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

Made-to-measure galaxy models - II Elliptical and Lenticular Galaxies

We take a sample of 24 elliptical and lenticular galaxies previously analysed by the SAURON project using three-integral dynamical models created with Schwarzschild's method, and re-analyse them using the made-to-measure (M2M) method of dynamical modelling. We obtain good agreement between the two methods in determining the dynamical mass-to-light (M/L) ratios for the galaxies with over 80% of ratios differing by < 10% and over 95% differing by < 20%. We show that (M/L)_M2M is approximately equal to (M/L)_Sch. For the global velocity dispersion anisotropy parameter delta, we find similar values but with fewer of the made-to-measure models tangentially anisotropic by comparison with their SAURON Schwarzschild counterparts. Our investigation is the largest comparative application of the made-to-measure method to date.Comment: 14 pages, 8 figures and 5 table

Made-to measure galaxy models - I Methodology

We re-derive the made-to-measure method of Syer and Tremaine 1996, for modelling stellar systems and individual galaxies, and demonstrate how extensions to the made-to-measure method may be implemented and used. We illustrate the enhanced made-to-measure method by determining the mass-to-light ratio of a galaxy modelled as a Plummer sphere. From the standard galactic observables of surface brightness and line-of-sight velocity dispersion together with the h_4 Gauss-Hermite coefficient of the line-of-sight velocity distribution, we successfully recover the true mass-to-light ratio of our toy galaxy. Using kinematic data from Kleyna et al 2002, we then estimate the mass-to-light ratio of the dwarf spheroidal galaxy Draco achieving a V-band value of 539 \pm 136 M_{\odot} / L_{\odot}. We describe the main aspects of creating a made-to-measure galaxy model and show how the key modelling parameters may be determined.Comment: 18 pages, 15 figure

Tracking the evolution of cancer cell populations through the mathematical lens of phenotype-structured equations

Background: A thorough understanding of the ecological and evolutionary mechanisms that drive the phenotypic evolution of neoplastic cells is a timely and key challenge for the cancer research community. In this respect, mathematical modelling can complement experimental cancer research by offering alternative means of understanding the results of in vitro and in vivo experiments, and by allowing for a quick and easy exploration of a variety of biological scenarios through in silico studies. Results: To elucidate the roles of phenotypic plasticity and selection pressures in tumour relapse, we present here a phenotype-structured model of evolutionary dynamics in a cancer cell population which is exposed to the action of a cytotoxic drug. The analytical tractability of our model allows us to investigate how the phenotype distribution, the level of phenotypic heterogeneity, and the size of the cell population are shaped by the strength of natural selection, the rate of random epimutations, the intensity of the competition for limited resources between cells, and the drug dose in use. Conclusions: Our analytical results clarify the conditions for the successful adaptation of cancer cells faced with environmental changes. Furthermore, the results of our analyses demonstrate that the same cell population exposed to different concentrations of the same cytotoxic drug can take different evolutionary trajectories, which culminate in the selection of phenotypic variants characterised by different levels of drug tolerance. This suggests that the response of cancer cells to cytotoxic agents is more complex than a simple binary outcome, i.e., extinction of sensitive cells and selection of highly resistant cells. Also, our mathematical results formalise the idea that the use of cytotoxic agents at high doses can act as a double-edged sword by promoting the outgrowth of drug resistant cellular clones. Overall, our theoretical work offers a formal basis for the development of anti-cancer therapeutic protocols that go beyond the 'maximum-tolerated-dose paradigm', as they may be more effective than traditional protocols at keeping the size of cancer cell populations under control while avoiding the expansion of drug tolerant clones. Reviewers: This article was reviewed by Angela Pisco, Sébastien Benzekry and Heiko Enderling

Individual-based and continuum models of phenotypically heterogeneous growing cell populations

T.L. gratefully acknowledges support from the MIUR grant “Dipartimenti di Eccellenza 2018-2022” (Project no. E11G18000350001). F.R.M. gratefully acknowledges support from the RSE Saltire Early Career Fellowship ‘Multiscale mathematical modelling of spatial eco-evolutionary cancer dynamics’ (Fellowship No. 1879).Existing comparative studies between individual-based models of growing cell populations and their continuum counterparts have mainly been focused on homogeneous populations, in which all cells have the same phenotypic characteristics. However, significant intercellular phenotypic variability is commonly observed in cellular systems. In light of these considerations, we develop here an individual-based model for the growth of phenotypically heterogeneous cell populations. In this model, the phenotypic state of each cell is described by a structuring variable that captures intercellular variability in cell proliferation and migration rates. The model tracks the spatial evolutionary dynamics of single cells, which undergo pressure-dependent proliferation, heritable phenotypic changes and directional movement in response to pressure differentials. We formally show that the continuum limit of this model comprises a non-local partial differential equation for the cell population density function, which generalises earlier models of growing cell populations. We report on the results of numerical simulations of the individual-based model which illustrate how proliferation-migration tradeoffs shaping the evolutionary dynamics of single cells can lead to the formation, at the population level, of travelling waves whereby highly-mobile cells locally dominate at the invasive front, while more-proliferative cells are found at the rear. Moreover, we demonstrate that there is an excellent quantitative agreement between these results and the results of numerical simulations and formal travelling-wave analysis of the continuum model, when sufficiently large cell numbers are considered. We also provide numerical evidence of scenarios in which the predictions of the two models may differ due to demographic stochasticity, which cannot be captured by the continuum model. This indicates the importance of integrating individual-based and continuum approaches when modelling the growth of phenotypically heterogeneous cell populations.Publisher PDFPeer reviewe

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

Effects of an advection term in nonlocal lotka-volterra equations

Nonlocal Lotka-Volterra equations have the property that solutions concentrate as Dirac masses in the limit of small diffusion. In this paper, we show how the presence of an advection term changes the location of the concentration points in the limit of small diffusion and slow drift. The mathematical interest lies in the formalism of constrained Hamilton-Jacobi equations. Our motivations come from previous models of evolutionary dynamics in phenotype-structured populations [R.H. Chisholm, T. Lorenzi, A. Lorz, et al., Cancer Res., 75, 930-939, 2015], where the diffusion operator models the effects of heritable variations in gene expression, while the advection term models the effect of stress-induced adaptation

Cell population heterogeneity and evolution towards drug resistance in cancer: Biological and mathematical assessment, theoretical treatment optimisation

Background Drug-induced drug resistance in cancer has been attributed to diverse biological mechanisms at the individual cell or cell population scale, relying on stochastically or epigenetically varying expression of phenotypes at the single cell level, and on the adaptability of tumours at the cell population level. Scope of review We focus on intra-tumour heterogeneity, namely between-cell variability within cancer cell populations, to account for drug resistance. To shed light on such heterogeneity, we review evolutionary mechanisms that encompass the great evolution that has designed multicellular organisms, as well as smaller windows of evolution on the time scale of human disease. We also present mathematical models used to predict drug resistance in cancer and optimal control methods that can circumvent it in combined therapeutic strategies. Major conclusions Plasticity in cancer cells, i.e., partial reversal to a stem-like status in individual cells and resulting adaptability of cancer cell populations, may be viewed as backward evolution making cancer cell populations resistant to drug insult. This reversible plasticity is captured by mathematical models that incorporate between-cell heterogeneity through continuous phenotypic variables. Such models have the benefit of being compatible with optimal control methods for the design of optimised therapeutic protocols involving combinations of cytotoxic and cytostatic treatments with epigenetic drugs and immunotherapies. General significance Gathering knowledge from cancer and evolutionary biology with physiologically based mathematical models of cell population dynamics should provide oncologists with a rationale to design optimised therapeutic strategies to circumvent drug resistance, that still remains a major pitfall of cancer therapeutics. This article is part of a Special Issue entitled “System Genetics” Guest Editor: Dr. Yudong Cai and Dr. Tao Huang

From a discrete model of chemotaxis with volume-filling to a generalized Patlak–Keller–Segel model

Funding: The authors gratefully acknowledge support of the project PICS-CNRS no. 07688. F.B. acknowledges funding from the European Research Council (ERC, grant agreement No. 740623) and the Université Franco-Italienne.We present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased random walk. In order to take into account possible alterations in cellular motility observed at high cell densities (i.e. volume-filling), we let the probabilities of cell movement be modulated by a decaying function of the cell density. We formally show that a general form of the celebrated Patlak–Keller–Segel (PKS) model of chemotaxis can be formally derived as the appropriate continuum limit of this discrete model. The family of steady-state solutions of such a generalized PKS model are characterized and the conditions for the emergence of spatial patterns are studied via linear stability analysis. Moreover, we carry out a systematic quantitative comparison between numerical simulations of the discrete model and numerical solutions of the corresponding PKS model, both in one and in two spatial dimensions. The results obtained indicate that there is excellent quantitative agreement between the spatial patterns produced by the two models. Finally, we numerically show that the outcomes of the two models faithfully replicate those of the classical PKS model in a suitable asymptotic regime.PostprintPeer reviewe

From a discrete model of chemotaxis with volume-filling to a generalized Patlak–Keller–Segel model

We present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased random walk. In order to take into account possible alterations in cellular motility observed at high cell densities (i.e. volume-filling), we let the probabilities of cell movement be modulated by a decaying function of the cell density. We formally show that a general form of the celebrated Patlak–Keller–Segel (PKS) model of chemotaxis can be formally derived as the appropriate continuum limit of this discrete model. The family of steady-state solutions of such a generalized PKS model are characterized and the conditions for the emergence of spatial patterns are studied via linear stability analysis. Moreover, we carry out a systematic quantitative comparison between numerical simulations of the discrete model and numerical solutions of the corresponding PKS model, both in one and in two spatial dimensions. The results obtained indicate that there is excellent quantitative agreement between the spatial patterns produced by the two models. Finally, we numerically show that the outcomes of the two models faithfully replicate those of the classical PKS model in a suitable asymptotic regime