Tissue-specificity metrics

tspex provides twelve distinct tissue-specificity metrics, which differ in their assumptions, scale and properties. Broadly, these metrics can be divided into two groups1:

  • General scoring metrics: Summarize in a single value how tissue-specific or ubiquitous is a gene across all tissues.
  • Individualized scoring metrics: Quantify how specific is the expression of each gene to each tissue.

For the following equations x_i represents the gene expression in tissue i and n is the number of tissues. Some metrics give results that are not in the [0,1] interval, such as the Simpson index, that lies in the [\frac{1}{n},1] range. For these metrics, we provide transformed versions, denoted by the ' symbol, that range from 0 (perfectly ubiquitous) to 1 (perfectly tissue-specific).

General scoring metrics

The general scoring metrics included in tspex are: Counts2, Tau3, Gini coefficient4, Simpson index5, Shannon entropy specificity6, ROKU specificity7, Specificity measure dispersion (SPM DPM)8, and Jensen-Shannon specificity dispersion (JSS DPM).

Counts

\begin{align} & t(x_i) = \begin{cases} 1, & \text{if } x_i > threshold \\ 0, & \text{if } x_i \le threshold \end{cases} \\[9pt] & \operatorname{Counts} = \frac{n-\sum_{i=1}^n(t(x_i))}{n-1} \end{align}

Tau

\begin{align} & \widehat{x_i} = \frac{x_i}{\max\limits_{0\le i \le n}(x_i)} \\[9pt] & \operatorname{Tau} = \frac{\sum_{i=1}^n(1-\widehat{x_i})}{(n-1)} \end{align}

Gini coefficient

\begin{align} & X = (x_1,x_2,\ldots,x_i,\ldots,x_{n-1},x_{n}) \\[9pt] & \qquad \;\; x_1 \le x_2 \le \ldots x_i \le \ldots \le x_{n-1} \le x_{n} \notag\\[9pt] & \operatorname{Gini} = \frac{\sum_{i=1}^n(2i-n-1)x_i}{n\sum_{i=1}^n(x_i)} \\[9pt] & \operatorname{Gini'} = \operatorname{Gini} \frac{n}{n-1} \end{align}

Simpson index

\begin{align} & p_i = \frac{x_i}{\sum_{i=1}^n(x_i)} \\[9pt] & \operatorname{Simpson} = \sum_{i=1}^n(p_i^2) \\[9pt] & \operatorname{Simpson'} = \frac{\operatorname{Simpson} - \frac{1}{n}}{1 - \frac{1}{n}} \end{align}

Shannon entropy specificity (HS)

\begin{align} & p_i = \frac{x_i}{\sum_{i=1}^n(x_i)} \\[9pt] & H = -\sum_{i=1}^n(p_i \log_2(p_i)) \\[9pt] & \operatorname{HS} = log_2(n) - H \\[9pt] & \operatorname{HS'} = \frac{\operatorname{HS}}{log_2(n)} \end{align}

ROKU specificity

\begin{align} & M = \operatorname{median}_{0\le i \le n}(x_i) \\[9pt] & S = \operatorname{median}_{0\le i \le n}(|x_i - M|) \\[9pt] & u_i = \frac{x_i - M}{5S + 10^{-4}} \\[9pt] & w(u_i) = \begin{cases} (1-u^2)^2, & \text{if } |u_i| \le 1 \\ 0, & \text{if } |u_i| > 1 \end{cases} \\[9pt] & t = \frac{\sum_{i=1}^n(x_i w(u_i))}{\sum_{i=1}^n(w(u_i))} \\[9pt] & X' = (|x_1-t|,|x_2-t|,\ldots,|x_i-t|,\ldots,|x_{n-1}-t|,|x_{n}-t|) \\[9pt] & p_i = \frac{x_i'}{\sum_{i=1}^n(x_i')} \\[9pt] & H = -\sum_{i=1}^n(p_i \log_2(p_i)) \\[9pt] & \operatorname{ROKU} = log_2(n) - H \\[9pt] & \operatorname{ROKU'} = \frac{\operatorname{ROKU}}{log_2(n)} \end{align}

Specificity measure dispersion (SPM DPM)

\begin{align} & \operatorname{\overline{SPM}} = \frac{\sum_{i=1}^n(\operatorname{SPM_i})}{n} \\[9pt] & \sigma = \sqrt{\frac{\sum_{i=1}^n(\operatorname{SPM_i} - \operatorname{\overline{SPM}})^2}{n-1}} \\[9pt] & \operatorname{SPM\ DPM} = \sigma \sqrt{n} \end{align}

Jensen-Shannon specificity dispersion (JSS DPM)

\begin{align} & \operatorname{\overline{JSS}} = \frac{\sum_{i=1}^n(\operatorname{JSS_i})}{n} \\[9pt] & \sigma = \sqrt{\frac{\sum_{i=1}^n(\operatorname{JSS_i} - \operatorname{\overline{JSS}})^2}{n-1}} \\[9pt] & \operatorname{JSS\ DPM} = \sigma \sqrt{n} \end{align}

Individualized scoring metrics

The general scoring metrics included in tspex are: Tissue-specificity index (TSI)9, Z-score10, Specificity measure (SPM)11, and Jensen-Shannon specificity (JSS)12.

Tissue-specificity index (TSI)

\begin{align} & \operatorname{TSI_i} = \frac{x_i}{\sum_{i=1}^n(x_i)} \end{align}

Z-score

\begin{align} & \overline{x} = \frac{\sum_{i=1}^n(x_i)}{n} \\[9pt] & \sigma = \sqrt{\frac{\sum_{i=1}^n(x_i - \overline{x})^2}{n-1}} \\[9pt] & \operatorname{Z-Score_i} = \frac{x_i - \overline{x}}{\sigma} \\[9pt] & \operatorname{Z-Score_i'} = \frac{\operatorname{Z-Score_i} + \frac{(n-1)}{\sqrt{n}}}{2 \frac{(n-1)}{\sqrt{n}}} \end{align}

Specificity measure (SPM)

\begin{align} & X = (x_1,x_2,\ldots,x_i,\ldots,x_{n-1},x_{n}) \\[9pt] & \operatorname{SPM_i} = \frac{x_i^2}{x_i*||X||_2} \end{align}

Jensen-Shannon specificity (JSS)

\begin{align} & X = (x_1,x_2,\ldots,x_i,\ldots,x_{n-1},x_{n}) \\[9pt] & X' = (0,0,\ldots,x_i,\ldots,0,0) \\[9pt] & p_i = \frac{x_i}{\sum_{i=1}^n(x_i)} \\[9pt] & q_i = \frac{x'_i}{\sum_{i=1}^n(x'_i)} \\[9pt] & \operatorname{JSS_i} = 1 - \sqrt{\frac{\sum_{i=1}^n(p_i \log_2(p_i))}{2} - \sum_{i=1}^n\left(\frac{p_i+q_i}{2} \log_2\left(\frac{p_i+q_i}{2}\right)\right)} \end{align}

References

  1. Kryuchkova-Mostacci, Nadezda, and Marc Robinson-Rechavi. "A benchmark of gene expression tissue-specificity metrics." Briefings in bioinformatics 18.2 (2017): 205-214. ↩

  2. Duret, Laurent, and Dominique Mouchiroud. "Determinants of substitution rates in mammalian genes: expression pattern affects selection intensity but not mutation rate." Molecular biology and evolution 17.1 (2000): 68-070. ↩

  3. Yanai, Itai, et al. "Genome-wide midrange transcription profiles reveal expression level relationships in human tissue specification." Bioinformatics 21.5 (2004): 650-659. ↩

  4. Ceriani, Lidia, and Paolo Verme. "The origins of the Gini index: extracts from Variabilità e Mutabilità (1912) by Corrado Gini." The Journal of Economic Inequality 10.3 (2012): 421-443. ↩

  5. Simpson, Edward H. "Measurement of diversity." Nature 163.4148 (1949): 688. ↩

  6. Schug, Jonathan, et al. "Promoter features related to tissue specificity as measured by Shannon entropy." Genome biology 6.4 (2005): R33. ↩

  7. Kadota, Koji, et al. "ROKU: a novel method for identification of tissue-specific genes." BMC bioinformatics 7.1 (2006): 294. ↩

  8. Pan, Jian-Bo, et al. "PaGeFinder: quantitative identification of spatiotemporal pattern genes." Bioinformatics 28.11 (2012): 1544-1545. ↩

  9. Julien, Philippe, et al. "Mechanisms and evolutionary patterns of mammalian and avian dosage compensation." PLoS biology 10.5 (2012): e1001328. ↩

  10. Vandenbon, Alexis, and Kenta Nakai. "Modeling tissue-specific structural patterns in human and mouse promoters." Nucleic acids research 38.1 (2009): 17-25. ↩

  11. Xiao, Sheng-Jian, et al. "TiSGeD: a database for tissue-specific genes." Bioinformatics 26.9 (2010): 1273-1275. ↩

  12. Cabili, Moran N., et al. "Integrative annotation of human large intergenic noncoding RNAs reveals global properties and specific subclasses." Genes & development 25.18 (2011): 1915-1927. ↩