Does this equation have a typo? (Systems Biology, singular value decomposition) [closed]

Question

I'm reading a systems biology paper, and I'm suspecting there's a typo in an equation, but I want to make sure.

The article is "Systemic metabolic reactions are obtained by singular value decomposition of genome-scale stoichiometric matrices." (Famili & Palsson 2003, http://www.ncbi.nlm.nih.gov/pubmed/12900206 / http://www.sciencedirect.com/science/article/pii/S0022519303001462). In equation 5, I wonder if the final subscript u_kr shouldn't be u_km, as the vector u_k^T is a column vector in the matrix U, which has dimensions m x m. Of course, some elements in u_k might be zero, in which case there would be less than m products in the sum, but I see no reason why there would be maximally r non-zero elements either (where r is the rank of the matrix S which has been decomposed). I worked out a simple example on paper where r < m and still got vectors with m non-zero elements. Can someone give me a second opinion on this?

Edit: Here are the main equations. A model of a metabolic reaction network is described by a stoichiometric matrix S of dimensions m x n and rank r. For the metabolites in the model, changes in metabolite concentrations is equal to S multiplied with the vector of reaction rates, v:

${\frac{{dx}}{{dt}} = Sv}$

The singular value decomposition of S is given by

$S = U\sum {V^T}$

Multiplying on the left side with $U^T$ (The transpose of U) we get:

${U^T}\frac{{dx}}{{dt}} = \sum {V^T}v$

After matrix multiplication, we have for each row in $U^T$ and $V^T$, where $\sigma_k$ is the k'th singular value

$\frac{{d(u_k^Tx)}}{{dt}} = {\sigma _k}(v_k^Tv){\text{ k = 1}},...,{\text{r}}$

And finally, equation 5:

${\text{u}}_k^T \cdot x{\text{ = }}{{\text{u}}_{k1}}{x_1} + {u_{k2}}{x_2} + ... + {u_{kr}}{x_m}{\text{ (5)}}$

Answer 1

You are right this is a typo. Equation 5 is just a restatement of the definition of the dot product of two vectors: $$\mathbf{u}_k^T\cdot \mathbf{x} = \sum_{i=1}^m u_{ki}^Tx_i$$ or written in another way $$\mathbf{u}_k^T\cdot \mathbf{x} = u_{k1}x_1+u_{k2}x_2+\cdots+u_{km}x_m.$$

In the text Equation 5 is stating that one can find a linear combination of the metabolites described in $\mathbf{x}$ whose time evolution is described simply by a linear combination of the flux rates described in $\mathbf{v}$. The coefficients of the first linear combination are given by a row vector $\mathbf{u}_k^T$ in $U^T$. From this perspective, since there are $m$ possible metabolites ($\mathbf{x}$ is a $m\times 1$ vector), there have to be $m$ coefficients, so that the indices must range from 1 to $m$ in Equation 5. You are correct that $u_{ki}^T$ might be 0 for some $i$. But that doesn't change the fact that there should be $m$ (possibly vanishing) coefficients.