The chaperonin GroEL-GroES, a machine which helps some proteins to fold, cycles through a number of allosteric states, the $T$ state, with high affinity for substrate proteins (SPs), the ATP-bound $R$ state, and the $R^{primeprime}$ ($GroEL-ADP-GroES$) complex. Structures are known for each of these states. Here, we use a self-organized polymer (SOP) model for the GroEL allosteric states and a general structure-based technique to simulate the dynamics of allosteric transitions in two subunits of GroEL and the heptamer. The $T to R$ transition, in which the apical domains undergo counter-clockwise motion, is mediated by a multiple salt-bridge switch mechanism, in which a series of salt-bridges break and form. The initial event in the $R to R^{primeprime}$ transition, during which GroEL rotates clockwise, involves a spectacular outside-in movement of helices K and L that results in K80-D359 salt-bridge formation. In both the transitions there is considerable heterogeneity in the transition pathways. The transition state ensembles (TSEs) connecting the $T$, $R$, and $R^{primeprime}$ states are broad with the the TSE for the $T to R$ transition being more plastic than the $Rto R^{primeprime}$ TSE. The results suggest that GroEL functions as a force-transmitting device in which forces of about (5-30) pN may act on the SP during the reaction cycle.