Bayesian Inference: The Comprehensive Approach to Analyzing Single-Molecule Experiments

Abstract

Biophysics experiments performed at single-molecule resolution provide exceptional insight into the structural details and dynamic behavior of biological systems. However, extracting this information from the corresponding experimental data unequivocally requires applying a biophysical model. In this review, we discuss how to use probability theory to apply these models to single-molecule data. Many current single-molecule data analysis methods apply parts of probability theory, sometimes unknowingly, and thus miss out on the full set of benefits provided by this self-consistent framework. The full application of probability theory involves a process called Bayesian inference that fully accounts for the uncertainties inherent to single-molecule experiments. Additionally, using Bayesian inference provides a scientifically rigorous method of incorporating information from multiple experiments into a single analysis and finding the best biophysical model for an experiment without the risk of overfitting the data. These benefits make the Bayesian approach ideal for analyzing any type of single-molecule experiment.

Keywords: cryo-EM; error propagation; kinetics; model selection; probability theory; scientific method.

Figure 1:. The analogy between the scientific method and Bayesian inference.

(a) The components of a single example of the scientific method (above) show a one-to-one correspondence with those of Bayesian inference (below), revealing how the latter is just a formal extension of the former to data analysis. (b) The analogy is reinforced in how repeated applications of both the scientific method and Bayesian inference extend the frontier of knowledge and certainty, respectively. The area in tan shows a scientist’s knowledge (or certainty) gained by the latest application of the scientific method (or Bayesian inference), which itself is built upon previous applications.

Figure 2:. The role of models in science.

Representations of simulated data (left) and a corresponding model (right) for common single-molecule studies, including (a) an electron micrograph probing the structure of a biomolecule (a ribosome) and the corresponding model of its structure (PDB ID: 6UZ7), (b) a current versus time trajectory probing the conformational dynamics of a biomolecule and the corresponding model of its conformational transitions, (c) a force versus extension curve probing the unfolding of a biomolecule and the corresponding model of its unfolding transitions, and (d) a single particle track probing the diffusion of a biomolecule and the corresponding model of the diffusion coefficient. The blowout of (a) shows that while scientists only aim for, and report, a portion of the model (red hexagon), the model is complex and includes noise as well as other background information (red ovals).

Figure 3:. Bayesian model selection.

(a) Representation of a typical E_FRET trajectory (top) and the corresponding 2-state (middle) and 3-state (bottom) HMMs for the trajectory, as analyzed by vbFRET. (b) The log of the ELBOs for HMMs with increasing number of states (as calculated by vbFRET) shows a peak at the 3-state model (above), and decays slowly as more states are added. Upon using these ELBOs to calculate the posterior probability for these models (below), it is clear that the 3-state model is overwhelmingly more probable than the others.