Bayesian Learning from Marginal Data in Bionetwork Models

Abstract

In studies of dynamic molecular networks in systems biology, experiments are increasingly exploiting technologies such as flow cytometry to generate data on marginal distributions of a few network nodes at snapshots in time. For example, levels of intracellular expression of a few genes, or cell surface protein markers, can be assayed at a series of interim time points and assumed steady-states under experimentally stimulated growth conditions in small cellular systems. Such marginal data on a small number of cellular markers will typically carry very limited information on the parameters and structure of dynamic network models, though experiments will typically be designed to expose variation in cellular phenotypes that are inherently related to some aspects of model parametrization and structure. Our work addresses statistical questions of how to integrate such data with dynamic stochastic models in order to properly quantify the information-or lack of information-it carries relative to models assumed. We present a Bayesian computational strategy coupled with a novel approach to summarizing and numerically characterizing biological phenotypes that are represented in terms of the resulting sample distributions of cellular markers. We build on Bayesian simulation methods and mixture modeling to define the approach to linking mechanistic mathematical models of network dynamics to snapshot data, using a toggle switch example integrating simulated and real data as context.

Approximate Bayesian Computation with the Wasserstein Distance

“…We borrow the system biology 'toggle switch' model that was used in Bonassi et al (2011) and Bonassi and West (2015), inspired by studies of dynamic cellular networks. This provides an example where the design of specialized summaries can be replaced by the Wasserstein distance between empirical distributions.…”

“…5 in Bonassi and West (2015). We compare our method using p = 1 with a summary-based approach using the 11-dimensional tailor-made summary statistic from Bonassi et al (2011) and Bonassi and West (2015). Since the data are one dimensional, the Wasserstein distance between data sets can be computed quickly via sorting.…”

Approximate Bayesian computation via regression density estimation

“…For both spherical and ellipsoidal inclusion models we additionally constructed a likelihood estimate by simply fitting a mixture of multivariate normals to the samples ( S ( i ) , θ ( i ) ) and then conditioning on θ . This is equivalent to our regression density approach without the transformation ( S , θ ) → ( U , θ ), or the method of Bonassi et al () but conditioning on θ to obtain a likelihood, rather than on S to produce a posterior. For both models this approach performed reasonably well when the 3 semi‐automatic statistics were used (results not shown).…”

“…The approximation of the mean and covariance is performed within a Markov chain Monte Carlo (MCMC) scheme, and through clever choice of summary statistics and quantile transformations it may be possible to improve the approximation to normality. The approach we propose is also related to that introduced by Bonassi et al (), although this method does not directly approximate the likelihood. They propose to use mixtures of multivariate normals to estimate a joint distribution for ( θ , S ) where the summary statistics are simulated from an informative prior, and then condition on the observed summary statistic in the estimated mixture model.…”

What Population Reveals About Individual Cell Identity: Single-Cell Parameter Estimation of Models of Gene Expression in Yeast

“…Such models assume that the different observations arise from different realizations of the same stochastic process and, therefore, are still based on the notion of a virtual mean-although noisy-cell. In comparison, and despite recent methodological developments [27,28], few attempts have been made to infer extrinsic noise models from data, see [4,10,23,29,30] and our previous work [31]. We refer the reader to Karlsson et al [24] for a detailed discussion of these works.…”