WORKSHOP ON ALGEBRAIC AND TOPOLOGICAL
METHODS FOR BIOLOGICAL NETWORKS


TIME:                       
Friday, January 15, 2016, 9:00 AM - 3:45 PM with reception to follow
Saturday, January  16, 2016, 9:00 AM - 2:00 PM

PLACE:                   
Glandt Forum

SPEAKERS AND ABSTRACTS:
Pablo Camara (Columbia University)

Applied topology reveals developmental progression at a single-cell level

Understanding the molecular transitions that underlie cellular differentiation and linage specification is a key problem in developmental biology, having great significance in the elaboration of genetic disease models. Due to its continuous, branched and asynchronous nature, cellular differentiation posses large experimental and computational challenges. The recent development of efficient single-cell RNA sequencing has opened a new experimental window into transcriptional studies of cellular differentiation. Computational methods that preserve the continuous and branched character of the process are required to analyse this data. Building upon topological data analysis, we present a novel mathematical and computational framework for analyzing single-cell RNA-seq data conducted as a function of time. We apply this method to the directed differentiation of murine stem cells into motor neurons in vitro, allowing us to dissect the complexity and timing of molecular transitions from pluripotent cells to neural precursors, progenitors and motor neurons.

Tomas Gedeon (Montana State University)

Database for Dynamics: a new approach to model gene regulatory networks

Experimental data on gene regulation is mostly qualitative, where the only information available about pairwise interactions is the presence of either up-or down- regulation. Quantitative data is often subject to large uncertainty and is mostly in terms of fold differences. Given these realities, it is very difficult to make reliable predictions using mathematical models. The current approach of choosing reasonable parameter values, a few initial conditions and then making predictions based on resulting solutions is severely subsampling both the parameter and phase space. This approach does not produce provable and reliable predictions.

We present a new approach that uses continuous time Boolean networks as a platform for qualitative studies of gene regulation. We compute a Database for Dynamics, which rigorously approximates global dynamics over entire parameter space. The results obtained by this method provably capture the dynamics at a predetermined spatial scale.

Robert Ghrist (University of Pennsylvania)

Applied Topology for Neuroscience: Theory & Computation

This talk will give a quick, gentle introduction to the ideas of applied topology that are fueling novel developments in neuroscience: homology, persistence, and barcodes. These tools are nontrivial to define, interpret, and, especially, compute. We will present new results of Henselman & the speaker on memory-efficient computation of persistent homology, closing with what the future holds for this exciting subject.

Heather Harrington (Oxford University)

Parameter-free approaches for comparing biochemical networks

Systems biology aims to understand the molecular interactions that turn genes on/off, ultimately regulating cellular decisions. These interactions may be described by a mathematical model that is a polynomial dynamical system. Generally these interactions are unknown, leading to multiple models; therefore it is desirable to compare models with experimental data (e.g., steady-state concentrations of proteins). Often model parameter values are unknown, and data is limited (subset of measurable variables, often with noise). An emerging field, `algebraic systems biology', offers algebraic approaches to study problems systems biology. We present an algebro-geometric method for ruling out models with limited information and apply it to a biological system known to dysfunction in many colorectal cancers. We are currently extending the framework to include dynamics (i.e., time course data) using differential algebra elimination and will present preliminary results.

This talk presents work that is joint with Helen Byrne (University of Oxford), Elizabeth Gross (San Jos\'e State University), Kenneth Ho (TSMC Technology), Adam MacLean (Imperial College London), Nicolette Meshkat (Santa Clara University), Zvi Rosen (University of Pennsylvania), and Bernd Sturmfels (University of California, Berkeley).

Kathryn Hess Bellwald (École Polytechnique Fédérale de Lausanne)

Topological analysis of digital reconstructions of neural microcircuits

A first draft digital reconstruction and simulation of a microcircuit of neurons in the neocortex of a two-week-old rat was recently published by the Blue Brain Project. Since traditional graph-theoretical methods may not be sufficient to understand the immense complexity of the network formed by the neurons and their connections, we explored whether methods from algebraic topology could provide a novel and useful perspective on the structural and functional organization of the microcircuit. 

Structural topological analysis revealed that directed graphs representing the connectivity between neurons are significantly different from random graphs, even those taking into account distance-dependent connection probabilities or constructed according to Peter's Rule, and that there exist an enormous number of simplicial complexes of dimensions up to seven, representing all-to-all directed connections within groups of neurons, the most extreme motif of neuronal clustering reported so far in the brain. Topological analysis of neuronal activity, based on data from simulations, confirmed the interest of this new approach to studying the relationship between the structure of the connectome and its emergent functions.  In particular, functional responses to different stimuli can be readily classified by topological methods. This study represents the first algebraic topological analysis of connectomics data from neural microcircuits and shows promise for general applications in network science.  

(Joint work with P. Dlotko, R. Levi, M. Nolte, M. Reimann, M. Scolamiero, K. Turner, E. Muller, and H. Markram)

Vladimir Itskov (Pennsylvania State University)

Topological inference of intrinsic structure in neural activity.

The brain represents information via patterns of neural activity.  Neurons in the brain often possess receptive fields, i.e. stimulus-response maps that determine neuronal response to particular stimuli.  When the space of relevant stimuli is well-understood, receptive fields can be easily computed from neural activity. However, for many areas of the brain the stimuli structure is largely unknown. One such example is olfaction, where the structure of the “space of smells” remains enigmatic. This raises a natural question: Can infer structural properties of the represented stimuli from neural activity alone?

We introduce a topological method for inferring structure of the underlying stimuli space from the neural activity. This method extracts features of the data invariant under non-linear monotone transformations. These features can be used to detect both random and geometric structure and also open new avenues for inferring more detailed information, such as dimension. Not any neural activity has a “geometric structure”, i.e. topological features compatible with convex receptive fields in Euclidean space.  We used our techniques and found geometric structure of neural activity in rat’s hippocampus during the behavior and sleep.

Eleni Katifori (University of Pennsylvania)

Topological fingerprints of biological distribution networks
 
Biological transport webs, such as the blood circulatory system in the brain and other animal organs, or the slime mold Physarum polycephalum, are frequently dominated by dense sets of nested cycles. The architecture of these networks, as defined by the topology and edge weights, determines how efficiently the networks perform their function. To characterize the structure of these weighted cycle-rich network architectures, we develop an algorithmic framework inspired by techniques in computational topology, which analyzes how the cycles are nested. Using this algorithmic framework and an extensive dataset of more than 180 leaves and leaflets, we show how the hierarchical organization of the nested architecture is in fact a distinct phenotypic trait, akin to a fingerprint, that characterizes the vascular systems of plants and can be used to assist species identification from leaf fragments. In the last part of the talk, we show how these methodologies can be generalized to non-planar graphs, such as the mammalian vasculature.

PROGRAM:
Friday, January 15th

8:30: Coffee
9:00: Opening Remarks
9:10: Talk: Robert Ghrist
10:10: Coffee and Discussion
10:45: Talk: 
Pablo Camara
11:45: Lunch (catered)
1:00: Talk:
 Heather Harrington
2:00: Coffee and Discussion
2:45: Talk: Tomas Gedeon
3:45: Reception

Saturday, January 16th

8:30: Coffee
9:00: Talk: Kathryn Hess Bellwald
10:00: Coffee and Discussion
10:45: Talk: Vladimir Itskov
11:45: Lunch
1:00: Talk: Eleni Katifori

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