  Thorne-$\.Z$ytkow objects (T$\.Z$O) are potential end products of the merger
of a neutron star with a non-degenerate star. In this work, we have computed
the first grid of evolutionary models of T$\.Z$Os with the MESA stellar
evolution code. With these models, we predict several observational properties
of T$\.Z$Os, including their surface temperatures and luminosities, pulsation
periods, and nucleosynthetic products. We expand the range of possible T$\.Z$O
solutions to cover $3.45 \lesssim \log \left(T/K\right) \lesssim 3.65$ and
$4.85 \lesssim \log \left(L/L_{\odot}\right) \lesssim 5.5$. Due to the much
higher densities our T$\.Z$Os reach compared to previous models, if T$\.Z$Os
form we expect them to be stable over a larger mass range than previously
predicted, without exhibiting a gap in their mass distribution. Using the GYRE
stellar pulsation code we show that T$\.Z$Os should have fundamental pulsation
periods of 1000--2000 days, and period ratios of $\approx$0.2--0.3. Models
computed with a large 399 isotope fully-coupled nuclear network show a
nucleosynthetic signal that is different to previously predicted. We propose a
new nucleosynthetic signal to determine a star's status as a T$\.Z$O: the
isotopologues $^{44}\rm{Ti} \rm{O}_2$ and $^{44}\rm{Ti} \rm{O}$, which will
have a shift in their spectral features as compared to stable
titanium-containing molecules. We find that in the local Universe (~SMC
metallicities and above) T$\.Z$Os show little heavy metal enrichment,
potentially explaining the difficulty in finding T$\.Z$Os to-date.
