In the present paper we have developed a concept of parallel ordinary (O) and mirror (M) worlds. We have shown that in the case of a broken mirror parity (MP), the evolutions of fine structure constants in the O- and M-worlds are not identical. It is assumed that E_6-unification inspired by superstring theory restores the broken MP at the scale sim 10^{18} GeV, what unavoidably leads to the different E_6-breakdowns at this scale: E_6 to SO(10)times U(1)_Z - in the O-world, and E_6 to SU(6)times SU(2)_Z - in the M-world. Considering only asymptotically free theories, we have presented the running of all the inverse gauge constants alpha_i^{-1} in the one-loop approximation. Then a `quintessence scenario is discussed for our model of accelerating universe. Such a scenario is related with an axion (`acceleron) of a new gauge group SU(2)_Z which has a coupling constant g_Z extremely growing at the scale Lambda_Zsim 10^{-3} eV.
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