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Kelly Street, Workshop 500: Trajectory inference across conditions: differential expression

Kelly Street
Research Fellow в Dana-Farber Cancer Institute
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Kelly Street, Workshop 500: Trajectory inference across conditions: differential expression
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500: Trajectory inference across conditions: differential expression and differential progression

Kelly Street (Dana-Farber Cancer Institute) Research Fellow

Koen Van den Berge (University of California, Berkeley) Postdoc

Hector Roux de Bezieux (University of California, Berkeley) Ph.D. Student

11:00 AM - 11:55 AM EDT on Friday, 31 July

WORKSHOP

In single-cell RNA-sequencing (scRNA-seq), gene expression is assessed at the level of single cells. In dynamic biological systems, it may not be appropriate to assign cells to discrete groups, but rather a continuum of cell states may be observed, e.g. the differentiation of a stem cell population into mature cell types. This is often represented as a trajectory in reduced dimension.

Many methods have been suggested for trajectory inference. However, in this setting, it is often unclear how one should handle multiple biological groups or conditions, e.g. constructing and comparing the differentiation trajectory of a wild type versus a knock-out stem cell population.

In this workshop, we will explore methods for comparing multiple conditions in a trajectory inference analysis. We start by integrating datasets from multiple conditions into a single trajectory. By comparing the conditions along the trajectory's path, we can detect large-scale changes, indicative of differential progression. We also demonstrate how to detect subtler changes by finding genes that exhibit different behaviors between these conditions along a differentiation path.

The "Differential Topology" vignette from the Slingshot package provides a more complete problem description and proposes a few analytical approaches, which will serve as the basis of our workshop.

Moderator: Erica Feick

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Kelly Street
Research Fellow в Dana-Farber Cancer Institute
Koen Van den Berge
Postdoctoral Researcher в University of California
Hector Roux de Bezieux
Ph.D. student в UC, Berkeley

I am a Postdoctoral Scholar at UC Berkeley and Ghent University, supervised by Sandrine Dudoit and Lieven Clement, where I am developing statistical methods to analyze biological high-throughput sequencing data, e.g. (single-cell) RNA-seq data. My research interests include normalization, dimensionality reduction, differential (expression) analysis and multiple testing. I support open science and open source software.

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So welcome everybody. My name is Kelly Street and this is the trajectory and friends across conditions Workshop. Like I said, you'd reject Tree in France across conditions Workshop, differential expression, and differential progression, just kind of a term that we made up. So we'll be sure to explain that very thoroughly. My name is Cody Street and I am happy to be joined by my fantastic collaborators. Coon Panda burger and Hector rutabagas you joining us. Some

last-minute changes of plans. But both of these guys are excellent collaborators. So you should all take any chance, you have to talk to you or work with them. And then down at the bottom, we have a link to our GitHub page which has the full work flow that we're going to be presenting today. Movie, khoon also posted in the chat link to the instance of our Workshop if you want to play with it yourself. So I'm going to start by presenting the data that we're going to be working with today, which is just one

example of the type of data set that you might be working with and that you might apply these methods to but it's very straightforward example and it fits very well with what we want to talk about. So this is sort of a simplified version of this experiment where we're studying EMT epithelial-mesenchymal transition. So, they take these epithelial cells plate them, and let them grow for a certain amount of time as they divide and differentiate it, turn into a ball. And then from

the second plate, be collected, both an inner and outer sample in her sample tended, to be more epithelial cells. Outer tend to be more mesenchymal. The original paper, by the way, I've listed at the bottom of gold single cell, genetic screening, identifies regulatory checkpoints in the Continuum of epithelial. Tissue is very well-written and definitely worth reading and they present their own methods for dealing with these sorts of analyses. What we're interested in is what to do when you have this sort of experimental design

where you're basically doing that twice. We've got this single, Continuum of cells that feeling all too well, but we've got it under two different conditions. We've got an untreated ormoc group, and we've got another group that's been treated with some sort of drug is tgf-beta and some were interested in comparing the development under these two conditions for the most part because this is not the main focus of our Workshop, but all of these steps are important and worth considering and you know, very much not like decided. So these are choices

that we made along the way to get to the part that we wanted to talk about, but each of these could individually be debated for a long time. So we're using Sarat for both normalization and batch integration. So I showed the experimental design a minute ago. We had to collect today that we had to collect the cells from the two different conditions, separately and so one of the first things we have to do after normalization is to integrate those two data sets so that they somehow match or be represented in the same space. An introductory in France, we want to identify

the primary access of epithelial to mesenchymal transition and we're going to use sling shot for that because I develop slingshot. So it would be weird if I use anything else. So one of the first steps in our workflow that we're going to talk about more in detail in a minute is differential topology. And by this, we mean, looking for differences, in the large structure of the conditions. And so you can see in the top left, ear panel a, we've got a single trajectory and two conditions that are sort of evenly mixed

throughout. But we could just, as easily have a situation like that, I'll be where the two conditions diverge, and they end up in different places. And so, if something like that happens, it might not be appropriate to fit a single trajectory to both conditions. This might not be appropriate to fit Branching trajectory, because the fact that you do an early cell is condition b means that it's already determined, it will end up in this Branch not in this branch and so it might not be appropriate to model that

as a branching trajectory but rather to suffer trajectories that happened overlap, another problem that we're going to tackle is differential expression and this is going to be in the context of continuous to do time. And we have a package for this called trade seek that we think is very well suited to handle this differential condition problem as well. And so this will allow us to answer questions, like, are these jeans exhibiting Dynamic expression over to do time in either condition and are they exhibiting different patterns over suit

of time between the two conditions? And so these are very similar to the types of questions that we used to ask of branching trajectories where we can say is a gene following the same pattern in this Branch versus that Branch. Now we're just talking about is it the same pattern between a condition? I think a very good fit for the types of models that we had already developed. So with that I'm going to switch over, I'm going to point out, I get home page of anybody wants to play around with anything there, but for the most part, Some stuff keeps getting in the way. Sorry, I'm just

going to present from our website which is a sweet 13 bios, e2020 trajectories. And this is the website generated automatically by our repository paper, which if you're interested in, this is a very interesting paper, very much worth reading a presents for the monocle approach to these questions. And yeah, we're basically just showing we used our own methodology, but we have similar results to there's so something in going to skip over introductory stuff. Because all of that stuff

that I just talked about was introduction. This is our Upstream workflow which like I said we're not going to talk about a whole lot but here's all of the code that we used to run. SE transforma normalization, what's here to do? The latest and greatest Sriracha integration and then work converting between cirrhotic spare and objects and single cell experiment objects using these as a single cell experiment and functions. So we're not actually running this code in our Workshop

are actually cheating and just importing the data afterwards, but all of this only took less than 10 minutes to run on my laptop, it's actually surprisingly fast. So, feel free to ask questions, and thought any of that game. We didn't include because like I said, we're focused on. So do we see between the two conditions, very Divergent, sorts of topologies where you know they might be completely distinct, they might be two, lineages are completely unexpected or they can

have one Connecting Point and then Branch off or be perfectly overlapping. And so what we see here and sort of are you map represent station of the data, is it? We do have pretty good overlap between our two conditions. The mock in the TGIF beta presented throughout this lineage And then I mentioned the beginning, we've got two different areas of the plate from which cells recollect inner and outer. And we can kind of see that there's more in ourselves towards

the left more out ourselves towards the right. And so that gives us an idea of the orientation of this trajectory that probably indicates that the left side is our early pseudo time and the right side is are late and so this is What we're going to be talking about when we talk about differential topology we're looking basically at neighborhoods around each cell and asking just a composition of cells in that neighborhood match the overall composition of the conditions. So we've got a more or less even split between our two conditions,

each local neighborhood around a given cell. We want to see you tomorrow unless you've been divided between the two conditions. If you're familiar with kvet k v e t metric for evaluating normalization methods. That's a very similar sort of metric. We do some smoothing should point out the function that we use here is imbalance or Gives us the neighborhood size round. Each point and smooth is a smoothing parameter to say, how much we should smoothies because you can

see if they're. There has been some smoothing the spot down here. And what we see is that there are a few sort of small neighborhoods where there's imbalance, but for the most part we see a lot of purple. And it seems like we've got pretty good mix between the two conditions and that's what we saw earlier. When we were looking at the the red black conditions blond. And so with that information, we feel pretty comfortable putting a single trajectory. It doesn't seem like there's any obvious distinction. We're like being in the normal versus

the treat, a crib, determines your outcome. It seems like the start and end position is more or less the same regardless of which treatment condition you're coming from. So that gets us into trajectory and friends which is going to be a pretty straightforward application of slingshot. The only interesting thing here that I want to point out is that slingshot works on a minimum spanning tree of clusters and so often times we would include a clustering step at this point where we just do something very simple, maybe like three bananas

case. We don't actually have to do that because we've got this inner and outer information already and we know ahead of time that the inner group cells could be earlier. And see, what time the outer group should be later in pseudo-time. So what we did here is we treated those as are clusters and we just made a minimum spanning Tree on those two pastors, which is, of course, straight line which we can supervise and say, okay. We know that the inner group should be our starting cluster and that ensures that we have the correct orientation Dora. Trajectory, which is shown here.

And then just looking at the distributions of cells collected from the inner and outer portions of the plates. We do see that the inner cells are sort of skewed towards beginning and the outer cells are sort of skewed towards the end. And so we've already covered a lot. I think this was one of the places where we wanted to take a break and see if anybody had any questions from the polls. The Thirst. There's a few people already asking questions about later steps and I suggest we do those when we're at those respective

there's the question of do you have a suggested by my Chanel to induction to take for the fusion that versus you met? That said we try to use PCA first if I can, but that's not always possible because sometimes you got more complex is generally pretty good results with Diffusion maps for the other one that was mentioned. I mean maybe I'm just not using them correctly but I find little stringy pieces. Highly linear branches off of the main cluster of cells. So

I don't know. I think it's more of an empirical thing that I've had better luck trying to do something with that tries to keep that the global structure of your data. Probably melts something like Disney, which favors local, or the original Rock comes with a batch Factor. I'm asking, because the batch corrected values from integration should generally not be used for quantitative. So I guess that this is already kind of pointing at the later differential expression analysis. And there we are not using

the patch correct itself, use for differential expression because we've also seen that these are not recommended to be used, but we do the best correction, or are the integration for the trajectory in front and to get the trajectory, right? There's one more question when assigning questers using known data parameters, are you studying the centroids for those data transfer using the centroid? Basically? It doesn't matter in this case because there's only two

extra points. We're really just using it because we know that we only want a single trajectory, so it's fine. If I'm going to run spanning tree is just a line and be the inner vs. Outer. That's the way to order or like, set the orientation of that line because we want to make sure that we don't end up with the lines. Going backwards to sort of the first test, actual formal test that we run. This is what we're calling differential progression to the question here is,

if we compared our two conditions along this. Now, trajectory framework, that we've established. Do we see any differences between the condition? We wanted something that sounded like differential expression. So, just looking at this density estimate of the two conditions, you can see that there definitely is some sort of difference between the pound. We see this interesting sort of Tri Moto structure in the untreated group and then the treated group is sort of missing or certainly has a lot fewer of that early cell type. It seems like they differentiate a lot

faster or a lot more and you expect some sort of differences here. And essentially, all we're doing is comparing these two distributions. There are a lot of different ways that sells could show different Behavior, along a trajectory. And so we wanted to be as general as possible. It might be the case that in a 1-1 condition with more bimodal. So it's either all at the beginning, didn't differentiate it all, or all at the end of a differentiated, where is another? One is sort of more uniform in the middle. So things like a, a cheetah or anything else that, you

know, is testing some sort of centrality perimeter, we didn't think would be super appropriate. And so what we went with was he Smirnoff test, which is attached to compare these distributions. And ask are these coming from the same CDF. And unfortunately, this output is blocked. Just decided that it was going to put anything in our final version. Every version of until our final version had the output. I'm just going to switch over real quick to a version that I just recompiled on my computer and show you that. We do get that good old classic 2.2 x 10 to the negative

16. Female you're here, these are very much different distributions that were saying and that's surprising. That's really just sort of it a sanity check, but I think this is a good General framework for comparing condition conditions along a single trajectory, however, it's kind of crude, it's really just asking One single question and it doesn't have a lot of fine little detail. And so if you want to ask some more detailed or potentially more interesting questions, that's where we're going to start looking at things at the gene level and

asking about differential expression. And so this is a pretty I think good application of the trade secret package which all three of us worked on. Developing is made for comparing different branches, or assessing differential expression along a single branch of a victory. What we're going to be doing is fitting negative, binomial, General additive models or and began models, and pecans, which is going to be for each gene along a trajectory, some sort of curve that represents the average gene expression.

And these curves are based on smoothing splines. And so the first thing that we have to do is determine a number of knots which is essentially a number of degrees of freedom for the smoothing splines. So how adaptive we want them to be. Answer for that purpose, we develop the evaluate K method, which lets you put in a range of possible nuts and it'll print out some summary statistics that you can use to, hopefully determine an appropriate number. And so this can be a little bit of occasionally burdensome, which is why

we've only tested five, how use here? If you're going any lower than 3:00 then, you know, maybe a quadratic model would be better for you. There's not a lot of flexibility at lower values, but we could have, certainly extend it a little bit further off the end than 7. If we had had sort of more computational resources to work with but misses the idea that it's going to show how well, the the trade seat model fits for various members of knots. And what we're looking at here is just a few different summaries. I'm going to focus on the average, and I see that's the same as the relatively icy,

or it's just relative to the highest value. Are sea. Urchins information criteria is hopefully familiar to some of you, but essentially, we're going to see better and better fits lower and lower values here as we get more knots. It's not surprising them or not. So you have the lower Brenda certain point, we start to see sort of diminishing returns on increasing the number of knots and so you can think of this similar to a scrape lot or you're trying to

pick the number for what sort of looking for an elbow Point here and I think it's the spot wasn't so very sort of see it around 5. And like I said, you know when we're actually running our analysis we might go out to 10 or 15 or higher number, if not. So hopefully we can really see that this pattern continues after 5. But 5 is the number of months that we decided to go with business analysis. You could certainly make a case for several other numbers, but I just wanted to describe our thinking about how to go about that. So once we've chosen a number of knots, we're going

to use the fit yam function actually fit on model to all of the genes. And I'm going to make special note of the conditions argument. So this is not available in the standard version of trade. Secret version is currently owned by a connector but it is on our get a job under the conditions, Branch. So right now, this is still very much and development. These are fairly new methods for us and so you have to install via to get a job if you want to access this

functionality. But otherwise all we're doing is like I said, before we began model, we have the original counts are not particularly interesting to the single trajectory that there is an enemy of the conditions and the number of this is going to fit our model which we will then use to assess Dynamic expression and differential expression between Question, which is again, a classic trait seek sort of question is, do we see James that have Dynamic expression patterns? We would certainly expect us, because we're seeing different cell types. So, as cells are changing types,

we would certainly expect to see some jeans changing expression. We're going to use the association test for that senses testing for any association between gene expression and suicide doesn't have to be monotonic. It could be or any other pattern that are spinning. Spines are capable of picking up. Resetting a cough on a log to pull change so that we sort of pick up the bigger changes. This is not looking specifically at the conditions on the we are. Fitting it separately. No, I'm sorry for fitting it once, but then, adjusting the P values within

condition. And so we've got separate certain age, condition. And what we see is that we get quite a few. Almost 2400, Dynamic James, or James that are associated with, in the condition, and a little less than half as many in the tgf-beta condition. And then, this is an upset plot that shows the overlap between those two sets of genes. And we see that there's very strong overlap, almost all of the jeans that were dynamic in the treatment condition, also dynamic in the Box condition and then there's a bunch of

extra ones in that muck condition visualize the pattern of gene expression. Using an insecure, we're looking at gene expression over to do time. And these are the jeans from the mock condition. So they may see a few different sort of broad patterns there genes that are highly expressed early into a time and then shut off their James that are off early and then turn on late. And then various patterns of Austin on then off again, at different points, along the linnaean, particularly in this middle section and this sort of 3/4 And if you wanted to do, I should also mention.

If you wanted to do clustering on expression patterns, that's another functionality that. We don't really exploring this pen yet but that we that trade seat does have. So if you want to Custer, we offer some sort of methods for clustering on the coefficients of this moving splines which can be an efficient way to Cluster based on expression. Animated some ginseng Richmond analysis on those genes? Which again is not the main focus here. So I'm going to be over in a little bit of epidermis development of

material cell differentiation. A lot of things that we would expect to see from the sort of experiment And then the final question that were interested in answering is, do we see different patterns between the two conditions in? This is really the Crux of the workshop. Do we see in our genes that are behaving differently? Despite there being a single trajectory? What are the more subtle differences that we can identify between the two conditions? So just sort of

an exploratory analysis which one, which is mentioned in the paper and sort of being similar between the two and it's probably the same sort of pattern up. And then down there's a little bit of a difference really on. Crv3 is the same thing. Hi, to Lo pretty similar between two conditions original paper. Also, mentioned a few other jeans that are very different. Between conditions fn1 is almost completely off and untreated and she's very high and increasing expression and they treated similarly, but less obvious ch2 is almost entirely off

in the untreated and shows fairly robust expression. So that's all exploratory to use this new condition test which again is only available in this latest version of trade secret. I do with a full change parameter to set requirement. And this is going to do something very similar to the pattern test. For those of you who are familiar with the curves between the two conditions and rascally do these have the same pattern of expression. And so smoothing splines are based on coefficients. I mentioned earlier we're basically testing

are those coefficients the same between the two conditions and so this is ultimately a weld test under the hood. Sorry, I should mention it. We're running that test and then doing a few value adjustment and we've got almost 2,000. Differential Express jeans, differential in terms of showing different patterns between the two conditions. Are most significant Gene is part of deer and you can see it's it's very strongly difference between the two and it's not surprising. This is sort of

one of those that you would certainly help you would pick up. Similarly are least significant Jane pretty similar pattern. It's mostly off throughout both conditions. And so, just as a paycheck like this makes sense. This is probably not differential expression. Again, we can do it, keep up to look at the different patterns and this time, we're comparing the two conditions. So we include both the tgf-beta and the lock conditions. And we're ordering by this hierarchical clustering of the tgf-beta patterns. And before

I go, without, talking about any of the specifics, you can clearly see that pattern is very strongly degraded in the mock and that's good. That's what we were hoping to see because we're we're looking specifically here for jeans that show different patterns between the two conditions, Sophie's to heat Maps were identical, that would be worrying because that would mean you're picking up James that had the same pattern in the two conditions while looking for James have different patterns. And so we see the same sorts of things for James that are highly

expressed and then turn off games that are off and then turn on and intermediate James there on at some point along the way. And for almost all of them, you can sort of see similar patterns in the mock, but it's it's very much appreciative that pattern. It's definitely not as clear. There's some streaks of blue throughout that are just jeans that are off the hallway. And so this is fairly reassuring that we are. In fact picking up James that show different patterns between the two conditions.

And I'm just in the last time we once again ran some Gene set enrichment analysis and found some markers for Locomotion. In cell motility things that we think are related to the treatment and similar to the types of conclusions drawn from the original paper. And so, with that again, for me to our website, or are GitHub repository, and we will open it up to questions. Thank you. Thanks Kelly. I'm sure there's a few more questions. First one, when combining data sets, Avenue General, in

further to check trees will be drawn between unrated sells simply because they are in the same thing and I would say a lot of it depends on the experiment self. Do you expect that to check three biologically or not? If you're looking at, for example, the day we expect the kind of dynamic change when serial cells, develop into mesenchymal cells. Expect the same thing for, for example, stem cells, develop into mature cells. But if you would say take a sample of a liver tissue, I wouldn't know. Even try to do to fix

reject me because they're you would rather expect kind of discrete discreet cell populations. I would also add just as a minor point that we actually something I've been thinking about a little bit recently. Thanks to Erin Lund in the context of putting projector. He's in sunshot. We have been working on I have better ways to separate. Distinct populations. It's always sort of something that I've been aware of. And so, I think we've, we've got a rule of thumb

to slingshot for how separated two clusters have to be. Before we say that they're not on the same trajectory Works in some cases pretty well in some cases, not as well as we would hope there's at least some consideration of that you can try to make sure that things that don't belong on the trajectory aren't included. actor, if you have anything to ask, if you want to share some views, I agree that the integration can sometime, It's tricky. Sometimes you can see the trajectory in every data, sets individual before,

merging them. And that's a pretty good indication that something is really popping up. The next question is, how do you prioritize jeans for inputs to V can function? Most valuable? I would recommend some kind of low-level filtering where you would, maybe just remove the jeans that have low expression. In general. We do realize that fit can function can be compassionate and see if you need to estimate a lot of a lot of cheese. If you want to filter jeans then filtering the most valuable name. I think may

not be the best idea, because it may not be in Independence, filtering criteria. So, in that case, I would rather try to Maybe filter on the mean, rather than the variance. Because I would think that filtering on the variance might inflate false positive group, I can share the chest. So, next question, can you explain more about the directionality? Would you get a different trajectory with different supervised? Directional assignments. Good to you. My faults have arrows. The directionality

doesn't actually. It doesn't actually affect the the final trajectory, it's just so that we know it's going left to right, rather than right to left. In this case since it's just a single lineage. We're basically just putting a principal curve which is a nonlinear generalization of pc-1. Basically, it's a line that tries to put the middle of the data. Directionality. And then put the game of life. Well, if you want to switch the time, just like putting a negative in front, you would get exactly the same results.

The next question. Can you please explain the concept of knots in reference to pseudo time to check to see what is just mouthful. Branching points did Nipsey the time to get to the Third Kind of do not really relates to be to check 3 that's been inferred with more relates to how complex we expect, the gene expression presence to be over soon. Stay safe. If you have more Nazi will be able to fit. More complex patterns will not be able to pick them up from jeans.

Suddenly increased quite drastically in the expression, if you will need, you know, a few notes to really pick that pick up that Peak and then go back down again. So they're not really relate to how complex the gene expression patterns are in the data sets. And that there is no immediate link with the number of branching points in a bit of time to Victory. So next question is, is it possible to include to temporal dimension, for example, treats and treated at different time points? If yes, would you

specify them in some way when we leave the pursuit of time or to check trees take care of it, in an unsupervised fashion? I'm not sure if I'm following the question sounds like they're asking, how would we handle multiple times points, which is his kind of wood that inner and outer is I would I would think of the way that we handle inner and outer as being analogous to how we would handle points because usually you know, if you're collecting single cells at

discreet 10 points, there will be some overlap in the types of cells that you get at each time, clients differentiating synchronously. So you'll get different distributions of cell types and so you can you can treat the time as A useful way of supervising, the trajectory and friends. But I would I would use the The suit of time rather than directly incorporating the variable next question. Can you differentiate special methods? Be extended to multiple records for Monica tgfbi. This is a good question and it's something we will be looking into for now.

We will allow you to add fix. The facts are the facts to fix the fact to the model, which is okay, if if your treatment effect is you know, if you need to do the differential expression within sample, then I fix the fact so we'll be appropriate You're comparing for example, healthy versus deceased patient samples. Then you, you may want to consider something more complicated than simply a fixed sample. Effects, we have not yet to implement, we do recommend you

to evaluate the number of Nazi want to use for each trajectory separately because it being expression patterns might be pretty difference between trajectory although we have found that the exact number of Kay doesn't make a huge difference and typically the most data sets are between four and eight notes that that seems to be like well I would I think that is what we call a trajectory The self-taught from one single coming in faster, and then can go into multiple branches. We should go lineages. So, if you want to bet you like that, the Cannabis tree structure, then you

would just need to run evaluate k1000 point. Then we would we commandeer any value at Kay for every Five Points, Independence Day next? Maps of jeans, de between conditions independently Rose Guild. So yes they are all on the same color scale. The former could make it more difficult to compare prices. Similar with one showing weaker than they are. Guilt. So, we used to kill each gene to have. The means you do and units variance. But they are also indeed separately scale for each condition to make more clear.

So, Yep, that's what we're doing. I'm not sure if you checked the heating up sweetheart to scaling. I think I can see what they're saying, where if you had similar patterns, but it was sort of a, a shift between the conditions that that might not show up in the heat back up, but that would still be something that that we picked up with the test. No problem. It's do scale their signatures to say. It's really nothing. But it's a good question about the next question. Is

it still changing? Then others. Is there a point where using one value of K is suboptimal for capturing a reasonable fraction of the data? Maybe this doesn't happen by Logic Peter so, I would answer to that that Read me can easily handle that, because if someone reaches our Dynamic, and others are on the way to smoothing works, is that there's shrinkage and shrinkage is identified through across from the day she approached. so if it's useful to, you know, fit a

complex pattern for one language than, you know, we will move to a higher K value when we can evaluate cable. That doesn't mean that the functions become overly Wiggly for for another week because we have the shrinkage. I think a second client that the most dynamic, an adjustable tend to be longer because they will impact more than veggies. I mention And so this is going to also be respected in and try to seek. The last question, I'm curious. If this

is suitable for solid tissues of different stages of the disease, like, an nafld or Nash. N, a s h, we can still use straight. See their rights to understand trajectories for specific cell types. since I think this could be one of the first questions is, when do we have a 2.3? When do you not have a trajectory? Again a lot of it, I think depends on the biology. If you have different stages of those diseases, she mentioned, you expect, you'll be able to capture them into decent resolution. There you go from one state to another state, and there's

some kind of dynamic switch or dynamic developments between them. Then it could be possible to look at your jack trees, but the love of, this is pragmatic a Time access, but this works for any So type of question of cell types that exists along a Continuum. It doesn't have to be a developmental lineage of the day. For example, just before Divergence of a language is difficult, could it be used at the position of the not-so-pretty fault? We use. We separate. North's according to quantiles. because, you know, if you have a lot of state, are you knew

you might want to get something else there, too, to pick up the items that are Maybe Shown by those data. This is indirectly related, I guess, two mediums customers because you have loved. So I think that's all the questions. Just a bit farther than that for now and Tracy could not allow users to buy where they want enough to be. Just the number of not enjoy the rest of the conference, Las Vegas.

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