A transformation approach to modelling multi-modal diffusions

• Tractable multi-modal models are obtained by transformation of simple diffusions.
• The new models have state-dependent diffusion coefficients.
• Estimators for observations with and without measurement error are derived.
• We apply the models to estimate the folding rates of the small Trp-zipper protein.
• Estimates may be severely biased if the effect of measurement error is ignored.

This paper demonstrates that flexible and statistically tractable multi-modal diffusion models can be attained by transformation of simple well-known diffusion models such as the Ornstein–Uhlenbeck model, or more generally a Pearson diffusion. The transformed diffusion inherits many properties of the underlying simple diffusion including its mixing rates and distributions of first passage times. Likelihood inference and martingale estimating functions are considered in the case of a discretely observed bimodal diffusion. It is further demonstrated that model parameters can be identified and estimated when the diffusion is observed with additional measurement error. The new approach is applied to molecular dynamics data in the form of a reaction coordinate of the small Trp-zipper protein, from which the folding and unfolding rates of the protein are estimated. Because the diffusion coefficient is state-dependent, the new models provide a better fit to this type of protein folding data than the previous models with a constant diffusion coefficient, particularly when the effect of errors with a short time-scale is taken into account.