In this study a novel framework for inverse modelling of cloud condensation nuclei (CCN) spectra is developed using Köhler theory. The framework is established by using model-generated synthetic measurements as calibration data for a parametric sensitivity analysis. Assessment of the relative importance of aerosol physicochemical parameters, while accounting for bulk-surface partitioning of surface-active organic species, is carried out over a range of atmospherically relevant supersaturations. By introducing an objective function that provides a scalar metric for diagnosing the deviation of modelled CCN concentrations from synthetic observations, objective function response surfaces are presented as a function of model input parameters. Crucially, for the chosen calibration data, aerosol-CCN spectrum closure is confirmed as a well-posed inverse modelling exercise for a subset of the parameters explored herein. The response surface analysis indicates that the appointment of appropriate calibration data is particularly important. To perform an inverse aerosol-CCN closure analysis and constrain parametric uncertainties, it is shown that a high-resolution CCN spectrum definition of the calibration data is required where single-valued definitions may be expected to fail. Using Köhler theory to model CCN concentrations requires knowledge of many physicochemical parameters, some of which are difficult to measure in situ on the scale of interest and introduce a considerable amount of parametric uncertainty to model predictions. For all partitioning schemes and environments modelled, model output showed significant sensitivity to perturbations in aerosol log-normal parameters describing the accumulation mode, surface tension, organic:inorganic mass ratio, insoluble fraction, and solution ideality. Many response surfaces pertaining to these parameters contain well-defined minima and are therefore good candidates for calibration using a Monte Carlo Markov Chain (MCMC) approach to constraining parametric uncertainties. A complete treatment of bulk-surface partitioning is shown to predict CCN spectra similar to those calculated using classical Köhler theory with the surface tension of a pure water drop, as found in previous studies. In addition, model sensitivity to perturbations in the partitioning parameters was found to be negligible. As a result, this study supports previously held recommendations that complex surfactant effects might be neglected, and the continued use of classical Köhler theory in global climate models (GCMs) is recommended to avoid an additional computational burden. The framework developed is suitable for application to many additional composition-dependent processes that might impact CCN activation potential. However, the focus of this study is to demonstrate the efficacy of the applied sensitivity analysis to identify important parameters in those processes and will be extended to facilitate a global sensitivity analysis and inverse aerosol-CCN closure analysis.