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

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

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

SilkDB: a knowledgebase for silkworm biology and genomics

The Silkworm Knowledgebase (SilkDB) is a web-based repository for the curation, integration and study of silkworm genetic and genomic data. With the recent accomplishment of a ∼6X draft genome sequence of the domestic silkworm (Bombyx mori), SilkDB provides an integrated representation of the large-scale, genome-wide sequence assembly, cDNAs, clusters of expressed sequence tags (ESTs), transposable elements (TEs), mutants, single nucleotide polymorphisms (SNPs) and functional annotations of genes with assignments to InterPro domains and Gene Ontology (GO) terms. SilkDB also hosts a set of ESTs from Bombyx mandarina, a wild progenitor of B.mori, and a collection of genes from other Lepidoptera. Comparative analysis results between the domestic and wild silkworm, between B.mori and other Lepidoptera, and between B.mori and the two sequenced insects, fruitfly and mosquito, are displayed by using B.mori genome sequence as a reference framework. Designed as a basic platform, SilkDB strives to provide a comprehensive knowledgebase about the silkworm and present the silkworm genome and related information in systematic and graphical ways for the convenience of in-depth comparative studies. SilkDB is publicly accessible at http://silkworm.genomics.org.cn