PDE models of adder mechanisms in cellular proliferation

Cell division is a process that involves many biochemical steps and complex biophysical mechanisms. To simplify the understanding of what triggers cell division, three basic models that subsume more microscopic cellular processes associated with cell division have been proposed. Cells can divide based on the time elapsed since their birth, their size, and/or the volume added since their birth-the timer, sizer, and adder models, respectively. Here, we propose unified adder-sizer models and investigate some of the properties of different adder processes arising in cellular proliferation. Although the adder-sizer model provides a direct way to model cell population structure, we illustrate how it is mathematically related to the well-known model in which cell division depends on age and size. Existence and uniqueness of weak solutions to our 2+1-dimensional PDE model are proved, leading to the convergence of the discretized numerical solutions and allowing us to numerically compute the dynamics of cell population densities. We then generalize our PDE model to incorporate recent experimental findings of a system exhibiting mother-daughter correlations in cellular growth rates. Numerical experiments illustrating possible average cell volume blowup and the dynamical behavior of cell populations with mother-daughter correlated growth rates are carried out. Finally, motivated by new experimental findings, we extend our adder model cases where the controlling variable is the added size between DNA replication initiation points in the cell cycle

Kinetic theories of state- and generation-dependent cell populations

We formulate a general, high-dimensional kinetic theory describing the internal state (such as gene expression or protein levels) of cells in a stochastically evolving population. The resolution of our kinetic theory also allows one to track subpopulations associated with each generation. Both intrinsic noise of the cell's internal attribute and randomness in a cell's division times (demographic stochasticity) are fundamental to the development of our model. Based on this general framework, we are able to marginalize the high-dimensional kinetic PDEs in a number of different ways to derive equations that describe the dynamics of marginalized or "macroscopic" quantities such as structured population densities, moments of generation-dependent cellular states, and moments of the total population. We also show how nonlinear "interaction" terms in lower-dimensional integrodifferential equations can arise from high-dimensional linear kinetic models that contain rate parameters of a cell (birth and death rates) that depend on variables associated with other cells, generating couplings in the dynamics. Our analysis provides a general, more complete mathematical framework that resolves the coevolution of cell populations and cell states. The approach may be tailored for studying, e.g., gene expression in developing tissues, or other more general particle systems which exhibit Brownian noise in individual attributes and population-level demographic noise.Comment: 13 pages, 1 figure, and 7 page mathematical appendi

A frequency-dependent pp-adaptive technique for spectral methods

When using spectral methods, a question arises as how to determine the expansion order, especially for time-dependent problems in which emerging oscillations may require adjusting the expansion order. In this paper, we propose a frequency-dependent $p$-adaptive technique that adaptively adjusts the expansion order based on a frequency indicator. Using this $p$-adaptive technique, combined with recently proposed scaling and moving techniques, we are able to devise an adaptive spectral method in unbounded domains that can capture and handle diffusion, advection, and oscillations. As an application, we use this adaptive spectral method to numerically solve the Schr\"{o}dinger equation in the whole domain and successfully capture the solution's oscillatory behavior at infinity

Why case fatality ratios can be misleading: individual- and population-based mortality estimates and factors influencing them

Different ways of calculating mortality ratios during epidemics have yielded very different results, particularly during the current COVID-19 pandemic. We formulate both a survival probability model and an associated infection duration-dependent SIR model to define individual- and population-based estimates of dynamic mortality ratios. The key parameters that affect the dynamics of the different mortality estimates are the incubation period and the time individuals were infected before confirmation of infection. We stress that none of these ratios are accurately represented by the often misinterpreted case fatality ratio (CFR), the number of deaths to date divided by the total number of confirmed infected cases to date. Using data on the recent SARS-CoV-2 outbreaks, we estimate and compare the different dynamic mortality ratios and highlight their differences. Informed by our modeling, we propose more systematic methods to determine mortality ratios during epidemic outbreaks and discuss sensitivity to confounding effects and uncertainties in the data.Comment: 17 pp, 6 figures + Supplementary Informatio

A Spectral Approach for Learning Spatiotemporal Neural Differential Equations

Rapidly developing machine learning methods has stimulated research interest in computationally reconstructing differential equations (DEs) from observational data which may provide additional insight into underlying causative mechanisms. In this paper, we propose a novel neural-ODE based method that uses spectral expansions in space to learn spatiotemporal DEs. The major advantage of our spectral neural DE learning approach is that it does not rely on spatial discretization, thus allowing the target spatiotemporal equations to contain long range, nonlocal spatial interactions that act on unbounded spatial domains. Our spectral approach is shown to be as accurate as some of the latest machine learning approaches for learning PDEs operating on bounded domains. By developing a spectral framework for learning both PDEs and integro-differential equations, we extend machine learning methods to apply to unbounded DEs and a larger class of problems.Comment: 21 pages, 5 figure

Learning unbounded-domain spatiotemporal differential equations using adaptive spectral methods

Rapidly developing machine learning methods have stimulated research interest in computationally reconstructing differential equations (DEs) from observational data, providing insight into the underlying mechanistic models. In this paper, we propose a new neural-ODE-based method that spectrally expands the spatial dependence of solutions to learn the spatiotemporal DEs they obey. Our spectral spatiotemporal DE learning method has the advantage of not explicitly relying on spatial discretization (e.g., meshes or grids), thus allowing reconstruction of DEs that may be defined on unbounded spatial domains and that may contain long-ranged, nonlocal spatial interactions. By combining spectral methods with the neural ODE framework, our proposed spectral DE method addresses the inverse-type problem of reconstructing spatiotemporal equations in unbounded domains. Even for bounded domain problems, our spectral approach is as accurate as some of the latest machine learning approaches for learning or numerically solving partial differential equations (PDEs). By developing a spectral framework for reconstructing both PDEs and partial integro-differential equations (PIDEs), we extend dynamical reconstruction approaches to a wider range of problems, including those in unbounded domains