Motivated by recent research on quantifying bacterial growth dynamics based on genome assemblies, we consider a permuted monotone matrix model $Y=ΘΠ+Z$, where the rows represent different samples, the columns represent contigs in genome assemblies and the elements represent log-read counts after preprocessing steps and Guanine-Cytosine (GC) adjustment. In this model, $Θ$ is an unknown mean matrix with monotone entries for each row, $Π$ is a permutation matrix that permutes the columns of $Θ$, and $Z$ is a noise matrix. This paper studies the problem of estimation/recovery of $Π$ given the observed noisy matrix $Y$. We propose an estimator based on the best linear projection, which is shown to be minimax rate-optimal for both exact recovery, as measured by the 0-1 loss, and partial recovery, as quantified by the normalized Kendall's tau distance. Simulation studies demonstrate the superior empirical performance of the proposed estimator over alternative methods. We demonstrate the methods using a synthetic metagenomics dataset of 45 closely related bacterial species and a real metagenomic dataset to compare the bacterial growth dynamics between the responders and the non-responders of the IBD patients after 8 weeks of treatment.