  We investigate the behavior of minimizers of perturbed Dirichlet energies
supported on a wire generated by a regular simple curve $\gamma$ and defined in
the space of $\mathbb{S}^2$-valued functions. The perturbation $K$ is
represented by a matrix-valued function defined on $\mathbb{S}^2$ with values
in $\mathbb{R}^{3 \times 3}$. Under natural regularity conditions on $K$, we
show that the family of perturbed Dirichlet energies converges, in the sense of
$\Gamma$-convergence, to a simplified energy functional on $\gamma$. The
reduced energy unveils how part of the antisymmetric exchange interactions
contribute to an anisotropic term whose specific shape depends on the curvature
of $\gamma$.
  We also discuss the significant implications of our results for studies of
ferromagnetic nanowires when Dzyaloshinskii-Moriya interaction (DMI) is
present.
