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Lindwall G, Gerlee P. Bayesian inference on the Allee effect in cancer cell line populations using time-lapse microscopy images. J Theor Biol 2023; 574:111624. [PMID: 37769802 DOI: 10.1016/j.jtbi.2023.111624] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/30/2023] [Revised: 09/08/2023] [Accepted: 09/13/2023] [Indexed: 10/03/2023]
Abstract
The Allee effect describes the phenomenon that the per capita reproduction rate increases along with the population density at low densities. Allee effects have been observed at all scales, including in microscopic environments where individual cells are taken into account. This is great interest to cancer research, as understanding critical tumour density thresholds can inform treatment plans for patients. In this paper, we introduce a simple model for cell division in the case where the cancer cell population is modelled as an interacting particle system. The rate of the cell division is dependent on the local cell density, introducing an Allee effect. We perform parameter inference of the key model parameters through Markov Chain Monte Carlo, and apply our procedure to two image sequences from a cervical cancer cell line. The inference method is verified on in silico data to accurately identify the key parameters, and results on the in vitro data strongly suggest an Allee effect.
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Analytical Solutions of Microplastic Particles Dispersion Using a Lotka–Volterra Predator–Prey Model with Time-Varying Intraspecies Coefficients. MATHEMATICAL AND COMPUTATIONAL APPLICATIONS 2022. [DOI: 10.3390/mca27040066] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
Discarded plastic is subjected to weather effects from different ecosystems and becomes microplastic particles. Due to their small size, they have spread across the planet. Their presence in living organisms can have several harmful consequences, such as altering the interaction between prey and predator. Huang et al. successfully modeled this system presenting numerical results of ecological relevance. Here, we have rewritten their equations and solved a set of them analytically, confirming that microplastic particles accumulate faster in predators than in prey and calculating the time values from which it happens. Using these analytical solutions, we have retrieved the Lotka–Volterra predator–prey model with time-varying intraspecific coefficients, allowing us to interpret ecological quantities referring to microplastics dispersion. After validating our equations, we solved analytically particular situations of ecological interest, characterized by extreme effects on predatory performance, and proposed a second-order differential equation as a possible next step to address this model. Our results open space for further refinement in the study of predator–prey models under the effects of microplastic particles, either exploring the second-order equation that we propose or modify the Huang et al. model to reduce the number of parameters, embedding in the time-varying intraspecies coefficients all the adverse effects caused by microplastic particles.
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Nakamura GM, Martinez AS. Hamiltonian dynamics of the SIS epidemic model with stochastic fluctuations. Sci Rep 2019; 9:15841. [PMID: 31676857 PMCID: PMC6825157 DOI: 10.1038/s41598-019-52351-x] [Citation(s) in RCA: 17] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/04/2019] [Accepted: 10/11/2019] [Indexed: 12/03/2022] Open
Abstract
Empirical records of epidemics reveal that fluctuations are important factors for the spread and prevalence of infectious diseases. The exact manner in which fluctuations affect spreading dynamics remains poorly known. Recent analytical and numerical studies have demonstrated that improved differential equations for mean and variance of infected individuals reproduce certain regimes of the SIS epidemic model. Here, we show they form a dynamical system that follows Hamilton’s equations, which allow us to understand the role of fluctuations and their effects on epidemics. Our findings show the Hamiltonian is a constant of motion for large population sizes. For small populations, finite size effects break the temporal symmetry and induce a power-law decay of the Hamiltonian near the outbreak onset, with a parameter-free exponent. Away from the onset, the Hamiltonian decays exponentially according to a constant relaxation time, which we propose as a metric when fluctuations cannot be neglected.
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Affiliation(s)
- Gilberto M Nakamura
- Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP), Universidade de São Paulo, Avenida Bandeirantes 3900, 14040-901, Ribeirão Preto, Brazil. .,Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos (INCT-SC), 22460-320, Rio de Janeiro, Brazil. .,Laboratoire d'Imagerie et Modélisation en Neurobiologie et Cancérologie (IMNC), Centre National de la Recherche Scientifique (CNRS), UMR 8165, Universités Paris 11 and Paris 7, Campus d'Orsay, 91405, Orsay, France.
| | - Alexandre S Martinez
- Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP), Universidade de São Paulo, Avenida Bandeirantes 3900, 14040-901, Ribeirão Preto, Brazil.,Instituto Nacional de Ciência e Tecnologia - Sistemas Complexos (INCT-SC), 22460-320, Rio de Janeiro, Brazil
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Johnson KE, Howard G, Mo W, Strasser MK, Lima EABF, Huang S, Brock A. Cancer cell population growth kinetics at low densities deviate from the exponential growth model and suggest an Allee effect. PLoS Biol 2019; 17:e3000399. [PMID: 31381560 PMCID: PMC6695196 DOI: 10.1371/journal.pbio.3000399] [Citation(s) in RCA: 51] [Impact Index Per Article: 10.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/19/2019] [Revised: 08/15/2019] [Accepted: 07/08/2019] [Indexed: 12/30/2022] Open
Abstract
Most models of cancer cell population expansion assume exponential growth kinetics at low cell densities, with deviations to account for observed slowing of growth rate only at higher densities due to limited resources such as space and nutrients. However, recent preclinical and clinical observations of tumor initiation or recurrence indicate the presence of tumor growth kinetics in which growth rates scale positively with cell numbers. These observations are analogous to the cooperative behavior of species in an ecosystem described by the ecological principle of the Allee effect. In preclinical and clinical models, however, tumor growth data are limited by the lower limit of detection (i.e., a measurable lesion) and confounding variables, such as tumor microenvironment, and immune responses may cause and mask deviations from exponential growth models. In this work, we present alternative growth models to investigate the presence of an Allee effect in cancer cells seeded at low cell densities in a controlled in vitro setting. We propose a stochastic modeling framework to disentangle expected deviations due to small population size stochastic effects from cooperative growth and use the moment approach for stochastic parameter estimation to calibrate the observed growth trajectories. We validate the framework on simulated data and apply this approach to longitudinal cell proliferation data of BT-474 luminal B breast cancer cells. We find that cell population growth kinetics are best described by a model structure that considers the Allee effect, in that the birth rate of tumor cells increases with cell number in the regime of small population size. This indicates a potentially critical role of cooperative behavior among tumor cells at low cell densities with relevance to early stage growth patterns of emerging and relapsed tumors.
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Affiliation(s)
- Kaitlyn E. Johnson
- Department of Biomedical Engineering, The University of Texas at Austin, Austin, Texas, United States of America
| | - Grant Howard
- Department of Biomedical Engineering, The University of Texas at Austin, Austin, Texas, United States of America
| | - William Mo
- Department of Biomedical Engineering, The University of Texas at Austin, Austin, Texas, United States of America
| | - Michael K. Strasser
- Institute for Systems Biology, Seattle, Washington, United States of America
| | - Ernesto A. B. F. Lima
- Institute for Computation Engineering and Sciences, The University of Texas at Austin, Austin, Texas, United States of America
| | - Sui Huang
- Institute for Systems Biology, Seattle, Washington, United States of America
| | - Amy Brock
- Department of Biomedical Engineering, The University of Texas at Austin, Austin, Texas, United States of America
- Department of Oncology, Livestrong Cancer Institute, Dell Medical School, The University of Texas at Austin, Austin, Texas, United States of America
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Dornelas V, Colombo EH, Anteneodo C. Single-species fragmentation: The role of density-dependent feedback. Phys Rev E 2019; 99:062225. [PMID: 31330753 DOI: 10.1103/physreve.99.062225] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2019] [Indexed: 11/07/2022]
Abstract
Internal feedback is commonly present in biological populations and can play a crucial role in the emergence of collective behavior. To describe the temporal evolution of the distribution of a single-species population, we consider a generalization of the Fisher-KPP equation. This equation includes the elementary processes of random motion, reproduction, and, importantly, nonlocal interspecific competition, which introduces a spatial scale of interaction. In addition, we take into account feedback mechanisms in diffusion and growth processes, mimicked by power-law density dependencies. This feedback includes, for instance, anomalous diffusion, reaction to overcrowding or to the rarefaction of the population, as well as Allee-like effects. We show that, depending on the kind of feedback that takes place, the population can self-organize splitting into disconnected subpopulations, in the absence of external constraints. Through extensive numerical simulations, we investigate the temporal evolution and the characteristics of the stationary population distribution in the one-dimensional case. We discuss the crucial role that density-dependence has on pattern formation, particularly on fragmentation, which can bring important consequences to processes such as epidemic spread and speciation.
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Affiliation(s)
- V Dornelas
- Department of Physics, PUC-Rio, Rua Marquês de São Vicente, 225, 22451-900, Rio de Janeiro, Brazil
| | - E H Colombo
- IFISC (CSIC-UIB), Campus Universitat Illes Balears, 07122, Palma de Mallorca, Spain
| | - C Anteneodo
- Department of Physics, PUC-Rio, Rua Marquês de São Vicente, 225, 22451-900, Rio de Janeiro, Brazil.,Institute of Science and Technology for Complex Systems, Rio de Janeiro, Brazil
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Ribeiro FL, Dos Santos RV, Mata AS. Fractal dimension and universality in avascular tumor growth. Phys Rev E 2017; 95:042406. [PMID: 28505817 DOI: 10.1103/physreve.95.042406] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/07/2016] [Indexed: 11/07/2022]
Abstract
For years, the comprehension of the tumor growth process has been intriguing scientists. New research has been constantly required to better understand the complexity of this phenomenon. In this paper, we propose a mathematical model that describes the properties, already known empirically, of avascular tumor growth. We present, from an individual-level (microscopic) framework, an explanation of some phenomenological (macroscopic) aspects of tumors, such as their spatial form and the way they develop. Our approach is based on competitive interaction between the cells. This simple rule makes the model able to reproduce evidence observed in real tumors, such as exponential growth in their early stage followed by power-law growth. The model also reproduces (i) the fractal-space distribution of tumor cells and (ii) the universal growth behavior observed in both animals and tumors. Our analyses suggest that the universal similarity between tumor and animal growth comes from the fact that both can be described by the same dynamic equation-the Bertalanffy-Richards model-even if they do not necessarily share the same biological properties.
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Affiliation(s)
- Fabiano L Ribeiro
- Departamento de Física, Universidade Federal de Lavras, 37200-000 Lavras, MG, Brazil
| | | | - Angélica S Mata
- Departamento de Física, Universidade Federal de Lavras, 37200-000 Lavras, MG, Brazil
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Ribeiro FL, Meirelles J, Ferreira FF, Neto CR. A model of urban scaling laws based on distance dependent interactions. ROYAL SOCIETY OPEN SCIENCE 2017; 4:160926. [PMID: 28405381 PMCID: PMC5383838 DOI: 10.1098/rsos.160926] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/24/2016] [Accepted: 02/21/2017] [Indexed: 06/07/2023]
Abstract
Socio-economic related properties of a city grow faster than a linear relationship with the population, in a log-log plot, the so-called superlinear scaling. Conversely, the larger a city, the more efficient it is in the use of its infrastructure, leading to a sublinear scaling on these variables. In this work, we addressed a simple explanation for those scaling laws in cities based on the interaction range between the citizens and on the fractal properties of the cities. To this purpose, we introduced a measure of social potential which captured the influence of social interaction on the economic performance and the benefits of amenities in the case of infrastructure offered by the city. We assumed that the population density depends on the fractal dimension and on the distance-dependent interactions between individuals. The model suggests that when the city interacts as a whole, and not just as a set of isolated parts, there is improvement of the socio-economic indicators. Moreover, the bigger the interaction range between citizens and amenities, the bigger the improvement of the socio-economic indicators and the lower the infrastructure costs of the city. We addressed how public policies could take advantage of these properties to improve cities development, minimizing negative effects. Furthermore, the model predicts that the sum of the scaling exponents of social-economic and infrastructure variables are 2, as observed in the literature. Simulations with an agent-based model are confronted with the theoretical approach and they are compatible with the empirical evidences.
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Affiliation(s)
- Fabiano L. Ribeiro
- Departamento de Física (DFI), Universidade Federal de Lavras (UFLA), Caixa Postal 3037, 37200-000 Lavras, Minas Gerais, Brazil
| | - Joao Meirelles
- Laboratory on Human-Environment Relations in Urban Systems—HERUS, École polytechnique fédérale de Lausanne (EPFL) Station 2, 1015 Lausanne, Switzerland
| | - Fernando F. Ferreira
- EACH—Universidade de São Paulo (USP), Av. Arlindo Bettio, 1000 (Vila Guaraciaba), 03828-000 São Paulo, SP, Brazil
| | - Camilo Rodrigues Neto
- EACH—Universidade de São Paulo (USP), Av. Arlindo Bettio, 1000 (Vila Guaraciaba), 03828-000 São Paulo, SP, Brazil
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