Session A47: Inference and Stochastic Processes in Biophysics

Quantifying the invisible: Bayesian approaches in fluorescence microscopy

Image datasets capturing the dynamics of large populations of moving objects are routinely acquired in fluorescent microscopy. Newest datasets contain quantitative and reliable observations with single molecule accuracy; however, despite of being obtained under state-of-the-art experimental procedures, their analysis remains a challenging task. Major difficulties are caused by: the governing photophysics which allow only limited fluorescent signals leading to very low signal-to-noise ratios; the optics of the experimental apparatus which limit the image resolution leading to significant blur even in the absence of other noise sources. Because of these characteristics, existing analysis methods, that are primarily developed for applications outside biophysics, can be used only sub-optimally as they typically rely on ergodic averages, thresholding, or otherwise artificial reduction of the available measurements. Instead, for biophysical applications, optimal analysis of the experimental datasets needs to utilize every data piece. This may be achieved only when the analysis is combined with simultaneous interpretation of the underlying physical system over the entire fluorescent population and over the entire time course available. In this presentation, I will describe a general framework that may be used for this task and I will walk through a number of example cases involving single molecule tracking.   

Biophysical inference mechanisms face a trade-off between external and internal noise resistance

Cells must extract relevant information from time-varying external signals while ignoring a sea of other irrelevant fluctuations in those signals. We find that biophysical mechanisms that are most effective at projecting out irrelevant external fluctuations are the most vulnerable to the internal noise originated from the mechanisms themselves. We show this trade-off relationship in circadian clocks, in gene regulation, and in biochemical receptors. We trace this trade-off to a fundamental tension in the geometry of dynamical systems. To be robust against external fluctuation, the geometry needs to be flat in the projected dimension of the signal and curved along the projected dimensions of external noise. However, this flat dimension makes the estimator vulnerable to internal fluctuations, which affect all dimensions. We conclude with experimental observations of S. Elongatus, and P. Marinus which have evolved distinct clock geometries in correspondence with their protein copy number as predicted by our theory.   

Inference of mechanical stresses within the actively migrating cell sheet

Collective cell migration is important in numerous physiological phenomena such as airway remodeling, cancer invasion and metastasis, and development. When epithelial cells migrate collectively they typically do so as a confluent monolayer, which is a form of active matter. As the monolayer migrates actively across a planar substrate, it develops internal stresses within the layer, some part of which are transmitted to that substrate and thus provide propulsive forces. These propulsive forces at the cell-substrate interface are called tractions. Our lab has shown that if the traction distribution is measured, then one can recover the internal 2D in-plane stresses within a monolayer[1]; this method is called Monolayer Stress Microscopy(MSM). The MSM approach assumes that the monolayer behaves as an elastic sheet, and stresses are calculated numerically with a finite elements scheme. To alleviate the need for a numerical scheme, and thereby simplify stress recovery, here we revisit the problem using a hydrodynamical formulation. We derive a novel 3D analytical solution that recovers internal stresses and velocities from tractions directly, and requires no numerics. We are currently comparing the analytical solution and experiments.

[1] D. T. Tambe et al., Nat. Mater. (2011).
  

Inference of Network Connectivity from Dynamics

Many biological systems of interest can be represented as networks
of many nodes that interact with one another. Often these systems
are subject to external influence or noise. One challenging
problem is to infer the interaction pattern of the system or the
connectivity structure of the network from measurements. We have
developed methods that can infer the connectivity structure of the network
using only time-series measurements of the dynamics of the nodes.
Our methods are guided by noise-induced mathematical relations
between the network connectivity structure and quantities that can
be calculated using solely the time-series measurements of the
dynamics of the nodes. These mathematical relations show clearly
that statistical covariance or correlation is not the appropriate
indicator for connectivity or interaction. Instead, the relevant
quantity is the product of the time-lagged covariance matrix and
the inverse of the covariance matrix for general directed networks
with unidirectional coupling or the inverse of the covariance
matrix for undirected networks with bidirectional coupling. We
have tested and verified our methods using simulated data from
different networks and dynamics.   

Adaptive trust in internal models alleviates trade-offs in biophysical inference of the environment

Living organisms often make unreliable measurements of their external environment while also making unreliable internal predictions of what their environment should be. The theory of Kalman filtering (or recursive Bayesian estimation) suggests that these two data should be combined using a time-dependent `trust’ factor that accounts for the relative unreliability of these two data. Using recent quantitative experimental measurements of circadian clocks in Synechococcus elongatus, we show that the coupling between the circadian clock and metabolism can naturally provide such an adaptive trust mechanism. Such adaptive trust allows the circadian clock to break sensitivity-robustness trade-offs on average, ignoring intensity fluctuations in natural light and yet responding quickly to phase changes.   

Bayesian nonparametrics for biophysics

As theorists, we draw trends and make predictions on protein function and interactions from models of protein dynamics. One route to modeling protein dynamics involves the bottom-up, molecular simulation, approach. Here we take a different route. Instead we present a top-bottom approach to building models of protein dynamics. The approach we present exploits a novel branch of Statistics -- called Bayesian nonparametrics (BNPs) -- first proposed in 1973 and now widely used in data science as the important conceptual advances of BNPs have become computational feasible in the last decade. BNPs are new to the physical sciences and use flexible (nonparametric) model structures to efficiently learn models from complex data sets. Here we will show how BNPs can be adapted to address important questions in protein biophysics directly from the data often limited by factors such as finite photon budgets as well as other data collection artifacts (e.g. aliasing, drift). More specifically, we will show that BNPs hold promise by allowing complex time traces (e.g. smFRET, photon arrivals) or images (e.g. single particle tracking) to be analyzed and turned into principled models of protein motion -- from diffusion to conformational dynamics and beyond.   

Progress in estimation of mutual information for real-valued data

Estimation of mutual information between (multidimensional) real-valued variables is used in analysis of complex systems, biological systems, and recently also quantum systems. The estimation is a hard problem, and universally good estimators provably do not exist. Kraskov et al (PRE, 2004) introduced a successful mutual information estimation approach based on the statistics of distances between neighboring data points, which empirically works for a wide class of underlying probability distributions. Here we improve their estimator in a number of ways. First, we use the reparameterization invariance property of mutual information to extend the estimator to work better for long-tailed and heavily skewed distributions. Second, we use subsampling techniques (more traditional resampling, such as bootstrap, produce biased results for mutual information estimation) to develop an estimate of the variance and the bias of the resulting estimator. We demonstrate the performance of our estimator on synthetic data sets, as well as on neuro- and systems biology datasets.   

The Limitations of Model-Based Experimental Design and Parameter Estimation in Sloppy Systems

Models of complex systems often involve a large number of unknown parameters to be estimated from data. In practice, these models are often sloppy, i.e., have parameter combinations with an exponential hierarchy of sensitivities as measured by eigenvalues of the Fisher Information Matrix. It has been argued that the extreme insenstivity of the model to changes in some parameters is what enables models to be predictive, analogous to the concept of irrelevance and effective theories in renormalizable systems. However, the unidentifiability of the irrelevant, sloppy parameters is often seen as a bottleneck for predictive modeling. It has been suggested that Optimal Experimental Design (OED) can be used to mitigate the effects of sloppiness and accurately estimate all of a model's parameters. We study models of two complex biological processes and show that when fit to "optimally designed experiments", they can have well-identified parameters, but that the models' predictive power is surprisingly greatly diminished. I interpret these results in the context of generalization error in machine learning and relevance and irrelevance for renormalizable systems.   

Data-Driven Inference for Jump-Diffusion Models of Neuroscientific Data

Detailed biophysical models are often used to describe neuroscientific data. They can, however, suffer from high-dimensional and poorly constrained parameter spaces. This makes it difficult to draw meaningful conclusions about the associated neural systems. Alternatively, a data-driven approach can be used where the observed fluctuations are captured by a single diffusion process. The resulting model is non-parametric, low-dimensional, and relies only on a minimal set of working assumptions. This approach has been applied to data such as membrane potential and EEG recordings. In some cases, however, these data exhibit abrupt jumps, or discontinuities, that must be disentangled from the diffusional fluctuations in order to fully understand the underlying dynamics. To address this, we develop an inference procedure that results in a fully specified jump-diffusion stochastic differential equation. This is done by first implementing a detection scheme for the jumps, taking into account the presence of false positives. The diffusion and drift functions are then obtained from the Kramers-Moyal coefficient and the differential Chapman-Kolmogorov equation, respectively. We successfully apply this procedure to data associated with membrane noise and active sensing in electric fish.   

Multilevel Bayesian Analysis of Biophysical Data in the Presence of Model Inadequacy and Measurement Error

Model inadequacy and measurement uncertainty are two of the most confounding aspects of inference and prediction in quantitative sciences. The process of scientific inference and prediction involve multiple steps of data analysis, hypothesis formation, model construction, parameter estimation, model validation, and finally prediction of the quantity of interest. This work seeks to clarify the concepts of model inadequacy, model bias, and measurement uncertainty, along with the two traditional classes uncertainty: aleatoric vs. epistemic, as well as their relashionships with each other. Starting from basic principles of probability, we build and explain a hierarchical Bayesian framework to quantitatively deal with model inadequacy and noise in data. We explain how this general approach can resolve many existing logical paradoxes that frequently appear in biophysical datasets. As an illustrative problem, we apply the methodology to an in-vitro dataset of the growth of C3A liver tumor cells subject to significant imaging background noise. We then explain how this general approach can retrieve the unknown quantities of interest from the available noisy data without any logical inconsistencies, such as obtaining negative values for quantities that are known to be inherently positive.   

Rational Ignorance: Simpler Models Learn More Information from Finite Data

We use the language of uninformative Bayesian prior choice to study the selection of appropriately simple effective models. We advocate for the prior which maximizes the mutual information between parameters and predictions, learning as much as possible from limited data. When many parameters are poorly constrained by the available data, we find that this prior puts weight only on boundaries of the parameter manifold. Thus it selects a lower-dimensional effective theory in a principled way, ignoring irrelevant parameter directions. In the limit where there is sufficient data to tightly constrain any number of parameters, this reduces to Jeffreys prior. But we argue that this limit is pathological when applied to the hyper-ribbon parameter manifolds generic in science, because it leads to dramatic dependence on effects invisible to experiment.   

Inference of Transition Rates in a Birth-Death Chain from Conditional Exit Times

Consider a birth-death process (BDP) of length N + 1 with general birth-death rates, which has a maximum population of N and becomes extinct when the population reaches zero. In this talk, a method of recovering the birth-death rates of the BDP from its extinction times (ETs) is presented. Given that the maximum site n reached by each trajectory is also known, we use the proportion of trajectories that do not exceed n and corresponding mean ET to recover the birth-death rates. Our method is based on properties of the characteristic polynomial of the BDP's infinitesimal generator. Given 50 million ETs of an 11-site birth-death chain, we can recover all the rates with a relative error about 3%. Our method could be used to infer potential landscape information for single-molecule spectroscopy experiments, or to compute the mutation rates of cancer cells undergoing sequential genotypic changes.   

Density of isolated particles and the hydrodynamic limit of generalized TASEP models: Application to mRNA translation rate inference

The Totally Asymmetric Exclusion Process (TASEP) is a classical stochastic model for describing the transport of interacting particles, such as ribosomes moving along the mRNA during protein translation. As recent experimental advances (called Ribo-seq) allow to examine position-specific densities of ribosomes, analytical tools for interpreting such data are needed. We hence revisit the TASEP theory and obtain new results. Motivated by possible biases in detecting ribosomes, we first study the density of isolated particles, by using the Matrix Ansatz to obtain exact formulas and a mean-field approximation for particles with arbitrary size (l-TASEP), both of which agree well with simulations. Second, we consider the hydrodynamic limit of the l-TASEP with heterogeneous rates. We derive and analyze the associated PDE, and obtain a phase diagram that generalizes previous results obtained on simpler models. Finally, we directly apply our theoretical study to Ribo-seq data. We infer the collision rate and the gap distance of nearby ribosomes leading to non-detection, and show that although the average number of ribosome per mRNA is widely used as a proxy for the flux, it actually leads to identification ambiguity for a substantial fraction of genes.