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+---
+title: "Kmer diff"
+author: "laurent modolo"
+format: 
+    revealjs:
+        transition: none
+        format:
+        highlight-style: monokai
+        theme: white
+        footer: "laurent.modolo@ens-lyon.fr"
+        slide-number: c/t
+        fontsize: 22pt
+revealjs-plugins:
+  - pointer
+---
+
+## Background
+
+The Delattre team’s studies, *Mesorhabditis worms*, some species of which present
+atypical reproduction mechanisms. 
+
+In a previous paper: *Males as somatic investment in a parthenogenetic nematode* [DOI: 10.1126/science.aau0099](https://doi.org/10.1126/science.aau0099),
+we characterized contigs of a *de novo* genome assembly of *M. belaris* as
+
+
+:::: {.columns}
+::: {.column width="40%"}
+- <span style="color:blue;">Autosomal</span> chromosomes
+- <span style="color:red;">X</span> chromosome
+- <span style="color:green;">Y</span> chromosome
+:::
+
+::: {.column width="60%"}
+![](./img/science_figure.png){width=350px}
+:::
+
+::::
+
+## Goal
+
+From raw sequencing data of male and female individuals, we want to identify $k$-mers corresponding to :
+
+- <span style="color:blue;">Autosomal</span> chromosomes
+- <span style="color:red;">X</span> chromosome, if present
+- <span style="color:green;">Y</span> chromosome, if present
+
+<div style="text-align: center">**We can study the chromosomal system without an assembly**</div>
+
+## Phylogeny
+
+:::: {.columns}
+::: {.column width="40%"}
+![](./img/phylogeny.png)
+:::
+
+::: {.column width="60%"}
+- *M. belari*
+- *M. monhystera*
+- *M. longespiculosa*
+- *M. spiculigera*
+:::
+::::
+
+## Kmer-diff
+
+:::: {.columns}
+::: {.column width="60%"}
+
+A nextflow pipeline to analyze the $k$-mer content of fastq files
+
+1. preprocess the fastq files
+2. count the $k$-mers of each file
+3. merge the counts to get a table of male and female $k$-mers counts
+3. test the sexual models
+4. identify the <span style="color:blue;">A</span>,<span style="color:red;">X</span> and <span style="color:green;">Y</span> $k$-mers
+
+:::
+
+::: {.column width="40%"}
+
+![](./img/flowchart.png){width=80%}
+
+:::
+::::
+
+## Kmer-diff
+
+:::: {.columns}
+::: {.column width="60%"}
+
+A nextflow pipeline to analyze the $k$-mer content of fastq files
+
+1. **preprocess the fastq files**
+2. count the $k$-mers of each file
+3. merge the counts to get a table of male and female $k$-mers counts
+3. test the sexual models
+4. identify the <span style="color:blue;">A</span>,<span style="color:red;">X</span> and <span style="color:green;">Y</span> $k$-mers
+
+:::
+
+::: {.column width="40%"}
+
+![](./img/flowchart.png){width=80%}
+
+:::
+::::
+
+## Count problem
+
+:::: {.columns}
+::: {.column width="75%"}
+![](./img/count_problem.png)
+
+:::
+::: {.column width="25%"}
+
+- The coverage is not the same between the male and the female
+
+:::
+::::
+
+## Preprocessing
+
+**preprocess the fastq files**
+
+Important for the clustering analysis:
+
+- **subsample each file to have the same number of reads between male and female**
+
+Important for the $k$-mer counting
+
+- **split the fastq files into manageable size files ($10ˆ6$ reads per files)**
+
+
+## Kmer-diff
+
+:::: {.columns}
+::: {.column width="60%"}
+
+A nextflow pipeline to analyze the $k$-mer content of fastq files
+
+1. preprocess the fastq files
+2. **count the $k$-mers of each file**
+3. merge the counts to get a table of male and female $k$-mers counts
+3. test the sexual models
+4. identify the <span style="color:blue;">A</span>,<span style="color:red;">X</span> and <span style="color:green;">Y</span> $k$-mers
+
+:::
+
+::: {.column width="40%"}
+
+![](./img/flowchart.png){width=80%}
+
+:::
+::::
+
+## $k$-mer count
+
+[fastqkmer](https://github.com/angelovangel/fastkmers)
+
+```
+fastkmers -k 12 file.fastq > file.csv
+```
+
+Run a **sliding window** of size $12$ by step of $1$ along the reads counting all the occurrences of each $k$-mers
+
+We have the letters: $A,C,T,G$ and $N$
+
+<div style="text-align: center">$244,140,625$ possible $k$-mers</div>
+
+We split the fastq files into $\sim$ $1400$ subfiles of $10^6$ reads.
+
+## Merging the $k$-mers
+
+$\sim$ $1400$ splits of $10^6$ reads $\rightarrow$ $1400$ csv files
+
+:::: {.columns}
+::: {.column width="50%"}
+
+- large number of $k$-mers
+- Unordered
+- not the same $k$-mers are present in every files
+
+build a **suffix tree** of the $k$-mer to store them
+:::
+
+::: {.column width="50%"}
+
+![](./img/sufixtree_a.png)
+
+:::
+::::
+
+## Merging the $k$-mers
+
+[mergekmer](https://gitbio.ens-lyon.fr/LBMC/Delattre/mergekmer) a small rust program that build a **suffix tree** of the $k$-mer
+
+:::: {.columns}
+::: {.column width="50%"}
+
+```
+merge fastkmers output
+
+Usage: mergekmer [OPTIONS] --output <OUTPUT>
+
+Options:
+  -c, --csv <CSV>...     list of csv files
+  -o, --output <OUTPUT>  merged csv file
+  -c, --collate          collate csv file
+  -h, --help             Print help
+  -V, --version          Print version
+```
+
+Each leaf of the tree contains the number of $k$-mers
+
+The tree traversal is easy to compute with a recursive function
+
+**We can merge all the counts file of a given sex and specie**
+
+:::
+::: {.column width="50%"}
+![](./img/sufixtree_b.png)
+:::
+::::
+
+## Merging the $k$-mers
+
+:::: {.columns}
+::: {.column width="50%"}
+
+```
+merge fastkmers output
+
+Usage: mergekmer [OPTIONS] --output <OUTPUT>
+
+Options:
+  -c, --csv <CSV>...     list of csv files
+  -o, --output <OUTPUT>  merged csv file
+  -c, --collate          collate csv file
+  -h, --help             Print help
+  -V, --version          Print version
+```
+
+In the `--collate` version each leave contains a list of the count of the $k$-mers in the female or the male of a specie.
+
+**We can fuse the counts of the male and female for each specie**
+
+
+:::
+::: {.column width="50%"}
+![](./img/sufixtree_c.png)
+:::
+::::
+
+## Test the sexual model
+
+**We have the following possible models**
+
+
+:::: {.columns}
+::: {.column width="33%"}
+
+- <span style="color:red;">X</span><span style="color:green;">Y</span> system
+
+![](./img/model_XY.png)
+    
+:::
+::: {.column width="33%"}
+
+- <span style="color:red;">X</span>O system
+
+![](./img/model_XO.png)
+:::
+::: {.column width="33%"}
+
+- OO system
+
+![](./img/model_OO.png)
+
+:::
+::::
+
+## Test the sexual model
+
+:::: {.columns}
+::: {.column width="33%"}
+
+- <span style="color:red;">X</span><span style="color:green;">Y</span> system
+
+![](./img/clustering_XY_naive.png)
+
+:::
+::: {.column width="33%"}
+
+- <span style="color:red;">X</span>O system
+
+![](./img/clustering_XO_naive.png)
+:::
+::: {.column width="33%"}
+
+- OO system
+
+![](./img/clustering_OO_naive.png)
+
+:::
+::::
+
+- A cluster with mean male $=$ mean female
+- A cluster above the diagonal
+- A cluster below the diagonal
+
+
+## Test the sexual model
+
+:::: {.columns}
+::: {.column width="25%"}
+
+**data**
+
+![](./img/model_XY.png)
+
+:::
+::: {.column width="25%"}
+
+**XY model**
+
+![](./img/clustering_XY_naive_XY.png)
+
+:::
+::: {.column width="25%"}
+
+**XO model**
+
+![](./img/clustering_XY_naive_XO.png)
+:::
+::: {.column width="25%"}
+
+**OO model**
+
+![](./img/clustering_XY_naive_OO.png)
+
+:::
+::::
+:::: {.columns}
+::: {.column width="50%"}
+
+
+Bayesian information criterion (BIC)
+
+![](./img/clustering_XY_naive_BIC.png)
+
+:::
+::: {.column width="50%"}
+
+Loglikelihood
+
+![](./img/clustering_XY_naive_loglik.png)
+
+:::
+::::
+
+
+## identify the <span style="color:blue;">A</span>,<span style="color:red;">X</span> and <span style="color:green;">Y</span> $k$-mers: Simple Model
+
+:::: {.columns}
+::: {.column width="50%"}
+
+![](./img/clustering_XY_naive_XY.png)
+
+:::
+::: {.column width="50%"}
+
+- Can be used to compare model
+- Not sensitive enough
+- Cannot be used to classify individual $k$-mers
+
+:::
+::::
+
+## identify the <span style="color:blue;">A</span>,<span style="color:red;">X</span> and <span style="color:green;">Y</span> $k$-mers: Bayesian version
+
+
+:::: {.columns}
+::: {.column width="75%"}
+![](./img/clustering_XY_bayse_XY.png)
+
+:::
+::: {.column width="25%"}
+
+- Can choose the prior for each cluster (mean and shape)
+- Can choose the prior for the proportion between <span style="color:blue;">A</span>,<span style="color:red;">X</span> and <span style="color:green;">Y</span>
+- Can choose the weight of each prior compared to the data
+
+:::
+::::
+
+## identify the <span style="color:blue;">A</span>,<span style="color:red;">X</span> and <span style="color:green;">Y</span> $k$-mers: Bayesian version
+
+
+:::: {.columns}
+::: {.column width="75%"}
+![](./img/clustering_XY_bayse_XO.png)
+
+:::
+::: {.column width="25%"}
+
+- Can choose the prior for each cluster (mean and shape)
+- Can choose the prior for the proportion between <span style="color:blue;">A</span>,<span style="color:red;">X</span> and <span style="color:green;">Y</span>
+- Can choose the weight of each prior compared to the data
+
+:::
+::::
+
+## identify the <span style="color:blue;">A</span>,<span style="color:red;">X</span> and <span style="color:green;">Y</span> $k$-mers: Bayesian version
+
+:::: {.columns}
+::: {.column width="75%"}
+![](./img/clustering_XO_bayse_XO.png)
+
+:::
+::: {.column width="25%"}
+
+- Can choose the prior for each cluster (mean and shape)
+- Can choose the prior for the proportion between <span style="color:blue;">A</span>,<span style="color:red;">X</span> and <span style="color:green;">Y</span>
+- Can choose the weight of each prior compared to the data
+
+:::
+::::
+
+
+## Count model
+
+We don't have nice Gaussian ellipses but 2D count data $(X_m, X_f)$
+
+Bi-Poisson distribution
+
+$$(X_m, X_f) \sim \mathcal{P}(\lambda_1,\lambda_2,\lambda_3)$$
+
+:::: {.columns}
+::: {.column width="40%"}
+
+- $X_m = X_1 + Z$
+- $X_f = X_2 + Z$
+
+:::
+::: {.column width="20%"}
+
+with
+:::
+::: {.column width="40%"}
+
+- $X_1 \sim \mathcal{P}(\lambda_1)$
+- $X_2 \sim \mathcal{P}(\lambda_2)$
+- $Z \sim \mathcal{P}(\lambda_3)$
+
+:::
+::::
+
+## Count model
+With the EM algorithm we can estimate $Z$ the hidden variable of the model
+
+**E step:**
+
+$$z_i = E(Z_i | X_m, X_f, \lambda_1^{(k)}, \lambda_2^{(k)}, \lambda_3^{(k)})$$
+$$z_i = \lambda_3^{(k)} \frac{\mathcal{P}\left(x_{m,i-1}, x_{f, i-1} | \lambda_1^{(k)}, \lambda_2^{(k)}, \lambda_3^{(k)}\right)}{\mathcal{P}\left(x_{m,i}, x_{f, i} | \lambda_1^{(k)}, \lambda_2^{(k)}, \lambda_3^{(k)}\right)}$$
+
+**M step:**
+
+- $\lambda_1 = \frac{\sum_i (x_{m,i} - z_i)}{n}$
+- $\lambda_2 = \frac{\sum_i (x_{f,i} - z_i)}{n}$
+- $\lambda_3 = \frac{\sum_i z_i}{n}$
+
+## Count model
+
+We can write the <span style="color:red;">X</span><span style="color:green;">Y</span> model as:
+
+$$P(h = i | (x_m, x_f), \theta)=\frac{\alpha_i f\left((x_m, x_f), \theta_i\right)}{\sum_{\ell \in \{A,X,Y\}} \alpha_{\ell} f\left((x_m, x_f), \theta_{\ell}\right)}$$
+
+With the EM algorithm we can estimate $H$ and $Z$ the two hidden variables of the model
+
+We can also add a bayesian prior on the $\alpha$'s with a Dirichlet distribution
+
+## Count data
+
+:::: {.columns}
+::: {.column width="50%"}
+
+![](./img/poisson_clustering_XY.png)
+
+:::
+::: {.column width="50%"}
+
+Works well for Poisson data:
+
+- Mean = Variance of the count
+- no overdispersion
+
+:::
+::::
+
+## Count data
+
+:::: {.columns}
+::: {.column width="50%"}
+
+![](./img/mbelari.png){width=65%}
+
+
+![](./img/mlongespiculosa.png)
+
+:::
+::: {.column width="50%"}
+
+![](./img/mspiculigera.png)
+
+![](./img/mmonhystera.png)
+
+:::
+::::
+
+## Count data
+
+:::: {.columns}
+::: {.column width="50%"}
+
+![](./img/mbelari_log1p.png){width=65%}
+
+
+![](./img/mlongespiculosa_log1p.png)
+
+:::
+::: {.column width="50%"}
+
+![](./img/mspiculigera_log1p.png)
+
+![](./img/mmonhystera_log1p.png){width=65%}
+
+:::
+::::
+
+## Remove the weird $k$-mers
+
+### mBelari
+
+:::: {.columns}
+::: {.column width="40%"}
+
+![](./img/mbelari_G_clust_m.png)
+
+![](./img/mbelari_G_clust_f.png)
+
+:::
+::: {.column width="60%"}
+
+![](./img/mbelari_G_clust_fm.png)
+
+:::
+::::
+
+## Remove the weird $k$-mers
+
+### mLongespiculosa
+
+:::: {.columns}
+::: {.column width="40%"}
+
+![](./img/mlongespiculosa_G_clust_m.png)
+![](./img/mlongespiculosa_G_clust_f.png)
+
+:::
+::: {.column width="60%"}
+
+![](./img/mlongespiculosa_G_clust_fm.png)
+
+:::
+::::
+
+## Remove the weird $k$-mers
+
+### mSpiculigera
+
+:::: {.columns}
+::: {.column width="40%"}
+
+![](./img/mspiculigera_G_clust_m.png)
+![](./img/mspiculigera_G_clust_f.png)
+
+:::
+::: {.column width="60%"}
+
+![](./img/mspiculigera_G_clust_fm.png)
+
+:::
+::::
+
+## Remove the weird $k$-mers
+
+### mMonhystera
+
+:::: {.columns}
+::: {.column width="40%"}
+![](./img/mmonhystera_G_clust_m.png)
+![](./img/mmonhystera_G_clust_f.png)
+:::
+::: {.column width="60%"}
+![](./img/mmonhystera_G_clust_fm.png)
+:::
+::::
+
+## Phylogeny
+
+:::: {.columns}
+::: {.column width="40%"}
+![](./img/phylogeny.png)
+:::
+
+::: {.column width="60%"}
+- *M. belari* <span style="color:blue;">A</span>,<span style="color:red;">X</span> and <span style="color:green;">Y</span>
+- *M. monhystera* <span style="color:blue;">A</span> and <span style="color:green;">Y</span>
+- *M. longespiculosa* <span style="color:blue;">A</span>
+- *M. spiculigera* <span style="color:blue;">A</span>,<span style="color:red;">X</span> and <span style="color:green;">Y</span>
+:::
+::::
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