Recently, in Gong et al (2008 Phys. Lett. B 663 147 [arXiv:0711.1597]) and Wang and Santos(2007 arXiv:0712.3938) we showed that the effective cosmological constant on each of thetwo orbifold branes can be easily lowered to its current observational value, by using thelarge extra dimensions in the framework of both M-theory and string theory onS1/Z2. In this paper, we study the current acceleration of the universe, using the formulaedeveloped in Wang and Santos (2007 arXiv:0712.3938). We first construct explicitly atime-dependent solution to the ten-dimensional bulk of the Neveu–Schwarz/Neveu–Schwarzsector, compactified on a five-dimensional torus. Then, we write down the generalizedFriedmann equations on each of the two dynamical branes, and fit the modelsto the 182 gold Supernova Ia data and the BAO parameter from SDSS, usingboth of our MINUIT and Monte Carlo Markov Chain (MCMC) codes. With thebest fitting values of the parameters involved as initial conditions, we integratethe generalized Friedmann equations numerically and find the future evolutionof the universe. We find that it depends on the choice of the radion potentialsV4(I) (I = 1,2) of the branes. In particular, when choosing them to be Goldberger–Wise potentials,V4(I) = λ4(I)(ψ2−vI2)2, we find that the current acceleration of the universe driven by the effectivecosmological constant is only temporary. Due to the effects of the potentials, theuniverse will finally be in its decelerating expansion phase again. We also study theproper distance between the two branes and find that it remains almost constantduring the whole future evolution of the universe in all the models considered.