Thanks very much. So I was charged as an NHGRI council member with describing the current efforts in functional genomics and really with the focus on scale here. There's a lot of functional genomics obviously going on throughout NIH and what can NHGRI bring to this. So just in terms of the broadest definition, and this is really the challenge I think and I was thinking about and putting together, this talk is that this area is really huge in breadth, right? Functional genomics, this is what I came up with. I added genomes, right, because now we really want to understand entire genomes in the context of what the whole genome is doing in chromosomes, et cetera, in addition to the genes and gene networks and their regulation and how to understand how this regulation affects cellular function, development, and obviously disease. And so these are the kinds of points to think about during the talk. What are the challenges, and I'll raise a few of those, that can be solved and the needs that are, that needs to be met relative to understanding the functional role of genomic variants in disease and health. You know, obviously NHGRI is only one of a number of ICs and has a budget that's limited in scope, so this has to be also kept in mind, and what would be the consequences of not having some functional genomics at scale at NHGRI. So when talking with staff about what their view of functional genomics within the context of the sequencing program, and there's a little asterisk here for caveat, there's really no existing sequencing programs that are directly pursuing functional genomics at scale with the caveat that there's obviously sequencing itself at large scale, right, and identifying variants in interesting genes is going to allow, for example, associating variants with common disease pathways. A beautiful example of the schizophrenia consortium where, for example, voltage-gated calcium channels are associated and a number of other obvious genes that are associated with significance in that disease based on large numbers, 37,000 individuals that were examined, and so only with this scale can you achieve the kind of power to begin to attempt to cluster and achieve some significance. So there's scale, but of course these are then associated with genes of known function. And also as you heard this morning all about the Mendelian Centers, you know, this is genetics, I mean geneticists, you're studying the functions of genes through genetics, and that, as you heard, can be scaled to some extent in terms of the numbers of Mendelian genes that exist and then the breadth of allelic variation within those, and that these individual genes are each an achievement, an interesting biology that comes out of them, but collectively they inform about human biology, right? So there's a level of scale here, and you heard an argument for an ongoing centers. So NHGRI does have large-scale functional genomics programs that are connected with interpretation of variants, and some of them are ENCODE, and I'll briefly mention some of the ENCODE work. Genomics of gene regulation, which maybe some of you haven't heard about because it's not yet funded, and also the functional variants program, which is also not yet funded. So ENCODE, I'm sure you're all aware of, is really the idea is to generate a comprehensive catalog of all functional elements in the human genome and also utilizing the genomes of selected model organisms, because the importance of identifying these variants and most of the variants that come out of whole genome sequencing are not in protein-coding genes. So associating the catalog of variants with the catalog of functional elements is going to be important for assigning those variants to elements that could have function. And so this is just taken from a recent P&S paper from the ENCODE group where it's a summary snapshot of the percentage of the genome that is covered by the different kinds of biochemical assays that are linked to the genome through particular elements. So transcripts, footprints, here's the percent, 100 percent. So some of these actually get quite large, and this has been debated in terms of the significance that I don't want to get into an argument about what function is. These are biochemical activities that are associated with the genome that can be useful to inform about variant associations. And so here are DNAs, footprints, transcription factor, binding sites. These are the things that you look at when you're trying to first link a variant that's in a non-coding region. You go to the ENCODE dataset and the roadmap epigenomics dataset, which I'm not going to discuss here, and associate variants with, for example, histone marks or these other annotations. So really, we're decorating the genome with biochemical information that can then be associated with coding or non-coding variants. So genomics of gene regulation, just to briefly say, it's not yet fun to get the aim of this program. And this isn't really at scale in the sense of sequencing, for example, at the level of the ENCODE project or the roadmap epigenomics project. This is more for understanding gene networks, largely focused on developing and validating models of gene regulatory networks to try to predict the functions of linking the different elements together into a regulatory network using, for example, RNA as an output. And really with a main goal of improving the current methodologies for regulatory network models, rather than incremental approaches. With the long-term goal, being able to look at sequence and accurately predict when and at what level a gene is expressed, that's a laudable goal. And in a particular cellular state, that's important. Genome sequences is, except for a few tissues, the same. And so trying to predict directly from the sequence is the goal. And functional variants also really can't be argued to be at scale. But these are tools that are going to be useful for understanding the variants as well. Computational tools, they aim to develop highly innovative computational approach for interpreting variants. And this is really an integration project, integration of data, phenotypes, patterns of variation to identify and narrow, this is really to narrow to focus on the set of variants that are likely to lead to disease states. So this is really to hone in to where, which variants to prioritize on. And accuracy in this program will be assessed by experiments. And then the common fund also has resources that have been contributed for interpretation of variants. One is the epigenomics project, which has focused largely on analysis of tissues. So tissue map, if you will, across human biology, looking at chromatin marks, hypersensitive sites, et cetera, DNA methylation throughout the genome, and developing a catalog in parallel, if you will, to the ENCODE project that's focused on tissues. There's also the GTEX project, which is a very large project to link genotype of individuals to gene expression patterns, to try and interpret the variants with gene expression, so QTL at a large scale. The LINX project is sort of a mixture of different technologies, transcript, analysis and perturbation, metabolomic projects, proteomics projects, where a system sort of an integrated systems approach to understand a network of genes and perturbation is the goal there. And then there's a new project, which is recently announced, which I think is quite exciting as part of the roadmap, it's the 4D-nucleome project, the idea being to understand a network, to develop technologies, there's a number of areas imaging molecular biology and disease as well. But this program is to enable how DNA is arranged, the study of how the DNA is arranged within cells in space and time, and how dynamic the changes are occurring within the nucleus, and to link these features to cell their states in normal and disease states. And really the goal here is to, is multi-fold, but it's to develop new technologies to be able to, and existing technologies can be applied to understand the DNA organization within the nucleus and its function. And so really you can put these kinds of projects into two bins, in terms of my thinking. One is resources for the interpretation of variants, where you're associating genome-wide information of a variety of types with a genomic location to help inform about a particular possible function of a variant. So that's what the ENCODE project is doing. That's what roadmap project and some of these other projects are linking biochemistry to those regions of the genome to help interpret variants. The other is a functional sort of genomics approach where you're validating the variants directly by assaying function in some way. And I'll touch on both of these first starting with the variants. So again, from the recent ENCODE description of function, the data can be organized in sort of a bull's eye fashion here in terms of the kinds of information that one can overlay. So genetic evidence from either mouse studies, from Mendelian project, etc., from any genetic evidence about gene function can be overlaid with this sort of three-tiered of, in terms of the activity level data from the biochemical data from ENCODE. And that's linked with both the protein coding information and then the rest of the genome outside here. So there's a large fraction of the genome that has activity that is linked to DNA hypersensitivity, etc., that lies outside, obviously, of the protein coding region. And then overlaid on that is the evolutionary evidence. And so this comes again from large-scale sequencing, comparative genomics, which will be one of the breakouts. And the question is, is this with continuing of this project be useful in terms of understanding more genomes in terms of evolutionary consequences, even understanding phenotypes from a variety of different model organisms. Overlaying that information essentially to try to understand variant function. So you can look at it in this sort of silo approach here where this is a canonical gene and they're the SNPs. The SNPs are associated here with gene expression. Okay, so you can link the degree of association, for example, in an EQTL study for the GTX data, let's say, to those SNPs. You can also overlay the evolutionary conservation. And each one of these data are not complete, because there are well-known regulatory sequences that have very low evolutionary conservation. And there are also sites of biochemical activity that don't overlap with these sites. So you can begin to overlay the sequence data from conservation, and these are still evolving in terms of being able to utilize new approaches for this. A variety of different types of assays for predicting TF motifs, either biochemically in vitro or in vivo, linking those to DNAs hypersensitive sites which themselves can identify footprints because they're high resolution and chip seek information. And a variety of other biochemical assays that then give you some probability. And this is what really needs to be, I think, a focus of confidence that a particular SNP is going to be a SNP to further evaluate in, let's say, a functional assay, and as Mike Snyder was discussing the possibility. Okay, so let's look at the challenges. If you go back and look at this model, this is a very simplified example, right? Where the SNPs are located next to something that is believed to be the gene that those SNPs might be associated with. But that's not really the case. Regulatory elements act over a long range, and this makes it very challenging to identify their target. So here's an example of one particular gene where obesity associated variants are connected at a very long distance, okay? Over 300 kilobases with a particular enhancer sequence that lies in another gene. So linking this information, we know that through biochemical experiments of a variety of types that looping occurs between enhancers and promoters. And that these distances can be very great. So that really creates an enormous challenge for identifying genes with no obvious function. So calcium channels linked to schizophrenia, a very good candidate. What if that was an unknown gene that had a high SNP with a high association? So we know that there are different layers and there's some beautiful recent work on using cryoEM to look at these different levels of organization of chromosomes. The gap, and I think the opportunity, is right here. Now, this is going on in a number of laboratories. But I think the scale is the issue to be addressed here. Is it worth scaling up the analysis of organization of enhancers and promoters and regulatory regions and SNPs at some scale? So I think there is an opportunity to explore these long-range interactions, carrying out genome-wide surveys, right, throughout a variety of cells. Identifying the general features of chromatin organization and dynamics on a large scale. Looking at local chromatin interactions between enhancers and promoters. And there's different technologies that I'll mention that you can do this. With really the goal of understanding the long-range regulatory element links to different genes. With the goal here of understanding in the context of NHGRI, the functions of SNP. So here's one assay that I think is really powerful developed by Yop-Dekker and Eric Landers group, where you can essentially capture the information from domains that are insist or potentially in trans, but are at a long distance from one another. By capturing the association through this proximity ligation assay. So it was published here. It won't go through the details, but allows you to associate large domains of interaction of proteins within the genome. And that's listed here by the different degrees of the heat map of the frequency, the higher frequency of interaction of domains that are closer and the relatively lower frequency of domains that are at a distance. And then through a number of studies and a number of laboratories, domains have been mapped out. For example, topological domains are so-called associated TAD domains between, which essentially represent these loops in the genome. But within these domains, you can also associate enhancers and promoters. And so for the study of enhancers, one can think of at least two approaches that are tractable, I think, and potentially could be scaled. One is the introduction of mutations into enhancers in their endogenous location to test for the expression of genes. This is a direct assay of the function of that enhancer on an associated gene. The con so far has been, it's been relatively, applied at a relatively low throughput level. And if it's going to be done in a model organism, maybe that SNP isn't necessarily going to be representative of an important one for human biology. The other approach is to exploit the natural variation in SNPs between two alleles. It's a global approach, right? So if you find SNP variation, and you can associate that with the expression of a gene, that requires that you have the haplotype of that genome. And so associating allele A with gene A1 and allele 2 with enhancer with gene A2. Approaches are possible, again, using high-seq in a method that was developed by Bing Ren's group and published this year, essentially of creating, using high-seq at a very high resolution to essentially make haplotype linkage maps. And so this is an example, this is unpublished work from taking all the variants in the H1, ES cell. And then phasing those variants, and then looking at the ratio of phase versus unfazed variants, and I'll zoom in on a region from this. So essentially using haploseq, you can phase both chromosomes and then begin to associate variation in enhancers with variation of nearby genes. And if there's information from the high-seq, you can have confidence that there's a particular enhancer promoter association. And so this is what some of the data looks like from those kinds of experiments where having that phased information, you can then link a variety of allele-specific information, transcripts, DNA methylation, chromatin marks of a variety of types. And this just shows one region where the RNA expression data here, increased expression, is linked in phase with reduced DNA methylation, increased marks of promoter on these genes. And this can also be done at a very long distance, right? So you can then, this is an example of linking some of those allele, so ES allele for mRNA, allele 1, allele 2 for methylation, et cetera, to chromatin marks at a distance. And so this is a distance of 150 kb. So this can be used, this kind of data can be used to link promoters, snip variants in enhancers through large-scale kinds of experiments, which really have been going on in individual labs, but can they be really scaled? The other approach is a functional validation of variants. And so a variety of approaches have been developed in a number of laboratories. To take particular, let's say, enhancer regions, carry out mutagenesis, or reorganize that information in a variety of ways, so mutagenesis. And then essentially synthesize those variants and then introduce those variants into a variety of vectors, either transient vectors or integrating vectors. And then either have readouts of sequencing at high throughput or fluorescence kinds of readouts to be able to understand at a very high level, that is a base, a nucleotide level resolution, the information that that variant can potentially produce in terms of regulation of gene expression. So really a direct test in a high throughput way of the association of a variant with gene expression. Okay, so this is a functional test that can be done at high throughput. And that kind of information, I think, will allow us to begin to understand when really scaled the grammar of gene expression. So what does it mean to have a particular motif with a slight variant? And this is a challenge to have a slight variation in that motif on gene expression function, okay, of that gene. Is it going to be functional? Is a particular residue going to be functional or non-functional? And I don't think we have the data at a very high throughput in terms of scale for this kind of information. There are a variety of other assays and contexts that one can use to understand the grammar of gene expression. But probably the most powerful tool for understanding function is through genome editing. And I think here is a real opportunity for NHGRI and potentially this can be scaled. I know many laboratories are using these tools, including ours, but really bringing this to analysis of variants at a large scale. I think hasn't really been done yet in a way that I think in NHGRI can do it. And so although there are a variety of editing tools such as ink fingers or talons, the Cas9 system, the CRISPR-Cas9 system has really provided, I think, some opportunities that are unique. This system first identified in bacteria, of yogurt bacteria, essentially for in terms of function, is to use a guide RNA to direct changes in the genome. That can be, for example, as crude as making an insertion and deletion by using a targeting sequence, a targeting RNA to a particular region in the genome to make an insertion or deletion. Then in trans, you can add any sequence that you want and get through homologous recombination, get a precise replacement of an edit. You can use two of these Cas9 or two CRISPR RNAs to direct the creation of large deletions or potentially rearrangements. And there are other biochemical assays that I won't go into here that make this a really useful system, I believe, for Lord scale editing. So one approach would be to validate, and I'll give an example here, of cis regulatory elements. And I think, again, this could be scaled. Here's an example from Bing-Ren's laboratory. An enhancer knockout can provide direct evidence of the function of that element for gene regulation. And you can test the effects on transcription. And you can test to see whether the effect is in cis or not by having that phased information. So you can have, if you have the linking information, plus the enhancer knockout, you can begin to link the loss of function or the editing function of that, the edited sequence and its effect on the neighboring gene. And so here's an experiment that the RAN lab carried out. They did a wide cross, created F1 mice, produced ES cells where they then created a deletion of particular enhancer in a particular context. And so this just shows the scenario here where you would use two different targeting RNAs, single guide RNAs, that flank the enhancer. In this case, it's a large enhancer of about 12KB. And these are the signals of the chromatin marks over that enhancer. And so this is the gene that is actually very well studied gene, SOX2, involved in reprogramming of pluripotent cells. And Bing's group identified a signal, suggestive of a enhancer that was distal to the SOX2 gene. And they were able to develop allele-specific probes for the two alleles of the mouse genetic background. So that they could analyze both the effect of the deletion on the gene expression as well as on the cis regulatory elements. And what they found was is that so they used that system to create mutations in either both alleles or only the 129 allele or the custanious allele. And then they looked at individual ES clones. And so when they found that both enhancers were knocked out, they almost eliminated the function. When they found, again, they were looking at allele-specific expression, they looked at the 129. In the 129, deletion, the expression of only that allele was affected in cis. And then you can have the opposite result where they deleted the custanious enhancer. And it only affected in cis, the custanious allele for succs 2. So I think these kinds of experiments will, in addition to the sort of high throughput mutagenesis experiments or surveys of nucleotide variation in a high throughput may really provide some powerful tools, I think, that can be capitalized if they're scaled. If there's a logic to being able to do these and to analyze large numbers of variants. So these tools can also be applied in a variety of context. And the challenge here is that this obviously can be scaled. So synthesizing variants on all agos, cloning those in large scale through into various plazas, creating lentiviral libraries. This has been published by a number of laboratories. And then using clever assays to assign function, selection, either forward, positive or negative selection, to really try and understand what the variants are doing to gene expression within different cellular contexts. So you can imagine, for example, in this case, taking a particular variant, introducing it into an ESL, differentiating, so replacing that allele in its context in a system where you can understand the variation on both alleles, so you have phase genome. And then reading out through different, so differentiating those cells into, for example, neurons, and then looking at function in those cases. You can also do this in the mouse. It's been done by a number of laboratories, Rudolf Janisch, has some beautiful work where he's replaced multiple alleles in the mouse. And so in this case, you can also inject the protein and the guide RNA as well to be able to target specific genes and then look at the effect in development. Can also do large scale and vivo experiments where lentiviral libraries can be injected into the tail vein to target different tissues. And this can be modified because there's a whole other area of biology that's being developed for targeting peptides to different organs in the body, some beautiful work at Scripps for doing this so one can potentially target other organs other than liver for assaying function. So I think there's a real opportunity for assessing the variants in terms of a phenotype. And now these are molecular phenotypes, and earlier we heard about organismal phenotypes. Equally, they're important, I think, in understanding the functions of these variants. And so I just point to one paper that I recently came out from the Ebert lab, which I think is a perfect example of this kind of Lord scale lentiviral screen where they identified five genes that result in a myeloid malignancy model. And you can extrapolate this to a variety of other disease models. The last thing I'll mention is an example here just briefly is what I think is my current favorite example of really beautiful work from the Kingsley lab in the challenge of understanding non-coding variants. And so there was studies that were published a number of years ago identified a variant for hair color. This is one of about eight that are add up to about 6% of the variation in blondness. And this is a population in northern Europe that has a high odds ratio for a particular allele, A to G allele. And this region that that SNP maps to. And so the Kingsley lab looked at all the data. I think this is a perfect example. They looked at the conservation using all of the organism sequencing data and found that the SNP lies right in the region of high conservation. They carried out transgenic experiments with different regions. And this gene is likely to be expressed in the follicles. And you can see that it is when you put that enhancer driving a LAC-Z promoter. They carried out both individual assays looking at SNPs where they've included a SNP next to a reporter gene that is either the ancestral SNP or the blonde allele or deletion allele. And they got a lot of variation. They were very clear about the fact that the deletion reduced this expression in the follicles. But the other alleles didn't have that consistent effect. They also used cell culture assays. They then used the data from ENCODE and linked a particular transcription factor left to a domain that right encompassed the SNP. They then went to the motif database and identified that this was a variant in the left motif. They then did transfection assays to show that that variant affected the expression of the kit gene. And then finally, they did a transgenic experiment where they replaced that allele in a safe harbor context, not in the kit gene in the mouse, where they included both alleles and they could demonstrate that a 20% variation in kit expression led to a change in the coat color of the mouse that was suggestive that this is in fact the allele that's causing the variation in people. Now, this study highlights why it's so difficult to identify causal variants for human traits. The SNP mapped more than 350 kb away from the kit gene. It acts in an anatomical site where the enhancers haven't been characterized yet by, for example, ENCODE. There hasn't been a survey yet of epidermal tissue. So then there's a lot more room for expansion of the catalog of regulatory elements in different tissues. The altered sequence doesn't perfectly match the left sequence, I didn't go into the details, and it only causes a 20% reduction that reads to this phenotype, and it's only in a subset of the tissues that kit affects, because the loss of function of kit is lethal, there are many other effects. So this is an enhancer that's tissue specific, but the study also illustrates how these difficulties can be overcome. The information from the population surveys allowed the identification of a variant that was actionable in this case, or at least was suggestive that could be actionable, linked to the region it was. The large scale genomic annotation projects provided the underlying data for assigning, for linking to conservation of biochemical activity. The transcription factor database allowed a hypothesis about a particular sequence, and the detailed functional tests of that enhancer in both cell lines and mice provided the biochemical, or sort of data, or genetic data that led to the pretty concrete finding that this is the causal variant of that variation. So I think that some of these kinds of assays that I've briefly discussed, I think can be scaled and can be useful for understanding the function of these non-coding variants, and the breakout session this afternoon will touch on these various topics. I won't go through any of this, what is the, starting with what is really the function at what level do you wanna study phenotype? Is it at gene expression? Is it at the cellular level? Is it at the organ level, or the organismal level? These are things that can be discussed. How do we interrelate function and variants on a large scale, and what are really the opportunities for NHGRI to be able to use these kind of approaches for leveraging the sequence variation that we now have? So I'll stop there, thanks. Thanks Joe, we have time for two, three, four questions. Ewan. Just in case, oh well. So this has been very focused on the non-coding area, which is incredibly important, and we know that this is, you know, a lot of stuff is happening in the non-coding area. I just wonder, are we, maybe this is something for us to discuss in the breakouts, but there is this business of us really knowing the impact of every amino acid change, and you know, the closer I've worked with people who are working in a clinical with panels, gene panels and stuff like that, these variants of unknown function are really annoying, and that's probably a British understatement, and getting a grip on those would be quite important. Have you thought about that? Has anybody else thought about that? I didn't mean to exclude those, there are challenges, you know, when you look through the data from the complete genomes and you say, wow, this should be, Mike Snyder should be dead if I'm looking at his genome. There's some obviously, there's some obviously, there's some. We've got a counterfactual on that. There's obviously some counter suppressor mutations, if you will, that are allowing those mutations that you expect have a devastating effect. So you could pick things out and say, hey, I'm gonna try and understand this, but in absent all of the other variants in the genome, it's a complex genetic problem to try and understand the function of that variant. You see what I'm saying? I do, I just wonder whether we shouldn't be trying to think more systematically about the function of every amino acid change, because we know, maybe even if you just restrict it in these panel genes, you know, genes, the iron channel. Yeah, I think that's an important goal. It's just, I think it's just a little more complicated because you've got protein complexes that can be affected by variants, et cetera. So, question over there? Dave, me? My question actually is not about iron channels, but when you say do this on scale, how many variants do you think it's realistic to conceive of studying? I mean, there are what, 250,000 or 500,000 rare non-synonymous SNPs in the coding region, let alone the non-coding region. How many of those do you think it's realistic to even contemplate addressing? I think there have all been studies to examine thousands, hundreds of thousands of variants, at least at the level of adding those variants to a marker and putting them in, transfecting them into cells and getting a readout. I think these experiments can be scaled to look at very large numbers of variants. I think it's like asking, how many genotypes can we collect in 1999, right? Do you think we could really do 10,000 at the same time, you know, today? Well, if you look at, you know, if you're thinking about the In vivo model where, you know, you're gonna make mice, that may be a little bit more challenging, but making stem cells, for example, that have variants, very large numbers of variants, and differentiating those to lineages that you believe that are important in those diseases, I think can be done at a large scale. Yeah. David Altschuler, I wanna connect a couple of the dots here. First of all, I wanna go back to what you and said. I think that you interpreted this question to mean reduced penetrance and some sort of second site suppressors. There is a very straightforward idea, which is to say, four genes for which we have, for example, a predictive assay already. For example, many Mendelian disease genes or other genes, simply functionally testing all the variants found in very large samples, sorting them into which are functional, which are not, and going back and doing the genotype-phenotype correlation is now practical at scale and is a way around all this, both variant of uncertain significance and reduced power. The second point is, my friend Jim Evans, who said, how do we translate all this stuff to drug discovery? We will continue to spend 50 or $60 billion here on drug discovery, even if it's hard. And the key challenge in that, or a key challenge, is having an in vitro or cellular assay or molecular assay that is predictive of clinical response in a patient. Using genetics to try and connect the assay to the patient, so the assay can then, through chemistry, being connected to the patient, is a key challenge, and that's gonna come through some sort of functional assays that are trained with genetics on the patient that they have a higher rate of success. I didn't mean that David, to leave out the drug part of, in fact, Richard Gibbs on a phone call really pointed this out, that you could couple these assays, obviously in stem cells in the right, even deriving maybe the right tissue type, to then test those in cellular assays with drugs and a readout. And let's not assume it's stem cells, let's just say predictive assays. Stem cells maybe means that end, or they may not. Sorry, yeah. Joe, so just to follow up on David and some of the points in your talk, one of the first things I'd like to point out is that the blonde hair story from Kingsley also illustrates that digging into function may give you a different answer in different populations. So blonde hair in Melanesian is actually controlled by a totally different gene. It happens to be an amino acid change. It also has a mouse phenotype. So it illustrates why you can't just stop in one population and why you need to deal with both the amino acids. Yeah, and failed to point that out that this is only one way to get blondness. There are much more. But the same will be true for lots of the phenotypes that people are interested in, right? I think the second point which goes to David is that doing every single amino acid change in the set of genes that folks are interested in, we can decide to prioritize that ACMG plus neonatal plus whatever may seem daunting today, but if we decide to come together and do it, that's totally feasible and doable, right? I think we really need to, you know, if we look back and say where would we be, obviously we didn't think we'd be where we are now in terms of sequencing. I think the same thing can happen. These tools, some of these recent tools really change the landscape for understanding the function of it. And it should be done in an unbiased way. And they come at exactly the right time. And we don't just need to focus on existing polymorphisms that are segregating in populations. In fact, one argument for going through and trying to do every site is that you sort of pre-computed it. And so, you know, even the patient that comes in with a mutation that you'd never seen before, you've already pre-computed it by doing this helioassay, it's there. All right, it sounds like it's just about time to break for lunch. It says here, setting the stage for the discussion. I think these four talks have set the stage for the discussion pretty well. There are some threads that are easy to tie together that I heard. There are some that are harder for me to tie together. I'm hoping that we can do some of that in the discussion if possible. So if you could be back here at about an hour and five minutes, 1.15, that would be great. Thanks very much.


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Open your mind to the cradle of civilization Mesopotamia, around 3500 BCE. Meet the Sumerians, the folks credited with crafting the first known writing system, Cuneiform. Starting as simple pictures representing objects or ideas, Cuneiform matured over time into abstract wedge-shaped characters. Think of a stylus pressing into clay tablets, then bake to keep the message fresh. These folks used Cuneiform for everything from grocery lists to divine texts, setting the stage for future civilizations to pen their own scripts. This is the kickoff to our series of firsts. Keep your eyes peeled for daily dives into history's premieres. Hit subscribe to stay in the know.


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Hi, I'm Zor. Welcome to a new Zor education. We will talk about linear inequalities to date, linear inequalities, not equations. Now, why I mentioned equations by mistake is because I was thinking, what can be simpler than linear inequalities, but only linear equations, right? All right. So let's talk about linear inequality. Well, kind of a general form can be something like this, or any other inequality sign like greater or less than equal, greater than equal at second. All right. So what's important? Well, obviously, this is important, a should not equal to zero because otherwise, it's not an inequality, it's some kind of, I don't know what it is. So, and the purpose is to find various of unknown variable x, which are necessary and sufficient condition for linear function a x plus b, to be in this case, less than zero. So that's what it means to solve an inequality. You have some kind of a condition on a function, in this case, linear function, and you have to find the corresponding condition on the argument of this function, which is x in this case, which is necessary and sufficient for the function to satisfy this condition. Well, it's easy, right? So we all know how to solve these type of things, and let me just make very, very brief remarks. Obviously, to solve algebraically inequality, you have to do the same thing, similar things to what you're doing when you're solving equations, which is invariantly, preferably invariantly, if not non-invariantly, but transform the original equality or inequality or equation, whatever you have, into some form, which really provides the information about the argument. Now, in this particular case, the first action which I'm sure you know how to do is add minus b to both sides of inequality. Now, adding any number, positive or negative, or zero, to both sides of inequality, is an invariant transformation, and the sign of inequality is retained. Whatever, if you have two numbers, one is less than another, you add something, doesn't matter what. Positive or negative to both sides of the same, of this inequality, will result in the sum on the left and sum on the right to be in the same relationship. So, if I add minus b to both sides, I will get ax plus b minus b less than zero minus b, which is ax less than minus b. All right, so that's the number one step, which we have done to transform our original inequality into form, which will give us the condition on the argument. Well, naturally, next step, considering a is not equal to zero, otherwise it would not be an inequality, we divide by a. So, on the left, we will have x. On the right, we will have minus b over a. a is not equal to zero, so there's nothing wrong with b over a. Well, question is what sign to put in? In the previous lecture, I was explaining different transformations of inequalities, and I was actually talking about multiplication and division of both sides of the inequality to be an operation which you have to do carefully. If you multiply or divide by positive number, then you retain the sign. If, however, you multiply or divide by negative number, you have to reverse the sign. So, this thing is like this, if a is positive and, sorry, reverse, if a is negative, and this is a solution. So, solution is a condition on the argument x in this case, which is a necessary and sufficient condition for this function to satisfy this condition. And it depends on the sign of the a. Well, basically there is nothing more which really is of any interest in talking about linear equations and their algebraic solution. And as I was thinking about how to make it a little bit more informative, I have decided to spend some time to analyze the graphical approach to this. So, let's talk about graphics. So, we have exactly the same type of inequality, and what I would like to do is basically find graphically the solution of this particular inequality. Well, to find it graphically, we have to construct the graph, right? So, let's construct the graph. Now, first, we start with function y equals x. That's our step number one to construct the graph. And, well, obviously I could construct it, right, you know, immediately, but I would like actually to do it step by step, so, you know, you will understand it better. Now, next step is I will construct the function y equals a times x. So, I have to multiply this graph by a. Now, how to multiply the graph by a? Basically, you have to multiply each y coordinate for each particular argument by the factor a. And here we have a slight difference. If you multiply by something like 2, for instance, then each y coordinate will be basically double in size. So, it will be something like this. If you multiply it by, let's say, 1 half, well, it will be shorter by 2. So, this is y equals 2x, this is y equals 1 half x. Now, what's common in all these graphs? They are all monotonically increasing, and that's very, very important. So, let's remember that whenever you multiply the graph y equals x by any positive number, less than 1 or greater than 1, it will move a little bit up or a little bit down, but it will still be directed upwards. If you go along the x-axis from minus infinity to plus infinity towards increasing of the axis, then the graph will also be increasing. It's monotonically increasing function. That's very important. If, however, you multiply this graph by a negative number, minus 2, minus 1 half, or minus whatever, then instead of being directed in this way, the graph will be directed this way. So, this would be y is equal to minus 2x. It's similar to, it's symmetrical to y equals x relative to the x-axis, right? So, we have to multiply, to multiply by minus 2x, we have to first multiply by 2, which will do this graph, and then multiply by minus 1, which means every positive y will go to a negative sign. So, that's why the direction will be this way. And again, for all negative multipliers, A. And all these graphs with negative multipliers are monotonically decreasing. So, as x is increasing from minus infinity to plus infinity, the y will be decreasing. And this is extremely important for further analysis of this inequality. All right, so we have concluded that multiplication of graph y equals x by positive number retains, although it changes in the scale, but it retains monotonically increasing behavior. And multiplying by negative factor A, we are making our function y equals x into a monotonically decreasing one. All right, so the first step is we multiply by A, and whatever the graph is, I will use the graph with a positive and something like this for negative A. So these are graphs for A positive and for A negative. Next is what we have to do is we have to add B. Now, what is add B? Add B means we have to move vertically, either upwards if B is positive or downwards if the B is negative. So that's how we build the graph AX plus B from the graph AX. So in this particular case, for instance, one more detail. If I'm shifting the graph upwards or downwards, it doesn't really matter. The behavior of the graph of being monotonically increasing or monotonically decreasing does not change. So if I move this particular graph up or down, it will still be monotonically increasing because it's directed upwards. And this one will be directed downwards again, regardless of where I move it up or down. So I will use, for instance, movement up, but it doesn't really matter. So on my next graph, I will put it this way. This would be my Y equals AX plus B with A greater than 0 and B is greater than 0. And I don't want even to draw the graph when B is equal less than 0. Now, in case of A less than 0, I will also move the graph upwards to the same value. This would be Y equals AX plus B where A is less than 0 and B is greater than 0, also greater than 0. So both graphs are moved in this particular up because the B is positive. Now, we can answer the question, what are the axis when this is less than 0? Well, in both cases, we can find the point where it's equal to 0. One graph is equal to 0 here, this one, and another graph is equal to 0 here. But now the question is, this is where this particular graph is equal to 0. It's monotonically increasing. That's important, which means that if X is increasing from this point to the right, graph will grow, it's monotonically increasing, which means it will be greater than 0, which is not good for our purposes. And only in case we move left towards minus infinity from this particular point, only in this particular point, only in this particular case, the graph will be less than 0 because the graph is decreasing if we go left. So what's important is, if you have a solution to AX plus B equals to 0 and solution is minus B over A, then in case A is greater than 0, our solutions are to the left of this point. And in case of A is negative, the graph is decreasing, monotonically decreasing, which means if I increase the X from this point towards plus infinity, the graph will decrease. And this is where it's equal to 0, which means it will do less than 0. So I have to move to the right from this point, which is a solution to this equation. So the answer is X should be greater than minus B over A to satisfy this particular inequality. If I move to the left, the graph will increase because when I'm moving to the left, monotonically decreasing function will increase the value. And the variable will be greater than 0, which is not good for this purpose. So I would like to basically conclude from the graphical representation of my linear function exactly the same thing which I did algebraically. When A is greater than 0, this goes to minus B, and I divide by A. And I will have exactly the same. I retain the sign less and less. When I divide both sides by A algebraically, my sign of inequality is the same, is retained. If, however, A is negative, my algebraic approach said that I have to change the sign. And as you see graphically, it basically gives you exactly the same result. So this is another explanation, if you wish, of the rule which you were talking about many times before, multiplication or division of the inequality by negative number must actually reverse the inequality sign while the positive number doesn't really change anything. OK, that's it for this particular lecture. I do recommend you to review the notes to this lecture on unizord.com. It's basically the first one in the linear equation category. And I will continue talking about linear inequalities in the next lecture. And well, it will be a slightly different approach to solving inequalities with absolute value and linear functions involved. And that's it for today. Thank you very much.