  QCD exhibits complex dynamics near S-wave two-body thresholds. For light
mesons, we see this in the failure of quark models to explain the $f_0(500)$
and $K_0^*(700)$ masses. For charmonium, an unexpected $X(3872)$ state appears
at the open charm threshold. In heavy-light systems, analogous threshold
effects appear for the lowest $J^P = 0^+$ and $1^+$ states in the $D_s$ and
$B_s$ systems. Here we describe how lattice QCD can be used to understand these
threshold dynamics by smoothly varying the strange-quark mass when studying the
heavy-light systems. Small perturbations around the physical strange quark mass
are used so to always remain near the physical QCD dynamics. This calculation
is a straightforward extension of those already in the literature and can be
undertaken by multiple lattice QCD collaborations with minimal computational
cost.
