  Our ability to calculate rates of biochemical processes using molecular
dynamics simulations is severely limited by the fact that the time scales for
reactions, or changes in conformational state, scale exponentially with the
relevant free-energy barriers. In this work, we improve upon a recently
proposed rate estimator that allows us to predict transition times with
molecular dynamics simulations biased to rapidly explore one or several
collective variables. This approach relies on the idea that not all bias goes
into promoting transitions, and along with the rate, it estimates a concomitant
scale factor for the bias termed the collective variable biasing efficiency
$\gamma$. First, we demonstrate mathematically that our new formulation allows
us to derive the commonly used Infrequent Metadynamics (iMetaD) estimator when
using a perfect collective variable, $\gamma=1$. After testing it on a model
potential, we then study the unfolding behavior of a previously well
characterized coarse-grained protein, which is sufficiently complex that we can
choose many different collective variables to bias, but which is sufficiently
simple that we are able to compute the unbiased rate dire ctly. For this
system, we demonstrate that our new Exponential Average Time-Dependent Rate
(EATR) estimator converges to the true rate more rapidly as a function of bias
deposition time than does the previous iMetaD approach, even for bias
deposition times that are short. We also show that the $\gamma$ parameter can
serve as a good metric for assessing the quality of the biasing coordinate.
Finally, we demonstrate that the approach works when combining multiple
less-than-optimal bias coordinates.
