Theory¶
Bayesian inference¶
Bayesian inference approaches for modeling conformational ensembles generally seek to model a posterior distribution \(P(X|D)\) of conformational states \(X\), given some experimental data \(D\). According to Bayes’ theorem, the posterior distribution is proportional to a product of (1) a likelihood function \(Q(D|X)\) of observing the experimental data given a conformational state \(X\), and (2) a prior distribution \(P(X)\) representing any prior knowledge about conformational states.
Here, the prior \(P(X)\) comes from theoretical modeling, while the likelihood \(Q(D|X)\) corresponds to experimental restraints, typically in the form of some error model reflecting how well a given conformation \(X\) agrees with experimental measurements. One can think of \(Q(D|X)\) as a reweighting factor for the population of each state \(X\), and the BICePs algorithm can be thought of as a way to reweight conformational populations to agree with experimental knowledge.
In BICePs, the error model reflects both uncertainty in the experimental measurements and heterogeneity in the conformational ensemble, (i.e. how likely are the observables for conformation \(X\) to be away from the ensemble-average measurement, either due to experimental noise or conformational heterogeneity). These uncertainties are usually not known a priori, and must be treated as nuisance parameters \(\sigma\) which can be modeled using some prior model \(P(\sigma)\):
Posterior sampling over \(X\) and \(\sigma\) using Markov Chain Monte Carlo (MCMC) can then be used to determine the conformational populations given the experimental restraints as \(P(X|D) = \int P(X,\sigma | D) d\sigma\). Similarly, the posterior distribution of the experimental uncertainty \(P(\sigma | D) = \int P(X,\sigma | D) dX\) gives information about how well the posterior conformational distribution agrees with the experimental restraints.
Reference potentials¶
The correct implementation of reference potentials is an importance advantage of the BICePs algorithm. The experimental data used to construct the likelihood function comes from some set of ensemble-averaged experimental observables \(\mathbf{r} = (r_1, r_2, ..., r_N)\). Such observables are low-dimensional projections of some high-dimensional state space \(X\), and therefore these restraints in the space of observables need to be treated as potentials of mean force. 1 2 3
The bracketed weighting function is now a ratio, with the numerator \(Q(\mathbf{r}|D)\) being a likelihood function enforcing experimental restraints in the space of observables, while the denominator \(Q_{\text{ref}}(\mathbf{r})\) reflects some reference distribution for possible values of the observables \(\mathbf{r}\) in the absence of any experimental restraint information, such that \(-\ln [Q(\mathbf{r}|D)/Q_{\text{ref}}(\mathbf{r})]\) is a potential of mean force. Without reference potentials, a great deal of unnecessary bias is introduced when many non-informative restraints are used.
As an example to illustrate the importance of reference potentials, consider an experimental distance restraint applied to two residues of a polypeptide chain (Figure 1). In the absence of any experimental information, we assume a reference potential \(Q_{\text{ref}}(\mathbf{r})\) corresponding to the end-to-end distance of a random-coil polymer with a chain length equal to that of the intervening residues. For residues near each other along the chain, a short-distance restraint may have \([Q(\mathbf{r}|D)/Q_{\text{ref}}(\mathbf{r})] \sim 1\), contributing little or no information to refine the conformational ensemble. If the residues are far apart along the chain, however, a short-distance restraint can be highly informative, with \([Q(\mathbf{r}|D)/Q_{\text{ref}}(\mathbf{r})]\) greatly rewarding small distances where the reference potential \(Q_{\text{ref}}(\mathbf{r})\) is small.
Figure 1.¶
BICePs scores for quantitative model selection¶
Another key advantage of the BICePs algorithm is its the ability to use the sampled posterior probability function to perform quantitative model selection. This is done through a quantity we will call the BICePs score.
Consider \(K\) different theoretical models \(P^{(k)}(X)\), \(k=1,...,K\), that we may wish to use as our prior distribution. Perhaps these models come from simulations using different potential energy functions. Or perhaps we want to compare a model shaped only experimental restraints, to a model using both simulation and experimental information. Which model is more consistent with the experimental data? To determine this, we compute the posterior likelihood of the model, \(Z^{(k)}\), by integrating the \(k^{th}\) posterior distribution across all conformations \(X\) and values of \(\sigma\):
The quantity \(Z^{(k)}\) can be interpreted as the total evidence in favor of a given model. Equivalently, we can think of this term as an overlap integral between the prior \(P^{(k)}(X)\) and a likelihood function \(Q(X) = \int [Q(\mathbf{r}(X)|D,\sigma)/Q_{\text{ref}}(\mathbf{r}(X)) ] P(\sigma) d\sigma\). The overlap integral quantifies how well the theoretical modeling agrees with the experimental data (Figure 2); i.e. the value of \(Z^{(k)}\) is maximal when \(P^{(k)}(X)\) most closely matches the likelihood distribution \(Q(X)\) specified by the experimental restraints.
Figure 2.¶
In Bayesian statistics, the so-called Bayes factor, \(Z^{(1)}/Z^{(2)}\) is a likelihood ratio that can be used to choose between competing models (1) and (2), similar to a likelihood-ratio test in classical statistics. In order to assign each model a unique score, we define a free energy-like quantity,
to compare model \(P^{(k)}(X,\sigma|D)\) against some reference model \(Z_0\). In practice, we choose the reference model to be the posterior for a uniform prior \(P(X)\), i.e. no information from theoretical modeling. We call \(f^{(k)}\) the BICePs score. The BICePs score provides an unequivocal measure of model quality, to be used for objective model selection. The lower the BICePs score, the better the model (Figure 2). This useful property means that the BICePs score can be used a metric for force field validation and parameterization. In addition, the BICePs score has a useful physical interpretation: it reflects the improvement (or disimprovement) of the posterior distribution when going from a conformation ensemble shaped only experimental constraints, to a new distribution additionally shaped by a theoretical model.
Summary of the key advantages of BICePs.¶
To put the BICePs algorithm in a larger context, we summarize the its key advantages as follows:
BICePs can be used with molecular dynamics (MD) or quantum mechanics (QM) methods. Conformational states can be individual conformations (like single-point QM minima) or collections of conformations (e.g. from clustering of trajectory data), as long as experimental observables can be attached to each conformational state.
BICePs performs reweighting of conformational states derived from modeling; it is currently a post-processing algorithm (MCMC) with no additional MD or QM required.
Bayesian inference offers a rigorous statistical framework for achieving the correct balance of theoretical modeling and experimental data.
BICePs correctly uses reference potentials, which is essential to proper weighing of experimental restraints
With proper reference potentials, BICePs scores can be used for unambiguous, objective model selection.
For more details about theory beneath BICePs, please check these work. 4 5
References¶
- 1
Olsson, S.; Frellsen, J.; Boomsma, W.; Mardia, K. V.; Hamelryck, T. Inference of Structure Ensembles of Flexible Biomolecules from Sparse, Averaged Data. PLoS One 2013, 8, e79439
- 2
Olsson, S.; Boomsma, W.; Frellsen, J.; Bottaro, S.; Harder, T.; Ferkinghoff-Borg, J.; Hamelryck, T. Generative Probabilistic Models Extend the Scope of Inferential Structure Determination. J. Magn. Reson. 2011, 213, 182−186.
- 3
Hamelryck, T.; Borg, M.; Paluszewski, M.; Paulsen, J.; Frellsen, J.; Andreetta, C.; Boomsma, W.; Bottaro, S.; Ferkinghoff-Borg, J. Potentials of Mean Force for Protein Structure Prediction Vindicated, Formalized and Generalized. PLoS One 2010, 5, e13714.
- 4
Voelz, V. A.; Zhou, G. Bayesian Inference of Conformational State Populations from Computational Models and Sparse Exper- imental Observables. J. Comput. Chem. 2014, 35, 2215−2224.
- 5
Yunhui Ge and Vincent A. Voelz, Model selection using BICePs: A Bayesian approach to force field validation and parameterization Journal of Physical Chemistry B (2018) 122 (21): 5610–5622