Numerical Analysis of the Immersed Boundary Method for Cell-Based Simulation

Mathematical modelling provides a useful framework within which to investigate the organization of biological tissues. With advances in experimental biology leading to increasingly detailed descriptions of cellular behavior, models that consider cells as individual objects are becoming a common tool to study how processes at the single-cell level affect collective dynamics and determine tissue size, shape, and function. However, there often remains no comprehensive account of these models, their method of solution, computational implementation, or analysis of parameter scaling, hindering our ability to utilize and accurately compare different models. Here we present an efficient, open-source implementation of the immersed boundary (IB) method, tailored to simulate the dynamics of cell populations. This approach considers the dynamics of elastic membranes, representing cell boundaries, immersed in a viscous Newtonian fluid. The IB method enables complex and emergent cell shape dynamics, spatially heterogeneous cell properties, and precise control of growth mechanisms. We solve the model numerically using an established algorithm, based on the fast Fourier transform, providing full details of all technical aspects of our implementation. The implementation is undertaken within Chaste, an open-source C++ library that allows one to easily change constitutive assumptions. Our implementation scales linearly with time step, and subquadratically with mesh spacing and immersed boundary node spacing. We identify the relationship between the immersed boundary node spacing and fluid mesh spacing required to ensure fluid volume conservation within immersed boundaries, and the scaling of cell membrane stiffness and cell-cell interaction strength required when refining the immersed boundary discretization. Finally, we present a simulation study of a growing epithelial tissue to demonstrate the applicability of our implementation to relevant biological questions, highlighting several features of the IB method that make it well suited to address certain questions in epithelial morphogenesis

Impact of implementation choices on quantitative predictions of cell-based computational models

‘Cell-based’ models provide a powerful computational tool for studying the mechanisms underlying the growth and dynamics of biological tissues in health and disease. An increasing amount of quantitative data with cellular resolution has paved the way for the quantitative parameterisation and validation of such models. However, the numerical implementation of cell-based models remains challenging, and little work has been done to understand to what extent implementation choices may influence model predictions. Here, we consider the numerical implementation of a popular class of cell-based models called vertex models, which are often used to study epithelial tissues. In two-dimensional vertex models, a tissue is approximated as a tessellation of polygons and the vertices of these polygons move due to mechanical forces originating from the cells. Such models have been used extensively to study the mechanical regulation of tissue topology in the literature. Here, we analyse how the model predictions may be affected by numerical parameters, such as the size of the time step, and non-physical model parameters, such as length thresholds for cell rearrangement. We find that vertex positions and summary statistics are sensitive to several of these implementation parameters. For example, the predicted tissue size decreases with decreasing cell cycle durations, and cell rearrangement may be suppressed by large time steps. These findings are counter-intuitive and illustrate that model predictions need to be thoroughly analysed and implementation details carefully considered when applying cell-based computational models in a quantitative setting

DANTSYS: a system for deterministic, neutral particle transport calculations

The THREEDANT code is the latest addition to our system of codes, DANTSYS, which perform neutral particle transport computations on a given system of interest. The system of codes is distinguished by geometrical or symmetry considerations. For example, ONEDANT and TWODANT are designed for one and two dimensional geometries respectively. We have TWOHEX for hexagonal geometries, TWODANT/GQ for arbitrary quadrilaterals in XY and RZ geometry, and THREEDANT for three-dimensional geometries. The design of this system of codes is such that they share the same input and edit module and hence the input and output is uniform for all the codes (with the obvious additions needed to specify each type of geometry). The codes in this system are also designed to be general purpose solving both eigenvalue and source driven problems. In this paper we concentrate on the THREEDANT module since there are special considerations that need to be taken into account when designing such a module. The main issues that need to be addressed in a three-dimensional transport solver are those of the computational time needed to solve a problem and the amount of storage needed to accomplish that solution. Of course both these issues are directly related to the number of spatial mesh cells required to obtain a solution to a specified accuracy, but is also related to the spatial discretization method chosen and the requirements of the iteration acceleration scheme employed as will be noted below. Another related consideration is the robustness of the resulting algorithms as implemented; because insistence on complete robustness has a significant impact upon the computation time. We address each of these issues in the following through which we give reasons for the choices we have made in our approach to this code. And this is useful in outlining how the code is evolving to better address the shortcomings that presently exist

DANTSYS/MPI: a system for 3-D deterministic transport on parallel architectures

Since 1994, we have been using a data parallel form of our deterministic transport code DANTSYS to perform time-independent fixed source and eigenvalue calculations on the CM-200`s at Los Alamos National Laboratory (LANL). Parallelization of the transport sweep is obtained by using a 2-D spatial decomposition which retains the ability to invert the source iteration equation in a single iteration (i.e., the diagonal plane sweep). We have now implemented a message passing version of DANTSYS, referred to as DANTSYS/MPI, on the Cray T3D installed at Los Alamos in 1995. By taking advantage of the SPMD (Single Program, Multiple Data) architecture of the Cray T3D, as well as its low latency communications network, we have managed to achieve grind times (time to solve a single cell in phase space) of less than 10 nanoseconds on the 512 PE (Processing Element) T3D, as opposed to typical grind times of 150-200 nanoseconds on a 2048 PE CM-200, or 300-400 nanoseconds on a single PE of a Cray Y-MP. In addition, we have also parallelized the Diffusion Synthetic Accelerator (DSA) equations which are used to accelerate the convergence of the transport equation. DANTSYS/MPI currently runs on traditional Cray PVP`s and the Cray T3D, and it`s computational kernel (Sweep3D) has been ported to and tested on an array of SGI SMP`s (Symmetric Memory Processors), a network of IBM 590 workstations, an IBM SP2, and the Intel TFLOPs machine at Sandia National Laboratory. This paper describes the implementation of DANTSYS/MPI on the Cray T3D, and presents a simple performance model which accurately predicts the grind time as a function of the number of PE`s and problem size, or scalability. This paper also describes the parallel implementation and performance of the elliptic solver used in DANTSYS/MPI for solving the synthetic acceleration equations

Time-dependent 3-D dterministic transport on parallel architectures using Dantsys/MPI

In addition to the ability to solve the static transport equation, we have also incorporated time dependence into our parallel 3-D S{sub {ital N}} code DANTSYS/MPI. Using a semi-implicit scheme, DANTSYS/MPI is capable of performing time-dependent calculations for both fissioning and pure source driven problems. We have applied this to various types of problems such as nuclear well logging and prompt fission experiments. This paper describes the form of the time- dependent equations implemented, their solution strategies in DANTSYS/MPI including iteration acceleration, and the strategies used for time-step control. Results are presented for a model nuclear well logging calculation

Topological Andr\'e-Quillen homology for cellular commutative SS-algebras

Topological Andr\'e-Quillen homology for commutative $S$-algebras was introduced by Basterra following work of Kriz, and has been intensively studied by several authors. In this paper we discuss it as a homology theory on CW $S$-algebras and apply it to obtain results on minimal atomic $p$-local $S$-algebras which generalise those of Baker and May for $p$-local spectra and simply connected spaces. We exhibit some new examples of minimal atomic $S$-algebras.Comment: Final revision, a version will appear in Abhandlungen aus dem Mathematischen Seminar der Universitaet Hambur

A clock and wavefront mechanism for somite formation

Somitogenesis, the sequential formation of a periodic pattern along the antero-posterior axis of vertebrate embryos, is one of the most obvious examples of the segmental patterning processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. In this article, we develop a mathematical formulation of a new version of the Clock and Wavefront model proposed by Pourquié and co-workers (Dubrulle, J., McGrew, M.J., Pourquié, O., 2001. FGF signalling controls somite boundary position and regulates segmentation clock control of spatiotemporal Hox gene activation. Cell 106, 219–232). Dynamic expression of FGF8 in the presomitic mesoderm constitutes the wavefront of determination which sweeps along the body axis interacting as it moves with the segmentation clock to gate cells into somites. We also show that the model can mimic the anomalies formed when progression of the wavefront is disturbed and make some experimental predictions that can be used to test the hypotheses underlying the model

The height of Denier Tournois minting in Greece (1289–1313) according to new archaeometric data

The years 1289–1313 witnessed particularly prolific minting activities at different southern and central Greek mints on behalf of different polities. The coin issues are of great economic and political relevance, and therefore of interest to modern historians. Our understanding of these is based on traditional sources, either numismatic (types and finds), or historical. This paper aims to investigate the possibilities of adding further details to the picture through archaeometry. Specifically, tournois pennies of the three main mints of the region (Clarentza, Thebes, Naupaktos) excavated at Ancient Corinth were analysed according to two different non-destructive methods, X-ray fluorescence spectrometry (XRF) and laser-induced breakdown spectroscopy (LIBS). The resulting relative silver percentages and the fingerprints of the trace elements have supported our attempts to put the different coin types in chronological order and to add detail to the context and intent for each one of these. A vivid picture of monetary production emerges. The different issuing authorities were usually intent on maintaining a decent standard while variously trying to put pressure on rivals or to harmonise their productions with their allies. All the analysed mints were commercial in character, though they were subject to the great political changes affecting Greece in this period, the ambitions of the Angevin dynasty, the various challenges which it faced in Athens, the Peloponnese and the western Mainland, and finally the destructive arrival of the Catalans. In times of need, specifically military, these same mints could therefore rely on further bullion which reached them through internal or external political channels

Subsoiling and surface tillage effects on soil physical properties and forage oat stand and yield

Much of New Zealand's agriculture integrates animal and crop production on poorly drained, easily compacted soils. We hypothesized that soil properties affecting forage oat (Avena sativa, cv Awapuni) establishment on land compacted by 15 years of conventional cropping might be influenced by various subsoiling and surface tillage combinations. Plots on a Moutoa silty clay (Typic Haplaquoil) were paraplowed (P), deep subsoiled (V), shallow subsoiled (5), or were left as non-subsoiled controls (C). Subsequently, the surface 15 cm was surface-tilled (T) using a power rotary-tiller and firmed with a Cambridge roller or were not tilled (N). Oats were then sown with a cross-slot drill. Subsoiling greatly reduced soil strength. Cone indices showed disruption to 40cm with P, 36 cm for V, and 30 cm for S. Approximately 60% of profile cone indices to a depth of 0.5 m from subsoiled treatments were less than 1.5 MPa, compared to approximately 30% for C. T slightly improved strength distribution in non-subsoiled controls but had little effect in subsoiled treatments. Subsoiling without T continued to show improved profile cone index cumulative frequency 233 days after subsoiling, Subsoiling after T in this high rainfall climate eliminated most of the separation in cumulative frequency of soil profile cone index values by two weeks after T. T reduced emergence from 142 to 113 plants per square meter and reduced yield from 5318 to 3679 kg ha-1. Forage yield increased from 3974 to 4674 kg ha-1 with subsoiling. Soil porosity, saturated and slightly unsaturated hydraulic conductivities (KSAT and K_40 ) and air permeability were highly variable but generally increased with subsoiling. Oxygen diffusion rate (ODR) (using Pt microelectrodes) was also variable, but N and C treatments had consistently lower ODRs than T or subsoiled treatments. Generally, subsoiling without T produced better soil conditions and oat crop performance than the prevailing New Zealand practice of T without subsoiling