# 5.1 - Linear Program

We interpret G with each edge $$e \in E$$ 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 $$e=(tail,head,exonic,T)$$ the contribution of the supporting transcripts $$\{t_1,\ldots,t_n\}\in T$$ to the flux $$X_e$$ observed along e can be described by a linear equation

$$\sum_{t_i\in T_e}(f_i t_i) \pm \Delta_e = X_e$$

(Equation 1)

where fi represents a factor that expresses the fraction of the respective transcript expression ti observed between taile and heade. In the trivial case, fi corresponds to the proportion of the interval [taile; heade] in comparison to the entire length of the processed transcript. The correction factor $$\Delta_e$$ 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 ti 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 $$e\in E$$. Subsequently, the linear equations spanned by a locus are resolved by the objective function

$$min(\sum_e \Delta_e)$$

(Equation 2)

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

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