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Metainference: A Bayesian Inference Method for Heterogeneous Systems
Massimiliano Bonomi, Carlo Camilloni, Andrea Cavalli, Michele Vendruscolo
TL;DR
Experimental data can be noisy and averaged across heterogeneous states, complicating model construction. Metainference addresses both issues with Bayesian replica-based inference, producing accurate ensembles and capturing protein conformational exchange.
Problem
Model construction must account for measurement errors and data averaged over multiple states when characterizing heterogeneous systems.
Method
Metainference uses Bayesian inference with replicas to model distributions of heterogeneous states while inferring experimental noise.
Results
Metainference produced accurate heterogeneous-system models and a ubiquitin ensemble that captured conformational exchange and downweighted less reliable chemical shifts.
Takeaways & Limitations
Metainference provides ensembles consistent with error-affected, state-averaged measurements for modeling heterogeneous and interconverting systems.
Takeaways & Limitations
The formulation assumes configurations are a priori independent and uses a Jeffreys prior for uncertainty parameters.
Abstract
from arXiv · showhide
Modelling a complex system is almost invariably a challenging task. The incorporation of experimental observations can be used to improve the quality of a model, and thus to obtain better predictions about the behavior of the corresponding system. This approach, however, is affected by a variety of different errors, especially when a system populates simultaneously an ensemble of different states and experimental data are measured as averages over such states. To address this problem we present a Bayesian inference method, called metainference, that is able to deal with errors in experimental measurements as well as with experimental measurements averaged over multiple states. To achieve this goal, metainference models a finite sample of the distribution of models using a replica approach, in the spirit of the replica-averaging modelling based on the maximum entropy principle. To illustrate the method we present its application to a heterogeneous model system and to the determination of an ensemble of structures corresponding to the thermal fluctuations of a protein molecule. Metainference thus provides an approach to model complex systems with heterogeneous components and interconverting between different states by taking into account all possible sources of errors.
Introduction
Metainference is introduced as a Bayesian method for modelling heterogeneous systems whose experimental data contain errors and average over multiple states. It combines prior information with experimental observations through replicas and connects Bayesian inference with maximum-entropy modelling.
- Method motivation: The method combines prior information with experimental data using a forward model and a noise model for deviations between predictions and observations.Bayesian inference quantifies agreement through uncertainty parameters, while the forward model predicts observations from conformational states.
- Method motivation: Metainference addresses experimental errors and ensemble averaging by modelling a finite sample of models as system replicas.The replicas represent the distribution of models and follow the replica-averaged maximum-entropy formulation.
- Method motivation: Metainference reduces to maximum-entropy modelling without data noise and to standard Bayesian modelling when data are not ensemble averages.These limiting cases establish the relationship between the two frameworks.
- Validation strategy: The method was benchmarked on a heterogeneous model system with synthetic measurements generated by averaging over discrete states at different noise levels.The benchmark tests accuracy under varying numbers of populated states and experimental noise.
- Validation strategy: Metainference was also applied to infer an ensemble of protein structures representing thermal fluctuations from experimental data.The paper presents this application as an example of modelling interconverting heterogeneous states while accounting for measurement errors.
SEM ( )
Metainference’s energy function incorporates replica-sampling, experimental, systematic, and forward-model errors. Its limiting cases recover replica-averaged maximum-entropy modelling or standard Bayesian modelling, while extensions handle multiple data points and outliers.
- Error model: Metainference’s energy function encodes statistical error from finite replicas, experimental and systematic errors, and forward-model errors.These contributions are represented in the associated energy function in units of kBT.
- Limiting cases: When data and forward-model errors vanish, metainference reduces to replica-averaged maximum-entropy modelling with a harmonic restraint coupling averaged observables to experimental data.In this limit, the restraint intensity scales with the number of replicas as N^2.
- Limiting cases: With nonzero data uncertainty, the restraint intensity scales as N and is modulated by the data uncertainty σ_B; when σ_SEM = 0, standard Bayesian modelling is recovered.The N-scaling applies in the presence of errors, whereas standard Bayesian modelling applies when measurements are not ensemble averages.
- Multiple data points and outliers: For multiple independent data points, metainference can model dataset-level uncertainty and marginalize individual error parameters while retaining one typical uncertainty parameter per replica.A long-tailed prior around a typical dataset uncertainty tolerates outlier data points.
Figures and Tables
The figures illustrate metainference’s treatment of experimental and interpretive errors, ensemble averaging, and heterogeneous systems. They show its application to synthetic heterogeneous-state data and protein structural fluctuations, alongside quantitative restraint-scaling analysis.
- Method overview: Metainference accounts for random and systematic measurement errors, inaccurate physico-chemical interpretation, and data dependence on multiple states and their populations.Figure 1 schematically summarizes these sources of uncertainty in generating accurate and precise models.
- Heterogeneous system: Equilibrium measurements of heterogeneous mixtures represent averages over ensembles rather than single species or conformations.Figure 2 compares metainference, maximum entropy, and standard Bayesian modelling on a synthetic discrete-state system.
- Scaling analysis: 0.99 is the average Pearson’s correlation coefficient for restraint-intensity scaling across 20 data points and replica counts from 8 to 256.The test used a five-state model system, Gaussian noise, and a prior accuracy of 16%; the restraint intensity scaled as N^2.
- Posterior formulation: Metainference models equilibrium ensemble averages using a finite sample of models represented by N replicas while incorporating uncertainty parameters for experimental and forward-model errors.The supplementary derivation defines the replica-based posterior for a single experimental data point.
SEM ( )
The SEM formulation marginalizes an unknown scaling factor under Gaussian data noise, extending metainference to multiple independent data points. Model-system simulations then assess how data noise, prior accuracy, replica number, and system size affect inferred-state accuracy.
- SEM: SEM marginalizes the unknown scaling factor as a Gaussian probability density, with σ_SEM encoding all sources of error.This yields the marginalized metainference posterior for Gaussian data noise.
- SEM: For multiple independent data points, metainference assigns separate scaling and uncertainty variables per replica and data point, then factorizes the data likelihood.The resulting expression is the metainference equation for multiple independent data points.
- Model-system simulations: The model-system benchmark varies state number, state populations, data-point count, and random or systematic noise to test competing modelling approaches.It uses 5-state systems with 2, 5, 10, or 20 data points and 50-state systems with 20, 50, 100, or 200 data points.
- Model-system simulations: Prior accuracy changes the data required for a given population accuracy: average population errors are 0.08 for high-accuracy priors and 0.16 for low-accuracy priors.The simulations define accuracy as the root mean squared deviation from exact state populations and average results over 300 independent simulations.
- Model-system simulations: Increasing the number of replicas improves metainference accuracy as finite-replica statistical error converges to zero.The benchmark uses 8, 16, 32, 64, and 128 replicas, with 50,000 Monte Carlo steps per simulation.