今天我們繼續掃盲，學習一些基礎的知識和概念。
Gene enrichment and covariation analysis
其實我們在做TCR分析的時候，應該也是實驗組 + 對照組進行分析，其中做重要的就是我們要尋找實驗組在接受病原刺激后TCR重排選擇基因的偏好性。Gene usage preferences were quantified by calculating a normalized Jensen–Shannon divergence (JSD) between the observed gene segment frequencies for each repertoire and background gene frequencies calculated from large-scale repertoire profiling studies，這里其實就是相對于正常的樣本，疾病樣本在TCR重排基因選擇的偏好性，當然，這里用到的是JS散度，大家可以參考文章KL散度、JS散度、Wasserstein距離，JSD 是 Kullback-Leibler 散度的對稱版本，further normalize the JSD values by dividing them by the mean Shannon entropy（香農熵，又叫信息熵，大家參考我之前的文章10X單細胞（10X空間轉錄組）基礎算法之KL散度） of the two distributions being compared, which helps to correct for variation in total gene number across segments。To set lower significance thresholds for the JSD heat maps(that is, the values below which the mapped colour is a uniform dark blue)。
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we compared the 2–4 different background repertoire datasets(這里就設置成我們的對照樣本) for each chain/organism to one another and took the largest observed JSD value across all comparisons.
Covariation（協變，協方差） between gene usage in different segments was quantified using the adjusted mutual information,a variant of the mutual information metric that corrects for the numbers and frequencies of the observed genes (mutual information between pairs of distributions tends to increase with the number of observation classes)。當然，這個在單細胞數據中其實應該用到的不多。
CDR3 motif discovery.
used a simple, depth-first search procedure to identify over-represented sequence patterns in the CDR3 amino sequences of each repertoire.Motifs were represented as fixed-length patterns consisting of fully-specified amino acid positions, wild card positions, and amino acid group positions,The score of a motif was calculated using a chi-squared formalism:
where ‘observed’ represents the number of times the motif was observed in the repertoire sequences and ‘expected’ represents an estimate of the expected number of observations based on a background set of TCR sequences with V and J gene compositions that match the observed repertoire（這里的背景我們設置為單細胞的對照樣本）。（這一部分才是最為關鍵的地方）。
Starting with two-position motifs scoring above a seed threshold, each motif was iteratively extended by adding new specified positions (that is, replacing an internal wild card or lengthening the motif at either end) that increased the motif score.The set of identified motifs were sorted by motif score and filtered for redundancy。Finally, motifs scoring above a threshold were extended to include near-neighbour TCRs using a stringent distance threshold; this allowed us to capture additional pattern instances that were not captured by our limited set of amino acid groupings. The final set of motifs for each repertoire were visualized using the TCR logo representation。（看來這才是TCR分析正確的打開方式）。
TCRdiv 多樣性的衡量(也很重要)
為了衡量多樣性，generalizes Simpson’s diversity index by accounting for TCR similarity as well as exact identity（關于辛普森多樣性指數，大家可以百度百科一下）。辛普森多樣性可以被認為是衡量從混合總體中抽取兩個獨立樣本中相同物種或類別的項目的概率，或者換句話說，如果樣本是返回 1 的兩個抽取樣本的函數的期望值 相同，否則為 0 。We instead estimate the expected value of a Gaussian function（高斯函數，確實需要很多的數學知識） of the inter-sample distance that returns 1 if the two samples are identical and exp(? (TCRdist(a,b) / s.d.)2) otherwise, where the s.d. was taken to be 18.45 for single-chain distances and twice that for paired analyses based on empirical assessments of receptor distance distributions for multiple epitopes。Taking the inverse of this estimate gives a diversity measure (TCRdiv) that can be interpreted as an effective population size for similarity-weighted sharing.(這部分有點難以理解，大家需要多一些耐心和學習了).
這部分的代碼在tcr-dist，作者已經都封裝好了，我們用一下就可以，感興趣大家可以多多學習一下。
到目前為止，算是把基礎說完了，接下來的分析，就要更上一層樓了。
生活很好，有你更好
