5.1 - Linear Program

We interpret G with each edge $//<![CDATA[ \begin{array}{l}e \in E\end{array} //]]>$ colored by read mappings as a flow network, considering the read volume assigned to every (super-) edge as a flux created by the expression of the underlying supporting transcripts T.Consequently, given an edge $//<![CDATA[ \begin{array}{l}e=(tail,head,exonic,T)\end{array} //]]>$ the contribution of the supporting transcripts $//<![CDATA[ \begin{array}{l}\{t_1,\ldots,t_n\}\in T\end{array} //]]>$ to the flux $//<![CDATA[ \begin{array}{l}X_e\end{array} //]]>$ observed along e can be described by a linear equation

where f_i represents a factor that expresses the fraction of the respective transcript expression t_i observed between tail_e and head_e. In the trivial case, f_i corresponds to the proportion of the interval [tail_e; head_e] in comparison to the entire length of the processed transcript. The correction factor $//<![CDATA[ \begin{array}{l}\Delta_e\end{array} //]]>$ in Eq.1 is to compensate for divergence from the expectation created by stochastical sampling intrinsic to RNA-Seq experiments.

The crux of the flux is that an RNA-Seq experiment provides a series of observations on the underlying expression level t_i along the transcript body. Following tradition in transportation problems, we model all of these observations as a system of linear equations by inferring Equation 1 on all $//<![CDATA[ \begin{array}{l}e\in E\end{array} //]]>$. Subsequently, the linear equations spanned by a locus are resolved by the objective function

Solving the linear program (Eq.2) imposed by a locus intrinsically provides an estimate for the expression level t_i of all alternative transcripts that are annotated.