Implementing vertex dynamics models of cell populations in biology within a consistent computational framework

The dynamic behaviour of epithelial cell sheets plays a central role during development, growth, disease and wound healing. These processes occur as a result of cell adhesion, migration, division, differentiation and death, and involve multiple processes acting at the cellular and molecular level. Computational models offer a useful means by which to investigate and test hypotheses about these processes, and have played a key role in the study of cell–cell interactions. However, the necessarily complex nature of such models means that it is difficult to make accurate comparison between different models, since it is often impossible to distinguish between differences in behaviour that are due to the underlying model assumptions, and those due to differences in the in silico implementation of the model. In this work, an approach is described for the implementation of vertex dynamics models, a discrete approach that represents each cell by a polygon (or polyhedron) whose vertices may move in response to forces. The implementation is undertaken in a consistent manner within a single open source computational framework, Chaste, which comprises fully tested, industrial-grade software that has been developed using an agile approach. This framework allows one to easily change assumptions regarding force generation and cell rearrangement processes within these models. The versatility and generality of this framework is illustrated using a number of biological examples. In each case we provide full details of all technical aspects of our model implementations, and in some cases provide extensions to make the models more generally applicable

Non-Euclidean geometry in nature

I describe the manifestation of the non-Euclidean geometry in the behavior of collective observables of some complex physical systems. Specifically, I consider the formation of equilibrium shapes of plants and statistics of sparse random graphs. For these systems I discuss the following interlinked questions: (i) the optimal embedding of plants leaves in the three-dimensional space, (ii) the spectral statistics of sparse random matrix ensembles.Comment: 52 pages, 21 figures, last section is rewritten, a reference to chaotic Hamiltonian systems is adde

Dynamic Matrix Ansatz for Integrable Reaction-Diffusion Processes

We show that the stochastic dynamics of a large class of one-dimensional interacting particle systems may be presented by integrable quantum spin Hamiltonians. Generalizing earlier work \cite{Stin95a,Stin95b} we present an alternative description of these processes in terms of a time-dependent operator algebra with quadratic relations. These relations generate the Bethe ansatz equations for the spectrum and turn the calculation of time-dependent expectation values into the problem of either finding representations of this algebra or of solving functional equations for the initial values of the operators. We use both strategies for the study of two specific models: (i) We construct a two-dimensional time-dependent representation of the algebra for the symmetric exclusion process with open boundary conditions. In this way we obtain new results on the dynamics of this system and on the eigenvectors and eigenvalues of the corresponding quantum spin chain, which is the isotropic Heisenberg ferromagnet with non-diagonal, symmetry-breaking boundary fields. (ii) We consider the non-equilibrium spin relaxation of Ising spins with zero-temperature Glauber dynamics and an additional coupling to an infinite-temperature heat bath with Kawasaki dynamics. We solve the functional equations arising from the algebraic description and show non-perturbatively on the level of all finite-order correlation functions that the coupling to the infinite-temperature heat bath does not change the late-time behaviour of the zero-temperature process. The associated quantum chain is a non-hermitian anisotropic Heisenberg chain related to the seven-vertex model.Comment: Latex, 23 pages, to appear in European Physical Journal

The Role of Intracellular Interactions in the Collective Polarization of Tissues and its Interplay with Cellular Geometry

Planar cell polarity (PCP), the coherent in-plane polarization of a tissue on multicellular length scales, provides directional information that guides a multitude of developmental processes at cellular and tissue levels. While it is manifest that cells utilize both intracellular and intercellular mechanisms, how the two produce the collective polarization remains an active area of investigation. We study the role of intracellular interactions in the large-scale spatial coherence of cell polarities, and scrutinize the role of intracellular interactions in the emergence of tissue-wide polarization. We demonstrate that nonlocal cytoplasmic interactions are necessary and sufficient for the robust long-range polarization, and are essential to the faithful detection of weak directional signals. In the presence of nonlocal interactions, signatures of geometrical information in tissue polarity become manifest. We investigate the deleterious effects of geometric disorder, and determine conditions on the cytoplasmic interactions that guarantee the stability of polarization. These conditions get progressively more stringent upon increasing the geometric disorder. Another situation where the role of geometrical information might be evident is elongated tissues. Strikingly, our model recapitulates an observed influence of tissue elongation on the orientation of polarity. Eventually, we introduce three classes of mutants: lack of membrane proteins, cytoplasmic proteins, and local geometrical irregularities. We adopt core-PCP as a model pathway, and interpret the model parameters accordingly, through comparing the in silico and in vivo phenotypes. This comparison helps us shed light on the roles of the cytoplasmic proteins in cell-cell communication, and make predictions regarding the cooperation of cytoplasmic and membrane proteins in long-range polarization.Comment: 15 pages Main Text + 8 page Appendi

The role of topology and mechanics in uniaxially growing cell networks

In biological systems, the growth of cells, tissues, and organs is influenced by mechanical cues. Locally, cell growth leads to a mechanically heterogeneous environment as cells pull and push their neighbors in a cell network. Despite this local heterogeneity, at the tissue level, the cell network is remarkably robust, as it is not easily perturbed by changes in the mechanical environment or the network connectivity. Through a network model, we relate global tissue structure (i.e. the cell network topology) and local growth mechanisms (growth laws) to the overall tissue response. Within this framework, we investigate the two main mechanical growth laws that have been proposed: stress-driven or strain-driven growth. We show that in order to create a robust and stable tissue environment, networks with predominantly series connections are naturally driven by stress-driven growth, whereas networks with predominantly parallel connections are associated with strain-driven growth

Computerized Analysis of Magnetic Resonance Images to Study Cerebral Anatomy in Developing Neonates

The study of cerebral anatomy in developing neonates is of great importance for the understanding of brain development during the early period of life. This dissertation therefore focuses on three challenges in the modelling of cerebral anatomy in neonates during brain development. The methods that have been developed all use Magnetic Resonance Images (MRI) as source data. To facilitate study of vascular development in the neonatal period, a set of image analysis algorithms are developed to automatically extract and model cerebral vessel trees. The whole process consists of cerebral vessel tracking from automatically placed seed points, vessel tree generation, and vasculature registration and matching. These algorithms have been tested on clinical Time-of- Flight (TOF) MR angiographic datasets. To facilitate study of the neonatal cortex a complete cerebral cortex segmentation and reconstruction pipeline has been developed. Segmentation of the neonatal cortex is not effectively done by existing algorithms designed for the adult brain because the contrast between grey and white matter is reversed. This causes pixels containing tissue mixtures to be incorrectly labelled by conventional methods. The neonatal cortical segmentation method that has been developed is based on a novel expectation-maximization (EM) method with explicit correction for mislabelled partial volume voxels. Based on the resulting cortical segmentation, an implicit surface evolution technique is adopted for the reconstruction of the cortex in neonates. The performance of the method is investigated by performing a detailed landmark study. To facilitate study of cortical development, a cortical surface registration algorithm for aligning the cortical surface is developed. The method first inflates extracted cortical surfaces and then performs a non-rigid surface registration using free-form deformations (FFDs) to remove residual alignment. Validation experiments using data labelled by an expert observer demonstrate that the method can capture local changes and follow the growth of specific sulcus

Graph-Theoretical Tools for the Analysis of Complex Networks

We are currently experiencing an explosive growth in data collection technology that threatens to dwarf the commensurate gains in computational power predicted by Moore’s Law. At the same time, researchers across numerous domain sciences are finding success using network models to represent their data. Graph algorithms are then applied to study the topological structure and tease out latent relationships between variables. Unfortunately, the problems of interest, such as finding dense subgraphs, are often the most difficult to solve from a computational point of view. Together, these issues motivate the need for novel algorithmic techniques in the study of graphs derived from large, complex, data sources. This dissertation describes the development and application of graph theoretic tools for the study of complex networks. Algorithms are presented that leverage efficient, exact solutions to difficult combinatorial problems for epigenetic biomarker detection and disease subtyping based on gene expression signatures. Extensive testing on publicly available data is presented supporting the efficacy of these approaches. To address efficient algorithm design, a study of the two core tenets of fixed parameter tractability (branching and kernelization) is considered in the context of a parallel implementation of vertex cover. Results of testing on a wide variety of graphs derived from both real and synthetic data are presented. It is shown that the relative success of kernelization versus branching is found to be largely dependent on the degree distribution of the graph. Throughout, an emphasis is placed upon the practicality of resulting implementations to advance the limits of effective computation