  This paper explores the concept of Watson-Crick conjugates, also known as
$\theta$-conjugates, of words and languages. This concept extends the classical
idea of conjugates by incorporating the Watson-Crick complementarity of DNA
sequences, from the perspective of DNA computing. Our investigation initially
focuses on the properties of $\theta$-conjugates of words. We then define
$\theta$-conjugates of a language and study closure properties of certain
families of languages under the $\theta$-conjugate operation. Furthermore, we
analyze the iterated $\theta$-conjugate of both words and languages. Finally,
we delve into the idea of $\theta$-conjugate-free languages and examine the
decidability problems surrounding $\theta$-conjugate-freeness for different
classes of languages
