Turing-like patterns in an asymmetric dynamic Ising model

Answering open questions in biology using spatial genomics and structured methods

Genomics methods have uncovered patterns in a range of biological systems, but obscure important aspects of cell behavior: the shapes, relative locations, movement, and interactions of cells in space. Spatial technologies that collect genomic or epigenomic data while preserving spatial information have begun to overcome these limitations. These new data promise a deeper understanding of the factors that affect cellular behavior, and in particular the ability to directly test existing theories about cell state and variation in the context of morphology, location, motility, and signaling that could not be tested before. Rapid advancements in resolution, ease-of-use, and scale of spatial genomics technologies to address these questions also require an updated toolkit of statistical methods with which to interrogate these data. We present a framework to respond to this new avenue of research: four open biological questions that can now be answered using spatial genomics data paired with methods for analysis. We outline spatial data modalities for each open question that may yield specific insights, discuss how conflicting theories may be tested by comparing the data to conceptual models of biological behavior, and highlight statistical and machine learning-based tools that may prove particularly helpful to recover biological understanding.

Boundary fluctuation dynamics of a phase-separated domain in planar geometry

Using phase-ordering kinetics and of renormalization group theories, we derive analytically the relaxation times of the long wavelength fluctuations of a phase-separated domain boundary in the vicinity of (and below) the critical temperature in the planar Ising universality class. For a conserved order parameter, the relaxation time grows like Λ^{3} at wavelength Λ and can be expressed in terms of parameters relevant at the microscopic scale: lattice spacing, bulk diffusion coefficient of the minority phase, and temperature. These results are supported by numerical simulations of 2D Ising models, enabling in addition calculating the nonuniversal numerical prefactor. We discuss the applications of these findings to the determination of the real timescale associated with elementary Monte Carlo moves from the measurement of long wavelength relaxation times on experimental systems or molecular dynamics simulations.

Lizard Skin Patterns and the Ising Model

The ocellated lizard (Timon lepidus) exhibits an intricate skin color pattern made of monochromatic black and green skin scales, whose dynamics of color flipping are known to be well modeled by a stochastic cellular automaton. We show that the late-time probability distribution of the pattern corresponds to the canonical probability distribution of the antiferromagnetic Ising model and can be generated by dynamics different from the commonly-used Glauber. We comment on skin scale patterns generated by the Ising model on the triangular lattice in the low-temperature limit.

A self-exciting point process to study multicellular spatial signaling patterns

Multicellular organisms rely on spatial signaling among cells to drive their organization, development, and response to stimuli. Several models have been proposed to capture the behavior of spatial signaling in multicellular systems, but existing approaches fail to capture both the autonomous behavior of single cells and the interactions of a cell with its neighbors simultaneously. We propose a spatiotemporal model of dynamic cell signaling based on Hawkes processes-self-exciting point processes-that model the signaling processes within a cell and spatial couplings between cells. With this cellular point process (CPP), we capture both the single-cell pathway activation rate and the magnitude and duration of signaling between cells relative to their spatial location. Furthermore, our model captures tissues composed of heterogeneous cell types with different bursting rates and signaling behaviors across multiple signaling proteins. We apply our model to epithelial cell systems that exhibit a range of autonomous and spatial signaling behaviors basally and under pharmacological exposure. Our model identifies known drug-induced signaling deficits, characterizes signaling changes across a wound front, and generalizes to multichannel observations.