Mathematical problem. Suppose we have $2n$ indeterminates $x_1,\dots,x_n$ and $y_1,\dots,y_n$ (which are denoted by $q$ with indices and called abundances below) and $m$ subsets $P_1,\dots,P_m$ of $\lbrace 1,2,\dots,n\rbrace$. Given real $r_1,\dots,r_m$, how to determine the indeterminates if they satisfy the system of linear homogeneous equations $$ \sum_{p\in P_i}x_p-r_i\sum_{p\in P_i}y_p=0 \quad\text{for } i=1,\dots,m $$ and the additional linear constraints $$ 0\le x_p\le 1 \quad\text{and}\quad 0\le y_p\le 1 \quad\text{for } p=1,\dots,n \quad? $$
Background. Proteins are important molecules in living organisms. In molecular biology you often want to deduce the fold change of abundance of the proteins in one sample versus another.
Proteins are built up of a sequence of molecules (amino acids). Peptides are parts of proteins, $i.e.$ part of the sequence. Using current technology (Mass spectrometry), you can only detect the peptides and ratios of these peptides.
Motivation. Normally if you want to deduce the ratio of a protein in sample A versus B, you just take the geometric mean of the ratios of the peptides specific to that protein. . However, you will observe peptides which are shared by proteins. It is assumed that the ratio of a peptide which is shared by multiple proteins is the sum of the abundances of the proteins in sample A divided by the sum of the abundances of the proteins in sample B. My question targets how to still be able to use these observations, and deduce ratios for proteins having only shared proteins.
Given
- $ P = \lbrace p_1,\...,p_n \rbrace $ ... Proteins
- $ Q = \left( P_1, ..., P_m \right), P_i \subseteq P $ ...proteins which share peptide i
- $ R = \left( r_1,\...,r_m \right) $ ... Observed ratios for peptides
find $q_{p,a},q_{p,b} \forall p \in P$ such that $$ r_i=\frac{\sum_{p \in P_i}q_{p,a}}{\sum_{p \in P_i}q_{p,b}}$$
The real abundances $q_{p,a},q_{p,b}$ are not known and not of interest, just their relation to each other.
My approach
Solve a nonlinear system of equations minimizing $$\log(r_i)=\log\sum_{p \in P_i}q_{p,a} - \log\sum_{p \in P_i}q_{p,b}, \forall r_i \in R$$
Is there a way to approach this in a linear system? Of course I can try solve $\sum_{p \in P_i}q_{p,a} - r_i \times \sum_{p \in P_i}q_{p,b} = 0, \forall r_i \in R$, but the ratios are not dealt with correctly.
