Session L15: Focus Session: Inferring Physical Models from Noisy Biological Data

Techniques for Statistically Scrutinizing Stochastic Model Assumptions Using a Single Noisily Measured Trajectory

The increased spatial and temporal resolution afforded by recent single-molecule experiments has inspired researchers to consider new techniques for quantifying molecular-level kinetics. Many researchers have contributed methods for improving the quality of estimators characterizing single-molecule kinetics, however techniques for checking the consistency of implicit distributional assumptions behind an assumed stochastic against a single experimental trajectory are under-developed. In this talk, likelihood-based goodness-of-fit testing and other model-based hypotheses tests accounting for the complexities of single-molecule trajectory analysis (heterogeneity, transient kinetic regime shifts, measurement noise, etc.) are discussed. Utility of the testing procedures are demonstrated on (i) single particle tracking (SPT) experiments characterizing mRNA motion in the cytoplasm of yeast cells and (ii) protein kinetics in the primary cilium of mammalian cells. In both cases, the testing procedures facilitated the discovery of new kinetic signatures of molecular motor facilitated transport not accounted for in traditional SPT models.   

Inferring cardiac phase response curve in vivo

Characterizing properties of biological oscillators with phase response cirves (PRC) is one of main theoretical tools in neuroscience, cardio-respiratory physiology, and chronobiology. We present a technique that allows the extraction of the PRC from a non-invasive observation of a system consisting of two interacting oscillators, in this case heartbeat and respiration, in its natural environment and under free-running conditions. We use this method to obtain the phase coupling functions describing cardio-respiratory interactions and the phase response curve of 17 healthy humans. We show at which phase the cardiac beat is susceptible to respiratory drive and extract the respiratory-related component of heart rate variability. This non-invasive method of bivariate data analysis for the determination of phase response curves of coupled oscillators may find application in other biological and physical systems.   

Form-Function Relationship in E. coli Chemotaxis

Cell-to-cell variations in protein abundance in clonal cell populations are ubiquitous in living systems. Because protein composition determines responses in individual cells, it stands to reason that the variations themselves are subject to selective pressures. However, the functional role of these cell-to-cell differences is not well understood. One way to tackle questions regarding relationships between form and function is to perturb the form (e.g., change the protein abundances) and observe the resulting changes in some function. We take on the form-function relationship from the inverse perspective, asking instead what specific constraints on cell-to-cell variations in protein abundance are imposed by a given functional phenotype [1]. We develop a maximum entropy based approach to posing questions of this type and illustrate the method by application to the well-characterized chemotactic response in {\it Escherichia coli}. We find that full determination of observed cell-to-cell variations in protein abundances is not inherent in chemotaxis itself but, in fact, appears to be jointly imposed by the chemotaxis program in conjunction with other factors (e.g., the protein synthesis machinery and/or additional non-chemotactic cell functions, such as cell metabolism). These results illustrate the power of maximum entropy as a tool for the investigation of relationships between biological form and function. \\[4pt] [1] Sayak Mukherjee, Sang-Cheol Seok, Veronica J. Vieland, and Jayajit Das, Proceedings of National Academy of Sciences {\bf 110} 18531 (2013).   

Inferring the Spatiotemporal DNA Replication Program from Noisy Biological Data

We generalize a stochastic model of DNA replication to the case where replication-origin-initiation rates vary locally along the genome and with time. Using this generalized model, we address the inverse problem of inferring initiation rates from experimental data concerning replication in cell populations. Previous work based on curve fitting depended on arbitrarily chosen functional forms for the initiation rate, with free parameters that were constrained by the data. We introduce a model-free, non-parametric method of inference that is based on Gaussian process regression. The method replaces specific assumptions about the functional form of initiation rate with more general prior expectations about the smoothness of variation of this rate, along the genome and in time. Using this inference method, we show that we can recover with high precision simulated replication schemes with data that are typical of current experiments. The method of Gaussian process regression can be profitably applied to a wide range of physical and biological problems.   

Distribution of population averaged observables in stochastic gene expression

Observation of phenotypic diversity in a population of genetically identical cells is often linked to the stochastic nature of chemical reactions involved in gene regulatory networks. We investigate the distribution of population averaged gene expression levels as a function of population, or sample size for several stochastic gene expression models to find out to what extent population averaged quantities reflect the underlying mechanism of gene expression. We consider three basic gene regulation networks corresponding to transcription with and without gene state switching and translation. Using analytical expressions for the probability generating function (pgf) of observables and Large Deviation Theory, we calculate the distribution of population averaged mRNA and protein levels as a function of model parameters and population size. We validate our results using stochastic simulations also report exact results on the asymptotic properties of population averages which show qualitative differences for different models. We calculate the skewness and coefficient of variance for pgfs to estimate the sample size required for population average that contains information about gene expression models. This is relevant to experiments where a large number of data points are unavailable.   

Multi-Scale Modeling to Improve Single-Molecule, Single-Cell Experiments

Single-cell, single-molecule experiments are producing an unprecedented amount of data to capture the dynamics of biological systems. When integrated with computational models, observations of spatial, temporal and stochastic fluctuations can yield powerful quantitative insight. We concentrate on experiments that localize and count individual molecules of mRNA. These high precision experiments have large imaging and computational processing costs, and we explore how improved computational analyses can dramatically reduce overall data requirements. In particular, we show how analyses of spatial, temporal and stochastic fluctuations can significantly enhance parameter estimation results for small, noisy data sets. We also show how full probability distribution analyses can constrain parameters with far less data than bulk analyses or statistical moment closures. Finally, we discuss how a systematic modeling progression from simple to more complex analyses can reduce total computational costs by orders of magnitude. We illustrate our approach using single-molecule, spatial mRNA measurements of Interleukin 1-alpha mRNA induction in human THP1 cells following stimulation. Our approach could improve the effectiveness of single-molecule gene regulation analyses for many other process.   

Outcome prediction in a mathematical model of immune response to infection

In clinical settings, it is of great importance to diagnose patients in the shortest amount of time and with the highest achievable accuracy. Current open questions concerning the modeling of the host response to infection include: How many measurements and with what frequency are needed to diagnose patients with a given accuracy? What is the effect of patient variation on the prediction accuracy? We employ machine-learning techniques to predict disease outcomes from data generated from a set of ordinary differential equations (ODE) used to model the immune response to infection. ODE models have the advantage that we can generate an unlimited amount of data, and we can easily simulate patient differences by varying model parameters. We explore the dependence of the prediction accuracy on data sets generated from the sets of ODEs as a function of the number of and spacing between measurements, number of measured variables, and the size of the patient variability.   

Information content and cross-talk in biological signal transduction: An information theory study

Biological cells respond to chemical cues provided by extra-cellular chemical signals, but many of these chemical signals and the pathways they activate interfere and overlap with one another. How well cells can distinguish between interfering extra-cellular signals is thus an important question in cellular signal transduction. Here we use information theory with stochastic simulations of networks to address the question of what happens to total information content when signals interfere. We find that both total information transmitted by the biological pathway, as well as its theoretical capacity to discriminate between overlapping signals, are relatively insensitive to cross-talk between the extracellular signals, until significantly high levels of cross-talk have been reached. This robustness of information content against cross-talk requires that the average amplitude of the signals are large. We predict that smaller systems, as exemplified by simple phosphorylation relays (two-component systems) in bacteria, should be significantly much less robust against cross-talk. Our results suggest that mammalian signal transduction can tolerate a high amount of cross-talk without degrading information content, while smaller bacterial systems cannot.   

Network topological analysis reveals the functional cohesiveness for the newly discovered links by Yeast 2 Hybrid approach

Protein-protein interaction maps and interactomes are the blueprint of Network Medicine and systems biology and are being experimentally studied by different groups. Despite the wide usage of Literature Curated Interactome (LCI), these sources are biased towards different parameters such as highly studied proteins. Yeast two hybrid method is a high throughput experimental setup which screens proteins in an unbiased fashion. Current knowledge of protein interactions is far from complete. In fact the previous offered data from Y2H method (2005), is estimated to offer only 5\% of all potential protein interactions. Currently this coverage has increased to 20\% of what is known as reference HI In this work we study the topological properties of Y2H protein-protein interactions network with LCI and show although they both agree on some properties, LCI shows a clear unbiased nature of interaction selections. Most importantly, we assess the properties of PPI as it evolves with increasing the coverage. We show that, the newly discovered interactions tend to connect proteins that have been closer than average in the previous PPI release. reinforcing the modular structure of PPI. Furthermore, we show, some unseen effects on PPI (as opposed to LCI) can be explained by its incompleteness.   

Spatio-temporal dynamcis of a cell signal cascade with negative feedback

We studied the spatio-temporal dynamics of a system of reactio-diffusion equations that models a cell signal transduction pathway with six cycles and negative feedback. The basic cycle consists of the phosphorylation-dephosphorylation of two antagonic proteins. We found two regimes of saturation of the enzimatic reaction in the kinetic parameters space and determined the conditions for the signal propagation in the steady state. The trajectories for which transduction occurs are defined in terms of the ratio of the enzimatic activities. We found that in spite of the negative feedback the cell signal cascade behaves as an amplifier and produces phosphoprotein concentration gradients within the cell. This model behaves also as a noise filter and as a switch.