay, so you can say that, so first of all, again, the answer is even a very bad deleterious mutation can fix. It decays exponentially with the strength of how bad it is, but it can still fix. And so, okay, so basically this is exponentially small. So it is small, P fix. I write it here, but you know what I'm saying. It's exponentially small unless 1 minus y is much smaller than 1 over s. Okay? So unless you're already very close to winning, you're probably as a very bad deleterious mutation. You are going to go extinct, but still this is probability, so anything can happen and you may actually fix. So let's draw this. The probability of fixation as a function of the selection coefficient. Starting with 1 over n between minus 1 over s and 1 over s, you basically have, if you start with 1 over n individuals, so 1 individual in your population, you have a constant probability of fixation relative of whether you're good or bad. Here you have an exponentially decaying probability of fixation. Here is 1 where you're certain to fix and in this intermediate regime. This is for, again, n mu. So to connect this again to experiments, there was a pretty nice experiment from another guy who got annoyed at all this. All we think in these abstract terms about fitness landscapes and reproducibility and how do you evolve from point A to point B? He went until the lab and he just did it. So this is Dan Weinreich in paper from 2006 where he took a protein, beta-lactamase, whatever, that conveys, confers antibiotic resistance. So you've probably all heard about antibiotic resistance. As the general, we as society have been eating too much antibiotics and now if you're old and you go into hospital, the thing that is actually likely to kill you is the hospital because of the bacteria that, which doesn't mean I'm being recorded, so may I shouldn't be saying things like this, which doesn't mean you should not have your grandparents go or your parents go to hospitals. Hospitals are very good and all that. But antibiotic resistance is a problem and especially for the elderly and very young, so people are very, very interested in this. Okay, but essentially it doesn't matter for what's going to happen. So what they knew about this protein is that if it gets five mutations, so this is the fifth mutation over here, it's gonna be very resistant to antibiotics. And if it doesn't have any mutations, then it gets killed by antibiotics. Antibiotics is measured in terms of something called MAKE, which is called the Minimum Inhibitory Concentration, which is essentially the lowest concentration of antibiotics that prevents visible growth, so what essentially kills the bacteria, okay? So you put in different concentrations and at some point the bacteria dies and that's what this is. Just that sort of point of interest. So what they did is if you have five mutations, that means you have 120 mutational trajectories to get from the beginning to the end, right? You have 120 different ways in which these five mutations could have appeared. And so they built all these 120, sorry, they didn't build the mutational trajectories and you have two to the five, that means 32 different mutants, right? So you can have all the point mutants, that's five, then you have all the double mutants so those that have two out of the five and so on, okay? So they know exactly what these five mutations are and then they combinatorically built all possible versions of these mutants, okay? So that's 32. And then they measure the fitness to the two, sorry, they measure the fitness which means they measure how resistant they are to antibiotics. They put in different concentrations and they watch when they'll finally die and here is a table for all of these mutants so this is no mutations, one mutation so on and all five and then it gives you this concentration of the antibiotic at which they die so the higher the number, the more resistant they are. And the first thing they sort of notice is that the order of mutation, well the combinations of mutations actually matter. It's not an additive process. So if you have this mutation, you don't essentially get any resistance and if you have this mutation, you don't get any mutation either but if you have both of them, you suddenly get a huge increase, okay? So you go up from this 088 which is nothing to this which is something. So what you're saying here are clear interactions, right? You have plus and minuses so you think these are like spin variables so if you think about like an Ising model you have a J-I-J term. This is what this is telling you that the effect of having the two is not the additive effect of having each one of them separately. So in evolution this is called epistasis. Epistasis is just the word for interactions and you see that a lot. This is sort of another example of things like that. So basically the background on which the mutations happen matters a lot and it also turns out the order of these mutations matter and the way they figured this, well what they did then is they actually used the sort of formalism that we were just looking at and they said, well I can calculate the probability of go for every trajectory I can calculate the probability of this trajectory taking place. So they say mutations happen independently so if I want all five mutations to happen then I need all five of them to happen independently it factorizes and then they say, then they make a very strong assumption they worked in the weak mutation strong selection limits so here and they say, well in fact if the mutation is deleterious or neutral then the probability of it surviving is zero, okay? So that they basically equate this to zero because they know that all of these mutations well, okay, they don't know that but so that's what they do and they assume that everything else is strong and they consider two models where there's a constant fixation rate and one where it's proportional to one over the resistor it confers and doing that basically they find if they get rid of all the trajectories where you would have something be deleterious so that means the fitness goes down when you add that you're left instead of 120 trajectories with only 18 possible trajectories, okay? So the take home message of this is that not all of the 120 trajectories are possible and the reason they're not all possible is because of these epistases because of these interactions, right? So I mean this should be somehow running bells to you because if you think about the network graph, right? There's some moves that are forbidden because they would decrease your overall function which is here fitness so you can't go there so that severely constrains the number of paths you can take and in fact if you plot the cumulative probability of the probability of the trajectories half, if you want this cumulative probability to get to a half you don't need many trajectories so depending on the model two or four trajectories basically capture half of the total probability so they take out most of the space so this really, you can take this down even from 18 to essentially two most likely trajectories so okay so this is plotted on this graph where the yellow line is the most likely trajectory and these are all the 18 ones but basically the first and the second most likely trajectories take up most of the probability space of all trajectories so when we think about this I started off with telling you that evolution is the most random of all processes, right? Because all mutations is random whether they arise and so on but it turns out that if you're actually evolving something specific the number of ways of getting there is much more limited than you would have initially thought we got down from a hundred and twenty to essentially two possible paths okay now how are we doing on break time I still have some time okay so let me continue with telling so just there's another way of this is another antibiotic resistant experiment from 2015 where they build something called the mobidostat okay which is basically a bacterial killing machine and so as I you know as as in the previous experiment we saw we have these bacteria and if we give them and if we give them more concentration of antibiotic you know it so there's some concentration of an of antibiotic at which they'll die so these people said well we don't want them to die we want them to constantly be on the edge and so we're gonna put in some concentration of antibiotics and then let them evolve resistance in this and if they manage then we'll pump it up and give them more okay so the bacteria are constantly pressured to find new solutions and so they did this with three types of bacteria and you know unfortunately the the bacteria do very well in this experiment they do manage to find new and new solutions and this is you know in a way what's happening in in our world right we we've been giving ourselves more antibiotics and we bacteria have been finding new solutions but I won't go into all these details of these uh... graphs but what you actually see here is when they do replicates and then they sequence the final population uh... these bacteria in different replicates find a very similar solutions so again there's some notion of reproducibility okay so they uh... these are the way to read this is these are four replicates as one fourth of the pie chart and so this is time and as time goes on you see which fraction of the bacteria have found the same solution at the gene level okay they don't find exactly the same mutation but they say find the same phenotypic solution and you see that many of them in many cases they find solutions so these are different kind of uh... bacteria so it is and you see that the fitness goes up so they keep on finding things okay so this is for the next part uh... so before the break let's do one simple estimate and then we can uh... then we can take a break okay so this this this is supposed to do that you know this is supposed to be as sort of crazy estimate time and the question is how many cell divisions have there been from the beginning of life okay this is i'm sure i'm kicking you way out of your comfort zone because you're mostly theoretical physicists which means you like to solve equations and you don't like to do estimates because estimates are kind of bullshitty right so that's what we're gonna do okay beginning of life when did life begin who knows on this planet yeah life any life any ideas yes ability through three point five billion years but essentially ten to the nine years ago well done your your your checks in the mail uh... okay then the next thing we saw this is how you know when life began okay so how many cell divisions per year so okay so let's let's think about the ecology because that's the only thing we can think about uh... so the coli in the lab in the wild this is of course not true and we don't really know but it divides about twice per hour okay so two times means forty eight times per day which means forty eight times forty eight times three hundred and sixty-five days which is of the order of ten to the four per year okay this is what you get from multiplying this numbers if you say this is about ten and this is about ten to the three you get that or if you say this is ten squared and this is ten squared you get that but if you actually multiply these numbers you get something that's close to ten to the four not that it matters for this level of bullshitty estimate that we were doing so we have ten to the nine years times ten to the four divisions per year which gives us ten to the thirteen divisions since life began okay well sorry these are just the number of divisions for once one lineage so now we have to multiply this by the number of organisms we have on earth any ideas ten million yeah give me it's a log question i'll help you there i don't know the name of this number if this is a hint up up thirty five is that what you said yeah yeah yeah yeah that's a very good estimate you said you said twenty five okay so you're roughly in the ballpark people estimate ten to the thirty okay we can do we can do an estimate to where where the you know this is obviously a completely bullshit number let's agree on that right i mean okay you can count the number of in principle you could try to do this and i think they have done it in a in a better you know in some sort of sensible way but one way i'll give you away where you can sort of get as i said it's a log question so whether you say twenty five or thirty five it doesn't matter really right yeah yeah yeah yeah yeah yes yes of course yeah no you're it's no no you should be integrating an exponential probably okay i as i said let's okay let let me if you know like i'll get back to so let me tell you how you can estimate this number in a fast way and so because it'll give you fun numbers and so let's say let's let's think about bacteria okay because bacteria make up essentially ninety eight percent of say marine life so anyway we all agree that's more bacteria in the world than mammals or vertebrates right no no question about that okay so bacteria rule okay so so where do we find bacteria we find them in our gut so how many bacteria do you have in your gut right now after a few wonderful days of cafeteria food in place again but you said billion very close yeah about a billion ten billion ten to the tenth okay okay how many people on earth this one you should know all you really have to go to that sustainability session yeah okay so again we're in the same ballpark so that gives us already ten to the twenty bacteria and then i'll give you a number that you probably have no no idea of i mean yes you can estimate it from some weird ways or you could also do this estimate using like how many bacteria are there in one liter of ocean water and stuff like that but then you have to know that and uh... that's not a number okay so it turns out that humans account for ten to the minus four percent of a non marine biomass so ten to the minus four percent is ten to the minus six in fractions okay so if we multiply these two numbers uh... actually divide by that you get ten so ten to the minus six ten to the minus seven whatever ten to the twenty seven uh... bacteria okay so as i said order of magnitude it's quite close but we get of the order of ten to the thirty organisms modulo the problems of things expanding than the you know the expanding and so on i'm just gonna multiply ten to the thirty by ten to the fourteen which is the number of cell divisions and this will give us approximately ten to the forty three cell divisions since life began okay apparently a more clever estimate gives you ten to the forty six as i said completely doesn't matter i don't believe either one of these numbers but it just gives you an order of estimate right so if you were to sort of say from the top of your head it's like ten to the fifty cell divisions from since life began ten to the forty ten to the fifty but now the question is what if that number was ten to a hundred cell divisions okay what if it was you know power square of that number would we look different would life look different sure i mean the you know so that's that's your your thinking exactly or along the right ways i mean the honest answer again is we have no idea right uh... but that the problem is that we don't have and we don't have an idea because you know we haven't mastered partly because we haven't mastered the theory of evolution well enough so one thing i can tell you so think back to the lenski experiment yesterday so lenski's lenski's experiment has been going on for sixty thousand generations okay at around generation thirty thousand lenski bacteria ecoli learned to utilize citrate which is a food source uh... and this means nothing to you but i'll tell you that the definition of ecoli as a species is that ecoli are defined as a bacteria that cannot utilize citrate so generation thirty thousand one strain from the twelve strains of the lenski experiment learns to utilize this thing that they are by the name forbidden to do and so they form a new species right if we believe in bacterial species okay so they've done something funky it's like i don't know us growing a trunk and suddenly being able to take a shower by ourselves right okay so but then the question is should we be surprised how do we gauge the element of surprise okay that's and we know since that's what statistical physics is about right we estimate surprise we calculate probabilities that gives us a concrete estimate of surprise so on one hand you're gonna say yes i'm surprised because these bacteria we gave them a name that says they cannot do this so how did they learn to you know they've really done something out of the way but on the other hand you can ask the other question it was one in twelve lines of experiments why didn't more of them do it so should we be surprised at only one or should we be surprised at even one right and again we i'm not gonna answer this question for you because we kind of stuck at the theory but uh... that's you know that's what we'd like to be at least you know forget about the question of whether we would look different and so on but we would at least like to know whether we should be surprised by this one line of lansky i mean i i think that's an achievable goal alive okay so with that let's take a break and come back it's it's a j-con mat and i don't remember whether it's two thousand eleven two thousand twelve something like that okay so it's it's like because if you look into a biology textbook you're gonna have way too much details you don't probably don't want to do that there are textbooks that's the physical biology of the cell by rob phillips and uh... collaborators uh... but it's most it's more that the the physics is easy most of the time and you it's it's a it's this fake okay so you're not gonna just read it and it has a lot of most of the things we haven't talked about but again reading maybe an introduction to a relevant chapter may help you and then this bio physics by bill beallic uh... which in some of the things you'll hear here may you may find that i don't even know some of the things you'll hear from terry you may find that but again it's this fake and this one is actually aimed at physicists it has a lot of information okay but uh... and i have a feeling those are not the textbooks you're looking for so i would sort of recommend uh... checking out wikipedia seriously for like a basic idea of the biology quickly uh... okay so now we're going to talk about the rate of adaptation so organisms of all and they have all through this mutation selection and the question is how fast will the population of all so that basically means what is the speed with which the fitness changes uh... and we're going to think about the constant environment similar to lansky uh... there's another question of how similar it will these mutations be which i've sort of been telling you that in fact they may be more similar than you expect although you know on one hand the similar on the other hand and not because these lansky strange to different things so okay but it's well mine will talk about uh... that and so maybe another so we've already said the laterious mutations can fix right and others question uh... which you have not seen the answer to this one so this is really a question uh... did which is very nonintuitive but i can ask you a question if you increase the size of the population will more or less the laterious mutations fix okay so you have your say ten to the fifth pop bacteria now you make it ten to the seven and that will more or less of the bad guys fix less more i hear both who says more yes okay somebody says more so the answer is more but it's completely nonintuitive uh... you know the intuition should be good that less will fix because it's harder for them to sweep through the whole population but act in fact more will fix because they hitchhike with the good guys so you're still going to get more beneficial ones and the deleterious one is going to be next to the good one and it's just really going to hitchhike right it's gonna we go along with whatever happens to the good guy but it's completely not obvious and but this thing you can actually get out of of formalism okay so we're going to consider something like the lansky experiment constant population size and mutation rate you as a each we're going to say that each mutation is beneficial so we're going to make things simple and that they have a selection coefficient s we're not going to talk about recombination as i said completely unrealistic there and so the first thing we're going to do is we're going to consider the case case where mutations are rare so that's that's the same limit that we worked out the fixation probabilities and we're going to ask uh... how often does the first question is how often does a mutation occur in any individual or in some individual in any individual right then in some individual i'll add the other population size and so you have n individuals and you have a mutation rate how often does the mutation occur n times u so the time that you have to wait for one mutation is one over n u okay so for one over new time nothing happens so then they can uh... so then it occurs and most of them of course quickly go extinct due to genetic drift but what is that probability that the mutation actually survives genetic drift so it's so we we can say that if it survives mutating genetic drift it establishes so just for simplicity for the estimate let's say that it fixes what's the probability that a mutation fixed in this limit from the first part of the lecture today strong selection week mutation strong beneficial selection remember that result s that i underlined and said it's important now you know why it's important for other reasons but so after one over n u s well you know we have these ones here that go extinct but we'll have one here that will go above one and then it will drift against some more and it'll go it'll become larger than one over s after after some time and after this time it'll grow deterministically okay so how long it will so and then after some time ill so it will grow deterministically and then it'll fix some max value of that it'll find or essentially it will fix so can we calculate now this deterministic time okay so okay so the so deterministically when it grows how does it grow exponentially thank you so that means that the population the number of these back mutants which started off as one over s then it increases exponentially with a coefficient s and we want to ask what is the time after which they'll take over the whole population right which is n so we can solve this equation and get that this time deterministic time is one over s log ns so that's one picture this is the picture for rare mutations so the next question is when does this whole picture break down exactly when the mutation is not rare and so what does it mean that the mutation is not rare right so look okay I didn't draw this perfectly well the mutation is rare means that the time for a mutation to actually appear and come up to here is much longer than this time it takes for it to fix right because it remains that this happens very rarely and then this is deterministic so it'll go okay so that means that this time is much longer than this time that means that one of a new s is much larger than one over s cancel and we get that this means that n over u is one log this is what rare mutation means okay so so maybe first let's see how possible this is for this to break down okay so now we know you have the 10 billion a E colite in your gut and so let's say n is of order of 10 of a billion and let's say they divide more rarely than in the lab so let's say they divide 20 times a day um does that this actually doesn't matter okay so and the genetic mutation rate of E colite so u overall is about 10 to the minus 3 per genome if you're being conservative but not all of those mutations are beneficial in fact about one in 10,000 mutations are beneficial so if you want the u of beneficial mutations which is what you want then you have to divide 10 to the minus 3 by 10 to 1, 2, 3, 4 is that right uh yeah so you get 10 to the minus 7 um and if you multiply that now uh by n so n u is well okay seven yeah so let's say this is 10 to the 8 10 to the 9 this is 10 to 100 okay uh selection coefficients are about 0.1 to 0.01 and so this gives us an ns of about 10 to the 6 so 1 over log 10 to the 6 is 10 to the minus 2 okay so what we're comparing here is n u which we got to be 10 to 100 to be much less than 10 to the minus 2 okay so that's clearly not true so this is even being conservative uh what we get is that for typical numbers that we have in bacteria mutations are really not rare which in a way we've also seen from those experiments that's why we see those clonal interference patterns okay which sort of yeah that's a very good question in general what we mean is a mutation that will uh increase the growth rate of an organism but as you saw in the example earlier today it depends on the context so there can be mutation happening in a specific position that if it happens in a given background it's good and it'll increase the growth rate of the bacteria of the organism but if it happens in another background it'll be bad so I give it the same exactly the same mutation so the same going from A to G in one context can be good and another can be bad so beneficial and deleterious on top of that that's even in the same environment now reality happens in fluctuating environment so in different environments right the environment we find ourselves in is constantly changing so something that's good in one environment can be bad in another so I mean just as a simple thing you know imagine you find yourself in a in an environment where there's tons of one food source so if you develop the ability to use that food source that's good but then you finally find yourself in an environment where that food source doesn't exist at all and now you're really specialized in using this one food source in fact that mutation you got which may said I can only eat this like I can only you know eat milk and then you find yourself in rural China you're you're not going to be doing very well right in an environment where there is no milk okay so this is this yeah very so that's what people are doing now that's a design you know the question of how do how does a changing environment influence evolution this is I'll show you an experiment in a second but it's also theoretically that's what we're doing many other people are doing you know there's questions of matching of time scales of the environment to evolution all of those questions are out there so these are you know this is part of why we don't understand anything but these are all very good questions and this is also why the answer to you know in in simple terms deleterious means bad and beneficial means good but what is good and bad really depends both on the genomic context and on the environment on everything so it's a phenotypic definition it's an effective definition okay so yeah no sorry this this says mutations are rare we're first doing the limit of mutations are rare this is the number this is n of t n of t is the number of bacteria or whatever of it's the number of organisms because you've survived genetic drift right so you've you your population size has grown enough that you're less susceptible to the fluctuations coming from small numbers an estimate yeah so okay no this is yes well the another definition wait so okay the philosophy of this is I did an estimate for the time scales for what happens assuming mutations are rare okay then I asked the question what does it mean for mutations to be rare right how do I define the limit of rare mutations and I based on these estimates a mutation is rare if the time this determinist whatever happens deterministically is much smaller than for a new mutation to happen okay I think it'll become so now I'll ask the question of what is what happens when mutations are not rare so what happens biologically here we had a situation where one mutation appeared and it was left alone until it fixed and took over the whole population now if mutations are not rare what will happen yeah so what will happen is basically before I managed to fix this one a new mutation will appear right so I'll have many mutations at the same time so if this is not true I will have many mutations and that's what we've been seeing on on these pictures where we see these different colors right a mutation appears here but before it manages to sweep this one appears and this one and this one so there's new mutations coming in all the time so we're going to have many lineages right a new mutation forms a lineage and the lineages are going to come in and they're going to interfere with each other that's why it's called clonal interference because a clone is a lineage essentially okay so that's what we're going to work out now okay so before maybe we work out that out now we can still ask a question about this limit about what is actually the speed of in the rare mutation limit so what happens so if we think about this picture a bit differently now if we think about the fitness instead of thinking about time I started off with what I'll define the initial individual that had zero fitness this is the fraction of the population and I got a mutation and I started a new lineage that has a fitness of one right it has it has one mutation so in terms of mutations it's higher in mutations by one and it's better so what will happen is this one will grow and this one will decrease right this one is better so it'll overtake that one as time goes on so at a later time my graph will look like I drew with the green one okay and so what the speed of adaptation is is how quickly you you change from this picture to that picture okay so what is the speed so to calculate the speed you have to calculate the rate of new mutations establishing so that we've already determined right this is the rate at which they overcome drift and once they establish the population grows with rate s it's growth rate or selective advantage okay so that means that the speed of adaptation is nu s squared I'll write it down here in case I delete other things so in the rare mutation rate of new mutations establishing right this is when they overcome drift and this is just once they establish right they grow with this rate because this is their selective advantage so the overall speed of going from this picture to that picture is given by that it's on average right okay I should also say for those of you with a mathematical inclination or maybe a rigorous inclination what I'm doing now is I'm doing a back of the envelope calculations there is a rigorous calculation there's rigorous calculations to do here based on generating functions and they're very beautiful calculations but we just don't have time for them and I want to get to a result if you're interested I can send you I can I can point you to the sources but I am I am waving my hands and I am waving my hands because I have the power of the rigorous calculation behind me so I know what the answer should be okay okay so now let's go to the non-rare regime so if we look at that same picture now we start off with zero mutations we start off with zero mutations and then a first mutation establishes so same thing it starts off small but it starts to grow because it's better right so it starts to grow whereas this starts to decrease but as it grows mutations are no longer rare so a second mutation will come about right and this grows even faster than this one so it grows grows grows but before it and this one you know keeps on decreasing now and before this one gets very far a third one comes about right that's what means mutations are not ready key one happening all the time so at the end of the day and this one keeps so at the end you know so this one will well this one will grow even more so it this but at the end of the day this one will have shrunk this will you know you'll get some sort of picture like this and a fourth one and so on so essentially whoops I lost my chalk you will get something that looks like this okay because the ones at the top grow the most quickly the ones here shrink the most quickly and all of these grow slightly less than the average the arrow should be symmetric but you get the idea right so if you're in the mean we can now renormalize and call this zero fitness and then there's the ones that are better than the mean and the ones that are worse with the mean okay you intuitively see that this happens right and so you get this distribution which actually you can work out from mutation selection balance in the probabilistic set and it's roughly a Gaussian but the other thing you may notice that happens is that this never stops that this is a wave that will move like that because there'll always be a fit a mutation so we're assuming they'll always be a fit a mutation and there'll be a wave that travels through this fitness picture okay so you're gonna get this traveling wave you're always translating to higher make this longer and this is what we can now calculate the speed with which the wave moved so maybe one control question what happens if this population size will increase what happens to this wave to the Gaussian yeah okay so you start you know you establish these things that are more and more fit and the fitter they are the more they grow that's the definition of them being more fit okay so but the fitter they grow since we have constant population size that means the faster these ones have to die and if you it happens with time then you'll find yourself with this mean of the the most individuals are in the mean of the distribution there's the head of the distribution where you get the new ones that are very fit but they just came about not too long ago so they haven't had time to grow and the long the more they are in this direction the older they are so the more time they've had to grow but the slower they grow so this is there this is fitness and this is the fraction of population so you'll find that most of the population is well in not very fit but not very bad because then there's the ones that are very bad which are the ones that were here a very long time ago and eventually they go extinct okay so you get this shape and i'm not proving to you it's a Gaussian but you can you know it depends on whether for beneficial mutations yeah i mean in a limit it's a Gaussian it's more like a Poisson okay but you can you can do the calculation you can work out exactly what this distribution is but it looks it looks like a you know little Gaussian envelope that the important thing is it always it will continue moving there because we're making the assumption that there's no ceiling there's no optimal solution you can always get better so it'll continue moving and if the since it's the mutations that make it move the rate of mutations are constant so it'll move on average at the constant rates yeah but there's a different calculation you can do the same so in if all the mutations are deleterious then yeah then then you have something like that it's called mules ratchets and you have the same same thing so actually the the you can build the formalism and people have with both beneficial and deleterious and the okay right so now if the population size will increase right so what happens to the distribution it gets wider exactly because it'll take more time for you to kill the bad guys and okay what if you increase the mutation rate same thing right also gets wider because now you're going to be producing more and more of them more quickly okay so question what is the speed of this wave in 15 minutes so we're going to break this up into two regimes we're going to break it up into two regimes so the first thing we're going to notice we're going to call the fittest class q okay so we're going to call this distance q and we're going to first assume we know what q is and then we're going to calculate itself consistently but we're going to notice two things that there's in fact two regimes to this distribution there's the bulk where there's a lot of individuals and we can trade the bulk bulk deterministically because there's no stochastic effects here right everything's been around for a while everybody's there probably in fairly large numbers however there's a lot of non-linearity here because there's very i drew it like this but in fact there's very many lineages all the individuals here are not the same right because they came from some different ancestor so there's many lineages interacting with each other and competing so there's high non-linearity but no stochasticity and then we have the head where there's a lot of stochasticity because they have very small numbers they just arose but there's no non-linearity because we're probably dealing with one lineage so we're going to treat these two different regimes differently and we're going to we're going to couple them okay so this is q so then this will have an advantage of q s the head of the distribution now okay so let's assume we know q for now and let's calculate the so the number of individuals in the nose so in this head of the distribution at time t is just like the previous time deterministically given that they've established they have a fitness e to the q s and then they grow with rate e to the q st okay so the question is how long does it take for an even fitter individual so one that has q one plus s to arrive and so we know the answer to that because it's the same thing as we did before it's the mutation rate times the number of individuals times the probability that they establish okay so to do that that they grow to a certain length and we can ask okay for new fitness class to establish and we're going to do right so we're going to ask how long if they're this is the probability of them appearing um dt how long before they reach a significant number and we're going to say that the significant number is one it can be five if you want it really doesn't matter right it's just order of magnitude do like that so we plug in the results for the nose and so we have zero to tau the q s cancels e to the q st dt equals one which is e u to the q s e to the q s tau minus one is equal to one if q s tau is probably larger than one for any reasonable value of q so i can assume this is smaller if you see q s tau is larger than one and so tau equals one of the q s logarithm of q s okay so this is the time for a new fitness class to establish i'm going to mark this with a dot okay and so every tau steps we're going to get yet a new fitter one this is a separate thing just okay now let's try and calculate q from the fact that the mean also changes right because as this wave travels the mean also changes so after q so okay after q of these towels this will become the mean right because we will have clicked q times and what is here will now come here so on average if you're growing at a rate zero here and a great q here and it takes you tau times q to get from here to here on average you're growing at a rate of q tau in the time of q tau over two sorry q sorry q s over two during this time you're you go from being here from being at the nose to being at the middle we're going to demand that every time the nose increases by one the mean has to increase by one so that the wave is really traveling symmetrically and we're going to say well if i start of one over q s individuals just as i do in in the nose after q s t and i'm on average growing with this rate as time goes on so i'm averaging over the growth rate in the bulk how long what's the time i need for me to get to here where essentially i take up most of the population right this is the bulk so i take up nearly all of the population okay of course i take on really n minus the wing the integral of the wings but the majority of the population is still there okay so this time so we can say i'll write it out for you time to go from nose to the mean okay so this is just from solving this equation yeah yeah because we're saying i want to estimate a raft time it takes me to get from being the best to being average okay like you start a new trend you're cool and then everybody else is doing it right this is the time it takes for you to stop being cool on average you're growing in all of this i mean your growth rate here is q here's q minus what q sorry your growth rate here's q times s right your advantage is the number of mutations you have here if you're here then it's q minus 1s q minus 2s q minus 3s and so on so over this time on average you're growing with a rate of qs over 2 because here's 0 so assuming that that's your growth rate on average and you started off with 1 over qs individuals how long does it take you to basically dominate the population and if you solve it then you solve for t and you get that and this by definition is q tau yes okay it's a bad as i said i agree this should be n minus integral of the distino okay but if it's a peak distribution it's not that bad of an approximation look i as i said i'm doing this because i know it'll come out roughly okay but i know i agree with you it's it's a it's a it's a strong approximation okay but you're getting you'll see you'll see why it's not so bad of an approximation you're going from zero to some q and it has to fall off once you see what q is maybe you'll feel a bit more comfortable okay but first we have to calculate it but it's a self-consistent argument so if it was a bad approximation then i would have to go back and i would have to add this and maybe i would have to worry about whether i haven't cheated too much here and there's many points like that so that's um but you know that there's a hardcore calculation of this okay so this by definition of tau has to be equal to q times tau the time because we define the tau as how long a new fitness class establishes okay so the time that we've lost our superiority somebody else has established as a new fitness class okay so then we want so this is another estimate for tau well i'll write it out explicitly this gives us a new estimated for tau which is one over two q s log n q s and then we want these two to agree and from having these two estimates for tau agree we can we can calculate q okay and so from that i get q equal to well why don't i go up there again q equal to two log n q s over q s okay um so this gives us a solution for q so this is okay so this is what the tip of the distribution is so okay the only thing is that you see this is an implicit equation for q so how do we solve these kinds of equations graphically is one answer right and there's another way we can solve them which is even more brute brute force and you're going to like even less but we can make approximations right we can do it recursively so recursively means that we ignore the q uh since q depends logarithmically here and linearly here as a first approximation we ignore the logarithmic dependence sorry i i messed up didn't i uh no no it's okay okay as over you so two is in at zero of order we are we ignore the q dependence and uh we get that and then at first order we take this solution for q here and plug it in here and then get a first order solution for q and so on right and we can do this until it starts to converge but let's stop at zero of order it'll turn out it's already pretty good compared to doing the full calculation and if we look at the in-lab e-coli population of n to the six with s to 10 to the minus 2 and u 10 to the minus 4 that gives us a q of 2 log 10 to the 4 of log 10 squared which is 4 and uh i don't actually know if i have it but 4 is a pretty good estimate for what goes on in in the yeast in the lab okay i'm gonna i'm gonna go a little bit over but then i finish and then we can stop with this and not get back to it so that gives me q so then i can calculate the speed which is just the rate of the growth divided by tau right which is the rate at which this whole thing moves and so i can take tau from either one of these uh so i can take it from here but let me make the same approximation i made over there to get rid of the logarithmic dependence in q and plug in uh sorry uh i have a yeah so i have a q here and a q here so i'm gonna get rid of the logarithmic dependence because i'm gonna plug in my q from over there without the logarithmic dependence so i get one to s log as over u log ns and if i do this so i have one over tau so i have one two s squared logarithm of ns logarithm squared and as i said this turns out to be roughly right and now the punchline of all of this so if we plot the velocity the speed of adaptation as a function of the population size for rare mutations it's linear in n right so these are but for the interfering mutations for the non-rare mutations it's logarithmic in n okay so it's much smaller it's much smaller and if you do this correctly it goes up a bit if you include um yeah so but essentially it still goes as as the log of n so what that means is that if you want to adapt quickly you're actually better off having rare mutations than having non-remutations right and this is like the log of n and this is like n so the question is how can you do better than this if you're a bacteria and you do have many mutations so you're forced to be in the non-rare regime is there something you can do exactly you can recombine okay the answer is recombination in fact recombination doesn't get you as far as here it does do slightly better but it definitely does in speed increase your speed and so this is one of the reasons that people give for the emergence of recombination or sexual reproduction because it increases your speed of adaptation okay yes yeah so what if you have a distribution yeah you can do this with a distribution i mean it's not as easy but once once you go to this generating function framework that's behind all of this you can do that you can put in a distribution of mutation rates the general conclusion doesn't change okay and if you find this call and if you find these fitness waves call you can work out what is the speed of this what is the i mean you basically can write down an equation like a diffusion equation that we've been writing down and show explicitly that this is a traveling wave in the sense of a traveling wave so x minus vt and work out what is v and so on so if you like that kind of thing you can go ahead but it's like a tall different set of a 20 hour lecture to do all this so we won't do it okay so i'll see you after lunch


The World Bank has given kudos to the Nigeria Living Standard Survey and the Nigeria Labor Force Survey, set to kick off by the Nigeria Bureau of Statistics. Country director of the World Bank, Dr. Shubham Choudhury, said no country can manage its economy properly without accurate data and other economic indices that will aid government in planning and budgeting. Plus TV Emanuel Ahijane completes the report. Accolades have thrilled unveiling of the Nigeria Standard of Living Survey and that of the Nigeria Labor Force at the sensitization workshop in Nabuja, organized by the National Bureau of Statistics. In his welcome address, the NBS bus, Print Adeyemi Adinirong, said a lot of planning and training has been proved in place to ensure a successful NH3 exercise. Extensive training for the feed passenger for both services has also been conducted. We just completed the training of the operators for the NLSS this week with adequate feed practical and testing of the passenger to ensure that they are suitable and get ready to collect data across the state. The country director of the World Bank commended the National Bureau of Statistics for embarking on the surveys, while emphasizing the importance of data collection and the planning and budgeting by any government. And I think so, this whole idea of using data, robust data, not fake data, we're using robust data to inform government and to also hold government accountable for delivery. I think it has to be part of the solution or what it will take going forward to restore the trust and faith. In their Goodwill messages, major stakeholders commended the initiative by the NBS, stating that its application will be a useful indicator for macroeconomic accounting and planning, while calling for agency cooperation. Specifically, the Living Standards Survey served as a useful input into the compilation of economic indicators like the GDP and the Algebra Prize Index. The commission has also facilitated the success of this process because we've been partnering and providing the listing exercise, the building number and household exercise, providing the maps, the digital maps that have been used for the current national framework that is existing for the annual eradication for the country. So many of the people within the informal sector do not really pay revenue. They don't pay taxes, I mean. The ones will be pleased if you can look at that and see how some of these people can even be brought in within the tax net. The Nigeria Living Standards Survey kicked off on Friday, 25th of November, and it's expected to last for 12 months, while the Niger Labor Force Survey has started already with termination dates set for October of 2023. Hello. Hope you enjoyed the news. Please do subscribe to our YouTube channel and don't forget to hit the notification button so you get notified about fresh news updates.


Okay, good morning everyone. Welcome to a new week of sessions. Let's begin this time with the word of prayer. Sid, is it okay if you can lead us in prayer? Yes, pastor. Go ahead. Hallelujah, hallelujah, hallelujah, hallelujah. Father, thank you for this day. You have given us a lot, the word from your word, what we are going to learn from your word, not whatever we will be learning Lord, whatever the pastor Paul is going to teach us Lord, like it should not be just kept in our heart Lord, but it should be revealing us and not Lord. When other people see us Lord, they should see the change in our life Lord. They should see the blessing we are going through a Lord. We should reveal and manifest the godly character Lord. Whatever is APC Bible college is putting the seed in our heart Lord. They are keeping us to do the ministry Lord. Not everything they are doing Lord bless us Lord. Bless the teacher Lord bless the students who are learning Lord about your word Lord. Lord, we need your fear to learn about your word Lord. Lord give us everything. Lord give us wisdom. In Jesus name we pray. Amen. Amen. Amen. Thank you, Sid. All right, before we go ahead, let's just do a quick review of what we did last week. Last week we talked about understanding the blood covenant and we saw that the blood covenant is the strongest covenant that God has ever made. Right, and we looked at the verse where the entire chapter where you know God makes this blood covenant with Abraham. He says Abraham you have believed what I've said and what does God do? He makes him you know put an offering there and then finally God himself walks through the offering and he makes a blood covenant and after the blood covenant he says a sign will be given to you and then that sign was circumcision and and we see that a blood covenant we studied there last week that the blood covenant is the strongest bond on earth. It's like God was telling Abraham Abraham. I know you have a lot of questions.   But I am making this covenant not on my own words. Making this covenant not not because I'm just saying it but I'm making a blood covenant with you. Right, now when somebody enters a blood covenant when we read further on from the Levitical offerings blood covenant refers to life where you have to be willing to each party is supposed to be willing to give their life for the other person. So it's like God is telling Abraham I'm willing to do all I can to fulfill my side of the covenant and a blood covenant is a covenant of love. You know, only when we love one another is when we can do anything for them at any cost and God was basically telling Abraham that it is out of love that I'm making this covenant with you out of a promise that that will never go unanswered.  God tells Abraham no matter who you will have to fight no matter what happens ahead I'm going to take you to a land that you don't even know of. It's going to be new people new places. There may be, you know, battles that you will have to fight. Remember as this blood covenant that I'm giving you Abraham you will have protection from my side. And so Abraham we saw last week that he believed God and God made this wonderful covenant, the blood covenant. He changed his identity. He gave him a seal of circumcision and God tells everyone who ever is part of this seal or part of this covenant will be blessed. And then we looked at a few revelations of the blood covenant where he himself became the Jehovah Jireh and God himself backs up the covenant. And let's pick up from the blood covenant with Moses and Israel. Let's learn what happened. I'm sure we all know the story of how God used Moses to bring the people of Israel out of bondage from Egypt into the promised land which God had promised. So let's look at a few insights and also we will see some of the covenants that God made to Moses and which we also see in the levitical offerings. First one, Passover. Passover was a type of redemption. Now we all know the feast of the Passover. God made this promise to Moses and he says, Moses I'm going to bring you out on the last plague or pestilence that is going to hit Egypt.  Death will pass over and so to commemorate that even now the Passover is the most celebrated festival that the Jews follow. We are on page 63 on the Nodes territory. Blood covenant with Moses in Israel. So it was a type of redemption. So even now when we translate it in the New Testament, it is the blood of Jesus that keeps us from sin. It is the blood of Jesus that saves us from death.  Because of the blood of that lamb, death will pass by. None of you will taste death and that's what happened. It was the last plague. The blood was put, death came and none of those who had the blood of the lamb on the dopo's died but everyone else who did not have it died. It was like God saying they are part of the covenant and when you're part of the covenant there's protection. Right, let's read this portion. It's a long portion. Exodus chapter 24 was 1 to 8. Yes, could one of us please read that? Exodus 24, 1 to 8. Now he said to Moses, come up to the Lord, you and Aaron, Nadab and Abihu and 70 of the elders of Israel and worship from afar. And Moses alone shall come near the Lord but they shall not come near nor shall the people go up with him. So Moses came and told the people all the words of the Lord and all the judgments and all the people answered with one voice and said all the words which the Lord has said we will do. And Moses wrote all the words of the Lord and he rose early in the morning and built an altar at the foot of the mountain and 12 pillars according to the 12 tribes of Israel. Then he sent young men of the children of Israel who offered burnt offerings and sacrificed peace offerings of oxen to the Lord. And Moses took half the blood and put it in basins and half the blood he sprinkled on the altar. Then he took the book of the covenant and read in the hearing of the people. And they said all that the Lord has said we will do and be obedient. And Moses took the blood sprinkled it on the people and said this is the blood of the covenant which the Lord has made with you according to all these words. Amen. Thank you Rosalind. So we see here that God is you know the people have come out they're on the Exodus they're on the way God is giving very specific instructions to Moses he says Moses I brought you out of Egypt now you're in this place in the desert now here's what I want you to go and tell the people not to come up the mountain but take the blood of this lamb and take half of it put it on the altar use the other half sprinkle it over the people and this will be a blood covenant according to the words that I have spoken right and later on we see that Moses and then the Aaronic priesthood became mediators of that covenant Aaron became the first high priest and with this blood covenant already set in place they became the mediators so it was like you know Aaron is standing there the blood of the animals or later on we see that the offerings were bought to the high priest and the high priest would take those offerings and offer it to God as a mediator and I love what the book of Hebrews teaches us Jesus Christ is our high priest our mediator now everything in the Old Testament has you know sometimes when we read it we may think why all this you know so complicated but it has meaning because right now the Lord Jesus in heaven the book of Hebrews says that he's standing as our high priest and our mediator and what is he doing he is standing as a mediator through the blood through his own blood that's why we sing you know we enter his throne room not by the blood of rams or bulls but by the blood of Jesus so here we see that there were mediators high priest and now we've got this perfect mediator Jesus who is standing who has made the blood covenant eternal right and so all through from Exodus we see that every offering every sacrifice every feast that they celebrated when we read through Exodus and Leviticus numbers every aspect has a meaning to it right now let's look at a few sacrifices and feasts right that God initiated during the time of Moses right we know that God gave Moses the 10 commandments and the mosaic law he told Moses okay you build a tabernacle you build an ark you in the tabernacle you have these you have the outer court you have the inner court holy place and this is will be a place where you will offer the sacrifices and these are the kinds of offerings these are the kinds of sacrifices there's a whole list of things right and also he you know when we read the book of numbers God even tells the tribes where to camp it's not like they would you know walk in in the desert and say okay I feel this place is good let me camp here no God tells them exactly where should the tribe of Benjamin be where should the tribe of Judah camp and so everything was very meticulously done by the Lord by God because everything has a significance in the new covenant sometimes we read you know when I read this I think you know God used Moses to do such wonderful works but only because he hit that stone the second time what does God tell him God tells Moses Moses you will not enter the promised land because of your disobedience you will not enter the promised land you know so what about the others what about the other Israelites they did all kinds of sins they did everything possible but they all entered maybe Moses thought why you know you told me to you know speak to it but you know I hit it but there was a significance there was something that God wanted to portray and bring out and so for everything every feast every way of worship had a reason in the old covenant right and we can't say that all of the things in the old covenant are you know don't apply now in the new covenant that are certain you know we can take in certain aspects certain points which is which in the new covenant will relate to the old covenant so let's look at a few of them right first one the feast of the Passover we know that the blood was put on the doorpost and you know death passed by the same way Christ is our redeemer and he's our redemption right those people who were in the homes with the blood on the doorpost they were redeemed they were redeemed they were the ones who walked out of Egypt into the promised land the feast of the Passover is is one of the most beautiful feasts you know if we read the history of the Passover it is the most celebrated feast it's something like Christmas for us it's it's it's a time of joy it's a time of celebration it's a time where the Jews especially they say God overcame God helped us and brought us out of Egypt so it was a celebratory feast right and people would come from all over different parts of the world different parts of Europe they would come to Israel to Jerusalem to celebrate the feast of the Passover right that is why when we look at the book of Acts right why were there so many people in Jerusalem people of different languages there were people of Greek Aramaic Hebrew all kinds of people were there why because the Jews from different places would come to the main the center which is Jerusalem to celebrate the Passover right so it was a it was a joyous time right the second feast was the feast of the unleavened bread now let me give you a context on what the feast is so that we can relate it to what's what in the New Testament the feast of the unleavened bread is is nothing but they would the the Jews would take barley and they would you know make that barley unleavened right so it it wouldn't be mixed with anything it would just be barley and they would mix it and they would eat that for about seven days and it would be tasteless right it is there's no taste but the point of this feast of the unleavened bread was to commemorate that we were living in sin in Egypt but God in his mighty hand brought us out and this we are now not living in sin anymore and so we do want the things of this world to come into us and make us sinful right so the moment you put additional things into that unleavened bread it becomes a leave in bread it becomes you know something else so what they would do is they would not add anything into the flower of barley so they would leave it just as it is what does it commemorate okay we were living in sin right and and it is it commemorates that now we are cleansed we are washed away from sin sin is removed from us right and so that is what the feast of the unleavened bread is now when we read in the New Testament we see that the Lord Jesus removed our sins he is the bread of life right Jesus himself said that I'm the bread of life he was relating to that unleavened bread the bread that takes away the the sins or not takes away but covers the sin cleanses them I'm sure in the Lord's supper the last supper when Jesus said this is my body I'm sure the disciples may be wondering how how is this body how is this his body maybe they thought that but he was trying to say this bread when you eat of it it will cleanse away your sins it will wash away your sins now this unleavened bread again was is a time some some people would have it for seven days it would go on for many more days people could you know if they'd like they could go on for 30 days but again it is to commemorate that we are no longer living in sin right same thing the Lord Jesus did for us the feast next one the feast of the first fruits speaking of our new life in Christ and what the Lord Jesus did for us by offering himself for for us now the feast of the first fruits also is a very interesting feast this is a feast where it's a feast of redemption the feast of the first fruits usually the book of Ruth is read through that whole feast in in you know in the new cover when we see that during the time of Jesus and the early church they would read the book of Ruth as a redemption story but what is this feast of the first fruits those who are working those who have crops and cattle and vegetation what they would do is when the time of harvest is ready they would have to go right into the middle of their land and they would take out the best crops right take out the best crops and they would keep that aside go to the temple and offer them as a sacrifice to God right they are not to eat of it now if they eat of the first fruits they will have to go back and make a sin offering right it's not like you know you know oh I forgot I ate and now I have to you know go back and make a sin offering no they should consciously be aware okay the feast is coming up the fruits are ready the first batch usually what they would do is the corner sides of the field this I'm talking about during the old you know during history I'm talking about history where during the corner sides of the land they would the crops that are there they would not use that they would usually leave those corner sides of the plot vegetation and fruits or vegetables whatever they were growing they would leave that as you know something that they would donate to people or to bless other people right so the feast of the first fruits is the first fruits that you get from your labor right now mostly mostly it is vegetation right like crops pomegranate oil so these are the normally the things that they would grow and so so this feast again it is to tell God you have blessed me whatever I have here is the best of what you have given me right and so when we look at the new covenant as well we see it as tight right we give one tenth first of what God has blessed us with and then additionally we can give more as well but it is to commemorate this life that we have in Christ a life of giving a life of blessing and and a life of also offering ourselves to the Lord then comes the feast of the pentecost right now the feast of the pentecost was a feast the word penta means 50 right five pentecost 50 days so this feast usually lasted for a long time and what they would do is they would believe in in the power of God to flow in them there would be extended times of prayer there be extended times of you know spending time meditating God's word being in God's presence and they would wait on the Lord some some Jews would very seriously you know even avoid work during the season but we know that others will have to work so there were some who would fast and work but even as they work they will remember that this is a time where we're seeking God because you know the Jews during those times history says that they would work not as you know just to fill their stomachs that's it not to you know not for business or for other reasons and it was a time of just staying and meditating on God remembering God for everything that he is and so that's what the feast of the pentecost is they believed that the power of the Holy Spirit the power of God will you know really flow in them as they you know celebrate the feast of the pentecost and that is why in the New Covenant as well it's it's it's wonderful that God chose the day of pentecost itself to you know to pour out his Holy Spirit upon the people who are in the upper room the next feast is the feast of trumpets now this feast is a feast of victory and triumph now in in times in the in history and during the time of Moses what they would do is they would blow the horn of a chauffeur it's a it's an animal just like just like a bigger version of a deer and they would blow that horn right now what is the feast of the trumpets all about it is a feast to commemorate how God triumphantly brought us out of Egypt to remember that in his mighty hand he sent the plagues he sent all those he he'd parted the seas into two and we walked through the seas and he provided for manna there was water going with us and the feast of the trumpets trumpets is a is a feast which declares that you know they are victorious the word trumpet I mean the usage of trumpet is mostly to alert or even when there is a victory or when there's ready for battle a trumpet is blown so this this whole feast was to remember and commemorate that that Jesus sorry God was victorious now when you translate that to the New Testament Colossians the book of Colossians chapter one Paul is writing and he says he was victorious where he made a public spectacle of the enemy he overcame and he's victorious and and so all through the New Testament we see Christ victorious the book of revelations talks about how trumpets will be blown and the victory of the Lord will will be shown upon people right so again to commemorate that in the old covenant even in the new covenant the Lord Jesus fulfilled that victory then comes the day of atonement now the day of atonement is a day of mourning like nobody works so God told Moses listen this day of atonement is the day you will sit and you will pray and you know spend time meditating on the word of God and there is no work so they just have to sit and they would pray and say God forgive me for my sins forgive me for my sins right that's all the whole day they would pray for forgive me of my sins and so the whole day they would fast they would pray they would you know read scriptures they would sing songs it was not a celebration but it's about their own life they would say okay I'm a sinner but God is a holy God how can we make atonement for this so they would sit and pray but at towards the end of the day what they would do is they would go take a lamp or an animal unblemished take it go to the altar and they would sacrifice that and that blood would be smeared on the offering saying cry the the the blood has covered your sins and now you have no sin in you it has covered the day of atonement as those sacrifices made the blood is covered your sins so they would do that offering they would go back home that is the day of atonement now Christ's atoning work on the cross did not just cover sins but it washed away our sins now can you picture this John the Baptist there's 400 years of silence right after Malachi 400 years of silence God has sent prophet after prophet after prophet but the Israelites the Jews are just doing whatever they want to do 400 years silence they've been doing all these things feast of the Passover feast of the Unleavened Bread feast of the Pentecost feast from all these things they've been doing but what has happened here is it has become it's not something that they're doing it out of reverence for God but it's become a ritual just a practice so for example the day of atonement comes oh I'm sorry God you know I shouldn't have done this I'm a sinner I'm sorry they go cut the offering put it on the you know give it to the high priest the high priest will pray and then it's done they go back they'll sin again so it became you know like a routine and that is why when John John the Baptist comes Israel was in a complete mess I would say their religious condition was very bad the Jews were doing what they wanted to do the high priests were happy they were getting offerings the Pharisees were happy the Sadducees were happy the Jews were happy they were hardly there was no reverence for God because all of these became just a you know a ritual a practice it didn't have meaning in them anymore that is why you know John the Baptist says to the Pharisees to the Pharisees and the Sadducees God can raise up Abraham's from the stone itself and so during these times when during Moses's time it was done out of reverence and when we look through and read through the Old Testament as prophets came and went prophets came and went the entire you know the weight of these offerings and sacrifices was lost finally the last was the feast of the tabernacles now again this feast was a feast of enjoyment it is a feast where people would come from different parts again what they would do is it was to this feast was mainly to remember that when God brought us out of Egypt we were journeying through the desert God gave us rest we stayed in small huts and houses but God protected us and and so what they would do even right now is especially you know staunch Jews in Jerusalem what they would do is they would know they would go to somewhere away in the desert they would build you know small houses you know basically just rooms one side out of bamboo and then they would cover it with hay or sticks and the reason they did that was because they wanted to commemorate that when we came out of Egypt we were living in these kind of places these kind of huts I would say and when we lived there God still provided for us and God was with us through this whole journey into the promised land so it was to remember that so even now you know if you go to Jerusalem and certain places you will see that when especially they will go find some empty place they would you know build those you know small tents kind and use sometimes they leave the upper portion open so that when rain comes it should come into the tent signifying the blessings of God and so all of these are wonderful to learn and and how they you know relate to the new covenant the Lord Jesus himself you know gave us rest when we were what does the Bible say in the book of Romans that while we were yet sinners Christ died for us he made a way he gave us rest he provided for us right so we come to an end on this chapter and it's any anybody have any questions any thoughts on the offerings does it is it something that you're able to understand have any questions everyone okay yes okay I just don't know in what page we are okay we are at page 63 okay thank you okay so we're looking at we just looked at the sacrifices and the feasts so any questions on the sacrifices you know we've not gone into the you know what kind of sacrifice but when we read the book of numbers Leviticus numbers God tells Moses what kind of animal what kind of sacrifice which are the blood offerings which ones you you know you don't offer the blood which one so there's a lot of technical things that God gives them but then we're just looking at how these covenants were fulfilled through our Lord Jesus Christ right should we go ahead should we go to the next chapter all right okay let's go to the next chapter now we've we saw all of these things in the old covenant in chapter four people of the covenant the old covenant affected the people in their everyday lives right so even during the exodus even when they were in the promised land God told Abraham sorry Moses write down whatever happened during the exodus and then he wrote it down he said remember this remember that and Deuteronomy was a book of remembrance now as people of the old covenant everything God was there in every aspect of their lives right how the covenant affected them in their workplace or even in even those who are high priests those who are in business in every area family there was a covenant set in place now with that in mind let's look at how we you and I being part of the new covenant how are we to respond to what God has done for us what are the things that we can take as people who are in covenant with God right now remember that being in covenant with God God is the one who initiated this covenant right we did not say God please send your son into this world to die on the cross for us no God himself initiated it right so God has given us the opportunity to enter the covenant so as people of his covenant what is it that we can receive from him of course there are plenty of things but let's look at a few things right Exodus chapter 19 and verse 5 can one of us please read that Exodus 19 verse 5 Exodus chapter 19 verse 5 Amen let's also read the next one Exodus 34 10 I made the covenant before all your people and I will I will do mother mother have not been done in all the heart not not in my mission and all the people among you you have to see the work of the Lord for you are also me that I will be with you amen amen so we see here God is telling Moses Exodus 19 5 if you indeed obey my voice and keep my commandment then you will be a special treasure to me above all people for all the earth is mine now this covenant as we are part of God's covenant as well this holds true for us as well where God is saying if you obey my voice we keep you keep my commandment you will be a special treasure to him and he goes on to say I will do marvels that I've never done before because God is telling Moses that you know I'm going to do these great things for the nation of Israel for your people but as part of God's covenant we can receive this and say God I know that you will do marvels in my life as well that the work of the Lord and people will see the work of the Lord for it is an awesome thing that I will do with you these are promises that we can apply as people of his covenant right we see this reflected in Peter Sermon in Acts 3 25 he says you are the sons of the prophets and of the covenant which God made with our fathers saying Abraham and in your seed all the families of the earth shall be blessed so Peter is remembering you know Jesus has died and he's gone he's resurrected to heaven and they're ready to start the ministry Peter's remembering the covenant that God made long time back with Abraham saying in your seed all family shall be blessed so how do we apply it to our lives we say God as part of your covenant my family shall be blessed my the generation to generation remember the blood covenant goes on for generations and more so we can say God through this covenant that you've made my family shall be blessed my children my children's children and going on from generation to generation God draws a distinction between those who obey him and those who disobey and do not know him read through the Old Testament we see that God sent prophet after prophets but God knew the people who will follow him and God you know he knew the people who disobeyed and you know rejected him God draws a distinction let's read Exodus chapter 8 21 to 23 go ahead anyone Exodus chapter 8 was 21 to 23 or else if you will not let my people go behold I will send swarms of flies on you and your servants on your people and into your houses the houses of the Egyptians shall be full of swarms of flies and also the ground on which they stand and in that day I will set apart the land of Goshen in which my people dwell that no swarms of flies shall be there in order that you may know that I am the Lord in the midst of the land I will make a difference between my people and your people tomorrow this sign shall be Amen thank you Rosalind so we see here and then we read on the other portions as well we are seeing that God is telling the people now I'm going to make a difference if you do not let my people go these are the troubles that will come upon you later on he says the life stop of Israel will all sorry of Egypt will all die the land of Goshen the children of Israel will nothing will happen to them but there will be hail on the people of Egypt then there will be darkness on the people of Egypt and then we saw that the first born of Egypt will die God draws a distinction now why do we why does he do that because we see in in in the old testament God revealed his power and he gave people the opportunity to believe in him and so does he do even now in the new covenant as God has given each one of us we are part of his covenant he gives people the opportunity to come and to believe now in God's mercy he does not you know it's not like God says okay you don't believe in me you're going to die in two days or your life is going to be taken away from you or you're going to go through the sickness it is God's mercy that he doesn't love that but God draws a distinction he says these are my children these are my people right you are a chosen generation a royal priesthood a holy nation why does he say all that because when we are part of the covenant he draws a distinction God loves the world the same way John 3 16 loves the world that he gave us life but he draws a distinction for those who believe and those who do not believe great wonders he will do for you and me as people of his covenant so even as we go about in our ministry in our studies in our in a work workplace anything that we're doing remember that God sees us differently God sees us as part of his covenant the Lord is willing to do all these miraculous miraculous works that he has done in the old covenant he's willing to do that and more in the new covenant right next he says that you are a special people Deuteronomy chapter 7 verse 6 to 11 yes could one of us please read Deuteronomy chapter 7 6 to 11 go ahead Deuteronomy chapter 7 verse 6 to 11 for you are a holy people to the Lord your God the Lord your God has chosen you to be a people for himself a special treasure above all the peoples on the face of the earth the Lord did not set his love on you nor chose you because you were more in number than any other people for you are the least of all peoples but because the Lord loves you and because he would keep the oath which he swore to your fathers the Lord has brought you out with a mighty hand and redeemed you from the house of bondage from the hand of Pharaoh king of Egypt therefore know that the Lord your God he is God the faithful God who keeps covenant and mercy for a thousand generations with those who love him and keep his commandments and he repays those who hate him to their face to destroy them he will not be slack with him who hates him he will repay him to his face therefore you shall keep the commandment the statutes and the judgments which I command you today to observe them hey man thank you Rosalind so here in Deuteronomy he's reminding Moses is reminding the people listen you are a holy people to the Lord you are chosen you are a special treasure and he goes on to say that those who love him he will honor them and and those who try to you know who don't obey him and dishonor him he will bring trouble and wrath upon them right now I want to focus on this aspect of being a special treasure he says that when you are a special treasure above the people of the face of the earth maybe sometimes you may feel God I'm you know I'm there I'm I'm here I'm your child I'm part of the covenant but doesn't look like I am blessed doesn't look like people are honoring me you know people don't respect me maybe sometimes we feel that way but I want to encourage you though we can stand and we must understand that we are part of God's covenant and we have already received all of this people may not understand it right but you say who you are maybe in the workplace you you know people are mocking and people are making fun of you and you don't you see that you know you've been working sincerely but things are not going right you know and those who have been doing the wrong things have been growing and those who you know you've been sincere but nothing's happening remember God is a God of covenant you are a special generation a special treasure to him continue to trust in him I continue to believe because you're part of the covenant right you may feel God you know I'm part of the covenant but I'm not able to experience any of your blessings no he's there his blessings are there his grace is there his mercy is there he's extending it to us we have to just you know hold on and continue to trust in him right so let's take a break we'll pick up from overtaken with blessings after we take a break we'll come back at 11 o'clock let's take a break