Optimal Estimation of Diusion Coecients from Noisy Time-Lapse- Measurements of Single-Particle Trajectories Single-particle tracking techniques allow quantitative measurements of diusion at the single-molecule level. Recorded time-series are mostly short and contain considerable measurement noise. The standard method for estimating diusion coecients from single-particle trajectories is based on leastsquares tting to the experimentally measured mean square displacements. This method is highly inecient, since it ignores the high correlations inherent in these. We derive the exact maximum likelihood estimator for the diusion coecient, valid for short time-series, along with an exact benchmark for the maximum precision attainable with any unbiased estimator, the Cramer-Rao bound. We propose a simple analytical and unbiased covariance-based estimator based on the autocovariance function and derive an exact analytical expression of its moment generating function. We nd that the maximum likelihood estimator exceeds the precision set by the Cramer-Rao bound, but at the cost of a small bias, while the covariance-based estimator, which is born unbiased, is almost optimal for all experimentally relevant parameter values. We extend the methods to particles diusing on a uctuating substrate, e.g., exible or semi exible polymers such as DNA, and show that uctuations induce an important bias in the estimates of diusion coecients if they are not accounted for. We apply the methods to obtain precise estimates of diusion coecients of hOgg1 repair proteins diusing on stretched uctuating DNA from data previously analyzed using a suboptimal method. Our analysis shows that the proteins have dierent eective diusion coecients and that their diusion coecients are correlated with their residence time on DNA. These results imply a multi-state model for hOgg1's diusion on DNA.