So this is yet another talk about FHE but this time it's on the circuit privacy point of view So quick reminders first. What is FHE and what is it used for? So let's say you want to compute something on the cloud Let's say you want an online diagnostic because you are feeling symptoms and you don't know what they are You send your data, your medical data to a server, it runs some computation and returns the diagnosis The thing is you want your medical data to remain private and even the diagnosis you don't want it out on the internet So you encrypt your data and you want to encrypt it in a way that So that Algorithms can still be run on your data. That's what we call FHE An example of FHE which you might have seen a few times already at this conference is a gentry side Waters 2013 So the G Matrix here is not the way it is made is not really important What matters is how you encrypt and how you do operations so To encrypt you have a random part, which is the matrix A you multiply it by the secret S and you add a noise port so to Just quickly to do operations to do a sum you do just the sum of encryptions and the product is a bit more Intriguing in that to do a product you need to multiply by g minus one, which is not a matrix It's an algorithm Which is such that g times g minus one of something like g times g minus one of v is equal to v and G minus one has to be small So you're just some color coding orange means uniform and blue means small for the rest of the talk So that's gsw encryption. It was published in 2013 reworked a few times afterwards and The problem is that now you get FHE, but the error term this error terms is Modified during the operations. So by analyzing the outputs the user can guess or maybe at least get some information about the algorithm that was run on the on his data and Maybe the server is running some proprietary Algorithm and does not want it leaked. So the focus of this work is to See how we can hide the everything that's all the operations that were done on the data so Toy example to see how information is leaked. Let's say you have three Encryptions of zero. It's not very interesting But what you have three encryption of zero and the servers wants to sum two of them the plain text associated Would be zero because it's just the sum of zero plus zero But if he says let's say c1 and c2 the error term which the user will be able to recover Will be e1 plus e2 so he can know exactly which Cypher text the server has Summed and then he gains the information about the algorithm. He knows exactly what the user did So that's what we want to avoid We want that the evaluation of a function and on the bunch of suffer cypher text should reveal nothing But the odd part of the function on the associated plain text so this has been addressed not Not that many times but a few times already so in the seminar work of Gentry Let's say you have a ciphertext which is which is the result of an evaluation and the first solution proposed by Gentry was to add some noise Since the information is leaked in the noise if we add the amount of noise that is super polynomial in what already exists All the noise all the information will be destroyed So the advantage is that you destroy all the information That was leaked the disadvantage is that you need to add super polynomial amount of noise and since the noise needs to be smaller than Parameter q which is the modules the parameter q needs to be super exponential and Another problem is that you cannot do more operations on your ciphertext since you have already added a very high amount of noise More recent approach was by Duke and Staley in a Eurocrypt 2016 And it consisted in first adding some noise But this time it's only at the same size of the noise already in the ciphertext and Then doing a bootstrapping. So here it's a bit of a pedantic way to write it But you evaluate the decryption site the decryption circuit on an encryption of the secret key Which means that you get a new ciphertext of the same plain text, but with a known bounded amount of noise so This is not enough to hide the computations But we can repeat and each time we repeat this process We add some entropy to the noise and if we repeat it a number of time linear in the security parameter Then all the noise will be randomized So this works with polynomial modules since we don't have to add a to make too much noise We can do more operations. So this is multi-op And the problem is that it requires bootstrapping. So this requires a circular security Assumption since we have an encryption of secret key under itself this is an assumption that is not very well known and It requires a linear number of time to be strapped a linear number number of time But strapping can be quite expensive. So this adds a lot of computation Then there is our approach approach. So still same ciphertext We add a noise that is practically the same size at the previous noise and we do nothing else the trick here is that our noise is Not blue is green Which means that it's not just small. It's a small Gaussian. It has to come from a Gaussian distribution with appropriate parameters and What we did in our paper is to analyze in depth the growth of the noise to show that It's in fact just adding a small noise is enough to hide everything So as before we had a small noise. So we have a polynomial modulus We don't need any circular security here because we don't do bootstrapping and we can do multi-op The problem is that since this is for This applies to branching programs It only works for NC1 which in itself is not really an issue because there is no FHC for All circuits without bootstrapping So as long as we don't use bootstrapping. It's normal that we are only for NC1 circuits another issue that it only works for the Encryptions that I have shown previously. There are other FHC encryption schemes, but this one is a Taylor made for JSW encryption and It leaks the size of the function or at least a bound on the size of a function So now how do we do this exactly? So we will have to go a bit more in depth in the branching programs. So a branching programs is a succession of states state v0, v1, v2 And To progress from one state to the other there is we apply permutation According to choice variable Which is to say here if the variable x1 is equal to 1 we will go from this state to this one if x2 is equal to 0 we will go from this state to this one and at the end of the Branching program evaluation if the state we reach is the first one the right top rightmost one The branching program returns one otherwise it returns zero So how do we write this down? This is a multiplexer multiplexer So let's say that vt of i will be this state It will be equal to this one if x2 is equal to 1 and it will be equal to this one if x2 is equal to 0 We follow the Incoming arrows, which is to say that vt minus 1 is a multiplexer of the variable choice The choice variable and the two previous locations So this can be rewritten as an polynomial equation like xt times vt minus 1 plus 1 minus 60 times vt minus 1 and Now this is for branching programs on plaintext and we want to do the same thing on ciphertext And it's the exact same thing so instead of having plaintext States we have encryptions so we used to have like one zero zero here We will have an encryption of one year an encryption of zero and encryption of zero and We will have encrypted choice beats and to do the this operation we do the exact same thing on ciphertext So vt of i will be ct, which is the encrypted choice beats to multiply. We need to apply g minus one Times vt minus one of j plus g which is an encryption of one It's just a noise less encryption of one minus ct times g minus one Apply to vt minus one of k. So exact same thing as before but on ciphertext And this is what we want to analyze to show that it does not really leak any information So what do we do? We will rewrite the choice ciphertext to be equal to a matrix plus The choice bit times g Just how the ciphertext is formed and here you just have to trust me, but if we reorder the terms we get this So what does this correspond to exactly? This you can see is a previous step of computation. This is vt minus one. So the previous state Taken at the right place According to the choice bit So this will be the previous ciphertext and this will be an additional noise that comes from the operation we have made So if we should show that the additional noise does not leak any information about the operation that was made at this step We can just do an induction on each step and show that no Nothing no information is leaked at all during the branching program So that's what we will be intent on doing And to do this we I will show the core lemma of our paper, which is gsw re-randomization So let's say we have an encryption of zero So just a ciphertext of zero With an error vector e then we show that for any matrix v just any matrix if we do c times g minus one of v plus zero times z It is statisticallyinguishable from c prime. I will explain what everything is So first observe that g minus one is no longer blue. It's green, which means that g minus one now has to be a Gaussian so Previously all we wanted was that g times g minus one Of v it was equal to v now we want g minus one of v to be a Gaussian condition on this It's not hard to do it has been Done by a gen 3 breaker turn by kutuna turn and it's just sampling a Gaussian on a lattice And the second thing we do is we add a small Gaussian noise in the form of z And just by doing this We can show that re-randomizing a ciphertext like this gives a fresh encryption of zero So it gives a new ciphertext of zero, which has a Gaussian parameter norm of e So it leaks The norm of the previous error But it is completely independent of the on v. It does not matter at all what v is in this context So how do we modify our branching program evaluation? This is our branching program evaluation First we take the Gaussian g minus one. We replace the g minus one by the Gaussian one And then we add noise. So here we added two times because we have two times something times g minus one So if we reorder we have bt Which here is a ciphertext of an encryption of zero Since as we have seen before ct is called bt plus xg So this is an encryption of zero times g minus one of something Plus a small error and second time the same thing Which means that both of these can be replaced by ciphertext That's encrypt zero and are completely independent of vt minus one and Of j and vt minus one of k So everything that is leaked Is only the only leakage is dependent on the choice bit ct So we already hide all the prime mutations in the branching program And what we show is that it does not really matter that we leak some information about ct because we only leak We only some New ciphertext with which depends on ct. So let's see how the noise evolves So we have our first encrypted state which has noise zero We start with the noise less encryption Then we do one round of operations So the noise on the new Encrypted states will be the noise on the previous one plus something that depends on c1 So a Gaussian that depends on the error in c1 So it only depends on this Gaussian Then we add we do the second step. It will be the previous noise plus a Gaussian that depends on c2 So it will be the sum of two gaussians one that depends on c1 one that depends on c2 And then we just repeat this until the last step So at the last step We will get something that the noise in the final Cipher the final state will only be the sum of Of gaussians that depend on the all the choices bit So what we can simply do is Pad the branching program Just add use less steps a few like polynomial amount of them so that the Each choice bit would be the same use the same number of time This way we do not leak all all we leaked was the number of time each choice bit was used Now they are all used the same number of time. So all we leak is the size of the branching program And like this we have achieved Secret privacy since we do not no longer leak either the permutations nor the choice bits used at each step And that's it


So today, actually, I'll be talking about what's near and dear to my heart where I've been spending almost half of my life, 25 years in technology commercialization. Hopefully by telling you a little bit about my story, you get a little bit inspired and see that it's all about creating value. So that's why the title I didn't get to change it anymore in the sense that it's all about being able to move your technology to the point where the value is created. Just like everyone, I'd like to establish my credential as a researcher first so that at least you'll believe half of what I'm going to say afterwards. Actually, I did get started in research after finishing my undergrad here and I went in biotechnology for the first few years. In 1989-93, I worked for a company in Boston and then moving onwards after coming back here for a short break in 1993 where I got acquainted with the technology. I started on that research as well and moved that from the bench being on my PhD thesis all the way to a commercial company and going from a development time of about 17 years. Again, as I mentioned, I was a Balik scientist in 2013 after having sold my company in 2012 to Johnson & Johnson. Of course, I've been always an advocate of technology commercialization and intellectual property. So hopefully you'll believe of half what I'm going to say after this. Just hoping. Okay, from a research area, I've always liked to impart to the group that before you actually start the commercialization cycle, you're normally involved with a lot of different things. Just like what Dr. Dalisa said earlier, humanities, other areas of studies actually fill you up to the point that you're ready to go to the next step. It helps. Definitely the humanities. I was sorry I'm not a UP graduate. I went to a Philippine science where Porsche, over there's my classmate in Batch 83, but I went to Baguio, a university, a Catholic university in Baguio called St. Louis. People asked me in grad school actually in 1989, what's the most important engineering course that you've taken? I said it's religion. Why? Because I ended up talking to my fellow grad students in Rensideer from India, from Korea, from Japan about the religion as a icebreaker. That's where the humanities, the soft touch came in. To just break the ice, I said I know about Shinto, I know about Buddhism, and really these are nothing to do with engineering. But that's the value of humanities. I really have for me or your earlier talk, Dr. Dalisa, thanks. But going into deep science now, as I got involved in medical product development after my research, so you'll never know when you'll end up. Again, I deal with doctors on a daily basis during my company, so dealing with UP Manila at this point is like par for the force. I tried to avoid when I came back here to go back to healthcare, but I got invited into the Global Health Forum and actually NIH, Dr. Eva, invited me as well. This is actually my third engagement in UP this year. I was in biotech and loss banyos and tried to look for... There they are. So some of these slides may be familiar, so bear with me, but I have a surprise slide at the end for you. So with that background, as you can see, I only post this so that people will get acquainted that technology commercialization takes a while. It's almost like raising a kid. So my career actually span about 25 years now and I'm still at it, actually. I'm still excited. Moving forward, may the technologies may have changed, but it's that kind of dedication, almost like vocation, if you want to call it that way. But clearly I went through several cycles and what I want to relate is every step of the way there's value creation and that's how you kind of chain of events that happen in your life as a researcher. So how did I get started? I was deep in research. I was a PhD in chemical engineering and I got started into it by simply doing one thing. I filed a patent. I discovered that the way to make my material, normally they say you can't. Once you've inoculated it, I'm a biochemical engineer, so it's like a fermentation process. Once you inoculate it, don't touch it. And I said, what if, instead of not touching it, I rotate it into the media and it created a new process of material and they said, how do you do that? And that's how I got started and I ended up filing a patent and this was actually during my break when I was involved. How many of you still remember the 1993 boom of Nada de Coco? So I'm in kindred spirits here. So I was there. I was here actually. I remember Willie Padulina who was the OSD secretary at that point talking about how we can make it sustainable at that time. There was a boom. Japan imported $28 million worth of Nada de Coco. They had a craze, a health craze about eating it. Same way, they had a craze on muscar bought in cheese from Italy. They bought the whole years in the house of gloves. So the same thing happened and Dr. Padulina said the way to make it sustainable at that point was we had to develop applications. We had to be able to know. And then do an honest-to-goodness development of product. So when I saw him, when I came back here in 2013, we were a little bit involved with the Pecari startup at that point in time, I told him hi Dr. Padulina, I did follow exactly what you did and I spent 17 years trying to commercialize the technology. So that's how it all got started by filing a patent and presenting my patent in one of these booths where an alumni of the university who just sold his diet for business approached me and he saw that material this material which is to a lot of people very familiar, that's nata di ko ko in a very thick state but it's not much different from what we actually eat in our halo-halo So he saw one of these and then that's what we got started talking in 1995 and end up starting a company and that's the birth of silos the one in the corner and it's all about a technology called microbial cellulose it's a cellulose material not much different from the table and the cloth that you're wearing it's cotton, except it's about 100 times finer than cotton and it absorbs about 100 times its weight So for a normally un characteristic layman they said oh, can I use it as a diaper super adsorbent but we ended up developing a different but as a researcher I had to be able to shift in terms of technical so being a researcher in the university is different when you already have a company where you have to think and swim, you have to raise money you have to find your counterparts, that's why I'm encouraging people here I'm daring them actually because just like anything else I always tell people you can read all the books about farming but until you take dirt you're not a farmer so the same thing with entrepreneurship and technology you can dabble and you can try to do it as a professor and do a little bit of commercialization here and there but to me the best way is to jump in and swim is what my daughter just did two weeks ago so with that kind of one thing you have to know exactly where you're jumping into and this I just want to illustrate that in my case I had to shift completely from research to more of what I need even being able to learn how to conduct clinical trials and dealing with the regulatory agencies like the FDA because my product was an FDA regulated product so those are the skill sets but what good thing about it is being an entrepreneur I can tell you I always have a different problem every month so never the same so I've always either worried about the patient worried about the sales worried about raising venture capital but it's a constant grind but it is very enriching so I dare you guys to do it but at the same time there are startup challenges which we're going through right now here in UP and we just went through one I'm happy to say that it was worth it one that went through our parents opinion report and been signed by the the technology and I'm hoping that within the next couple of years that now that the runway is open that we'll have more spinoffs coming up another area is on being able to complete your team I'm telling people just like an entrepreneurial company I'm encouraging all the CUs actually of UP commercialization team in that way they can deal with the upcoming surge in intellectual property and commercialization activity that we're trying to promote from the system level with Louisieson so clearly I just want to have for you guys to have a flavor what it takes and also be careful a lot of people ask me money is always a problem it's not actually if you have a very good idea and a clear view on where you want to take your technology money comes naturally there's enough money even in the Philippines I've been consulted by people in the business sector which I'm very intimately acquainted with and are asking me where's the technologies that you're supposed to line up for us this is what CC was asking me every year every time we see each other for dinner and I said it's not yet ready because I'm not going to put my name on it until it's good and ready and unfortunately when Dr. Benny Pacheco invited me to do a colloquium early 2013 and looked at the UP technologies and patent holders including Dr. Henry Ramos Dr. Patasanza and a host of other patents I tried in my earnest to push them forward towards commercialization unfortunately at that point in time it was not yet ready and there were additional activities that need to be done I think Professor Ramos was able to get his technology out on license but it still hasn't gone through that 50-year cycle that would make it sustainable so we're still going through the challenges so as you can see it does take time but in the event that you do get money be careful what you wish for because the people that fund startups they'll ask at your first board at least they'll ask for milestones and this is the kind of milestone activity that we're also implementing in our grant making in U.S. aid stride which I'm also a consultant I think Dr. Carla knows that because I'm handing her development research for stride we establish milestones before even starting with a project and then we monitor the milestones the same way a VC giving you money this is actually my first $1 million funding when I got it and this are the marching orders very specific but it also makes you very focused in that whatever you want to achieve it's already listed because they already know and that's exactly what we're trying to do with some of the technologies that we have here and I'm hoping that by the time we're done coming over the patent filings to disclosures to the patent filings to the commercialization strategy in UP I'm also once ready for venture capital funding and I hope that Chancellor Menchit in Dileman in Manila would have three definitely from Las Baños I'm expecting more five with all the technologies that are there already even have FPA applications cleared already in UP Dileman I'm also hoping that we have at least three three to five as well so by the time we all combined it including Visayas would have a share at least if we can achieve 20 commercialization efforts in earnest will probably at least have a real case load because I'm trying to I just visited the Rutgers University with Louis in May last year and we spoke to Mark Robson and their technology center they file about 215 patents a year and they have about 110 licensing agreements moving forward so this is one little university in New Jersey and our metrics are way small compared to that but I think we'll get there I think we'll achieve 100 disclosures and at least maybe 20-25 patents file this year and we're going to scale up and I think VCREX over there is not in his head I hope we can do that with Rick also for Visayas so moving forward on that one kind of thing that you have to adapt not everyone is geared up for being able to commercialize the technology but I always encourage the young researchers the guys in the back the young inventors usually it's not the national scientists like Dr. Toronto over here that tend to do it in their mid-career now he's doing it after emeritus career it's the young ones that have the energy and the risk taking ability so I do encourage you because it does require you to have a host of skill sets that are quite different from being a research scientist but if you if you really wanted to do it I think you'll pick it up much easier in terms of being able to do all of these things but you have to follow it from each from pitching what do you have elevator pitch we always have this rehearsal that if you happen to meet Bill Gates on the elevator and he ask you hi Al, what do you do? ah, I develop medical devices I actually build body parts to replace breaking parts in the human body oh really? how do you do that? ah, use this different material that's not been used before for these applications and we can replace human body parts as well as animal body parts that you typically use talk to me later so you should be able to concisely pitch every single time that you are available or every chance you get you see President Pascual walking there or the self-conception I need money how about an EIDR okay, then tell me what's your EIDR I'm able to digest data from the big data from Dex from Department of Education to be able to direct them where to put their supplies and their evacuation centers maybe a simple elevator pitch like that would be come in handy, you'll never know so always prepare always prepare and know your technology and what is your elevator pitch and then going to that being able to tell them why should I be interested what's the competitive edge as well as going all the way down to the clinical this word gets bloody when you have to really prove what your proof of concept is and that's what I'll stress to everyone because when I came here I also was a Balik scientist and I looked up that portfolio of DOST and if there's anything and Dr. Nuna was the EIDT head at that time if there's anything that was very challenging for us at DOST was being able to articulate the value proposition of all the technologies we funded that was the biggest challenge we have a filter, we have a dengue of the lab result of it and yet we can't say that connect that technology to how many lives did we save yesterday or today and that's the kind of technology that I wanted portion alloy from UP Manila on the RX box to be able to vocalize we now have 1000 RX boxes and it will save thousands of lives every month if this are properly put in the hands of the clinicians out there in the rural areas that's the kind of pitch that we need to be able to tell the DOI secretary the president, the senators so clearly just because you're a researcher doesn't mean you can't pitch you have to learn how to be able to tell them exactly what you're all about so with that I think just to give you a time perspective I usually use a metric I just posted the former president of the AUTM this is the association of university technology managers in the United States his name is John Pressure he was here in September and I took them to meet the system people this last week I made them meet the Dileman and UP Manila group and he always in that talk of his he always mentioned that even the most advanced technology manager in the United States like Colombia it does take time there was a video clip that he showed before the 3 by 7 it says that from the time that we filed our first patent takes us about 3 years before one third of that group of patents that we filed gets noticed for licensing and only about 70% of them gets licensed in 7 years license pa lang yan na and remember the real product development only starts at the beginning of the licensing wala pamproducto yan so imagine how long it will take to develop a product so I'm talking about like raising a kid almost before you can let them go at 18 I guess but hopefully it's not that going to be always the case but in my case 2, 5 years and 3 marketing partnerships to be able to launch a first product and a lot of people discount marketing as being just ah, laway lang yan and bila gay mo sa bag tapos get out there, sell it dali lang yan but a lot of people don't realize that this is the most difficult part of the process and this is where a lot of the technologies die because of not having the right marketing partner and I 3 of them before I could actually launch nationwide in the United States luckily the ones that we talked was able to do a good job and we were able to go to Europe within 2 years and 3 years later they wanted to buy the whole business so that's the kind of relationship that you are on the hunt for but just like anything else after you've done your first round time to scale up that's exactly what I'm worried about the RX back because it has scaled up from 100, now you're 1,000 the full scale is 45,000 if I'm correct so I'm hoping that it will come I'm not questioning that it will it's just when I hope we'll get there sooner rather than later maybe 3 years but that's a kind of iteration that we need to go through and so clearly and you also have to have a vision in my case I was there all about eliminating the risk of using animal and human materials so that was my value proposition and then of course applications there's a lot of applications in any technology but I always tell people focus only on one in software development lingo apps so in my technology there was a handful we were developing products in all these areas but the first two were the one that's fully commercialized in the wound care and the neurosurgical that took us about 8 years to develop I think I'll show that here so that's all we're doing really just combining the chemical processes as well but the one critical point that I want to make is let the users design the product for you be intimately associated with them because they'll be the ones who's going to be adopting it so in our case it was the doctors we lay out our prototypes every time they ask for it but it is them that eventually will decide what works again just to highlight this is the value proposition example of what I was replacing human and animal sourced materials so I had to identify what's the difference with my material so and just to even make it more difficult we are the first company that develop a material for implant permanent implantation of microbial cell use but it took us almost 9 years in about 20 million dollars to do it but unfortunately we got to the finish line we had our FDA approval in 2012 but it was a good time to sell the company as well but I had at least a very patient venture capital group that backed me up but it took that long so at the end of my 17 years this is the value we had 7 FDA approved product we had about I had 10 US granted patents in my name about 13 internationally I had been able to market the product for both the US and Europe with a CE mark and it was a natural point in time to sell also because again the venture capitalists that funded you want their money back and then some so you better be able to give them back with interest plus plus but what it did was it opened up more opportunities so to me it was a good time also 2012 was a good time for me to take a break and come back here I did race car driving for a year after I sold the business and my father-in-law said you can only chase so many sunsets with your Porsche I'll come back here and help the scientists in the Philippines you may be able to give them a few pointers and that's exactly what I did so in my first talks if you've seen me talk before there's one word that I always tell people to remember if you have a technology you have to trim it tabasa nyo and trim is address all those particular foods focus your technology deal with regulatory have intellectual property and yung tatlong eme yung tatlong eme nito is manufacturing, marketing and money so if you have all these ingredients then come back and form your business plan so that's my requirement elevate and usually if you do calculate your ROI it's probably science fiction in the first time around but try it anyway and have an impressive number if you can so as a personal counsel to entrepreneurs that I give because I do mentor students in enterprise with Louie I always dare the young ones and the old ones and dare them to dream about hopefully commercializing their technologies remember you can't do it all so form a team and a lot of people are hung up in the Philippines about control they think they should own 100% or 51% or super majority at 66% all these kinds of no, it's an illusion you're never in control because you don't have the money if you want to pay for everything, yes you can be in control but if you're trying to raise with venture capital investors and dealing with the FDA and the funding agencies you're never in control you're in control of technology and the business but remember you can try your best even if the venture fails or the business fails which 9 out of 10 it will yan ang stats sorry to say it this is the scary part only 9 out of 10 or even worse stats enterprise solicitation 128 business plans 8 got funded in Chicago only 2 were alive a year later so that's why we don't have a technology in UP coming out or a successful business because our technology is not that many we started when Benny was we had about 15-18 doctor Pacheco so we tried to move 3 and we didn't succeed we got to a certain point but there was not enough so we need more that's what we've been trying to do is to plant more seeds essentially but even if you fail the first time around don't worry because the business may fail but you as a person you don't fail you actually improve you get better you get harder so do it anyway so I think for those people who have seen me talk you've seen it to this one this is the news slide so at the end of all of this talk what does it take being a race car driver this is what I came up with you want to commercialize the technology drive deliver you'll promise if you're going to make early detection early cure deliver give me your scientific based proof set or reduce my risk if I'm going to adopt it the more that you have FBA or FBA approval the better because my risk as an investor is less I I think President Pasqual touched on this one innovate solve a real problem get out there know what the problems are and then innovate to solve it I always ask the professors how often do you meet industry industry I see them every now and then but no get out there because only then will you really know what industry needs and wants and then go back to your labs and then invent or innovate to be able to solve their problems then you get adoption and of course V is the value I was talking about all this time it's a value chain every step of the way every time you wake up in the morning you ask how can I make or create value for my technology today same way how am I going to serve my government or serve UP today and if you're a good researcher you can do both and last but not the least effort nipuwaiting ni nga skogun matagali ni it takes time in my case it's been 25 years it's been fun though and I encourage you to try it sometime thank you very much


Thanks, Adam. It's really a delight to be here. I want to thank Eric for inviting me and also wish you a happy birthday and another many years in the future of great success with the NISC. So why population genomics? What are we referring to here? This is simply taking the set of questions that population genetics addresses and taking it to the genomic level. So population genetics is interested in trying to infer the balance of the roles of the mutation, drift, migration, and selection to make a statement about the way evolution works. In other words, it's not just a description of evolution. It's sort of an attempt to really understand the mechanism behind evolutionary change. Unfortunately for population genetics, this means having to drag armies of students through very tedious lectures dealing with these nasty statistics like pi and theta and rho. I'm going to spare you much of this, but at least give you a little bit of the flavor of where we can do this at a genomic level. Another question that one can address with population genomics is having to do with past population demography. It turns out a lot of the issues with the HATMAP project, with inference about complex traits and identifying genes associated with complex traits are contaminated by confusion about the demographic history and population structure, so we need to understand that. In addition, having genome-wide data from multiple individuals allows us to do sort of scan statistics, looking across the genome for heterogeneities in these various statistics about levels of nucleotide diversity, recombination, and so forth. So the basic building block for modern population genetics is built on the idea of having complete data, so sequence data from multiple individuals ascertained in some uniform and random way, and so that ideal data set would look like this, where you have complete sequence with no errors, of course, no missing data, and that's the ideal. Of course, what we actually usually get is something far from that, and what we want to do from those data in any case is to estimate some of the primary parameters of how much variability is there, what's the sort of cause of that level of variability. The primary parameter that arises from every which direction, if you look at theoretical population genetics at all, it's astonishing the degree of convergence on this parameter, theta, which is four times the population size times the mutation rate. You can derive this as being a primary determinant of the level of variability in a population forward in time with a Fisher-Wright model, backward in time with a coalescent from infinite sites, from infinite alleles model. It's an amazingly convergent sort of statistic as being a primary parameter. So one of the things we'd like to do is to estimate that from various sorts of genome data. The particular kind of genome data I'd like to talk about is the very exciting short read data that we've been hearing a little bit about already. We've heard several talks dealing with short read data and the interest in using these short read technologies to recover the sequence of the individual. Often this is for the purpose of identifying the individual mutation or the set of mutations in that individual, getting high accuracy for an individual. And we've seen in those circumstances that it's necessary to go to reasonably high read depth, 10, 20, 30 as a read depth for those individuals. I want to look at the opposite situation where the data that are available are actually rather sparse. So this is ten different individuals where you're looking at a coverage of well under one X in each individual, but you have many individuals. Can you do any kind of useful analysis with that? And it turns out in the population genetic setting, the issue of estimating these sorts of genome-wide parameters, this sort of data works very well. And I'm going to show you some of the sort of directions to approach this. This was actually a problem that arose at first, the first that I was aware of it, with the celerogenomics data when they had sequences from six individuals and it sort of fell upon my lap to estimate some things like nucleotide diversity and so forth from those six individuals. And we did a rather crude ad hoc method. At the end it wasn't published because they weren't releasing the snips at the time, but it was a problem that I kicked around with Rasmus Nielsen right at the beginning and he was sort of enchanted by the problem. It's actually quite interesting, not only is the data very gappy, but of course there's relatively high error rate, a typical single sequence error rate for Selexa reads might be on the order of a percent or two. There's also site specific errors, so particular regions of the genome might have different sequence error rates. And in addition, if you're sampling from deployed individuals, you're actually sampling from the two alleles of that individual. So there's a binomial sampling from each individual. It's a very interesting sampling problem. And along with Rasmus Nielsen, Anas Hellman, we have a paper submitted that actually derives a water synestimator in the face of all those errors. And all you need to do is you need to have a very good error model. You have to know what are the determinants of error, what's the rate of error, what's the sort of base neighborhood for those error rates. If I can specify an error rate model and have that kind of data, I can estimate theta actually very well. So the particular test data set that we're going to talk about was actually funded by Adam Felsenfeld. It's a pilot project for one of those interminable committees one sits on for NIH. And this was one that was dealing with selecting genome sequences, and in particular the idea of using short read technologies in that context came up for SNP finding across the whole genome. How does one most effectively identify polymorphic sites across an entire genome without having to fund another HapMap project for every organism? And so the idea of just throwing them into 454 or Selexa was very appealing. And we proposed doing this with just ten lines of fly, six from North Carolina, four from Africa. 454 sequencing was done at the Wash U Genome Center. Elaine Martis was our primary contact, and she was terrific. And this was back in the days of the GS20. So one run was done for each of the ten lines. It gave 3.4 million reads, about 351 megabase pairs total of sequence. That alignment did look like this. This is actually a region of chromosome 2 of the real data. And you can see that there are regions that actually are gaps in the data, other regions where there's a depth of about 2.5 average across the whole project. About 74% of all those reads had a unique fit. This mosaic is one of the assemblers I'll talk about in just a second, but it's a pretty interesting start to the data. The first thing to ask about is how homogeneous is the depth? Are we sampling some regions of the genome better than others? Are there gaps and so forth? The North Carolina population, there were six lines, remember, so the depth for North Carolina is always going to look greater than the depth of Africa, where there are only four lines. And it was quite homogeneous across the, this was the X chromosome, except for occasional spikes one way or the other, presumably for some kind of repetitive element. In some cases, we could clearly see that's what it was. For these data, there was a reasonably good fit to the Landau-Waterman equation for coverage from a whole genome assembly, whole genome shotgun. I have to say however, since doing this, realization was that the power for discrimination of the goodness of fit that Landau-Waterman was quite poor for these data, because the read depth was so low, you saw from Richard Gibbs' slide quickly flashed by for the Jim Watson data that there was actually a bimodal distribution. There's also been something like 7x coverage of the C. elegans genome done by 454, and there's a pronounced excess fatness of the tails of the coverage distribution. So there are too many regions of the genome with insufficient coverage, too many regions with excess coverage, and those partures from the Landau-Waterman equation are really crucial for use of these short read data for inferences of things like expression level by counting methods. So I think this is a really important problem we need to get a handle on with these methods, but what really determines coverage. The average coverage for the African lines is about 40% of the genome. The North Carolina line is about 60%, so the pooled was about 3 quarters. Most of the regions that were covered by one were covered by the other and so forth. So we could look at particular regions of the genome that had particularly poor coverage, and so this is 10 kb fragments that had less than half coverage, and again it's quite spiky. Particular regions of the genome are looking like they're falling in those sort of gappy regions. So the real problem then, the primary problem of this whole pilot was to infer polymorphic sites. Where are their snips in the genome, and can we devise a sort of inferential method that would have a relatively low false positive rate and a reasonably good accuracy of determining those snips. So this first began, so we actually turned to some folks who had actually already been thinking about this, Gabor Marth and Erin Quinlan, that Boston University have been sort of working on problems of this sort with saying or sequencing, and recently turned their attention to short read technologies, and they realized right off the bat that the critical thing was to understand error. So one of the lines that we sequenced was actually the ISO-1 stock, the stock that was done to something like 14x coverage by Sanger Sequence, Sanger Resolfo Molana-Gaster. That gave lots and lots of reads that did have errors in them, and we could then build this error model. PyroBase is then their base caller that not only called bases but also called the confidence in the bases that they devised, that's actually currently online, you can download this and start to play with it. They're going to town with it. So anyway, they produced this multi-alignment of the 10 strains all across the reference genome, which is off of the Lana-Gaster, and called some 660,000 snips across the whole genome. About 1,200 of them were submitted for validation to the Washoe Genome Center, and those 1,200 had a posterior Bayesian probability of being a snip of about 90% and 92% in fact validated. So it's actually looking pretty good. So this is not the 99% confidence of snips in each individual. Is this a polymorphism in that position of the genome? And that accuracy is pretty good. So there's actually a quite strong correlation of the nucleotide diversity in particular regions. If it's very low diversity in Africa, it will be very low diversity in North Carolina and so forth. That's sort of expected. These are populations that are derived one from the other. Basically, the fly population pretty much followed the human population in migrating out of Africa, so we expect them to show that kind of state. Divergence between species, so Lana-Gaster versus Simulans is also correlated with this level of diversity within Africa. So what we're asking here then is is there simply heterogeneity in mutation rate? Are the regions with as high diversity driving that high diversity due to high elevated mutation rate in that part of the genome? If so, you would expect there to be an elevated divergence between species, and in fact you do see this to some extent. However, this correlation is much stronger than this one, and so mutation doesn't drive all of that difference. Something else must be going on. And one of the things that you can do to get at what's going on is to actually look at the ratio of polymorphism to divergence. So here's that level of polymorphism ought to be predicted by this parameter theta. The divergence is determined by twice the mutation rate times the time since the divergence between the two species, and the ratio of those has mu divide out as you can see, and so we ought to get something that is a distribution that depends only on those other factors, namely effective population size and the time since divergence. And when you do this, you have to simulate then to get what's the expected level, and the expected level is shown here under a neutral simulation. This is on this axis the diversity to divergence ratio across the whole genome, and those 10 kb chunks across the whole genome, we get this distribution. What we actually observe has a much greater variance. In other words, some regions of the genome have much more diversity than expected under that neutral simulation. Others have much less than expected. So that means the other parameters, effective size and time since divergence must be what are differing between those different regions of the genome, and those are precisely parameters that are driven up and down by things like natural selection by correlations with recombination rate, and so forth. So there's some interesting heterogeneity across the genome in these parameters of evolution. So one of the things that's often done with data that address polymorphism across the whole genome is to look for signatures of natural selection. We've seen this certainly many times over now with the human genome, and it's still an area of interesting research going on, but we can detect selective sweeps by troughs in diversity. If there's a favorable mutation, it's going to drag that particular variant up in frequency, replacing all the others, and hence reducing the local diversity around that particular adaptive mutation. And we can look at this then across the whole genome and ask, are there particular regions where there are big dips in nucleotide diversity, and it doesn't jump out at you very whoppingly for this particular data set, although there are regions where there's a curious dip for both the African and the North American sample, and it is actually significant at the genome-wide scale. Some of them are actually regions of the genome that have otherwise shown signatures already. This gene bag of marbles on the X chromosome is one interesting candidate that's being pursued in Chip Iquodra's lab, for instance. We've also seen in the human data efforts to look at differences between populations as being a means of identifying particular regions that might have undergone region-specific natural selection. We can do the same with flies. This is a difference between African and North Carolina diversity, so are there regions where there's a big spike up or down in the diversity, and in fact we see them. Those are then again candidates that are nominated for potentially interesting region-specific selection. On the issue of demography, flies show the same sort of demography as humans. Namely, there was an ancestral smaller population that grew rapidly at some point in the past, and in the African population there was not particularly much change since that time, but a very narrow bottleneck and expansion into Europe and the Americas. This is seen in the site frequency spectrum in African versus European flies. European flies have a big excess of rare variants because of the nature of the variation that would make it through that bottleneck would show this skew in the site frequency spectrum. So that was already published. Do we see this with these data? It's not nearly so clear because we only have a depth of two and a half on average, but what you can ask is, are there differences between different parts of the genome with respect to the relative frequency of different polymorphic sites? So one of the things you see is that there's a consistently reduced diversity in North Carolina compared to Africa. Now remember there were only four African lines, six North Carolina lines, and nevertheless there's more variability in the African lines. Again, for this demographic region, reason is well known. African populations have a larger effective size, look like they're more diverse. If you look at the X versus the autosomes, the mean for the X chromosome is about .004. The mean for the autosomes is about .006 for the North Carolina populations, and you see that X is less diverse than the autosomes. This is seen in almost every organism. The reason for this, of course, is there are fewer X chromosomes than autosomes. So the X chromosome has a smaller effective size in that mutation selection balance. Beta ought to be smaller, and you end up with less diversity, even if it were strictly neutral, you'd end up with less diversity. But in fact, the theoretical expectation is the X ought to have three quarters the diversity of the autosomes, and it's more like half in this case. Well, it turns out there's good theory for this. In a population that's undergone a bottleneck, you actually expect to see more severe reduction in diversity in the X compared to the autosomes. This is a paper by John Poole and Rasmus Nielsen. If you compare them now, in Africa, the X to autosome ratio is about 65%, which is, again, already lower than the 75% expectation if it were neutral. For North Carolina, it's about 50%. North Carolina is a derived population, and you can see in fact there was a greater reduction in diversity on the X compared to the autosomes. Nicely consistent with that expectation. The final sort of point I wanted to illustrate from these data at a sort of genome-wide scale is that regions of low recombination, particularly around centromeres, show dramatically reduced heterozygosity. You see that both in Africa and North Carolina. If one actually looks then at the local recombination rate, intensity, estimated as centimorgans per megabase pair. This is estimated again from many of the mapping experiments done with flies over the years. For the given local recombination rate, what's the diversity in Africa and in North Carolina? You see a pronounced positive correlation. This is widely described in the literature as being attributable due to the fact that there's a thing called the Hill-Robertson effect. Regions of very low recombination are going to suffer from the fact that a positively favored mutation is going to drag down diversity, so lower recombination will make a larger region swept to fixation and drop the diversity more than a region of low recombination. And also regions of low recombination, if there's negative selection, so deleterious mutations are occurring, it will also reduce the effective population size to a greater extent when there's lower recombination rate, so called background selection model. So some combination of those two is driving this positive correlation. You see it in these completely independent samples of four individuals here and six individuals here. We see this positive correlation between recombination rate and level of diversity. One thing that might drive that is if recombination itself were slightly mutagenic, so that when recombinations occur you also get mutations, that would drive a positive correlation between divergence and diversity. And in fact in flies we do not see that. This is the recombination rate again on this axis against the melanogaster, so there's no convergence and there's no correlation. So it seems like recombination is not inducing this positive correlation. It really is a Hill-Robertson like effect, the local sort of environment force adaptive mutations is favoring a greater reduction in diversity in regions of low recombination. This made a paper back in 1999 with just 12 data points, here are 30,000 data points, it seems to be true still, and so that was begun in a quadro back in 2000. I wanted to take the last couple of minutes to sort of shift gears, and this is another project that was funded by NHGRI to look at sort of comparative genomic lessons that we learned from a dozen different Drosophila genome sequences. And this is a particularly sociologically interesting project that featured data from all the genome centers except Stanford, I think, including adjunct biosciences, I guess NISC didn't contribute to it, but many, many different groups over many years contributed to these data. The choice of these 12 species was made on the basis of the fact that there's a huge diversity in the ecologies and sort of lifestyles of these different flies. It's over 400 million years of evolution spanned by that tree, so there's phenomenal saturation of mutation at many, many sites in the genome. Manolis Kellis was in charge of the analysis of the sort of annotation of the melanogaster genome and how we could improve the annotation of the melanogaster genome using these data. And this is just to illustrate something that I think you all know very well, which is that in protein coding regions, of course, you expect to see more substitutions between species at synonymous sites, because after all, they would still retain the same amino acid sequence. You expect to see substitutions that preserve the reading frame, and you expect to see substitutions that replace one amino acid with another one that has very similar properties. So a number of different software tools, Exonify and so forth, are very, very good at finding exons in the genome based on these sorts of signatures. So we can color code them based on their sort of attributes through these substitutions along this 12 species alignment. So this is a 12 fly species. Do those substitutions smell like they are protein coding substitutions and color them green? If there are substitutions that are radical, non-synonymous changes or frame shifts that are less likely to be protein coding, we can color them in red. When we do this, we can identify regions of the genome that are more likely to be protein coding, and we can use it then to improve our annotation of any given genome. This is just to indicate the difference between that kind of strategy and the strategy that's based on just simple conservation. So this is the track that's from the UCSC browser, the Fastcons track, showing this sort of degree of conservation. It's a very nice metric for overall conservation of the sequence, and it's showing again, so here's the region of this particular gene, CG9945, the boxes being, again being exons, and you see high conservation for most of those exons. But if you look carefully, you see there are regions where there actually is high conservation, but we failed to annotate an exon. Subsequently, we see that it also has a very high probability of being a codon based on this evolutionary model, and in fact, we go back and see that, in fact, that was another transcript there is, in fact, an exon in that region of the genome. So that's a novel sort of annotation to the melanogaster genome that came about because of our comparison to these 12 species, giving us much more power to detect these things. We also see things like this, where FlyBase was right that says there's not an exon there, but we see very high conservation. When we look at the protein coding signal, in fact, the substitutions that do occur in that region look very exon-like, and in fact, there's not an exon there. So this is, again, where there's a big difference between simple conservation and the protein coding region. So these sorts of methods resulted in many new annotations of the melanogaster genome, particularly in protein coding regions. Some 413 cases of translation start changes, 912 cases of different splice signals, 240 cases of polycystronic genes and so forth. So we gain considerable power and annotation of a genome by looking at this sort of comparative approach. Just two other stories with a sort of comparative evolution analysis, one of them has to do, again, with this X versus autosome comparison. Now, I made it sound simple. Comparing X and autosome, the X has a smaller effective size, so it looks like it ought to have more drift than the autosomes, because it's a smaller effective size. Things bounce around stochastically more in a smaller effective size. On the other hand, the X chromosomes hemizagis in males. So any mutation that occurs that's expressed in males, even if it's a recessive mutation, it's immediately expressed. So there's no masking in the sense of being hidden, for a rare allele being hidden in males, there's no such thing as recessivity. And so one expects then that deleterious mutations ought to be more effectively screened by natural selection. Recessive, advantageous mutations ought to be more effectively and more quickly identified by natural selection and dragged up in frequency. So these things lead in opposite directions. The latter results in an expectation that the X chromosome ought to be evolving faster. And so across the set of papers that have addressed this, the evolution of X versus autosome in different organisms, it's a wildly chaotic literature with very poor consistency. And if you look at the neutral divergence, so this is at synonymous sites, the divergence at non-synonymous sites, now with full genome data and all these species, again you see this sort of pattern where some lineages, it looks like the X is faster, other lineages, the X is slower, and you can see why the literature has been so confused. Because even with whole genome data, the pattern is still rather on Knife's edge, depending on changes in the demography and other things the X or the autosome looks like it's going to be evolving more quickly. The one thing that's absolutely universal is codon bias, which is the degree of codon bias is greater on the X chromosomes than the autosomes all the time. Now codon bias refers to the differential use of the synonymous codons, the fact that the codon bias is greater in the X means that the population is better able to discriminate between these very weak differences in selection between different alternative forms. Now those are almost certainly recessive differences, the fact that the X chromosome is detecting them is saying that again it's probably because of this increased efficacy of natural selection to see variants on the X chromosome and on autosomes. And remarkably consistent on every branch of this tree except for a couple where there's just insufficient power. One final point has to do with the evolution of innate immunity. This is the idea of taking a pathway for sort of any process you can imagine and having it descend down this 12 species phylogeny it's kind of exciting to imagine. So how is the pathway tuned? How is there pressure from pathogens exerting itself on this pathway? Do we see accelerated evolution and recognition or effector molecules? How does this go? And this is work of Tim Sackton in my lab and a number of collaborators of paper that's still actually just in review of nature genetics. And one of the punch lines that came from this is centered around the gene relish which is one of the transcription factors that results in launching of transcription of a number of the antimicrobial peptides. There's an inhibitor domain on relish that's joined by a linker. That particular linker region this is showing again sort of signature of positive selection on this axis against position along the gene. Most of the signatures of positive selection are right in that linker region. Other proteins that are actually involved in cleaving that linker dread has a caspase domain that actually does that cleavage. And again the caspase domain shows this excess signature of positive selection. So it's an intriguing case where in just the melanogaster lineage we're seeing multiple signatures of positive selection on just that part of the pathway. So this is a paper with over 244 authors. It's going to be in the November 8th issue of nature. It's been a tremendous sort of thrill and opportunity and privilege really to be working with this group. And at that I'll just close for questions. Thanks. Questions from the floor. So Andy, what are the challenges to take these approaches and then apply them to studies of human populations? Obviously flies are going to be much simpler and clearly we want to be able to do the kinds of comparisons you are showing with conservation and better annotation of proteins. But it must not be an easy generalization. Well, I mean there are a number of things that are different in the human situation because we're sort of coming at it in a different context. We have so much data from the million base pair SNP typing platforms and so forth that I think doing something like short read sequencing on top of those million SNPs will really, really sort of leverage each other in a very exciting way. Some of the primary issues of estimation of these parameters are less important perhaps for human than for sort of model organism studies. We're much more interested in the medical sort of questions where it's so important to get real accuracy of individual calls. But that's where we have in the context of the HapMap project and understanding that haphap, haphap, haphap type background, these methods that allow one to impute missing data. If you combine that sort of imputation with the short read sequencing I think they could really, really fuel each other.


Hi, I'm Zor. Welcome to Unizor Education. Continuing talking about graphs, I would like to devote this particular lecture to a completely different way of graphing dependency between two variables on the plane. We used to have rectangular or cartesian as they're called sometimes, covenants on the plane. So you have one axis, which is x-axis, another perpendicular to it y-axis. Both are linear dimensions basically. They are represented the points where any number is basically represented by a certain point on this axis and point on this axis and something like the dependency which you have is a set of all points, x, y, on this particular plane where x is projection to the x-axis and y is projection to the y-axis and this particular equation is satisfied. So this is the rectangular or cartesian system of covenants. Now, I would like to introduce a completely different approach to represent a point. Here it is. Let's just assume that the position on the plane is not defined by a relative position to two perpendicular axes, but you have only one axis and it's not really an axis, it's an array. I mean it can be called an axis, but it's only half line. Now, every position on the plane can be characterized by two things. Let's draw this particular vector to this point. So this is called pole, this point pole. This is called polar axis. This I'm not sure that it can be called as a polar vector if you wish and this angle can be called polar angle. Now, position of this point is defined by how much this particular vector in angular dimensions is rotated from this position along the polar axis. So if you take this particular direction, how much do you have to turn to get to this particular direction? Now, obviously for any point on the plane, there is this particular angle which is basically from 0 to 360 degrees or from 0 to 2 pi in gradients and the angle defines direction towards the point. Now, how far this point is from the center is a second variable which we need which is basically the distance from the center. So two things. We have a distance from the center which is usually denoted by letter r and the angle from the polar axis angle of the vector two words our point from the polar axis. It's usually denoted by letters either phi or theta. I'll use probably phi, it doesn't really matter. In my notes I think I'm using theta. Both are equally acceptable. Obviously you can use anything but this is this tradition. The r and phi or r and theta are traditionally for these two things. So knowing these two variables is sufficient to determine the position of the point on the plane. So instead of two perpendicular axis and y axis we are using the polar axis there are still two different variables which we need to determine the position of any point relative to the polar axis. So let's just give a couple of examples. What would be coordinate of this point on the polar axis itself on the distance let's say a. Well coordinates would be r is equal to a and phi is equal to 0. Since we are on this polar axis we don't have to move that's why phi is equal to 0 and the distance we have to move from the pole itself is equal to the a as I was just saying so r is equal to a and phi is equal to 0. So coordinates at this point are a 0. Okay how about this point I shouldn't say minus. I'm talking about distance so this is also distance a so the continuation of the polar axis to the left from the pole to the opposite direction I shouldn't again say left because there is no left and right. This is a positive direction and that's the only we have. So I'm going to the negative direction against the direction of the polar axis also on the distance a. So what would be coordinates of this particular point well again the distance is still a how about the angle but from this position we have to turn to this. So it's 180 degree or usually it's in regions which is phi. By the way I'm using gradients, angles and other vectors and other terminology here well obviously you understand that we are talking about graphs we are talking about something which is in the middle between algebra and geometry so that's why I'm using for instance the region measure of the angle if you're not familiar with this you have to go to geometry lecture which where I explained what actually region measure of the angle is so I will use it basically as it is a known term and hopefully you do know if you don't there is a reference the same Unisor.com contains the lecture about the angles and how they are supposed to be measured. So as I was saying coordinate of this point are same a note this is a distance which is always positive there is no negative direction on this particular axis this is only one positive direction so everything else is achieved through turning in this case by pi so that's coordinate of this. How about perpendicular angle on the distance a well let's think about the distance again is a and what's the angle? angle is 90 degree or half of the particle so this is a pi over 2 and on this side if it's also distance a same thing r is equal to a well we can actually turn 3 quarters of the full circle but this is not again traditional measurements. Traditional measurements is going to the opposite direction by a quarter of a circle so anything which is on this half of the plane is usually measured in a negative value of the angle because the positive value is counter clockwise as we know again that's the geometry measure of the angle is always positive when you're moving to the counter clockwise direction. Moving from here to here is clockwise and that's why it's negative so the coordinate of this particular angle is again r is equal to a and pi is equal to minus p over 2 so that's the coordinates of this one at this point and you know if it's something like on a bisector of this angle at 45 degrees or p over 4 it will be correspondingly a and p over 4 etc. So this is how the polar system of coordinates is built again we have the angle and we have the distance from the origin from the pole. Well let's do a couple of examples well example number one is r is equal to is this a function remember the function I'm not using x and y anymore the function would be described as this and again I shouldn't really say a function a dependency between r and pi is expressed in this particular way is this an example of this dependency yes absolutely it means pi can be anything so no matter what it is but r minus a is equal to 0 because that's what it is this is our function so for any phi for any angle any point which is distanced from the pole pi a would satisfy our our equation now obviously a should be greater or equal to 0 because r is a distance right so only equation of this type or this type where a is non-negative makes sense so let's draw the graph of this particular equation well this is our pole this is our polar axis and what we are saying is that any point on a distance a which is this point this point regardless of the angle regardless of the angle any point which is distanced by a from the pole would lie on the graph of this particular equation so what is it well obviously this is a certain so this is an equation of the circle in polar coordinates now recall that the equation of the circle in Cartesian coordinates is x squared plus y squared is equal to x a little bit more complex right than this one so circle is much more simply represented in the polar coordinates than in Cartesian well another example by the way this is a circle with a center coinciding with the pole itself of the polar coordinate system if center is not coinciding the equation is much more complex but let's not go into this we are talking about just very simple things now how about straight line which is going through the pole well let's just consider any point on this line has exactly the same angle phi in this particular case any point on this piece of the line this is the same angle plus 5 plus 180 degree so it's a combination of two pieces one piece is determined by phi is equal to let's say alpha this is an angle alpha and I presume that alpha is from 0 to to phi and this piece is phi is equal to alpha plus phi so union of these two graphs basically of these two graphs represents the whole line I mean this one represents half of the line and this one represents another half of the line so how to unionize well the previous lecture I was talking about unionize two different graphs you can use for instance the product equal to 0 right so phi minus alpha times phi minus alpha minus phi equals to 0 this might be an equation of the line r is not part of this r doesn't depend on r it depends on the phi so the phi can be either equal to alpha or equal to alpha plus phi so for any alpha within this range that's true last example which I wanted to present is this a spiral how can I draw a spiral look I'm talking about right now the spiral which is called our comedians spiral for Greek mathematician archived I don't know how to pronounce well in any case as you see I'm starting in the beginning so when my phi is equal to 0 my r is equal to 0 then as phi increasing my r is also increasing so as I move my vector this way I'm moving away from the center from the pole and that's what actually makes the spiral whenever I make a one circle in this particular case phi is equal to basically 360 degrees right 2 pi so if I have an equation which looks like this what does it mean well again if phi is equal to 0 r is equal to 0 then my phi increasing my angle increasing so I'm going this way and my r is increasing by the same thing now by the end of one circle my phi is equal to 2 pi right 1 circle 360 degrees so this particular segment is equal to 2 pi in lengths but then I can increase my even further now when it goes to second rotation I will have point here which is 4 pi now 2 pi and again 2 pi so it's growing this way this is how the spiral actually is increasing in ranges so this is called the Archimedes spiral we can actually have it a little bit more complicated instead of just playing r is equal to phi let's see what happens if I have a slightly more interesting equation if I have a linear dependency r is equal to a plus b phi what's this well let's think about let's just draw this we start again from our polar axis so at phi is equal to 0 my r is equal to 8 so h should be greater than 0 so this is 8 now let's consider that b is positive so as phi is increasing r is also increasing so it will be further so if I will draw let me just get a little bit smaller scale here what if a is here this is a ok so if I will start from this this is the point with phi is equal to 0 r is equal to 8 now my increasing phi if b is greater than 0 I'm moving further from 8 so if this is my circle of the radius a my point would be further from this circle and further and further and further again and again so I will increase the radius for positive b as phi is increasing if b is negative what's interesting is then as phi is increasing I'm subtracting basically something from 8 so instead of going outside of this circle of the radius 8 I'm going inside so it goes this way closer and closer and closer to 0 and at some point I will reach r is equal to 0 when phi is equal to if this is 0 so it's minus a over b right so if b is negative a is positive so that's why it's a positive way it's minus and b is negative so it's positive but this is the maximum value of phi which I can afford because at this particular value r is equal to 0 and my graph goes to the center of the pole and there is nowhere else to go r cannot be negative r cannot be negative so for a negative b my spiral is going inside and basically going tighter and tighter around 0 around point 0 and finally it does reach in a finite actually number of rotations if b is positive it goes outside of the circle bigger and bigger and bigger and that's to infinity alright so that's basically what the spiral is the Archimedean spiral is it's just an interesting example of how simple certain very complicated curve and this spiral is a complicated curve how simple it can be expressed in polar coordinates something which is simple in Cartesian coordinates like for instance the equation for the line it's just a linear function between x and y is a little bit more complex in the polar coordinate I have to basically combine two pieces together etc and vice versa something which is simple in polar coordinates like for instance Archimedean spiral is a very difficult curve in the Cartesian coordinates so that's why people are using both at certain you know different cases by the way speaking about polar coordinates and it's usage actually there is an equivalent in a three-dimensional space we are using as you know the longitude and latitude on the surface of the herbs basically these are spherical coordinates which is kind of extension of the polar coordinates to three-dimensional space alright so what's next is we would like to see how polar coordinates are related to Cartesian coordinates in some simple case well let's consider you have Cartesian coordinates and polar coordinate which is with a polar axis coinciding with x axis because that's the simplest way right so the polar coordinates are on the same plane this beginning of the Cartesian coordinates is the pole and the positive direction of the x axis is the polar axis and now I have a point which has on one hand coordinates r and phi in a polar coordinate system and x and y in the Cartesian coordinate system the same point on the plane now my task is to express one set of coordinates through the other well now again we are talking about graphs so it's in geometry and algebra in this case it's trigonometry which is kind of extension of geometry towards algebra so I will use trigonometry and again I will use it as is I would basically urge you to go to the corresponding lectures or whatever the educational material you have to learn about it but I will use trigonometry as given so let's just draw the perpendicular here so this is x and this is y right because the coordinates are x y in Cartesian system now this is r and this is phi so let's just consider this particular picture how to express let's say x y in terms of r and phi well basically because x is equal to r cosine of phi and y is equal to r sine of phi basically it's from the definition of the sine and cosine almost now how to do it in reverse how to find out what my r and phi are if we know x and y well r is easy because this is the Pythagorean theorem right so since r square is equal to x square plus y square r is equal to square root of this arithmetic square root positive value x square and y square are also positive so it's always correctly defined regardless whether the point is here or here or here or here with a phi is a little bit more difficult here is y basically from the definition of tangents which is another function of trigonometry tangent of phi is equal to y over x this is practically from the definition most at least for the phi within the range from 0 to 90 degree to p p over 2 that's basically true so from this we can always find phi using an inverse function called arc time so this is I would say the beginning of the process of pressing r and phi well actually the phi I already finished the phi because now we have some problems first of all in this particular case x cannot be equal to 0 right so what if x is equal to 0 but let's just think about this x is equal to 0 is a special case it means all the points where x is equal to 0 are here right so for all these points phi is either equal to pi over 2 90 degree or minus pi over 2 minus 90 degree right if point is here it's 90 degree pi over 2 if point is in this particular part of the y axis the angle phi is equal to minus p over 2 so I can actually consider all these cases so if x is equal to 0 if y is equal to greater or equal to 0 then phi is equal to pi over 2 if y is less than 0 then phi is equal to minus pi over 2 so we have finished case x is equal to 0 now we can define we can divide y over x whenever we want to and we will have no problems now what if x is now greater than 0 or less than 0 in this particular part or in that particular part then it's easy it's only one case and we can say this formula actually is correct arc tan y over x now why is it correct if y is also positive then it would be positive value and we will have the positive value of phi from 0 to pi over 2 if y is negative the result would be symmetrical but with a minus sign and that actually corresponds exactly to this picture so if x is less than 0 that actually presents a little problem if less than 0 it means it's here so the points are either here or here so let's think about if point is here then this angle is the same as this angle plus pi so I can always say that if y is positive for 0 then my phi is equal to arc tan y over x plus pi and finally if x is less than 0 and y is also less than 0 so it's somewhere here I can always say that this is the same as this plus pi I mean sorry we have to go negative, negative pi from this point we go backwards because all these guys have negative phi so it's equal to the same arc tan y over x minus pi so it might actually look a little complicated but this is the main thing so as soon as you have defined it for the first quadrant of the coordinate plane everything else is just basically mechanics but the problem is that you have to know a little bit trigonometry to convert one to another and basically that's it I mean if you know just a little bit of trigonometry then the conversion will be very easy so this is conversion from polar to cartesian and this is a conversion from cartesian to polar, r is very simple but phi is, it depends considering many different cases so basically this is all I wanted to talk about the polar system of coordinates again as I was saying under certain circumstances polar coordinates are more convenient than the cartesian however I should really say that cartesian are much more often used in start of mathematics but it's interesting because some very nice curves like archimedean curve can be expressed, spiral can be expressed in this particular way alright thank you very much and good luck