Learning unknown ODE models with Gaussian processes

In conventional ODE modelling coefficients of an equation driving the system state forward in time are estimated. However, for many complex systems it is practically impossible to determine the equations or interactions governing the underlying dynamics. In these settings, parametric ODE model cannot be formulated. Here, we overcome this issue by introducing a novel paradigm of nonparametric ODE modelling that can learn the underlying dynamics of arbitrary continuous-time systems without prior knowledge. We propose to learn non-linear, unknown differential functions from state observations using Gaussian process vector fields within the exact ODE formalism. We demonstrate the model's capabilities to infer dynamics from sparse data and to simulate the system forward into future.Comment: 11 pages, 2 page appendi

In vivo kinetics of transcription initiation of the lar promoter in Escherichia coli. Evidence for a sequential mechanism with two rate-limiting steps

<p>Abstract</p> <p>Background</p> <p>In <it>Escherichia coli </it>the mean and cell-to-cell diversity in RNA numbers of different genes vary widely. This is likely due to different kinetics of transcription initiation, a complex process with multiple rate-limiting steps that affect RNA production.</p> <p>Results</p> <p>We measured the <it>in vivo </it>kinetics of production of individual RNA molecules under the control of the lar promoter in <it>E. coli</it>. From the analysis of the distributions of intervals between transcription events in the regimes of weak and medium induction, we find that the process of transcription initiation of this promoter involves a sequential mechanism with two main rate-limiting steps, each lasting hundreds of seconds. Both steps become faster with increasing induction by IPTG and Arabinose.</p> <p>Conclusions</p> <p>The two rate-limiting steps in initiation are found to be important regulators of the dynamics of RNA production under the control of the lar promoter in the regimes of weak and medium induction. Variability in the intervals between consecutive RNA productions is much lower than if there was only one rate-limiting step with a duration following an exponential distribution. The methodology proposed here to analyze the <it>in vivo </it>dynamics of transcription may be applicable at a genome-wide scale and provide valuable insight into the dynamics of prokaryotic genetic networks.</p

Cell-to-cell diversity in protein levels of a gene driven by a tetracycline inducible promoter

<p>Abstract</p> <p>Background</p> <p>Gene expression in <it>Escherichia coli </it>is regulated by several mechanisms. We measured in single cells the expression level of a single copy gene coding for green fluorescent protein (GFP), integrated into the genome and driven by a tetracycline inducible promoter, for varying induction strengths. Also, we measured the transcriptional activity of a tetracycline inducible promoter controlling the transcription of a RNA with 96 binding sites for MS2-GFP.</p> <p>Results</p> <p>The distribution of GFP levels in single cells is found to change significantly as induction reaches high levels, causing the Fano factor of the cells' protein levels to increase with mean level, beyond what would be expected from a Poisson-like process of RNA transcription. In agreement, the Fano factor of the cells' number of RNA molecules target for MS2-GFP follows a similar trend. The results provide evidence that the dynamics of the promoter complex formation, namely, the variability in its duration from one transcription event to the next, explains the change in the distribution of expression levels in the cell population with induction strength.</p> <p>Conclusions</p> <p>The results suggest that the open complex formation of the tetracycline inducible promoter, in the regime of strong induction, affects significantly the dynamics of RNA production due to the variability of its duration from one event to the next.</p

lgpr: an interpretable non-parametric method for inferring covariate effects from longitudinal data

Motivation: Longitudinal study designs are indispensable for studying disease progression. Inferring covariate effects from longitudinal data, however, requires interpretable methods that can model complicated covariance structures and detect non-linear effects of both categorical and continuous covariates, as well as their interactions. Detecting disease effects is hindered by the fact that they often occur rapidly near the disease initiation time, and this time point cannot be exactly observed. An additional challenge is that the effect magnitude can be heterogeneous over the subjects. Results: We present lgpr, a widely applicable and interpretable method for non-parametric analysis of longitudinal data using additive Gaussian processes. We demonstrate that it outperforms previous approaches in identifying the relevant categorical and continuous covariates in various settings. Furthermore, it implements important novel features, including the ability to account for the heterogeneity of covariate effects, their temporal uncertainty, and appropriate observation models for different types of biomedical data. The lgpr tool is implemented as a comprehensive and user-friendly R-package.Peer reviewe

Learning unknown ODE models with Gaussian processes

In conventional ODE modelling coefficients of an equation driving the system state forward in time are estimated. However, for many complex systems it is practically impossible to determine the equations or interactions governing the underlying dynamics. In these settings, parametric ODE model cannot be formulated. Here, we overcome this issue by introducing a novel paradigm of nonparametric ODE modelling that can learn the underlying dynamics of arbitrary continuous-time systems without prior knowledge. We propose to learn non-linear, unknown differential functions from state observations using Gaussian process vector fields within the exact ODE formalism. We demonstrate the model’s capabilities to infer dynamics from sparse data and to simulate the system forward into future.Peer reviewe

Learning unknown ODE models with Gaussian processes

In conventional ODE modelling coefficients of an equation driving the system state forward in time are estimated. However, for many complex systems it is practically impossible to determine the equations or interactions governing the underlying dynamics. In these settings, parametric ODE model cannot be formulated. Here, we overcome this issue by introducing a novel paradigm of nonparametric ODE modelling that can learn the underlying dynamics of arbitrary continuous-time systems without prior knowledge. We propose to learn non-linear, unknown differential functions from state observations using Gaussian process vector fields within the exact ODE formalism. We demonstrate the model’s capabilities to infer dynamics from sparse data and to simulate the system forward into future.Peer reviewe

PairGP: Gaussian process modeling of longitudinal data from paired multi-condition studies

Publisher Copyright: © 2022High-throughput technologies produce gene expression time-series data that need fast and specialized algorithms to be processed. While current methods already deal with different aspects, such as the non-stationarity of the process and the temporal correlation, they often fail to take into account the pairing among replicates. We propose PairGP, a non-stationary Gaussian process method to compare gene expression time-series across several conditions that can account for paired longitudinal study designs and can identify groups of conditions that have different gene expression dynamics. We demonstrate the method on both simulated data and previously unpublished RNA sequencing (RNA-seq) time-series with five conditions. The results show the advantage of modeling the pairing effect to better identify groups of conditions with different dynamics. The pairing effect model displays good capabilities of selecting the most probable grouping of conditions even in the presence of a high number of conditions. The developed method is of general application and can be applied to any gene expression time series dataset. The model can identify common replicate effects among the samples coming from the same biological replicates and model those as separate components. Learning the pairing effect as a separate component, not only allows us to exclude it from the model to get better estimates of the condition effects, but also to improve the precision of the model selection process. The pairing effect that was accounted before as noise, is now identified as a separate component, resulting in more accurate and explanatory models of the data.Peer reviewe

Bayesian metabolic flux analysis reveals intracellular flux couplings

Motivation: Metabolic flux balance analysis (FBA) is a standard tool in analyzing metabolic reaction rates compatible with measurements, steady-state and the metabolic reaction network stoichiometry. Flux analysis methods commonly place model assumptions on fluxes due to the convenience of formulating the problem as a linear programing model, while many methods do not consider the inherent uncertainty in flux estimates. Results: We introduce a novel paradigm of Bayesian metabolic flux analysis that models the reactions of the whole genome-scale cellular system in probabilistic terms, and can infer the full flux vector distribution of genome-scale metabolic systems based on exchange and intracellular (e.g. 13C) flux measurements, steady-state assumptions, and objective function assumptions. The Bayesian model couples all fluxes jointly together in a simple truncated multivariate posterior distribution, which reveals informative flux couplings. Our model is a plug-in replacement to conventional metabolic balance methods, such as FBA. Our experiments indicate that we can characterize the genome-scale flux covariances, reveal flux couplings, and determine more intracellular unobserved fluxes in Clostridium acetobutylicum from 13C data than flux variability analysis.Peer reviewe