From Birds to Bacteria: Generalised Velocity Jump Processes with Resting States.

There are various cases of animal movement where behaviour broadly switches between two modes of operation, corresponding to a long-distance movement state and a resting or local movement state. Here, a mathematical description of this process is formulated, adapted from Friedrich et al. (Phys Rev E, 74:041103, 2006b). The approach allows the specification any running or waiting time distribution along with any angular and speed distributions. The resulting system of integro-partial differential equations is tumultuous, and therefore, it is necessary to both simplify and derive summary statistics. An expression for the mean squared displacement is derived, which shows good agreement with experimental data from the bacterium Escherichia coli and the gull Larus fuscus. Finally, a large time diffusive approximation is considered via a Cattaneo approximation (Hillen in Discrete Continuous Dyn Syst Ser B, 5:299-318, 2003). This leads to the novel result that the effective diffusion constant is dependent on the mean and variance of the running time distribution but only on the mean of the waiting time distribution.

Thymic epithelial cell fate and potency in early organogenesis assessed by single cell transcriptional and functional analysis.

During development, cortical (c) and medullary (m) thymic epithelial cells (TEC) arise from the third pharyngeal pouch endoderm. Current models suggest that within the thymic primordium most TEC exist in a bipotent/common thymic epithelial progenitor cell (TEPC) state able to generate both cTEC and mTEC, at least until embryonic day 12.5 (E12.5) in the mouse. This view, however, is challenged by recent transcriptomics and genetic evidence. We therefore set out to investigate the fate and potency of TEC in the early thymus. Here using single cell (sc) RNAseq we identify a candidate mTEC progenitor population at E12.5, consistent with recent reports. Via lineage-tracing we demonstrate this population as mTEC fate-restricted, validating our bioinformatics prediction. Using potency analyses we also establish that most E11.5 and E12.5 progenitor TEC are cTEC-fated. Finally we show that overnight culture causes most if not all E12.5 cTEC-fated TEPC to acquire functional bipotency, and provide a likely molecular mechanism for this changed differentiation potential. Collectively, our data overturn the widely held view that a common TEPC predominates in the E12.5 thymus, showing instead that sublineage-primed progenitors are present from the earliest stages of thymus organogenesis but that these early fetal TEPC exhibit cell-fate plasticity in response to extrinsic factors. Our data provide a significant advance in the understanding of fetal thymic epithelial development and thus have implications for thymus-related clinical research, in particular research focussed on generating TEC from pluripotent stem cells.

Modeling Osteocyte Network Formation: Healthy and Cancerous Environments.

Advanced cancers, such as prostate and breast cancers, commonly metastasize to bone. In the bone matrix, dendritic osteocytes form a spatial network allowing communication between osteocytes and the osteoblasts located on the bone surface. This communication network facilitates coordinated bone remodeling. In the presence of a cancerous microenvironment, the topology of this network changes. In those situations, osteocytes often appear to be either overdifferentiated (i.e., there are more dendrites than healthy bone) or underdeveloped (i.e., dendrites do not fully form). In addition to structural changes, histological sections from metastatic breast cancer xenografted mice show that number of osteocytes per unit area is different between healthy bone and cancerous bone. We present a stochastic agent-based model for bone formation incorporating osteoblasts and osteocytes that allows us to probe both network structure and density of osteocytes in bone. Our model both allows for the simulation of our spatial network model and analysis of mean-field equations in the form of integro-partial differential equations. We considered variations of our model to study specific physiological hypotheses related to osteoblast differentiation; for example predicting how changing biological parameters, such as rates of bone secretion, rates of cancer formation, and rates of osteoblast differentiation can allow for qualitatively different network topologies. We then used our model to explore how commonly applied therapies such as bisphosphonates (e.g., zoledronic acid) impact osteocyte network formation.

On the figure of elastic planets I: gravitational collapse and infinitely many equilibria.

A classic problem of elasticity is to determine the possible equilibria of an elastic planet modelled as a homogeneous compressible spherical elastic body subject to its own gravitational field. In the absence of gravity, the initial radius is given and the density is constant. With gravity and for small planets, the elastic deformations are small enough so that the spherical equilibria can be readily obtained by using the theory of linear elasticity. For larger or denser planets, large deformations occur and the general theory of nonlinear elasticity is required to obtain the solution. Depending on the elastic model, we show that there may be parameter regimes where there exist no equilibrium or arbitrarily many equilibria. Yet, at most two of them are dynamically stable with respect to radial disturbances. In some of these models, there is a critical initial radius at which spherical solutions cease to exist. For planets with larger initial radii, there is no spherical solution as the elastic forces are not sufficient to balance the gravitational force. Therefore, the system undergoes gravitational collapse, an unexpected phenomenon within the framework of classical continuum mechanics.

Simulated ablation for detection of cells impacting paracrine signalling in histology analysis.

Intra-tumour phenotypic heterogeneity limits accuracy of clinical diagnostics and hampers the efficiency of anti-cancer therapies. Dealing with this cellular heterogeneity requires adequate understanding of its sources, which is extremely difficult, as phenotypes of tumour cells integrate hardwired (epi)mutational differences with the dynamic responses to microenvironmental cues. The later comes in form of both direct physical interactions, as well as inputs from gradients of secreted signalling molecules. Furthermore, tumour cells can not only receive microenvironmental cues, but also produce them. Despite high biological and clinical importance of understanding spatial aspects of paracrine signaling, adequate research tools are largely lacking. Here, a partial differential equation (PDE)-based mathematical model is developed that mimics the process of cell ablation. This model suggests how each cell might contribute to the microenvironment by either absorbing or secreting diffusible factors, and quantifies the extent to which observed intensities can be explained via diffusion-mediated signalling. The model allows for the separation of phenotypic responses to signalling gradients within tumour microenvironments from the combined influence of responses mediated by direct physical contact and hardwired (epi)genetic differences. The method is applied to a multi-channel immunofluorescence in situ hybridisation (iFISH)-stained breast cancer histological specimen, and correlations are investigated between: HER2 gene amplification, HER2 protein expression and cell interaction with the diffusible microenvironment. This approach allows partial deconvolution of the complex inputs that shape phenotypic heterogeneity of tumour cells and identifies cells that significantly impact gradients of signalling molecules.

DNA double-strand break repair: a theoretical framework and its application.

DNA double-strand breaks (DSBs) are formed as a result of genotoxic insults, such as exogenous ionizing radiation, and are among the most serious types of DNA damage. One of the earliest molecular responses following DSB formation is the phosphorylation of the histone H2AX, giving rise to γH2AX. Many copies of γH2AX are generated at DSBs and can be detected in vitro as foci using well-established immuno-histochemical methods. It has previously been shown that anti-γH2AX antibodies, modified by the addition of the cell-penetrating peptide TAT and a fluorescent or radionuclide label, can be used to visualize and quantify DSBs in vivo. Moreover, when labelled with a high amount of the short-range, Auger electron-emitting radioisotope, (111)In, the amount of DNA damage within a cell can be increased, leading to cell death. In this report, we develop a mathematical model that describes how molecular processes at individual sites of DNA damage give rise to quantifiable foci. Equations that describe stochastic mean behaviours at individual DSB sites are derived and parametrized using population-scale, time-series measurements from two different cancer cell lines. The model is used to examine two case studies in which the introduction of an antibody (anti-γH2AX-TAT) that targets a key component in the DSB repair pathway influences system behaviour. We investigate: (i) how the interaction between anti-γH2AX-TAT and γH2AX effects the kinetics of H2AX phosphorylation and DSB repair and (ii) model behaviour when the anti-γH2AX antibody is labelled with Auger electron-emitting (111)In and can thus instigate additional DNA damage. This work supports the conclusion that DSB kinetics are largely unaffected by the introduction of the anti-γH2AX antibody, a result that has been validated experimentally, and hence the hypothesis that the use of anti-γH2AX antibody to quantify DSBs does not violate the image tracer principle. Moreover, it provides a novel model of DNA damage accumulation in the presence of Auger electron-emitting (111)In that is supported qualitatively by the available experimental data.