Before revision:  By combining Tikhonov--Fenichel small parameter with scaling analysis  , we derive improved upper bounds on the  error for the sQSSA, rQSSA and tQSSA. We find that the condition for the validity of the rQSSA is far less restrictive than was previously assumed, and we derive a new small parameter that determines the validity of this approximation.
After revision:  By combining Tikhonov--Fenichel small parameter with scaling analysis , we derive improved upper bounds on the error for the standard, reverse and total quasi-steady-state approximations. Furthermore, previous analyses suggest that the reverse quasi-steady-state approximation is only valid when initial enzyme concentrations greatly exceed initial substrate concentrations. However, our results indicate that this approximation can be valid when initial enzyme and substrate concentrations are of equal magnitude. Using energy methods, we find that the condition for the validity of the rQSSA is far less restrictive than was previously assumed, and we derive a new small parameter that determines the validity of this approximation.