The dynamical diquark model assumes that exotic hadrons can be formed from colored diquarks. This model asserts a multiquark exotic state composed of a compact diquark δ and antidiquark $\overline{\delta }$ for a tetraquark interacting through a gluonic field of finite extent. Using Born-Oppenheimer (BO) approximation and BO potential calculated numerically on the lattice, we study mass spectra of S-wave $cc\overline{b}\overline{b}$, $bc\overline{c}\overline{c}$, and $bc\overline{b}\overline{b}$ tetraquark systems in the basis of diquark spins. We assume that color-antitriple diquark and color-triplet antidiquark form tetraquark state. The predicted mass spectrum for ground state of $cc\overline{b}\overline{b}$ is found to be lower then their corresponding two-meson thresholds. This system may be a candidate for bound state. Masses of $bc\overline{c}\overline{c}$ and $bc\overline{b}\overline{b}$ tetraquark states are found to be above than the corresponding two-meson thresholds.