Inferring the MYC GRN
Andreas Tjärnberg
andreas.tjarnberg@scilifelab.se
Sonnhammer group meeting; 2015-03-26
Why?
Specific
- Myc is a very important cancer-related transcription factor.
- Thousands of genes have been shown to be targets of Myc
General
- Gene Regulatory networks gives insight in to information flows in living cells
- Complex regulatory systems responses could be predicted, controlled and altered
What?
The Data Properties
- siRNA knockdown data
- 45 genes (40)
- 3\(^*\) replicates (\(\cdot\) 2 technical) single gene knockdowns
- single set of experiments with double knockdowns
- SNR\(_v= 0.56\)
- kondition number, \(\kappa \approx 70\)
The Data Properties
| \[ \begin{array}{r c l} \text{SNR}_v &\equiv& \arg\min_i \frac{||\boldsymbol{y}_i||}{\sqrt{\chi^{-2}(\alpha,M)\lambda}}\\ \\ \\ \\ \\ \end{array} \] |
The Data Properties
| \[ \begin{array}{r c l} \text{SNR}_v &\equiv& \arg\min_i \frac{||\boldsymbol{y}_i||}{\sqrt{\chi^{-2}(\alpha,M)\lambda}}\\ \\ \\ \\ \kappa &\equiv& \frac{\sigma_1}{\sigma_N}\\ \end{array} \] |
The Data Properties
| \[ \begin{array}{r c l} \text{SNR}_v &\equiv& \arg\min_i \frac{||\boldsymbol{y}_i||}{\sqrt{\chi^{-2}(\alpha,M)\lambda}}\\ \\ \\ \\ \kappa &\equiv& \frac{\sigma_1}{\sigma_N}\\ \end{array} \] |
The Model
\[ \begin{array}{r c l} \dot{x}_i(t) &=& \sum_{j=1}^N a_{ij}x_j(t) + p_i(t) - f_i(t)\\ \\ \\ \\ \end{array} \]
The Model
\[ \begin{array}{r c l} \dot{x}_i(t) &=& \sum_{j=1}^N a_{ij}x_j(t) + p_i(t) - f_i(t)\\ \\ \\ y_i(t) &=& x_i(t) + e_i(t)\\ \end{array} \]
How?
Model fitting
\[ \boldsymbol{Y} - \boldsymbol{E} = -\boldsymbol{A}^{-1}(\boldsymbol{P} - \boldsymbol{F}) \]
Model fitting
\[ \boldsymbol{Y} - \boldsymbol{E} = -\boldsymbol{A}^{-1}(\boldsymbol{P} - \boldsymbol{F}) \]
Model fitting
\[ \begin{array}{r c l} \boldsymbol{Y} - \boldsymbol{E} &=& -\boldsymbol{A}^{-1}\boldsymbol{P}\\ &\Downarrow&\\ \boldsymbol{\hat{Y}} &=& -\boldsymbol{A}^{-1}\boldsymbol{P}\\ &\Downarrow&\\ \boldsymbol{A}\boldsymbol{\hat{Y}} &=& -\boldsymbol{P}\\ &\Downarrow&\\ \boldsymbol{A}\boldsymbol{\hat{Y}} + \boldsymbol{P} &\approx& \boldsymbol{0}\\ \end{array} \]
Model fitting
Least squares
\[ \begin{array} \boldsymbol{A_{ls}} = \arg\min_{\boldsymbol{A}} || \boldsymbol{A}\boldsymbol{\hat{Y}} + \boldsymbol{P}||_{\ell_2}\\ \text{subject to:}~~ \boldsymbol{A}\boldsymbol{\hat{Y}} = -\boldsymbol{P}\\ \end{array} \]LASSO
\[ \boldsymbol{A_{\zeta}} = \arg\min_{\boldsymbol{A}} || \boldsymbol{A}\boldsymbol{\hat{Y}} + \boldsymbol{P}||_{\ell_2} + \zeta||\boldsymbol{A}||_{\ell_1} \]Bootstrap Regularization
- Statistical method for estimating statistics
- Applied to the LASSO formulation by Bach 2008
Bootstrap Regularization
Procedure
- Sample uniformly from data \([\boldsymbol{\hat{Y}}, \boldsymbol{P}]\) a new data set \([\boldsymbol{\hat{Y}_{bs}}, \boldsymbol{P_{bs}}]\)
- Apply LASSO on the new data set so that \[ \boldsymbol{A_{reg}}(\zeta) = \arg\min_{\boldsymbol{A}} || \boldsymbol{A}\boldsymbol{\hat{Y}_{bs}} + \boldsymbol{P_{bs}}||_{\ell_2} + \zeta||\boldsymbol{A}||_{\ell_1} \]
- Repeat for \(n\) number of bootstraps.
- A link \(s_{ij} \in \boldsymbol{A_{bs}}\) has bootstrap support \(s_{ij} = \frac{e_{ij}}{n}\cdot 100\%\) where \(e_{ij}\) is the times we have seen \(a_{ij}\) in \(\boldsymbol{A_{reg}}(\zeta)\)
Bootstrap Regularization
Prediction
\[
\begin{array}{c c c c}
% & {\Large{\boldsymbol{\hat{Y}}}} & & {\Large{\boldsymbol{Y}}}\\
\text{Predicted} & & \text{Measured} & \\
{\Large{\text{RSS}}} = &
\begin{vmatrix} \begin{pmatrix} \begin{bmatrix}
\hat{y}_{1,1} & \hat{y}_{1,2} & \cdots & \hat{y}_{1,n} \\
\hat{y}_{2,1} & \hat{y}_{2,2} & \cdots & \hat{y}_{2,n} \\
\vdots & \vdots & \ddots & \vdots \\
\hat{y}_{m,1} & \hat{y}_{m,2} & \cdots & \hat{y}_{m,n}
\end{bmatrix} & - &
\begin{bmatrix}
y_{1,1} & y_{1,2} & \cdots & y_{1,n} \\
y_{2,1} & y_{2,2} & \cdots & y_{2,n} \\
\vdots & \vdots & \ddots & \vdots \\
y_{m,1} & y_{m,2} & \cdots & y_{m,n}
\end{bmatrix} \end{pmatrix}^{.2} \end{vmatrix} \\
\end{array}
\]
Prediction
\[
\begin{array}{c c c c}
% & {\Large{\boldsymbol{\hat{Y}}}} & & {\Large{\boldsymbol{Y}}}\\
\text{Predicted} & & \text{Measured} & \\
{\Large{\text{RSS}}} = &
\begin{vmatrix} \begin{pmatrix} \begin{bmatrix}
\hat{y}_{1,1} & \hat{y}_{1,2} & \cdots & \hat{y}_{1,n} \\
\hat{y}_{2,1} & \hat{y}_{2,2} & \cdots & \hat{y}_{2,n} \\
\vdots & \vdots & \ddots & \vdots \\
\hat{y}_{m,1} & \hat{y}_{m,2} & \cdots & \hat{y}_{m,n}
\end{bmatrix} & - &
\begin{bmatrix}
y_{1,1} & y_{1,2} & \cdots & y_{1,n} \\
y_{2,1} & y_{2,2} & \cdots & y_{2,n} \\
\vdots & \vdots & \ddots & \vdots \\
y_{m,1} & y_{m,2} & \cdots & y_{m,n}
\end{bmatrix} \end{pmatrix}^{.2} \end{vmatrix} \\
\end{array}
\]
\[ {\large{\text{RSS} \sim \chi^2(mn)}} \]
Prediction
\[
\begin{array}{c c c c}
% & {\Large{\boldsymbol{\hat{Y}}}} & & {\Large{\boldsymbol{Y}}}\\
\text{Predicted} & & \text{Measured} & \\
{\Large{\text{RSS}}} = &
\begin{vmatrix} \begin{pmatrix}
\begin{bmatrix}
\hat{y}_{1,1} & \hat{y}_{1,2} & \cdots & \hat{y}_{1,n} \\
\hat{y}_{2,1} & \hat{y}_{2,2} & \cdots & \hat{y}_{2,n} \\
\vdots & \vdots & \ddots & \vdots \\
\hat{y}_{m,1} & \hat{y}_{m,2} & \cdots & \hat{y}_{m,n}
\end{bmatrix} & - &
\begin{bmatrix}
y_{1,1} & y_{1,2} & \cdots & y_{1,n} \\
y_{2,1} & y_{2,2} & \cdots & y_{2,n} \\
\vdots & \vdots & \ddots & \vdots \\
y_{m,1} & y_{m,2} & \cdots & y_{m,n}
\end{bmatrix}
\end{pmatrix}^{.2} \end{vmatrix} \\
\end{array}
\]
\[ {\large{\text{RSS} \sim \chi^2(mn)}} \]
Results
When?
Errors in variables
Total Least Squares (TLS)
Remember:
Errors in variables
Total Least Squares (TLS)
Remember:
Errors in variables
Total Least Squares (TLS)
\[ \boldsymbol{A}(\boldsymbol{Y} - \boldsymbol{E}) = -(\boldsymbol{P} - \boldsymbol{F}) \]
Errors in variables
Total Least Squares (TLS)
\[ \boldsymbol{A}(\boldsymbol{Y} - \boldsymbol{E}) = -(\boldsymbol{P} - \boldsymbol{F}) \]
Errors in variables
Total Least Squares (TLS)
\[ \begin{array} \boldsymbol{A_{tls}} = \arg\min_{\boldsymbol{A}} || [(\boldsymbol{A}\boldsymbol{\hat{Y}} + \boldsymbol{\hat{P}})~ (\boldsymbol{\hat{Y}} + \boldsymbol{A}^{-1}\boldsymbol{\hat{P}})]||_{\ell_2}\\ \text{subject to:}~~ \boldsymbol{A}\boldsymbol{\hat{Y}} = -\boldsymbol{\hat{P}}\\ \end{array} \]
Markovsky and Van Huffel 2007
Errors in variables
Total Least Squares (TLS)
\[ \begin{array} \boldsymbol{A_{tls}} = \arg\min_{\boldsymbol{A}} || [(\boldsymbol{A}\boldsymbol{\hat{Y}} + \boldsymbol{\hat{P}})~ (\boldsymbol{\hat{Y}} + \boldsymbol{A}^{-1}\boldsymbol{\hat{P}})]||_{\ell_2}\\ \text{subject to:}~~ \boldsymbol{A}\boldsymbol{\hat{Y}} = -\boldsymbol{\hat{P}}\\ \end{array} \]
Markovsky and Van Huffel 2007
- To incorporate in pipeline, we need a structure constraint TLS algorithm (fast)
- A LASSO with TLS would also be nice.
Not implemented yet!
Thanks