Limit of the accuracy of parameter estimation for two molecules moving in close proximity

Biomolecular interactions are central to biological processes and typically take place at nanometer scale distances. They often involve molecular motion which is known to affect the accuracy of the parameter estimates. Therefore, in this paper, we consider a case of two closely spaced molecules with planar trajectory and present a general expression of the Fisher information matrix in terms of their trajectory from which the benchmark for the accuracy of the parameter estimates is obtained. Through simulations, we show its application in the case of two moving objects and another case where only one of the two objects is moving. It is shown that the deterioration of the limit of the accuracy is not only dependent on the proximity of their starting position but also on their speed and direction of movement. The effect of differing photon emission intensities on the limit of the accuracy of parameter estimation is also investigated.