Ill-Posed? Not a Problem: Inferring Mechanisms of Action from Amyloid Formation Kinetics using Approximate Bayesian Computation

Abstract

Amyloid formation is implicated in a number of human diseases, and is thought to proceed via a nucleation-dependent polymerization mechanism wherein small concentrations of “nuclei” form initially, followed by the rapid, highly favorable addition of further monomer to nuclei/fibril ends. This type of behavior results in highly cooperative, sigmoidal fibrillization kinetics. In many cases, experimenters wish to relate changes in the observed kinetics, for example in response to small molecule effectors, to specific mechanistic steps along this pathway. However, fitting kinetic fibril formation data to a complex model including explicit rate constants results in an ill-posed problem, with a vast number of potential solutions. The amount of uncertainty remaining in parameters calculated from these models, which arises both from experimental noise and high levels of covariance in parameters, is often unclear.Here, we demonstrate the utility of combining these explicit mathematical models with approximate Bayesian computational (ABC) techniques to assign mechanistic roles to small molecule effectors of amyloid fibril formation. We show even when exact rate constants cannot be extracted from bulk amyloid formation kinetic data, the ratios and products of these rate constants can be recovered and used to assign mechanistic effects and their relative magnitudes with confidence. Furthermore, ABC provides a robust method for visualizing uncertainty remaining in the model parameters, regardless of its origin. We apply these methods to the problem of heparin-mediated tau polymerization, which displays complex kinetic behavior not amenable to analysis by more traditional methods. Our studies indicate that heparin's effect cannot be explained by enhancement of nucleation alone, as has been previously proposed. These methods are applicable to a wide range of systems, as models can be easily adapted to account for new reactions and reversibility.