Quantitative estimation of calcium dynamics from ratiometric measurements: a direct, nonratioing method

Measuring variations of intracellular free calcium concentration through the changes in fluorescence of a calcium-sensitive dye is a ubiquitous technique in neuroscience. Despite its popularity, confidence intervals (CIs) on the estimated parameters of calcium dynamics models are seldom given. To address this issue, we have developed a two-stage model for ratiometric measurements obtained with a charge-coupled device (CCD) camera. Its first element embeds a parametric calcium dynamics model into a fluorescence intensity model and its second element probabilistically describes the fluorescence measurements by a CCD camera. Using Monte Carlo simulations, we first show that the classical ratiometric transformation gives reliable CIs for time constants only and not baseline calcium concentration nor influx. We then introduce a direct method, which consists of fitting directly and simultaneously the fluorescence transients at both wavelengths, without any data ratioing. This approach uses a probabilistic description of the camera, leading to the construction of meaningful CIs for the calcium parameters. Moreover, using approaches inspired by constrained linear regression, we can take into account the finite precision on calibrated parameters (such as the dye dissociation constant in the cell). These key features are illustrated on simulated data using Monte Carlo simulations. Moreover, we illustrate the strength of the direct method on experimental recordings from insect olfactory interneurons. In particular, we show how to handle a time-dependent buffer concentration, thereby considerably improving our goodness of fit. The direct method was implemented in the open-source software R and is freely distributed in the CalciOMatic package.