Mathematical Models for Cell Migration with Real-Time Cell Cycle Dynamics

The fluorescent ubiquitination-based cell cycle indicator, also known as FUCCI, allows the visualization of the G1 and S/G2/M cell cycle phases of individual cells. FUCCI consists of two fluorescent probes, so that cells in the G1 phase fluoresce red and cells in the S/G2/M phase fluoresce green. FUCCI reveals real-time information about cell cycle dynamics of individual cells, and can be used to explore how the cell cycle relates to the location of individual cells, local cell density, and different cellular microenvironments. In particular, FUCCI is used in experimental studies examining cell migration, such as malignant invasion and wound healing. Here we present, to our knowledge, new mathematical models that can describe cell migration and cell cycle dynamics as indicated by FUCCI. The fundamental model describes the two cell cycle phases, G1 and S/G2/M, which FUCCI directly labels. The extended model includes a third phase, early S, which FUCCI indirectly labels. We present experimental data from scratch assays using FUCCI-transduced melanoma cells, and show that the predictions of spatial and temporal patterns of cell density in the experiments can be described by the fundamental model. We obtain numerical solutions of both the fundamental and extended models, which can take the form of traveling waves. These solutions are mathematically interesting because they are a combination of moving wavefronts and moving pulses. We derive and confirm a simple analytical expression for the minimum wave speed, as well as exploring how the wave speed depends on the spatial decay rate of the initial condition.     

MSMBuilder: Statistical Models for Biomolecular Dynamics

MSMBuilder is a software package for building statistical models of high-dimensional time-series data. It is designed with a particular focus on the analysis of atomistic simulations of biomolecular dynamics such as protein folding and conformational change. MSMBuilder is named for its ability to construct Markov state models (MSMs), a class of models that has gained favor among computational biophysicists. In addition to both well-established and newer MSM methods, the package includes complementary algorithms for understanding time-series data such as hidden Markov models and time-structure based independent component analysis. MSMBuilder boasts an easy to use command-line interface, as well as clear and consistent abstractions through its Python application programming interface. MSMBuilder was developed with careful consideration for compatibility with the broader machine learning community by following the design of scikit-learn. The package is used primarily by practitioners of molecular dynamics, but is just as applicable to other computational or experimental time-series measurements.     

Subharmonics and Chaos in Simple Periodically Forced Biomolecular Models

This article uncovers a remarkable behavior in two biochemical systems that commonly appear as components of signal transduction pathways in systems biology. These systems have globally attracting steady states when unforced, so they might have been considered uninteresting from a dynamical standpoint. However, when subject to a periodic excitation, strange attractors arise via a period-doubling cascade. Quantitative analyses of the corresponding discrete chaotic trajectories are conducted numerically by computing largest Lyapunov exponents, power spectra, and autocorrelation functions. To gain insight into the geometry of the strange attractors, the phase portraits of the corresponding iterated maps are interpreted as scatter plots for which marginal distributions are additionally evaluated. The lack of entrainment to external oscillations, in even the simplest biochemical networks, represents a level of additional complexity in molecular biology, which has previously been insufficiently recognized but is plausibly biologically important.     

Simulation of Cell Patterning Triggered by Cell Death and Differential Adhesion in Drosophila Wing

The Drosophila wing exhibits a well-ordered cell pattern, especially along the posterior margin, where hair cells are arranged in a zigzag pattern in the lateral view. Based on an experimental result observed during metamorphosis of Drosophila, we considered that a pattern of initial cells autonomously develops to the zigzag pattern through cell differentiation, intercellular communication, and cell death (apoptosis) and performed computer simulations of a cell-based model of vertex dynamics for tissues. The model describes the epithelial tissue as a monolayer cell sheet of polyhedral cells. Their vertices move according to equations of motion, minimizing the sum total of the interfacial and elastic energies of cells. The interfacial energy densities between cells are introduced consistently with an ideal zigzag cell pattern, extracted from the experimental result. The apoptosis of cells is modeled by gradually reducing their equilibrium volume to zero and by assuming that the hair cells prohibit neighboring cells from undergoing apoptosis. Based on experimental observations, we also assumed wing elongation along the proximal-distal axis. Starting with an initial cell pattern similar to the micrograph experimentally obtained just before apoptosis, we carried out the simulations according to the model mentioned above and successfully reproduced the ideal zigzag cell pattern. This elucidates a physical mechanism of patterning triggered by cell apoptosis theoretically and exemplifies, to our knowledge, a new framework to study apoptosis-induced patterning. We conclude that the zigzag cell pattern is formed by an autonomous communicative process among the participant cells.