MFBM

Topological and network analyses for data

eSMB2020 eSMB2020 Follow Tuesday at 11:15am EDT
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Organizers:

John Nardini and Maria-Veronica Ciocanel

Description:

The era of big data has challenged researchers to develop novel methods to succinctly summarize and analyze complex datasets. Such methods include network and topological data analysis (TDA), which use methods from graph theory, dynamical systems, and topology to describe the patterns underlying a given data set over different scales. These areas of data science have proven successful for analyzing many biological phenomena, including disease transmission, ecological swarming, medical diagnostics, and within-cell protein interactions. In this minisymposium, the speakers will 1. introduce scholars to the basic principles behind persistent homology (a popular and useful technique of TDA) and 2. highlight current challenges and key areas of using TDA to inform mathematical modeling. In particular, their talks will highlight recent studies that have shown that this area of data science can provide informative summaries for mathematical models mimicking biological phenomena, including pattern formation and random rod networks.



Samuel Heroy

University of Oxford, United Kingdom, samuel.heroy@maths.ox.ac.uk
"Rigidity percolation in random rod networks"
In certain classes of both biological (e.g. actin) and material-based (e.g. nanocomposites) networks, the underlying system undergoes a transition in the mechanical strength at a critical system density. For instance, in a composite material composed of rigid interacting monodisperse particles randomly dispersed in a soft polymer matrix, the system experiences a phase transition at a critical particle density, whereas this transition may depend for instance on the mean number of filaments per contact in an actin network. This experimental phenomenon, termed rheological percolation, has been shown to occur in many systems at a density that is beyond the contact percolation threshold, demonstrating that a more complex mechanism is responsible for the observed mechanical gains. In this study, we construct a network model in which sphereocylinders are randomly dispersed in a medium and contact at intersection points (supposing penetrability). Idealizing these sphereocylinders (rods) as attractive particles that stay fixed at but can rotate about their points of contact (hinge-like connections), we posit that the rheological transition occurs when the rods form a spanning component that is not only connected, but connected in such a way as to remove all nontrivial degrees of freedom in the component (rigidity percolation). We build on results from two dimensions (see the paper https://epubs.siam.org/doi/abs/10.1137/17M1157271) to develop an approximate algorithm that identifies such spanning components through hierarchically identifying and compressing provably rigid motifs—contact patterns by which rigid components interact to form larger rigid components. We apply this algorithm to networks we generate at various density/system size, using a finite size scaling approach to rigorously estimate a rigidity percolation transition point bound, which we show agrees fairly well with a simple mean field estimation. We also estimate the transition point (and critical exponents) for networks with different rod aspect ratios, and find that the transition point scales with the square of the aspect ratio. In this study, we construct a network model in which sphereocylinders are randomly dispersed in a medium and contact at intersection points (supposing penetrability). Idealizing these sphereocylinders (rods) as attractive particles that stay fixed at but can rotate about their points of contact (hinge-like connections), we posit that the rheological transition occurs when the rods form a spanning component that is not only connected, but connected in such a way as to remove all nontrivial degrees of freedom in the component (rigidity percolation). We build on results from two dimensions (see the paper https://epubs.siam.org/doi/abs/10.1137/17M1157271) to develop an approximate algorithm that identifies such spanning components through hierarchically identifying and compressing provably rigid motifs—contact patterns by which rigid components interact to form larger rigid components. We apply this algorithm to networks we generate at various density/system size, using a finite size scaling approach to rigorously estimate a rigidity percolation transition point bound, which we show agrees fairly well with a simple mean field estimation. We also estimate the transition point (and critical exponents) for networks with different rod aspect ratios, and find that the transition point scales with the square of the aspect ratio.


Yu-Min Chung

UNC Greensboro, United States, y_chung2@uncg.edu
"On the morphology of mitochondria via a multi-parameter persistent homology approach"
Mutations in autophagy-gene Optineurin (OPTN) are associated with Primary Open Angle Glaucoma (POAG) and amyotrophic lateral sclerosis, but the pathophysiological mechanism is unclear. The E50K OPTN mutation is associated with glaucoma. Recent studies have shown that OPTN may play an important role in regulating mitochondrial networks and interacting with parkin as part of the mitophagy pathway. We hypothesized that loss of normal OPTN function disrupts mitochondrial morphology. To investigate and quantify the phenomena, we use multi-parameter persistent homology on confocal images of cells from transgenic mice with the E50K mutation and genetic knockout of optineurin. In particular, we combine methods in mathematical morphology to form a multi-parameter filtration. We will show that such filtration contains both topological and geometric information about the mitochondria, and will demonstrate ways to extract meaningful features from it. Preliminary results support the hypothesis. This is a joint work with Chuan-Shen Hu at National Taiwan Normal University, Emily Sun, and Dr. Henry C. Tseng at Duke Eye Center.


Alexandria Volkenning

Northwestern University, United States, alexandria.volkening@northwestern.edu
"Topological data analysis of zebrafish skin patterns"
Wild-type zebrafish feature black and yellow stripes across their body and fins, but mutants display a range of altered patterns, including spots and labyrinth curves. All these patterns form due to the interactions of pigment cells, which sort out through movement, birth, and competition during development. Using an agent-based approach, we have coupled deterministic cell migration by ODEs with stochastic rules for updating population size to reproduce stripe pattern development and predict cell interactions that may be altered in mutant patterns. Within a single zebrafish mutant, however, there is a lot of variability, and this makes it challenging to first identify the features of a pattern that we are trying to reproduce and then judge model success. Moreover, agent-based models have many parameters, and empirical descriptions of zebrafish patterns are largely qualitative. To help address these challenges, we draw on topological data analysis to develop a set of methods for automatically quantifying pattern features in a fully interpretable, cell-based way. We apply our techniques to both simulated data and real fish images, and we show how to quantitatively distinguish between and characterize different patterns.


Ashish Raj

UCSF, United States, ashish.raj@ucsf.edu
"Inference on models of network spread and protein aggregation in Alzheimer’s and dementia"
Alzheimer’s disease, Parkinson’s and other related dementias involve widespread, stereotyped and progressive deposition of misfolded proteins. There is mounting evidence for “prion-like” trans-neuronal transmission, whereby proteins misfold, trigger misfolding of adjacent same-species proteins, and there- upon cascade along neuronal pathways, giving rise to networked spread along white matter projections. The question of how protein aggregation and subsequent spread lead to stereotyped progression in the brain remains unresolved. We present here mathematically precise parsimonious modeling of these patho- physiological processes, extrapolated to the whole brain. We model monomer seeding and production at specific seed regions, aggregation using Smoluchowski equations; and networked spread using our prior Network-Diffusion model, whereby anatomic connections govern the rate at which two distant but con- nected brain regions can transfer pathologic proteins. These models involve several unknown parameters, some or all of which might be individual-specific. Hence parameter inference is a challenging problem, without which downstream applications of such models in disease diagnosis, prognosis and therapy will not be possible. We present several inference strategies we have explored in our lab, including nonlinear cost function minimization, variational Bayesian inference, and purely machine learning methods like support vectors and neural nets. Unlike most previous theoretical studies of protein aggregation, using these techniques our theoretical models are able to be fitted to and validated by experimental in vivo imaging and fluid protein measurements from large datasets.




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