Kenneth L. Ho

A Fast Direct Solver for Structured Linear Systems by Recursive Skeletonization

We present a fast direct solver for structured linear systems based on multilevel matrix compression. Using the recently developed interpolative decomposition of a low-rank matrix in a recursive manner, we embed an approximation of the original matrix into a larger but highly structured sparse one that allows fast factorization and application of the inverse. The algorithm extends the Martinsson--Rokhlin method developed for 2D boundary integral equations and proceeds in two phases: a precomputation phase, consisting of matrix compression and factorization, followed by a solution phase to apply the matrix inverse. For boundary integral equations which are not too oscillatory, e.g., based on the Green functions for the Laplace or low-frequency Helmholtz equations, both phases typically have complexity

Construction and analysis of a modular model of caspase activation in apoptosis

\mathcal{O} (N)

Bistability in apoptosis by receptor clustering.

in two dimensions, where

Parameter-free model discrimination criterion based on steady-state coplanarity

N

A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions

is the number of discretization points. In our current implementation, the corresponding costs in three dimensions are

Butterfly Factorization

\mathcal{O} (N^{3/2})

A Fast Semidirect Least Squares Algorithm for Hierarchically Block Separable Matrices

and

Numerical algebraic geometry for model selection and its application to the life sciences

\mathcal{O} (...

Fast Spatial Gaussian Process Maximum Likelihood Estimation via Skeletonization Factorizations

BackgroundA key physiological mechanism employed by multicellular organisms is apoptosis, or programmed cell death. Apoptosis is triggered by the activation of caspases in response to both extracellular (extrinsic) and intracellular (intrinsic) signals. The extrinsic and intrinsic pathways are characterized by the formation of the death-inducing signaling complex (DISC) and the apoptosome, respectively; both the DISC and the apoptosome are oligomers with complex formation dynamics. Additionally, the extrinsic and intrinsic pathways are coupled through the mitochondrial apoptosis-induced channel via the Bcl-2 family of proteins.ResultsA model of caspase activation is constructed and analyzed. The apoptosis signaling network is simplified through modularization methodologies and equilibrium abstractions for three functional modules. The mathematical model is composed of a system of ordinary differential equations which is numerically solved. Multiple linear regression analysis investigates the role of each module and reduced models are constructed to identify key contributions of the extrinsic and intrinsic pathways in triggering apoptosis for different cell lines.ConclusionThrough linear regression techniques, we identified the feedbacks, dissociation of complexes, and negative regulators as the key components in apoptosis. The analysis and reduced models for our model formulation reveal that the chosen cell lines predominately exhibit strong extrinsic caspase, typical of type I cell, behavior. Furthermore, under the simplified model framework, the selected cells lines exhibit different modes by which caspase activation may occur. Finally the proposed modularized model of apoptosis may generalize behavior for additional cells and tissues, specifically identifying and predicting components responsible for the transition from type I to type II cell behavior.

A Technique for Updating Hierarchical Skeletonization-Based Factorizations of Integral Operators

Apoptosis is a highly regulated cell death mechanism involved in many physiological processes. A key component of extrinsically activated apoptosis is the death receptor Fas which, on binding to its cognate ligand FasL, oligomerize to form the death-inducing signaling complex. Motivated by recent experimental data, we propose a mathematical model of death ligand-receptor dynamics where FasL acts as a clustering agent for Fas, which form locally stable signaling platforms through proximity-induced receptor interactions. Significantly, the model exhibits hysteresis, providing an upstream mechanism for bistability and robustness. At low receptor concentrations, the bistability is contingent on the trimerism of FasL. Moreover, irreversible bistability, representing a committed cell death decision, emerges at high concentrations which may be achieved through receptor pre-association or localization onto membrane lipid rafts. Thus, our model provides a novel theory for these observed biological phenomena within the unified context of bistability. Importantly, as Fas interactions initiate the extrinsic apoptotic pathway, our model also suggests a mechanism by which cells may function as bistable life/death switches independently of any such dynamics in their downstream components. Our results highlight the role of death receptors in deciding cell fate and add to the signal processing capabilities attributed to receptor clustering.