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• 5.1 - Linear Program

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Section

We interpret G with each edge

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

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Column

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

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(Equation 1)

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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

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body $\Delta_e$
in Eq.1 is to compensate for divergence from the expectation created by stochastical sampling intrinsic to RNA-Seq experiments.

Section

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

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body $e\in E$
. Subsequently, the linear equations spanned by a locus are resolved by the objective function

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body $min(\sum_e \Delta_e)$

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(Equation 2)

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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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