The spider family. Da-da-da-da-da-da-da-da-da. What's going on, guys? My name is Jack The Busch. You can call me that. We'll go back to a brand new video. Today, we have us with 16 of our Demon Slayer Watchers named Letting Something Else Go First. Interesting title choice, like many of them. Now, guys, last episode, we were introduced to this mountain. What was the mountain called? Mount Natagomo? Now, Mount Natagomo had a very interesting characteristic. There's a lot of spiders. And spider people, apparently. I think five of them. But we found out that a Demon Slayer Squad and Demon Hunter Squad was sitting in first 10 people and they were all taken down. Except for one dude. The others were being controlled by webs. And now, Tanjido and Onosuke have this very headstrong mission. I would say he's introduced to, but he was too scared to go to the mountain so he's left behind. And then he realized Nezuko was with them so he started to run in. I don't know if he'll catch up with them this episode, or if he'll have his own little misadventures. Guys, we're gonna hop right into it. But if you need a channel, drop a sub, hit that big blue bell so you don't miss any future episodes. So if you've got a like on this one, let's just hop right into it. And these episode titles are so long. They're always very specific though. Like it makes sense at the end. My man is struggling. Where did they go? Oh, maybe he'll fight one of the other five. That'd be sick. He's gotta pass out first though. He'll pass out from exhaustion or fear. I didn't even see any spiderwebs around him. I don't think he's even remotely close. This guy's got such a soft heart. Whoa, his face is cut. Morata, actually some redemption. This guy's face. He's got some of the best like visual comedic effects. He's got some kind of plan. It's funny how clear the threads are now. Man, is that of him just dragging him? Just send him over after the spider lady. He knows exactly where she is now. He's gotta climb the mountain. Hey, faster you kill her, take her down, the faster you can punch him. He like insulted him. What do you call a manchino? He's blood thirsty.  I got deeper into the spider's nest. My goodness. Oh, another. I thought it was gonna be the big dude. Oh my God. You gotta watch yourself do that. So why have some of them like knocked out? Are they dead, dead? No, cause they wouldn't slice them. Some of them are knocked out, but like she's like straight up completely conscious. Imagine, yeah. Oh, that was the dude that was on the wire. Why is she so weird? So I'm scared her. What? Uh. Is that a threat? You're gonna tattle? Wait, so that's the mother. I don't wanna meet the father. I'm guessing the father's that big dude. Yeah, I'm assuming the mother and father don't have a great relationship based off of this. Maybe some marital issues. May I suggest spider counseling? Possibly. This little kid's just threatening her. And now she's gonna fight emotionally and freak out. Now she doesn't have confidence anymore. God, if she's scared of the father, I don't wanna meet him. God, she's gonna like break her. Yeah, oh my God. Oh. Horrible. Yeah, cause I guess she can still use it. But would it be as effective if the bones are broken? God. Oh yeah, cause some demons have like their own special, blood art or something. So that must be theirs.   I love how on those cases ready for it. She's not that far gone though. Such a tonchidow thing to say.  He's like, that's sick. He just wants us to kill him. How could she even see? You have to go to the source though. Just run past. Maybe wind him up so much the way she's running. I think he wants to like wind him around a tree or something. But it looks like they're like coming from the sky. He just wants to get him stuck, baby. Oh, just hang him up. That's pretty smart. Look, they could have gone horribly wrong, but it paid off. A method without killing them? He has to do everything Tonchidow does, but better. I love a nose case so much, bro. It's just a competition to him. Sky won.  I missed it. He's definitely just saying that. Uh-oh, she's not happy. Oh, she's a secret doll, a special one. I don't think that was the big dude though. Pretty sure the big dude was like a part of the family, I would imagine. I wouldn't imagine they would keep one of the family as a doll. I swear to God, Zenitsu. My man is struggling, going through it. He didn't even call me in. He knows, Sky, called a wild boar. Oh. Oh, the bird. Is it like it's the bird's duty? Like he has to come with him? Second only to Nezuko. Every time we check in with Zenitsu, I feel like it was gonna set up a fight with one of the other five. He's like, I didn't see that one either, bud. He's got the bloated walk. Try not to be violent. Uh-oh. Oh my God. Whoa, she's so ruthless. Dang, poor Tanji though. Try his best to keep him alive. Oh, he mad. Oh my God, Inosuke was intimidated, bro. It's like I've never seen this power before. My God, Tanji, that scares me, bro. I thought they already had her location based off Inosuke.  Well, that was her special doll. I think that was the same big dude that we saw, though. He's got giant, oh, he's missing a head. Oh. Wait, how do they take it down then? Cause he's literally just a doll. It's not alive. So cute. Oh my God. My man's like, probably went an entire different way. It's gotta be setting something up. Something's moving behind him, then. Oh. Got anger now. I think someone's gonna be following him. Why else would the birds move? Oh, yep. Nope, don't like that. Not in the slightest, not even remotely. That's probably one of the five. Guy gets arms off, or actually the webs. If you get the webs off, she can't control it unless she reattaches them. Monk's robe cut. I mean, he cut an enormous rock at one point, so. Inosuke's too headstrong. Oh. I don't know anything that happened in Inosuke's cause he's too overconfident. Thank you. I think it's terrifying looking. Yeah, the big shadowy figure had a head, as far as I know. He's like, I am eternally grateful. I think that effect is like to signify like his development as a character. Cause I happened when he thought back to that old lady. Someone actually cares about him. Wait, where's Nesco? Is she gonna come out? Oh, there you go. That was actually way easier than I expected. I'm just gonna reattach him with the webs. Wait, is he gonna do water wheel? Is that water wheel? Oh, no, water wheel's like seven. Let's take all the limbs off.  He just thinks quickly. So the water is actually, it's just like there is an effect, right? It's something completely different. I think it's actually like kind of taking notes from him. Uh-oh, it actually went way better than I was expecting. No, it's fading away. I think that was actually like a demon, like maybe one of their family members? I don't know. What happened? I think he's just learned. Oh. What are you doing? Is he just that overjoyed? What's happening? He's throwing him to her. He sees the line. Wait, are they that close? Yeah, that didn't last as long as I expected. Rui. Rui's the small boy. Small spider boy. Got her eyebrows. He's just coming, flying it. One of the other spiders are gonna stop. The spider family members are gonna like fly in. God, the smile. Oh my God. Oh my God. That's so sad. He still has to do it. Oh, it's like a, wow, he recognized it. Right before he switched forms. Oh man, Tanjiro has honor, bro. That was so beautiful. I keep forgetting there's humans, like behind every demon. Wait, so, oh, this is the father. Oh, that's the big dude. So we've only seen three of them. Oh. That's probably the one that's after Zenitsu. It's like close to the ground. Ugh. Oh my God. And then there's the fifth. Oh, it's another little girl. Now we see all five. I like Tanjiro's recognition in it though. I feel like it looks sad. It made me feel bad for every single demon that's been killed in this show so far. Except the swamp one. I didn't like that one. They give life to like every single one. That's probably the father, yeah. 100%. Dang dude, I can't get over that shot whenever it slowed down just like the sunbeams came in. That was fricking sick. That was beautiful. Yeah, cause the Suzumi demon used to be, but he got demoted out of the 12, right? Well, well, well, and down goes one of the spider family. Like I said, it's crazy how easily they're able to switch it and just make me feel bad for them at the end. But it kind of already made me start to whatever he warned her like, hey, I'm gonna tell dad if this doesn't work out.  And as soon as that happened, I was like, oh, that's right.  We don't know their whole backstory. And of course it gave it to us. And my goodness, all she could think of was when she was a regular person. And the one person that looked at her with kind eyes wondered how they were doing. They just want a simple life. She literally recognized as she was about to die. She's like, oh, maybe this, maybe, maybe I do want this. It would release me from all this pain and the fear I have.  It turns it very quickly. But guys, that's been episode 16 of our Demon Slayer. Watch, let me know what you thought of the episode down in the comments below. And as always, I hope you guys have a wonderful rest of your day. As per usual, I'll catch you in the next one. Peace base.


They will mock and humiliate you. They will show a lot of anger towards you and they may even physically abuse you.  The rest of the family joins in on the abuse to win over the narcissist to survive or to get their own power and many people who join in on the abuse are also narcissistic as the scapegoat of your narcissistic family. You were blamed for the wrong doings mistakes and faults of others because although it was improper and immoral it was convenient. You were an easy target for your narcissistic family. It was easy for them to take advantage of you because you were very naive. You showed a lack of experience wisdom and judgment and you were too trusting. You believed that they were honest and sincere so you became their whipping boy. You were blamed and punished for the faults and incompetence of others. You were made to bear the blame for another's mistake which may have left you with a lot of anxiety. It may have caused you to develop issues with self-esteem. It may have caused you to experience depression. It may have caused you to engage in substance abuse or other forms of unrestrained and improper actions which was caused by suppressed and denied feelings and emotions resulting from the abuse because they used you to regulate their emotions because they're incapable of regulating their own anger, disappointment and frustration or not being able to get what they want which is why they need an easy target to vent their frustration and as the scapegoat you were that easy target because your mere presence calls up shame in the narcissist because deep down they know that what they're doing is not okay but they keep doing it even while they know that it makes them a horrible person but they cannot withstand the idea of them being a horrible person so then they abuse you even more but after a lifetime of being abused and being blamed for everything that went wrong at some point you decide that you've had enough and you decide to walk away and when you do that you begin to acquire insight into your own character you begin to learn more about yourself you begin to heal and you begin to live your best life because it's a life free from abuse when you leave your narcissistic family you may decide to set boundaries you may become more assertive and you may desire respect but when you do that you leave your abusive family behind they lose their emotional punching bag which leaves a void in the family it interrupts their settled condition because now you're no longer around them they're left in complete disorder they're left in confusion your absence has disrupted their systematic functioning and arrangement and now there's nowhere for them to dump their garbage but they're not going to try to deal with their destruction they will just lay in their own filth until they find a new way to regulate and they will continue to attack you while they're still able to call or email you which is why you have to go no contact you have to cut off all contact with your narcissistic family because they will only use that contact to abuse you they will only use that contact to regulate themselves but if you continue to stay out of reach from the narcissist at some point they will have to find a new person to abuse they will have to find a new scapegoat and sometimes even the very person who enabled the abuse eventually becomes the new scapegoat the very person who supported the narcissist because now you're no longer around your narcissistic family but while you were there you kept it together you made your narcissistic family seem more functional the reason why the narcissist still had people supporting them it's because you were their scapegoat they were able to blame and punish you for everything that went wrong and the narcissist reserved the abuse for you and spared the rest of the family but when you finally leave the narcissist has to abuse someone and that is when more and more people will see just how bad the narcissist behavior is they may even try to pull you back in so that they could be spared of the abuse but when you leave your narcissistic family they won't often attack you directly instead they will start a smear campaign against you they will triangulate you with other people they will criticize you publicly they may even make stuff up because it helps the narcissist to regulate and then they can continue to use you as their emotional punching bag even if you're no longer around them thank you for watching I hope this video resonate with you please like comment share and subscribe if you would like to donate my PayPal link is in the video description coaching inquiries you can email me at coaching at NARC Survivor at the Cody UK thank you for watching and I'll talk to you soon


Okay, so what we will do now is continue with the importance, continue discussing the importance of local rings. So again you know just to recall we started the variety capital X and the point small x, then you have the local ring of capital X at small x which contains the ring of regular functions, global regular functions namely regular functions that are defined on all of capital X and which in turn is contained in the function field of X, okay which turns out to be the quotient field of the local ring as well. And now the local ring so the point is that local rings contain a lot of information that is the idea. So what we saw in the last lecture is how local rings capture isomorphisms. Of course an isomorphism induces isomorphism at the level of local rings and conversely if a morphism induces isomorphism at the level of local rings and it is a homeomorphism then it is an isomorphism, okay that is what we saw. Now what I am going to show is I am going to discuss local rings and regular functions I am going to say that if you take an element here namely an element which is a rational function that means if the function is regular on an open set and suppose this element is here for every local ring then it is actually defined on all of X, so it is a regular function on all of X. So it is very clear that if you take a regular function on all of X then regular function is of course a rational function because after all for by definition a rational function is just a regular function defined on an open set and since if I take a global regular function is defined on all of X which is also an open set. So any regular function is always a rational function and any regular function is going to live in every local ring namely that regular function is going to give you its germ in each local ring. So something here is going to be in each local ring and of course it is going to be here as well but something here when is it here the condition is that this if that lives in every local ring then it is here, okay. So let me state that that is another important thing so to check that a function is defined everywhere if you can somehow check that it belongs to every local ring then it is defined everywhere, okay and the reason why I will do that is because all the local rings sit inside the quotient field, all the local rings sit inside the quotient field. So let me write that here local rings and regular functions so here is so here is a theorem so theorem is let X be a variety an element of KX that is in the sub ring OXX for all small X in capital X is actually in OX. So this is the theorem it says that you take an element here namely a rational function if it is in every local ring then it has to be a global regular function, okay. So for proof so what I want to tell you is that see if you think of it philosophically it is quite you can even think that this is obvious or even total logical, okay. Philosophically what you are saying is that you have a rational function namely function is defined on an open set but you are saying that it is equal to the germ of a regular function at every point which means you are saying that it lives in a neighborhood of every point and that is the same as saying that it is domain of definition that is a maximal open set where it is defined is all of X which means it is actually the open set where it is defined is not just a proper open subset of X but it is actually all of X which means it is a global regular function, okay. So that is the philosophy but then it involves a little bit of commutative algebra and that is what I want to tell you see the important thing is that you know trying to say that rational function is defined as a regular function in a neighborhood of a point is the statement that you know that this actually belongs to the sub ring, okay. The fact that an element here belongs to a sub ring like this is another way is actually the commutative algebraic way of saying that the rational function is actually defined in a neighborhood of the point small x, okay that is or it can be extended to a neighborhood of the point small x, okay. So you know trying to say that the rational function is defined at every point is if you want to say it commutative algebraically how do you say it you say it like this you say that the element of the rational function defines an element of kx and you are just requiring that it is in each of these sub rings I mean if it is in one such sub ring then that is equivalent to saying that it is defined in a neighborhood of the point small x, okay. So the problem with a rational function is that it could be meromorphic, okay namely it could have it could basically it could be a quotient of regular functions and it may not be defined where the denominator function is vanishing, okay and therefore it need not extend to all of x but the fact that it extends to a point is reflected by saying that it belongs to the sub ring, okay and that is the content of the theorem. So let me do the following thing so what I would first like to say is so we will use this fact namely any variety is covered by affine varieties, okay any variety is covered by open subsets and in fact finitely many of them each of which is isomorphic to an affine variety. So what I will do is I will go from x to you know I will first take the case when x is an affine variety, okay I will first solve the theorem for the case when x is an affine variety and then I will say that therefore it also solves the case when x is a general variety because a general variety is a finite union of affine varieties, okay. So first assume x is an affine variety say x equal to max spec A of x, A of x is equal to A say. So you know this is how you recover an affine variety from its co-ordinate ring. If x is an affine variety you have Ax it is affine co-ordinate ring and if you take the maximal spectrum then you get back the affine variety, okay so in fact so you know if you want Ax is sitting inside some affine space as an invisible closed subset and Ax is the co-ordinate ring of x, okay it is the it is the regular function of affine space namely polynomial ring in as many variables as a dimension of affine space modulo the ideal of the closed subset, invisible closed subset which is a prime ideal, okay. So A is an integral domain which is a finitely generated k algebra, right and now what happens is that you have so you have situation like this you have Ax which is equal to O x that is A for us, okay because if x is affine Ax is same as O x and that is A and that is sitting inside well O x x local ring, okay and mind you this local ring well I will rather write I will write it here, okay and this is sitting inside Ax this is the function field, right and how does all this translate well O x is Ax which is A and then the local ring is Am, okay where M is the maximal ideal of A that corresponds to small x. So after all small x is an element in capital X which is which can be thought of as a maximal ideal in Ax, okay which is A. So this is just Am, okay this is the expression for the local ring at a point on affine variety where the point corresponds to this maximal ideal M, okay and what about the quotient field this will be you will get Q of A you have the quotient field of A and that will be the same as the quotient field of X the function field of X, okay. So this and mind you Q of A is the same as Q of Am if you have an integral domain then the integral domain every localization of the integral domain is also an integral domain and the integral domain sits inside as a sub ring in its localizations and all the localizations have along and the integral domain itself they all have one and the same quotient field, okay. So this is A is an integral domain Am is localization of the maximal ideal M which means that you invert everything outside the maximal ideal and both A and Am have the same quotient field, okay Q of A, okay and that identifies with the function field of X, okay and now what is what is happening is that I have a so what is given to me is a rational function which so this is a little bit of simple commutative algebra but the point I want to make is that whatever commutative algebra little simple commutative algebra that we are doing is beautifully enough to give as this result, alright. So you start with so you start with the rational function here I mean you start with the rational function here by definition it is here so it is of the form f by g, okay so I start with an f by g here, okay I start with an f by g here because it is see because this rational function is an element here and this is identified with this I am easily able to write as f by g because it is the quotient field of A where f and g belong to A and what is given to me is that this f by g belongs to Am for every maximal ideal M that is what is given to me, okay so f by g is equal to well an element here, okay and an element here well let me write this as f sub M by g sub M in Am for every M maximal ideal, so max spec is the set of all maximal ideals away and for every maximal ideal I have this local ring and this f by g belongs to the local ring means the f by g is equal to an element in the local ring alright and an element in the local ring is written like this where the denominator is not in the maximal ideal, okay this is given to me what I will have to you know what I will have to do I will have to show that this f by g is actually in A, okay I have to show that this rational function I started with is actually here, okay so I have to show that f by g belongs to A which means I will have to show that you know I have to actually show that g I have to show that g divides f, okay I have to show I have to if I show that g divides f, okay then I am just that is the same as saying f by g is an element of A and that will show that f by g belongs to A, so I will have to show that g divides f alright, so we need to show f by g belongs to A that is that well g divides f in A or in other words you just have to show that f belongs to the ideal generated by g this is what we will have to show f is a multiple of g, so f by g is an element of A, okay that is what you will have to show. Now you see now let us analyze this of course you know if g is a unit there is nothing to prove g is a unit there is nothing to prove because then f by g is f by a unit, okay and f by a unit is also an element of A, okay so assume that g is not a unit, so assume g is not a unit since g is not a unit it means that you know every non-unit is contained in a maximal ideal, okay so there exist maximal ideals m which contain g, okay there exist maximal ideals m which contain g and so you know for such a maximal ideals m you look at this equation, now what you will get is you will get f gm is equal to g fm for such m implies what does this imply, so you see g is in the maximal ideal m, okay and therefore this right hand side is in the maximal ideal m therefore the left hand side is in the maximal ideal m but maximal ideal is prime and therefore one of them has to be in m but gm is not in m therefore f has to be in m, okay so this will tell you that well f will have to be in m for every m maximal which contains g you have this, okay and anyway but what is very very important is this condition even if I take an m which may or may not contain g, okay perhaps this is just an observation that comes from this equation but what is more important is this condition that for every maximal ideal I am getting an element outside that maximal ideal and the beautiful thing about that is that you know whenever you give me one element out of each maximal ideal then this collection of elements actually generates the unit ideal, okay that is a crucial topological fact it is the same as saying that all the basic open sets defined by all these gms they cover the whole space, okay see now so let us not worry about this observation for the moment, okay let us look at this see you take gm does not for every for every m in max spec a which is actually x there exists gm which is not in m that is what you have here, alright and now the claim is the set of all you take all these gms where m belongs to m is a maximal ideal of a the ideal generated by this is actually a it is a unit ideal, okay and the answer is I mean why this is true is very very simple because you see if you take the ideal generated by all these gms the ideal generated by gms if it is not a if it is not the unit ideal then it is a proper ideal, alright some an ideal which is not the unit ideal is a proper ideal and you know any proper ideal is always contained in a maximal ideal therefore this ideal will then be contained in an m knot, okay but then gm knot will be in this ideal which is contained in m knot but by definition gm knot is not supposed to contain to m so not supposed to contain be contained in m knot that is a contradiction, okay so let me repeat it if this is not a then this is contained in an m knot where m knot is a maximal ideal, okay and then the problem is that gm knot is in this collection and that will be contained in this which is contained in m knot but gm knot is not supposed to contain be contained in m knot you get a contradiction therefore this ideal generated by all these gms is a, okay this is this is this is just equivalent to so let me also say that this is equivalent to x is given by the union of all the d gms where m belong to max by k because what you should understand is you see you see if gm is not if gm is not in m, okay you look at d gm what is d gm d gm is the basic affine open set where the function gm does not vanish, okay so it is same as saying that the point corresponding to m is in the basic open set defined by gm, okay so but I have covered all the points I have taken all the maximal ideals so I have covered all the points and for each point I have this d gm so you know if you draw a picture this condition is actually like this I mean it is a bad picture which is very only set theoretically descriptive but it is not correct picture but it is correct enough to think a little so you have to give you some basic idea of how things are so you have x given any point m, okay you are thinking of a maximal ideal in A as a point of x, okay and for this every point you are able to find this d gm which is an open subset which contains that point, alright that is what it means to say that gm is not in m that is what it geometrically means and since you have covered every such point the union of all these d gms will be x, okay but then you know Zariski topology is quasi-compact therefore only finitely many gms have to cover x but the fact that finitely many gms cover x is the same as saying that those finitely many gms will generate the unit ideally, okay so which is if and only if x is equal to union of i is equal to 1 to l d gmi so here I will have those g it is ideal generated by d gm1 etc up to d I mean up to g gms of l. So if something is a unit ideal then 1 can be generated by finitely many gms call those gms as gm1 through gml then the ideal generated by gm1 through gml contains 1 so it is the full ideal a but then writing a is equal to the ideal generated by gm1 through gml is topological equivalent to writing max spec of a which is x is union of all the affine opens defined by these gmis that is what is the topology that is involved, okay so you can see side by side algebra and topology going hand in hand for the Zariski topology now if you look at it carefully what you get is well therefore what you get is therefore there exist fmi so let me use something hmi in a such that sigma hmi gmi is equal to 1 i equal to 1 to l this is what it means to say that the gmis generate 1, okay that means ring linear combination of the gmis is equal to 1 for a suitable ring linear combination, okay. Now you know but then you see now I can use this I can now use now I can take f by g is equal to f by g times 1 where you know I am doing all this computation here in the quotient field, okay whatever equations I am writing now are valid in the quotient field so f by g is f by g times 1 and the 1 the 1 in the a is the same as the 1 in the am is the same as the 1 in the quotient field, okay so this makes sense also in the quotient field so it is f by g sigma i is equal to 1 to l hmi gmi because after all this is 1 now I push it inside I will get sigma i is equal to 1 to l f by g times hmi g sub mi but you see I have f gmi is g fmi, alright so this is going to be sigma so you know if I can write a sigma i equal to 1 to l f gmi hmi by g but that is equal to sigma i equal to 1 to l f gmi is actually g fmi by g and well g cancels out and what you get is I get sigma i equal to 1 to l fmi hmi which is actually in a this is an element of a, okay so what I have done is I started with f by g in kx and I proved it is actually in a x so it is actually a regular function, okay so this settles the case when x is affine, okay now when x is not affine you know that x can be covered by finitely many open sets each of which is affine and what this argument will tell you is that on each of those affines the rational function will be a regular function, okay and what will tell you is that the regular function lives on all open sets of a cover, okay and it coincides and it will of course coincide on the intersections therefore it will give a global regular function, okay so that finishes the proof in the general case, okay so let me write that down this settles the case when x is affine, okay when x is general x is a finite union of open subsets open affine subsets, okay if x is a general variety then we have seen that any general variety is a finite union of open affine subsets and what we by what we just proved the rational function, okay the rational functions will not change whether you consider rational functions of x or rational functions of any open subsets, okay therefore what you will get is that the rational function is started with actually is a regular function on each affine open sets which all of which form a cover for x and of course on the intersections also it will be it will give the same function because everything is finally happening in the quotient field which is the same, okay so basically what will happen is that you will get a regular function on each affine open subset by the previous argument and all these regular functions will coincide on the intersection then therefore they will define a global regular function after all a global regular function is a global function which is locally regular, okay so to define a regular function you have to just define it on an open cover, okay so the case when x is general is also covered, okay and so that is the general case you know if you want maybe I can even write something in symbols so that you are little bit more convinced comfortably so you know x is union i say j equal to 1 to m xi where each xi inside x is open affine which means that each xi is an affine variety each xi is an open subset of x which is isomorphic to an affine variety and then you must understand that you know kx is the same as kxi for every xi, okay and o capital X small x is the same as o capital X small xi, o capital small x capital xi, small x if small x is an xi the you know the local rings and the function fields do not change if you go to open fields, okay so what will happen is that if you start with an element of kx it will belong to and it can be thought of as an element of kxi for each i but because each xi is affine and it is in each local ring by the previous argument it will be in o xi for every i, so you will have an element of kx which is in o xi for every i but then that will mean that it is actually in o x, okay so and a phi in kx is equal to kxi belongs to o xi for every i by affine k is above and thus to o x and that is the end of the proof, okay so after all you know to define a function you have to only define functions on open sets which agree on intersections, okay and similarly to define a regular function you have to give me a regular function on an open cover which agree on the intersections and that is what happens here, okay so this is a very nice fact that to conclude that rational function is actually a global regular function that it is defined everywhere you just have to check that it belongs to every local ring, okay so that is another that demonstrates another important power of the local ring, okay then the next thing I want to tell you about is something that is far more profound it is the following thing see the you can see that somehow the very word local ring will tempt you to think that it somehow encodes all the local information at the point but the problem is local information if you want to think of it is a information on an open set containing the point you must remember the open sets are huge for the Zariski topology because the open sets are after all every open set is irreducible and dense, okay therefore if you take an open neighbourhood of a point it is not a small neighbourhood it is not small in the sense of usual topology it is not like having a smaller and smaller disk or smaller and smaller interval of a point on a real on the real line or having a smaller and smaller disk of epsilon radius of a point on the plane it is not like that every open set is huge therefore the fact that the local ring stores such information should tell you that it is actually having much more than local information so the truth is the answer to that is yes it actually also stores information on a huge subset containing the point so that is the power of the local ring so what I am going to next state is this very beautiful result it says that take two varieties X and Y okay and take a point of one variety take a point of the other variety take a local ring at this point for this variety and take a local ring at that point for the other variety if just these local rings are isomorphic okay then there is a whole open subset containing this point in this variety and an open subset containing that point in the other variety which are isomorphic so if two local rings are isomorphic if the local ring of a variety at one point just the local ring alone is isomorphic as a K algebra to the local ring of another variety at some other point then a huge open neighbourhood of this point on this variety has to be completely isomorphic to a huge open neighbourhood of the other point. So even isomorphism of local rings is actually in this sense pretty global okay it tells you that if two varieties have two points which are whose local rings are isomorphic it means that these two varieties are on an open set on open sets they are actually isomorphic that means the only place where they are not isomorphic is a boundary okay so that is another powerful property of a local ring. So let me state that so local rings so I will call this local rings and birationality so local rings and birationality okay so let me define this notion of birationality. The two varieties are called birational if there is a non-empty open subset of each which are isomorphic okay so X varieties X and Y are called birational if there exists U open in X of course non-empty and V open in Y of course again non-empty such that we have an isomorphism of varieties U to V. So two varieties are birational if there is an open subset of one which is isomorphic to an open subset of the other see the varieties themselves need not be isomorphic but the point is there is a huge open set of this of each of these which are which look the same okay that is the concept of birationality. Now what you must understand is that this actually this notion of birationality is actually an equivalence because you know if X is if X and Y are birational that is of course symmetric okay X is birational to Y is the same as Y is birational to X and of course an isomorphism is always if two varieties are isomorphic they are of course birational because you can choose the open sets to be the whole varieties themselves. So the notion of birationality actually it is a relation which is an equivalence relation if X is birational to Y then Y is birational to X any variety X is always birational to itself by the identity map which is an isomorphism okay and if X is birational to Y and Y is birational to Z then X is birational to Z okay because any two open sets will always intersect okay so if you have an isomorphism if U inside X is isomorphic to V inside Y and V inside Y is isomorphic to W inside Z then you can get the isomorphism composed in on V intersection V prime which will be non-empty okay therefore this birationality of varieties is an equivalence relation okay and if you go if you take the set of all varieties if you take the category of all varieties and then you go modulo this equivalence you get what is called the set of birational equivalence classes of varieties and actually it is the first step in classification of varieties is to find in each birational equivalence class a variety with say good properties for example a variety which is smooth okay and so you know basically whenever we study objects in mathematics we always ask the question how many non-isomorphic objects are there okay so or in other words you try to describe the set of isomorphism classes of objects okay this is called the classification problem so you can ask the classification question for varieties you can ask you take all varieties and you go modulo isomorphism then you get the set of isomorphism classes of varieties what is that set okay now the fact is that that is not a very easy question okay and in fact the first answer is to instead of going modulo isomorphism which is very strong you try to first go modulo weaker equivalence relation like birationality so you first ask the question what is the set of birational equivalence classes of varieties okay that brings down the problem to a more tacklable level okay so so birationality is very important in the classification of varieties so let me write this down birationality it is easy for you to check that is an equivalence relation and now comes the theorem that I wanted to state theorem is let x belong to x small x be a point in capital X small y be a point in capital Y such that of course x and y are varieties OXX the local ring of capital X at small x is isomorphic to OYY isomorphism as K-algebra so I just have two varieties capital X and capital Y and points small x belonging to capital X small y belonging to capital Y such that the local ring of capital X at small x is isomorphic to the local ring of capital Y at small Y by an isomorphism which is a K-algebra isomorphism okay then the result is that there is an open set U in X in capital X which contains the point small x there is an open set capital V of Y which contains the point small y and there is an isomorphism of varieties from U to V which under pull back of regular functions at the local rings induces V okay so then there exists U in X open V in Y open with X belonging to U Y belonging to V and an isomorphism of varieties so let me use so let me use a different symbol for this so let me call this something let me call this a psi phi from U to V such that phi hash Y is actually psi so it is a very very powerful theorem what it says is that if one local ring of one variety is isomorphic to another local ring of another variety then these two varieties are birational they are the same on huge open sets they are the same okay and that tells you how much information the local ring contains just isomorphism we think of local ring as something that is concentrating attention at a point but this tells you it is much more than that just local ring at one point of one variety being isomorphic to local ring at another point of another variety is enough to say that these two varieties are the same on a huge open set except for a boundary they are isomorphic okay so that is the fact that gives you that the fact that local ring contains a lot of information even more or less global information as in the sense that it contains information on contains information on a big open set okay so one needs to prove this and again the beautiful fact is that this also involves lot of commutative algebra but pretty easy commutative algebra involving localizations and it is very it is proof is not very difficult but the point is that you can just use nice commutative algebra to capture these two open sets as nice basic open sets okay which are isomorphic basic open sets containing those two points which are isomorphic okay so let us do that in the next lecture.


human gut microbiome. Okay, well thanks Daniel and the organizers for inviting me here and I realize that I'm the man standing between you and lunch and it's getting late so I will try to be short and if I'm getting overtime please signal me and I will try to even be shorter. So I'm going to talk about this drug-bacteria interaction landscape of human gut microbiome. Just to give some background, I have background in modeling but today I'm not going to present much of the modeling work. Instead I'm going to try to play or try to be analogous to Tycopra who was a Danish astrologist who has been looking into planetary motions. He was one of the first ones who made very accurate maps of positions of the planets and their dynamics which eventually enabled basically the physical laws of motion that could be that could be eventually realized and I think the reason why my lab moved from mostly computational work to now actually mostly experimental work is because at one point I realized that we're hitting the wall when we try to understand microbial communities. We have reached the limit of how much we understand metabolism. In fact actually I don't have any longer illusion that the central metabolism is very well understood. I mean we hit so many points even with saccharomyces cerevisiae we don't realize what's going on. So we decided that we need more experimental systems, we need more data and this is part of the work that my lab at the moment is doing. So the question that I'm going to talk about among the various different when addressing at the moment is looking at the drug-pactra interactions in the human gut microbiome and this is really, really largely unknown. We can look into bacterial. People have been doing a lot of metagenomics studies making correlative analysis between for example how the obesity diabetes many different types of diseases are related to gut microbial composition, how diet can influence it, also some studies coming up how drugs can influence it, but these are mostly correlative studies and not causative. Of course there are some causative studies and sometimes I like to make a joke sometimes in private, sometimes in public, there's almost race now to relate everything to gut microbiome you know so eventually I will leave it on that. So the before I go to experimental results and since I may run out of the time I would like to first hand acknowledge the people that have been working on this people in my lab of course and the collaborator especially Nasus Tipas and Pierre Bork who are group leaders also at EMBL, Hadelberg and everything on the gut microbiome I'm going to talk about has been done in collaboration with them. So this is going to be my only slide that's based on published work and everything else that will come after would be unpublished work. This is just a motivation example so they have in two nature papers looking into the effect of type 2 diabetes on the gut microbial composition you find a very big signatures until it turns out that the signature that we have been observing is not the signature of related to anything to type 2 diabetes but it has because of the metformin which is the first line of the drug that people take as a treatment for type 2 diabetes and since most of the people take it there hasn't been enough statistical power to or people even didn't think about it so most of the changes in the microbiome that people associated with type 2 diabetes actually turned out that it was metformin right. This is just one example and there are only a couple of other examples that exist but how far this goes nobody knows so we decided that we should start some steps in direction so what I'm going to talk about is first of all this is going to be mostly all in vitro studies so we need to grow this gut bacteria so we need to grow them in reliable reproducible way and ideally in a fashion that in a defined media so we can actually go back in to see what these interactions are do molecular analysis and so on I will then talk about drug bug interactions and then bug drug interaction if the time permits I will also talk about bug bug interactions okay on media and the on the bacterial resources we have now approximately 100 representative gut bacteria that can grow in a defined medium I'm not going to the details but basically this bacteria have been chosen so as to represent there's a healthy gut abundance into the healthy gut we also added some known probiotics pathogens this is associated bugs also the bugs that have been known to associate with the drug in effects and so on we have been growing them into 15 defined out of this donor novel media that has not been described before and to four different rich media and basically on the phylogenetic side we get quite high coverage of the known diversity approximately 75 percent of the known abundance on the metabolic diversity or at least the metabolic diversity as we know of today we cover almost all of it okay so this is the let's say the heat smaps summarizing actually almost four years of work so what we see here is the different bacterial species slash trains over here different growth media rich media semi defined defined some even minimal media and basically the conclusion is that almost 85 percent of the species they grow in minimal or defined media and most of the species where at least actually two different defined media that we can grow and of course this has been actually in the beginning quite surprising for us and we have been doing a lot of follow-ups you know and then making sure that we don't have contamination we don't have media follow kind of a media contamination etc etc and this actually has been quite clean we have reproduced these results many many times now we also find just to give some highlights because i don't have time to go through all the results we find many novel musin degraders so actually we actually just double the number of species that have been known to colonize or to utilize musin and this has large implications for many diseases associated with the intestines because the colonization of the musin and weakening of the intestinal barrier is one of the first steps that gives to leads to many diseases and also immune responses etc quite interesting actually we don't not only see what are the metabolic requirements of different bacteria we see actually quite substantially inhibitory effects of amino acids and short chain fatty acids in particular for many different bacteria and this actually has been a very good lesson at least for us because when we think about growing difficult to grow bacteria we always think about what's missing and people tend to add and we also tended to add you know what's missing let add extra amino acids let's add this and this and this but it turns out that actually inhibitory effects can be actually quite so important so i think we should start not thinking not only thinking about what's missing but also watch too much and i think many times we don't get spaces growing because we just simply have too much of amino acids for example or short chain fatty acids these inhibitory effects can be simply too strong another message that we learned was the phylogenetic distance is not always a good predictor so we observed two different strains of the same bacteria that can have very very different metabolic properties they can prefer very different media on here you can see a genus level map actually this is clustering based on their growth preferences on different media and the color of different genuses if this orange genus for example just simply spread all over right so you can get actually this kind of differences at the level of species sometimes also at the the level of strains and simply because of this and also that this kind of resource has not been available before our ability to model this metabolism also has been limited so if you basically we took around 50 different or at least all available gut bacterial metabolic models we had in hand and try to see if they can just simply recapitulate this growth in defined media only eight percent of them could do this so it means that actually just from gene annotations we cannot accurately predict what the metabolic capabilities of different species are so I hope you can appreciate the value of such a resource and I think this is one of the first steps I think we need to do for many different community systems before we can go and model larger communities some others of course one of the things that reason we started doing such a resource because we want to move to to individual studies in a larger drug screens or even the community studies so another few interesting observation they had that there are some species actually they prefer defined media right so you give them everything they don't like it so much and there are some species that grow only in defined media and show very very poor growth in in the rich media right again is the question of the of the inhibitory effects and the bacteria that prefer defined media actually we find that they're at least significantly more prevalent right seeing that they have this kind of strategy that they can be in many different places they just require so they're basically living off into into niches where the resources are scarce what's also interesting is that relative growth in certain media correlate very well into relative abundances across different people this is the chart or here of course I'm not trying to say that this media are media are anyway by any means represent you of the good environment but this means that this media we can actually use for in vitro screenings at least with the growth relative growth rate differences we can to some extent or the effects that come because of that can be to some extent tackled and one of this media actually there are many defined media they do this and the MGM is one of the the rich media actually they also does it not shows good correlation okay so moving on from the resource to looking into the drug bug interaction landscape so what we have done is that we have screened 1200 different drugs basically these are from the press week library these are all ft approved drug compounds and we are screened them against the 40 different representative bugs again from the coming from the these are the subset of the hundred bugs that we have actually looked into and of course this study will continue and we will hopefully reach the whole 101 point so all in all we have tested 48,000 drug shrimp pairs and this has been done on all in biological triplicates and under anaerobic conditions basically what you do is that you throw different different drugs into different bacteria and then basically you record the growth curves okay again yeah many years of work summarizing to one slide what you see here I mean it's I don't have time to go into all the details just want like to present some highlights so the first thing actually that was quite I don't have proper word quite astonishing at least for me was that most of these who many of the human targeted drugs in this case almost 25 percent of them they're extensive impact on the gut bacteria so the inhibit gut bacteria of course you can see that antibiotics do the job as well but this is not surprising they're made to kill bacteria human drugs actually are made to target humans they have quite a large impact on gut bacteria and this is actually a conservative estimate because we are done our screen on one concentration of the for across all board which is 20 micro molar and if you look into estimates of the what's expected concentration of the different drugs inside our gut for example made from in the example I talked over here this is way above what we have tested right so if we for many of number of drugs actually we are now increased concentration and see what happens of course as you increase concentration you see starting effects of many many other drugs hitting gut bacteria uh another interesting observation is that the abundant commensals are more affected by the human targeted drugs that meaning that the effects that are going to have on the human microbial community and the effects thereafter side effects for example or the effects secondary effects on the other bacterial communities and eventually up to horse are going to be quite large and if you look into the drug side effects right so if you look into the drug side effect databases look into side effects that human drugs or horse targeted drugs have on the horse or in this case humans is quite interesting to see that if you have the drugs that show a hit in our screen meaning that these drugs are affecting commensal bacteria then they tend to show actually side effects that are very similar that you see with antibiotics right so this is kind of an let's say indirect in vivo relevance of our screen that we would actually expect that these drugs are affecting gut bacteria this will also show side effects that are similar to antibiotics which would also kill this bacteria which we also hope so now this resistance very interesting question is about need to rethink about antibiotic resistance and when we think about the antibiotic resistance of course we think about antibiotics but i think now we need to also start thinking about non-antibiotic drugs the human targeted drugs that can also lead to resistance and here is in just one of the few examples that we're following up so on this axis you see the number of antibacterial drugs with anti-common selectivity and then here you see the the number of human targeted drugs with anti-common selectivity here's the E coli wild type over here now if you make the the mutant of the tall C which is one of the the efflux pumps of the E coli so it's one of the common mechanism through which bacteria basically become resistance they just pump out the drugs they become resistance and now if you take make this E coli mutant it moves diagonally upright so it's not moving or like this over here so if it becomes resistant to antibiotics it's also becoming resistant to human targeted drugs right so this we are done also with not only E coli some of other bacteria over here and then we see that there are many times the mechanisms that through which the the back check can be resistant to host to antibiotics also apply to the to the mechanisms through which they can also be resistant to host targeted drugs which will mean that the wise first also will be true that if we are eating let's say a lot of certain drug that is affecting the commensal bacteria which we don't know about it can also lead to development of antibiotic resistance okay so moving on i would from moving from drug bug interaction i will move to drug interactions meaning that so far we have looked into how drugs affect the growth of bacteria what we're also doing of course on the smaller scale because of the analytical limitations looking into how bacteria affect the drugs itself right how they're for example metabolizing drugs so we are looking into 19 drugs that have been selected on based on different criteria of course the main criteria has been that we can analytically measure them easily into lab this has to be the drugs that are orally taken and sufficiently sufficient quantity so that there is some chance that they will end up into our gut i just mentioned that it's not necessary that you need to eat the drug so that it ends up into the gut because of the the hepatocirculatory system because of the circulatory system a lot of the drugs even if you inject them would end up into the gut as well okay and so we are in total around 19 drugs that cover many different classes i mean we have not put any chemistry filter on it right basically selected try to diversify whenever we can so i have many different chemistries many different therapy targets etc okay the screen again is conceptually very simple you grow the bacteria with the drug you measure the drug concentration in the beginning and you see at the end whether the drug concentration has decreased or not so one of the main conclusion again it was quite surprising at least for me that many of the drugs are affected or affected by bacteria use the word affected at the moment and it will clarify in a few slides why i don't say they're metabolized by the bacteria but many of the drugs are affected by bacteria i don't actually now recall the numbers i think 12 out of the 19 drugs actually we look into their concentration is reduced in the medium spectrum this also has been quite surprising for me because if you look into literature the number of known interactions that have been described are quite few and ours is a quite actually relatively small screen so just to extrapolate which i should not do but if i would increase the screen i would expect similar heat rate across the board if you look at the types of bacteria that are doing it these are mean we again have been screening only with the 20 bacteria so far but they have been selected for their diversity in terms of types of functions that they play in the gut ecology or their phylogenetic diversity we see all types of bacteria all the from commensal to potential pathogens to probiotic bacteria they are interacting with these drugs in this fashion so the reason i've been talking about affection and not metabolism so far is because at one point we realize that this what we are seeing partially is not only biotransformation which is going to be in zymatic modification or less a chemical modification of the drug but also to some extent actually bioaccumulation so what we have done basically is that now we do two different type of extraction protocols one is let's say looking only into supernatant other is the hard extraction where you break the cells and again to a very hard solvent extraction to get everything of the drug that we can and this way we can now distinguish between two different mechanisms one is the biotransformation or zeno metabolism other is the bioaccumulation and this is the summary of the results over here so what we are looking at is the different bacteria that we found some of at least showing one hit the depletion of the drugs that were here and of course in the bonus readout that we have is also how the drugs affect the bacterial growth as well in the same screen everything in the gray scale this gray scale thing you see these are our positive controls these are the drugs that have been known to be metabolized by the gut bacteria from the before you can actually easily see now they were why they were known for even decades because every bacteria you throw on them they do the same biotransformation on them so it has been one of the easy things to catch up everything that you see either black and green is some a novel finding that has been not been described before the black are the biotransformation hits and the green are the bioaccumulation hits bioaccumulation meaning that these drugs are being accumulated by the bacteria they're taken up but not metabolized surprisingly we find very little overlap on the two sides meaning that the bacteria depleting the drugs are either bioaccumulating or biotransforming actually it's very rare we found I think only four cases for interaction cases where the growth of bacteria is affected so in the most cases the growth of bacteria at least under our acid condition is completely unchanged since the bioaccumulation turned out to be so prevalent in our screen and at least actually quite an underappreciated mechanism of how drugs can be influenced by how bacteria can influence the drugs we did some mechanistic follow-up for some drugs and one of them is geloxetine which is an antidepressant unfortunately a lot of people take it in quite large quantities if you look at the over time period of the year or the time period of that people take this drug so what we have done managed to do is to put a click chemistry tag on the drug over here meaning that we can now pull it down with the biotin or steptavidin over here and here is the pull down as you can actually get quite clean pull down if you do mass spec then you can look at the enrichment of the which proteins are basically binding shrugs so what we also done I won't show the direct data here is the metabolomics also treat of the of the bacteria after treating them with with the geloxetine here is the summary map so every line here in this cake style map you see is the enzyme that we pulled down and the circles of the metabolites that actually are differentially expressed in the bacteria after treating with geloxetine and what I hope you can appreciate is that they actually lay mostly in the proximity with each other meaning that a lot of central carbometabolic enzymes actually binding geloxetine are leading to metabolic perturbations inside the cell that brings me to kind of conclusions of at least this part of the talk so basically what I would like to convey the message that we see really profound implications for for the drug the drug discovery and the usage of our findings for the first part we have very good resource for representative gut bacteria and growth media for in vitro studies we see a lot of collateral damage a substantial fraction of the human targeted drugs impact commensurate bacteria this of course has a reliance for drug side effects emergence of antibiotic resistance drug repurposing and also inciting to drug mode of action for example especially for antipsychotic antidepressant and it so many of these antipsychotics antidepressant which also hit in our screen on the both bug drug and drug bug side it actually doesn't if when people start taking it doesn't work immediately sometimes it takes weeks and sometimes in months before you see the mode of action and it's suspected the mode of action might be through the gut and which is has recently been actually shown for metformin case so the metformin if you inject it doesn't work you need to take it orally and the gut microbiota plays an important role in the mode of action for this drug okay and also the bioaccumulation seems to be a very widespread yet underappreciative phenomena and of course it has a direct relevance for the for the drug efficacious in pharmacokinetics and overall of course one can imagine that you should be quite relevant for drug design in general so for personalized medicine and if either I could stop here or I could have five minutes more so to the oh I have 15 more minutes okay I hope people are not too hungry then by then but okay I will try to be okay I will try to feed you some drugs so moving from the single species to community right so what I've shown you so far this is all single species work looking into you know one drug one bacteria interaction of course these effects can be quite complicated as soon as you can get emergent effects higher order effects as soon as you move from single bacteria to communities but it's important to keep in mind that if you don't understand how single bacterial metabolism works or how a drug affects single bacteria there is no way we are going to understand if the higher order level so we need to map these interactions just to give them kind of a flavor for what we might expect to see so even if you take a simple community with five bacteria that's actually the follow-up from our the drug bug screen again the duloxetting case the community effects can be very counter intuitive so this is the case where erectile which is the the rate bar over here so this is a five bacteria make sure you grow them as a community in a single shot either or actually you can also get them into stable formation you grow them without duloxetting or with duloxetting so this red guy over here is actually if you grow them alone with duloxetting it's very sensitive to duloxetting so duloxetting basically inhibits this growth very very strongly for this bacteria yet in a community context this bacterium actually is rescued by duloxetting and this basically happens because the same mechanism I showed you before so duloxetting the bacteria that bioaccumulate duloxetting their metabolism is shifted they start secreting nucleotides and many other metabolites which are actually beneficial for this year actually over here and that benefits that comes from metabolic cross-feeding is it basically masks the duloxetting effect so this means that the community effects actually can be counter intuitive and you can actually understand them based on the the single bacteria study we have done before this is of course a small community and not representative of the gut environment so one of the other resources that we're building now like we have the bacterial resource for the single bacteria yeah just a question in this case I don't know the top of my head we have measured them but total biomass I don't expect to change because the duloxetting is not reducing growth so much for the other bacteria mostly for for the year actually yeah and this is not the most dominant one again I know what I'm showing is the relative abundance so this is not the total biomass and also measuring total biomass is not so easy with gut bacteria many of them you know the flocculate cluster together so odys I mean is a barrier of approximation I think what we need to do either look into the total CO2 produce or basically just measure the the dry cell weight so like we have the single bacteria resource we also want to build a community resource that we can use studies like I've just shown in the slide before the first so this actually I forgot the name Alvaro Sanchez that did a very good job in introducing type of experiments we are doing basically start with the community you basically get them into particular media and then you basically keep the transplant every 48 hours exactly some experiment the difference being this is of course the gut bacterial community and run under an Arabic condition and this is the initial mixture is known right so we put this bacteria together so we know exactly where we start we know exactly who is there and so we can really reliably follow so we don't have the problems with the phylogenetic assignment over here and we know how each of these bacteria grows in this in this come in this particular media very well actually it was quite interesting to see that we get a very rapid assembly so this is basically the number of transfers each transfer is up to 48 hours you get exactly the same same result if you do transfer after 24 hours we do it 48 hours because it's well let's say experimentally more convenient to do so basically get establishment a very diverse subcommittee in both media as you can appreciate there is no competitive exclusion meaning that there's no one winner of the system you get actually mixture of the bacteria so we have indicating that there is either a strong niche partitioning and or there is also metabolic or other kind of niche creation going on in the system we have of course done this now on a much larger scale so we are starting with three different bacterial mixtures so they are either we exclude for example probiotics or pathogens or mix all of them together we are 15 different growth media defined growth media which are exactly the same that I showed you in the resource slide earlier and two different pH just rest to test how the pH would affect the system so 7 and 5.5 so 5.5 is closer to what you would expect into right after the after the the stuff comes out from the from the stomach we do nine transfer we know that four three transfer also sufficient just do nine transfer just to be sure and we do the barcode sequencing at the end so we get a range of defined communities as snapshot of it at very very different diversity you get as few as four or five bacteria you get up to 20 bacteria into the system it's just a pca plot I mean I don't want you to read through this I mean doesn't mean anything unless you look closely into it but this is basically just means that these are assemblies are media dependent different media different assemblies as one would expect what's interesting here is that you get lots of winners and losers so I mentioned that we also characterize in depth all these bacteria and the same growth media before what you can now do is that to comparisons like this for particular media look into monoculture growth rank and then we see the abundance sign into community how they do right so you can and then the winners are the one basically which grow very poorly in the same media when they're growing alone but in the community context they grow very very well right and the losers are the one for example equal I often turn the south to be the loser when you grow equal I alone is one of the best cross and we just you know buy for the easiest to grow and the best crossing across almost all the media that we look into it basically quite strongly loses out and even the many different media conditions so you can now if you make statistics of who is winning often for example like to Berserker say our cluster in Ramoza are winning under many of different conditions I got a lenta or this guys over here actually are losing out in many different conditions you look into so you can start now looking into who is winning from being in the community who is losing out into the community how to understand this so I don't turn again time to go through the details but what we are doing now is again the condition media type of experiments that you grow spaces one then you add take the condition media and the grow other spaces you get very interesting results just a snapshot these are different spaces these are commensals this is an opportunistic path version and this media it doesn't grow alone at all very barely detectable any growth use a condition media of a commensal bacteria and then it starts growing over here so the goal basically is going to be the reconstruct the metabolic interaction map the human gut bacteria this is computer simulation purely this is for the fun of it because we can do it we can simulate almost a hundred bacteria together take the genome scale models and and simulate every possible metabolic interaction that can grow between them basically you keep your supercomputer busy for as long as your institute would allow it you can start mapping into possible interaction landscape like this and the goal basically is going to be see how many of this turns out to be true in our experimental setup and with this actually this is the things my all everything about gut microbiota I realize that a lot of I mean I came only today which actually is a pity but had many family constraints so I could not attend the talks before but I realized that the niche generation and the cooperation mutualism seems to be quite strong interest in this in this audience so either I can stop here or I have few slides on the niche generation that are mapped through metabolic analysis so I think stop or continue continue okay so okay technically I still have 10 minutes so so this is the work entirely or turned by Olga Pomorova who has a PhD student with me and is being continued by Natalia her postdoc in the lab so we're interested into metabolic cross-fitting especially the question of niche generation and mutualism I mean I don't need to convince you these are interesting questions to look into and the reason so this whole project started with modeling right so we started doing modeling you know we actually see quite interesting patterns we started when we started measuring metabolites and even if I look into literature so there are lots of lots of implications of metabolic cross-fitting very few examples of metabolites have actually been measured and shown going from one space to other spaces these are hard experiments to do I mean you can look at the DNA RNA and you can say okay this DNA came from this space is this RNA came from this space is fine you cannot do that with metabolite the pyruvate when it leaves the cell bacterial cell fungal cell once into the medium this could belong to anyone right there is no way unless you do clever labeling experiments which are also very difficult to do in a community context you can't do that so there's a reason why it's so difficult to do and the Olga has been actually quite interested in in getting at least one very good well mapped community and what she has been looking into is metabolic dependencies between yeast and lactic acid bacteria why yeast and lactic acid bacteria these are one of the let's say microbial communities of actually they co-occur in many many different environments especially with food associated environments if you look into wine fermentations kimchi sourdough kefir cocoa fermentations bread fermentations everywhere almost everywhere you would naturally find saccharomyces cerevisia some other yeast almost always you'll also find lactic acid bacteria so all guys been puzzled by this there must be something going on so again this is summary of a lot of different work but basically we managed to make models model predicted that there must be some amino acid exchange between this community and we have done together with uesauer and marcos ralser uesauer from itia Zurich and marcos ralser from Cambridge a lot of extensive metabolomics experiment the idea have been again doing a condition media experiment just to give you a flavor so we have a media or actually many different media where lactic acid bacteria do not grow alone right so put them into won't they grow only in presence of each you can also do in condition media grow each remove each add lactic acid bacteria then they grow suggesting there is a flow of metabolites or some diffusible factor from each to lactic acid bacteria and if you do take now basically the samples during the each culture and then lactic acid bacteria culture which is after you remove the yeast basically what we are hunting for in the mass spec is the the ions that show this bell shaped profiles that this would imply that basically the metabolites that are secreted by yeast and taken up by lactic acid bacteria these are the results of untargeted mass spec profiling you see very very interesting profiles but the red ones actually the the ones that we're looking for there are lots of molecules and many of them actually assigned to the amino acid that we predict by the model are basically exchange between yeast and lactic acid bacteria and just to to drag your attention that these are not only so we have done come complementation assays and so on so forth to basically prove we also done quantitative metabolomics and supplementation assay in the same media and we basically can reliably show that this interaction are indeed taking place hey just again we are done also mutant screen and here's some example of the the four different mutants that we find interesting these two mutants are that each mutant that increase the growth of lactic acid bacteria two mutants that decrease the growth of lactic acid bacteria or at least one of the two bacteria we taste and here is the quantity amino acid secretion profiles and you can see this perfectly mirrors the growth of lactic acid bacteria meaning that you can also genetically manipulate your community and show show what's happening over here all these mutants are related to nitrogen catabolite repression which I will come in a second and just to convince ourselves that this is an active secretion not cell death basically this was a month of sleepless night sport for organi because we got troubled by the the thought that there is no active secretion yeast cells are simply lysing if you take different mutants there lies more and then basically there is no active secretion but simply not cooperation but you know cell death now we have done many different experiments that show that this is not the case many lines of evidence one is that if you would now make condition media of the different growth growth along the growth curve of the yeast they all show proportionate secretion and there's not much cell death going on over here you can actually also take this different mutant I showed before that showed different degree of the support to lactic acid bacteria we have done the live dead staining and basically first of all if you just calculate the number of days we observe they are by I mean they are far far away from explaining this lactic acid bacteria support that they agree and plus that if you try to make correlations between how much support different mutant use and the death rate there is simply no correlation this is indeed an active secretion and not cell death yes because I will come to it I mean again I don't have the data here because we also done gene expression analysis and the German gap one there's general amino acid permeates one expression is proportional to the amino acid that you secret into the medium so this is really dependent on it and you can see in the mutant that are not secreting much gap one expression is also going going down as I said I mean I don't have to go through all the experiments we have done a lot of also looking into mutants gene expression metabolomic etc but this is a summary side what we think is happening what basically we are seeing is a niche generation in action right so metabolic decisions of the each that are enabling lactic acid bacteria survival and I use the word decisions because they said this in each to actually make so here is the the summary model from the work so you have extracellular nitrogen sources amino acids ammonium etc so the the uptake mechanisms are you are the same for different I mean uptake mechanisms are not specific to amino acid they are general permeates the nitrogen basically comes in now if at the same time if you start making synthesizing nitrogen inside or the amino acid inside that can happen either because you're mutant or you're the torsive torsive signaling pathways I've activated for example adding rapamycin and forgot to mention that if you would add even slightest amount of rapamycin to your media the growth of lactic acid bacteria boosted many many folds right in the in the wild rubbish culture so you can manipulate by rapamycin then basically what you get is a nitrogen overflow because you have excess nitrogen coming in first of all east has all it needs but it starts to think either you have bad nitrogen source or some kind of other signaling if it's things that it's under nitrogen starvation it starts also synthesizing amino acid inside and this basically leads to a nitrogen overflow phenomena and part of that can also be toxic because you can't simply just accumulate nitrogen inside and this basically leads to amino acid secretion into the medium and until for a while actually people have been thinking the saccharomycesis is a bad amino acid secretor it's not bad amino secretor you just need to get under right conditions its secret is actually fairly good amount of amount of amino acids and basically this creates a niche for lactic acid bacteria now this interaction you can manipulate either by making mutant for example that these two mutant I showed or add rapamycin you can get a lot more lactic acid bacteria or you can make another mutant or till it change decrease it or you can reduce the nitrogen sources into the medium and this is actually quite proportional so if you would look at the how much amino acids are secreted at different nitrogen load you can basically make lines linear curve to it just a few thoughts on this and again another evidence that this is active decision of the east because if you look at the which amino acids are being secreted so these are the red ones over here amino acids that are secreted and these are the biosynthetic costs of the of different amino acids only the cheap amino acids are secreted right so each is not secreting something that costs a lot to make it secretes the cheap amino acids over here is this relevant into ecological context we are looked into different east isolates that come either from the grapes or from for example from the kefir and then you see whether they do the similar growth promotion for lactic acid bacteria in that they do this is our let's elaborate is s90 lactic acid bacteria also east and this is the growth of lactic acid bacteria so this has in the scale we have been looking up to now we can see for example candida californica kefir isolate is lot lot more growth support so they're actually secreting the same effect of the the growth support you also see in grape juice experiments so it doesn't need to be in the laboratory medium that we have designed and we also done metabolomics and the mutant analysis in the grape just showing that this is again the same mechanism through which you see this impact on the on the grape juice and those who know about wine fermentations or the beer fermentations lactic acid bacteria actually quite notorious to contaminate this fermentation sometimes we want to have them for mallactic fermentation but they can also contaminate quite heavily and once you get them actually it's quite hard quite difficult to get rid of them and you can see the reason for that each actually are creating niche for lactic acid bacteria over here and of course it has relevance for many other communities some of them were experimentally following up like kefir for example but I won't talk about that and that brings me to my last slide as a teaser now we have been looking into one way interaction right and this is of course is much harder to think ecologically why would each secret things and we actually quite convincingly shown that each is secreting things because it's profitable for each right it doesn't need a nitrogen it has plenty of nitrogen accumulating more nitrogen is toxic since it secretes it other bacteria can take up into it now how that situation actually can be precursor for mutualism or cooperation depending on how we use the terminology and in a very very simple experiment you can see this happens is simply replace glucose by lactose and then you choose lactic acid bacteria strain that is lactose positive and this is what we exactly get so these are the lactic acid bacteria isolated from kefir which is a milk drink and this is the wild type is so these are all each at the same od and the dilution is on this axis you can see now the each grows only closer to lactic acid bacteria and vice versa right and you see exactly the same into the liquid medium here's the monoculture versus co-culture monoculture you don't see almost no growth co-culture you see both lactic acid bacteria each grow this is the glucose control if you do the second mix when glucose each doesn't care about lactic acid bacteria each does its stuff like it lactic acid bacteria grow closer to the east as we saw in the previous experiment if you look into again we are done metabolomics you just show the the lactose consumption lactose is consumed only even to the co-culture okay and with that I would stop here and again once I can acknowledge I mean to all the people who have been into the lab who have been worked into this and the the corp and the collaborator labs as well and thank you for your attention