A Five Dimensional Perspective on Many Particles in the Snyder basis of Double Special Relativity

After a brief summary of Double Special Relativity (DSR), we concentrate on a five dimensional procedure, which consistently introduce coordinates and momenta in the corresponding four-dimensional phase space, via a Hamiltonian approach. For the one particle case, the starting point is a de Sitter momentum space in five dimensions, with an additional constraint selected to recover the mass shell condition in four dimensions. Different basis of DSR can be recovered by selecting specific gauges to define the reduced four dimensional degrees of freedom. This is shown for the Snyder basis in the one particle case. We generalize the method to the many particles case and apply it again to this basis. We show that the energy and momentum of the system, given by the dynamical variables that are generators of translations in space and time and which close the Poincar\'e algebra, are additive magnitudes. From this it results that the rest energy (mass) of a composite object does not have an upper limit, as opposed to a single component particle which does.Comment: 12 pages, no figures, AIP Conf. Pro

Rotationally resolved spectroscopy of (20000) Varuna in the near-infrared

Models of the escape and retention of volatiles by minor icy objects exclude any presence of volatile ices on the surface of TNOs smaller than ~1000km in diameter at the typical temperature in this region of the solar system, whereas the same models show that water ice is stable on the surface of objects over a wide range of diameters. Collisions and cometary activity have been used to explain the process of surface refreshing of TNOs and Centaurs. These processes can produce surface heterogeneity that can be studied by collecting information at different rotational phases. The aims of this work are to study the surface composition of (20000)Varuna, a TNO with a diameter ~650km and to search for indications of rotational variability. We observed Varuna during two consecutive nights in January 2011 with NICS@TNG obtaining a set of spectra covering the whole rotation period of Varuna. After studying the spectra corresponding to different rotational phases, we did not find any indication of surface variability. In all the spectra, we detect an absorption at 2{\mu}m, suggesting the presence of water ice on the surface. We do not detect any other volatiles on the surface, although the S/N is not high enough to discard their presence. Based on scattering models, we present two possible compositions compatible with our set of data and discuss their implications in the frame of the collisional history of the Kuiper Belt. We find that the most probable composition for the surface of Varuna is a mixture of amorphous silicates, complex organics, and water ice. This composition is compatible with all the materials being primordial. However, our data can also be fitted by models containing up to a 10% of methane ice. For an object with the characteristics of Varuna, this volatile could not be primordial, so an event, such as an energetic impact, would be needed to explain its presence on the surface.Comment: 6 pages, 5 figures, to be published in A&

The Spectrum of Pluto, 0.40 - 0.93 μ\mum I. Secular and longitudinal distribution of ices and complex organics

Context. During the last 30 years the surface of Pluto has been characterized, and its variability has been monitored, through continuous near-infrared spectroscopic observations. But in the visible range only few data are available. Aims. The aim of this work is to define the Pluto's relative reflectance in the visible range to characterize the different components of its surface, and to provide ground based observations in support of the New Horizons mission. Methods. We observed Pluto on six nights between May and July 2014, with the imager/spectrograph ACAM at the William Herschel Telescope (La Palma, Spain). The six spectra obtained cover a whole rotation of Pluto (Prot = 6.4 days). For all the spectra we computed the spectral slope and the depth of the absorption bands of methane ice between 0.62 and 0.90 $\mu$m. To search for shifts of the center of the methane bands, associated with dilution of CH4 in N2, we compared the bands with reflectances of pure methane ice. Results. All the new spectra show the methane ice absorption bands between 0.62 and 0.90 $\mu$m. The computation of the depth of the band at 0.62 $\mu$m in the new spectra of Pluto, and in the spectra of Makemake and Eris from the literature, allowed us to estimate the Lambert coefficient at this wavelength, at a temperature of 30 K and 40 K, never measured before. All the detected bands are blue shifted, with minimum shifts in correspondence with the regions where the abundance of methane is higher. This could be indicative of a dilution of CH4:N2 more saturated in CH4. The longitudinal and secular variations of the parameters measured in the spectra are in accordance with results previously reported in the literature and with the distribution of the dark and bright material that show the Pluto's albedo maps from New Horizons.Comment: This manuscript may change and improve during the reviewing process. The data reduction and calibration is reliable and has been checked independently using different reduction approaches. The data will be made publicily available when the paper is accepted. If you need them before, please, contact the autho

From individual-based mechanical models of multicellular systems to free-boundary problems

In this paper we present an individual-based mechanical model that describes the dynamics of two contiguous cell populations with different proliferative and mechanical characteristics. An off-lattice modelling approach is considered whereby: (i) every cell is identified by the position of its centre; (ii) mechanical interactions between cells are described via generic nonlinear force laws; and (iii) cell proliferation is contact inhibited. We formally show that the continuum counterpart of this discrete model is given by a free-boundary problem for the cell densities. The results of the derivation demonstrate how the parameters of continuum mechanical models of multicellular systems can be related to biophysical cell properties. We prove an existence result for the free-boundary problem and construct travelling-wave solutions. Numerical simulations are performed in the case where the cellular interaction forces are described by the celebrated Johnson-Kendalli-Roberts model of elastic contact, which has been previously used to model cell-cell interactions. The results obtained indicate excellent agreement between the simulation results for the individual-based model, the numerical solutions of the corresponding free-boundary problem and the travelling-wave analysis

Evolutionary Dynamics in Vascularised Tumours under Chemotherapy: Mathematical Modelling, Asymptotic Analysis and Numerical Simulations

We consider a mathematical model for the evolutionary dynamics of tumour cells in vascularised tumours under chemotherapy. The model comprises a system of coupled partial integro-differential equations for the phenotypic distribution of tumour cells, the concentration of oxygen and the concentration of a chemotherapeutic agent. In order to disentangle the impact of different evolutionary parameters on the emergence of intra-tumour phenotypic heterogeneity and the development of resistance to chemotherapy, we construct explicit solutions to the equation for the phenotypic distribution of tumour cells and provide a detailed quantitative characterisation of the long-time asymptotic behaviour of such solutions. Analytical results are integrated with numerical simulations of a calibrated version of the model based on biologically consistent parameter values. The results obtained provide a theoretical explanation for the observation that the phenotypic properties of tumour cells in vascularised tumours vary with the distance from the blood vessels. Moreover, we demonstrate that lower oxygen levels may correlate with higher levels of phenotypic variability, which suggests that the presence of hypoxic regions supports intra-tumour phenotypic heterogeneity. Finally, the results of our analysis put on a rigorous mathematical basis the idea, previously suggested by formal asymptotic results and numerical simulations, that hypoxia favours the selection for chemoresistant phenotypic variants prior to treatment. Consequently, this facilitates the development of resistance following chemotherapy

Bridging the gap between individual-based and continuum models of growing cell populations

Continuum models for the spatial dynamics of growing cell populations have been widely used to investigate the mechanisms underpinning tissue development and tumour invasion. These models consist of nonlinear partial differential equations that describe the evolution of cellular densities in response to pressure gradients generated by population growth. Little prior work has explored the relation between such continuum models and related single-cell-based models. We present here a simple stochastic individual-based model for the spatial dynamics of multicellular systems whereby cells undergo pressure-driven movement and pressure-dependent proliferation. We show that nonlinear partial differential equations commonly used to model the spatial dynamics of growing cell populations can be formally derived from the branching random walk that underlies our discrete model. Moreover, we carry out a systematic comparison between the individual-based model and its continuum counterparts, both in the case of one single cell population and in the case of multiple cell populations with different biophysical properties. The outcomes of our comparative study demonstrate that the results of computational simulations of the individual-based model faithfully mirror the qualitative and quantitative properties of the solutions to the corresponding nonlinear partial differential equations. Ultimately, these results illustrate how the simple rules governing the dynamics of single cells in our individual-based model can lead to the emergence of complex spatial patterns of population growth observed in continuum models

A stochastic individual-based model to explore the role of spatial interactions and antigen recognition in the immune response against solid tumours

Spatial interactions between cancer and immune cells, as well as the recognition of tumour antigens by cells of the immune system, play a key role in the immune response against solid tumours. The existing mathematical models generally focus only on one of these key aspects. We present here a spatial stochastic individual-based model that explicitly captures antigen expression and recognition. In our model, each cancer cell is characterised by an antigen profile which can change over time due to either epimutations or mutations. The immune response against the cancer cells is initiated by the dendritic cells that recognise the tumour antigens and present them to the cytotoxic T cells. Consequently, T cells become activated against the tumour cells expressing such antigens. Moreover, the differences in movement between inactive and active immune cells are explicitly taken into account by the model. Computational simulations of our model clarify the conditions for the emergence of tumour clearance, dormancy or escape, and allow us to assess the impact of antigenic heterogeneity of cancer cells on the efficacy of immune action. Ultimately, our results highlight the complex interplay between spatial interactions and adaptive mechanisms that underpins the immune response against solid tumours, and suggest how this may be exploited to further develop cancer immunotherapies

Modelling the Immune Response to Cancer: An Individual-Based Approach Accounting for the Difference in Movement Between Inactive and Activated T Cells

A growing body of experimental evidence indicates that immune cells move in an unrestricted search pattern if they are in the pre-activated state, whilst they tend to stay within a more restricted area upon activation induced by the presence of tumour antigens. This change in movement is not often considered in the existing mathematical models of the interactions between immune cells and cancer cells. With the aim to fill such a gap in the existing literature, in this work we present a spatially structured individual-based model of tumour–immune competition that takes explicitly into account the difference in movement between inactive and activated immune cells. In our model, a Lévy walk is used to capture the movement of inactive immune cells, whereas Brownian motion is used to describe the movement of antigen-activated immune cells. The effects of activation of immune cells, the proliferation of cancer cells and the immune destruction of cancer cells are also modelled. We illustrate the ability of our model to reproduce qualitatively the spatial trajectories of immune cells observed in experimental data of single-cell tracking. Computational simulations of our model further clarify the conditions for the onset of a successful immune action against cancer cells and may suggest possible targets to improve the efficacy of cancer immunotherapy. Overall, our theoretical work highlights the importance of taking into account spatial interactions when modelling the immune response to cancer cells

A hybrid discrete-continuum approach to model Turing pattern formation

Since its introduction in 1952, with a further refinement in 1972 by Gierer and Meinhardt, Turing's (pre-)pattern theory (the chemical basis of morphogenesis) has been widely applied to a number of areas in developmental biology, where evolving cell and tissue structures are naturally observed. The related pattern formation models normally comprise a system of reaction-diffusion equations for interacting chemical species (morphogens), whose heterogeneous distribution in some spatial domain acts as a template for cells to form some kind of pattern or structure through, for example, differentiation or proliferation induced by the chemical pre-pattern. Here we develop a hybrid discrete-continuum modelling framework for the formation of cellular patterns via the Turing mechanism. In this framework, a stochastic individual-based model of cell movement and proliferation is combined with a reaction-diffusion system for the concentrations of some morphogens. As an illustrative example, we focus on a model in which the dynamics of the morphogens are governed by an activator-inhibitor system that gives rise to Turing pre-patterns. The cells then interact with the morphogens in their local area through either of two forms of chemically-dependent cell action: Chemotaxis and chemically-controlled proliferation. We begin by considering such a hybrid model posed on static spatial domains, and then turn to the case of growing domains. In both cases, we formally derive the corresponding deterministic continuum limit and show that that there is an excellent quantitative match between the spatial patterns produced by the stochastic individual-based model and its deterministic continuum counterpart, when sufficiently large numbers of cells are considered. This paper is intended to present a proof of concept for the ideas underlying the modelling framework, with the aim to then apply the related methods to the study of specific patterning and morphogenetic processes in the future

Expected spectral characteristics of (101955) Bennu and (162173) Ryugu, targets of the OSIRIS-REx and Hayabusa2 missions

NASA's OSIRIS-REx and JAXA's Hayabusa2 sample-return missions are currently on their way to encounter primitive near-Earth asteroids (101955) Bennu and (162173) Ryugu, respectively. Spectral and dynamical evidence indicates that these near-Earth asteroids originated in the inner part of the main belt. There are several primitive collisional families in this region, and both these asteroids are most likely to have originated in the Polana-Eulalia family complex. We present the expected spectral characteristics of both targets based on our studies of our primitive collisional families in the inner belt: Polana-Eulalia, Erigone, Sulamitis, and Clarissa. Observations were obtained in the framework of our PRIMitive Asteroids Spectroscopic Survey (PRIMASS). Our results are especially relevant to the planning and interpretation of in-situ images and spectra to be obtained by the two spacecraft during the encounters with their targets.Comment: 22 pages, 11 figures. Accepted for publication in Icarus on May 11, 201