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Freire TFA, Hu Z, Wood KB, Gjini E. Modeling spatial evolution of multi-drug resistance under drug environmental gradients. PLoS Comput Biol 2024; 20:e1012098. [PMID: 38820350 PMCID: PMC11142541 DOI: 10.1371/journal.pcbi.1012098] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/17/2023] [Accepted: 04/23/2024] [Indexed: 06/02/2024] Open
Abstract
Multi-drug combinations to treat bacterial populations are at the forefront of approaches for infection control and prevention of antibiotic resistance. Although the evolution of antibiotic resistance has been theoretically studied with mathematical population dynamics models, extensions to spatial dynamics remain rare in the literature, including in particular spatial evolution of multi-drug resistance. In this study, we propose a reaction-diffusion system that describes the multi-drug evolution of bacteria based on a drug-concentration rescaling approach. We show how the resistance to drugs in space, and the consequent adaptation of growth rate, is governed by a Price equation with diffusion, integrating features of drug interactions and collateral resistances or sensitivities to the drugs. We study spatial versions of the model where the distribution of drugs is homogeneous across space, and where the drugs vary environmentally in a piecewise-constant, linear and nonlinear manner. Although in many evolution models, per capita growth rate is a natural surrogate for fitness, in spatially-extended, potentially heterogeneous habitats, fitness is an emergent property that potentially reflects additional complexities, from boundary conditions to the specific spatial variation of growth rates. Applying concepts from perturbation theory and reaction-diffusion equations, we propose an analytical metric for characterization of average mutant fitness in the spatial system based on the principal eigenvalue of our linear problem, λ1. This enables an accurate translation from drug spatial gradients and mutant antibiotic susceptibility traits to the relative advantage of each mutant across the environment. Our approach allows one to predict the precise outcomes of selection among mutants over space, ultimately from comparing their λ1 values, which encode a critical interplay between growth functions, movement traits, habitat size and boundary conditions. Such mathematical understanding opens new avenues for multi-drug therapeutic optimization.
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Affiliation(s)
- Tomas Ferreira Amaro Freire
- Center for Computational and Stochastic Mathematics, Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal
| | - Zhijian Hu
- Departments of Biophysics and Physics, University of Michigan, United States of America
| | - Kevin B. Wood
- Departments of Biophysics and Physics, University of Michigan, United States of America
| | - Erida Gjini
- Center for Computational and Stochastic Mathematics, Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal
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2
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Freire T, Hu Z, Wood KB, Gjini E. Modeling spatial evolution of multi-drug resistance under drug environmental gradients. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2023:2023.11.16.567447. [PMID: 38014279 PMCID: PMC10680811 DOI: 10.1101/2023.11.16.567447] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/29/2023]
Abstract
Multi-drug combinations to treat bacterial populations are at the forefront of approaches for infection control and prevention of antibiotic resistance. Although the evolution of antibiotic resistance has been theoretically studied with mathematical population dynamics models, extensions to spatial dynamics remain rare in the literature, including in particular spatial evolution of multi-drug resistance. In this study, we propose a reaction-diffusion system that describes the multi-drug evolution of bacteria, based on a rescaling approach (Gjini and Wood, 2021). We show how the resistance to drugs in space, and the consequent adaptation of growth rate is governed by a Price equation with diffusion. The covariance terms in this equation integrate features of drug interactions and collateral resistances or sensitivities to the drugs. We study spatial versions of the model where the distribution of drugs is homogeneous across space, and where the drugs vary environmentally in a piecewise-constant, linear and nonlinear manner. Applying concepts from perturbation theory and reaction-diffusion equations, we propose an analytical characterization of average mutant fitness in the spatial system based on the principal eigenvalue of our linear problem. This enables an accurate translation from drug spatial gradients and mutant antibiotic susceptibility traits, to the relative advantage of each mutant across the environment. Such a mathematical understanding allows to predict the precise outcomes of selection over space, ultimately from the fundamental balance between growth and movement traits, and their diversity in a population.
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Affiliation(s)
- Tomas Freire
- Center for Computational and Stochastic Mathematics, Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal
| | - Zhijian Hu
- Departments of Biophysics and Physics, University of Michigan, USA
| | - Kevin B. Wood
- Departments of Biophysics and Physics, University of Michigan, USA
| | - Erida Gjini
- Center for Computational and Stochastic Mathematics, Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal
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3
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Martín PV, Koldaeva A, Pigolotti S. Coalescent dynamics of planktonic communities. Phys Rev E 2022; 106:044408. [PMID: 36397572 DOI: 10.1103/physreve.106.044408] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/07/2022] [Accepted: 09/19/2022] [Indexed: 06/16/2023]
Abstract
Planktonic communities are extremely diverse and include a vast number of rare species. The dynamics of these rare species is best described by individual-based models. However, individual-based approaches to planktonic diversity face substantial difficulties, due to the large number of individuals required to make realistic predictions. In this paper, we study the diversity of planktonic communities by means of a spatial coalescence model that incorporates transport by oceanic currents. As a main advantage, our approach requires simulating a number of individuals equal to the size of the sample one is interested in, rather than the size of the entire community. By theoretical analysis and simulations, we explore the conditions upon which our coalescence model is equivalent to individual-based dynamics. As an application, we use our model to predict the impact of chaotic advection by oceanic currents on biodiversity. We conclude that the coalescent approach permits one to simulate marine microbial communities much more efficiently than with individual-based models.
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Affiliation(s)
- Paula Villa Martín
- Biological Complexity Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
| | - Anzhelika Koldaeva
- Biological Complexity Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
| | - Simone Pigolotti
- Biological Complexity Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
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4
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Benzi R, Nelson DR, Shankar S, Toschi F, Zhu X. Spatial population genetics with fluid flow. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2022; 85:096601. [PMID: 35853344 DOI: 10.1088/1361-6633/ac8231] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/02/2022] [Accepted: 07/19/2022] [Indexed: 06/15/2023]
Abstract
The growth and evolution of microbial populations is often subjected to advection by fluid flows in spatially extended environments, with immediate consequences for questions of spatial population genetics in marine ecology, planktonic diversity and origin of life scenarios. Here, we review recent progress made in understanding this rich problem in the simplified setting of two competing genetic microbial strains subjected to fluid flows. As a pedagogical example we focus on antagonsim, i.e., two killer microorganism strains, each secreting toxins that impede the growth of their competitors (competitive exclusion), in the presence of stationary fluid flows. By solving two coupled reaction-diffusion equations that include advection by simple steady cellular flows composed of characteristic flow motifs in two dimensions (2D), we show how local flow shear and compressibility effects can interact with selective advantage to have a dramatic influence on genetic competition and fixation in spatially distributed populations. We analyze several 1D and 2D flow geometries including sources, sinks, vortices and saddles, and show how simple analytical models of the dynamics of the genetic interface can be used to shed light on the nucleation, coexistence and flow-driven instabilities of genetic drops. By exploiting an analogy with phase separation with nonconserved order parameters, we uncover how thesegeneticdrops harness fluid flows for novel evolutionary strategies, even in the presence of number fluctuations, as confirmed by agent-based simulations as well.
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Affiliation(s)
- Roberto Benzi
- Department of Physics and INFN, University of Rome Tor Vergata, I-00133 Rome, Italy
| | - David R Nelson
- Department of Physics, Harvard University, Cambridge, MA 02138, United States of America
| | - Suraj Shankar
- Department of Physics, Harvard University, Cambridge, MA 02138, United States of America
| | - Federico Toschi
- Department of Applied Physics, Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
- CNR-IAC, I-00185 Rome, Italy
| | - Xiaojue Zhu
- Solar and Stellar Interiors, Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3, Göttingen 37077, Germany
- Center of Mathematical Sciences and Applications, and School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, United States of America
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5
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Labavić D, Loverdo C, Bitbol AF. Hydrodynamic flow and concentration gradients in the gut enhance neutral bacterial diversity. Proc Natl Acad Sci U S A 2022; 119:e2108671119. [PMID: 34969835 PMCID: PMC8740595 DOI: 10.1073/pnas.2108671119] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 11/08/2021] [Indexed: 01/23/2023] Open
Abstract
The gut microbiota features important genetic diversity, and the specific spatial features of the gut may shape evolution within this environment. We investigate the fixation probability of neutral bacterial mutants within a minimal model of the gut that includes hydrodynamic flow and resulting gradients of food and bacterial concentrations. We find that this fixation probability is substantially increased, compared with an equivalent well-mixed system, in the regime where the profiles of food and bacterial concentration are strongly spatially dependent. Fixation probability then becomes independent of total population size. We show that our results can be rationalized by introducing an active population, which consists of those bacteria that are actively consuming food and dividing. The active population size yields an effective population size for neutral mutant fixation probability in the gut.
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Affiliation(s)
- Darka Labavić
- CNRS, Institut de Biologie Paris-Seine, Laboratoire Jean Perrin (UMR 8237), Sorbonne Université, F-75005 Paris, France
| | - Claude Loverdo
- CNRS, Institut de Biologie Paris-Seine, Laboratoire Jean Perrin (UMR 8237), Sorbonne Université, F-75005 Paris, France;
| | - Anne-Florence Bitbol
- Institute of Bioengineering, School of Life Sciences, École Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland;
- SIB Swiss Institute of Bioinformatics, CH-1015 Lausanne, Switzerland
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6
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Guccione G, Benzi R, Toschi F. Strong noise limit for population dynamics in incompressible advection. Phys Rev E 2021; 104:034421. [PMID: 34654119 DOI: 10.1103/physreve.104.034421] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/17/2020] [Accepted: 08/31/2021] [Indexed: 11/07/2022]
Abstract
Genetic diversity is at the basis of the evolution process of populations and it is responsible for the populations' degree of fitness to a particular ecosystem. In marine environments many factors play a role in determining the dynamics of a population, including the amount of nutrients, the temperature, and many other stressing factors. An important and yet rather unexplored challenge is to figure out the role of individuals' dispersion, due to flow advection, on population genetics. In this paper we focus on two populations, one of which has a slight selective advantage, advanced by an incompressible two-dimensional flow. In particular, we want to understand how this advective flow can modify the dynamics of the advantageous allele. We generalize, through a theoretical analysis, previous evidence according to which the fixation probability is independent of diffusivity, showing that this is also independent of fluid advection. These findings may have important implications in the understanding of the dynamics of a population of microorganism, such as plankton or bacteria, in marine environments under the influence of (turbulent) currents.
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Affiliation(s)
- Giorgia Guccione
- Department of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, Netherlands.,Department of Physics and INFN, University of Tor Vergata, 00133 Rome, Italy
| | - Roberto Benzi
- Department of Physics and INFN, University of Tor Vergata, 00133 Rome, Italy
| | - Federico Toschi
- Department of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, Netherlands.,CNR-IAC, 00185 Rome, Italy
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Villa Martín P, Buček A, Bourguignon T, Pigolotti S. Ocean currents promote rare species diversity in protists. SCIENCE ADVANCES 2020; 6:eaaz9037. [PMID: 32832617 PMCID: PMC7439499 DOI: 10.1126/sciadv.aaz9037] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/18/2019] [Accepted: 05/29/2020] [Indexed: 05/23/2023]
Abstract
Oceans host communities of plankton composed of relatively few abundant species and many rare species. The number of rare protist species in these communities, as estimated in metagenomic studies, decays as a steep power law of their abundance. The ecological factors at the origin of this pattern remain elusive. We propose that chaotic advection by oceanic currents affects biodiversity patterns of rare species. To test this hypothesis, we introduce a spatially explicit coalescence model that reconstructs the species diversity of a sample of water. Our model predicts, in the presence of chaotic advection, a steeper power law decay of the species abundance distribution and a steeper increase of the number of observed species with sample size. A comparison of metagenomic studies of planktonic protist communities in oceans and in lakes quantitatively confirms our prediction. Our results support that oceanic currents positively affect the diversity of rare aquatic microbes.
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Affiliation(s)
- Paula Villa Martín
- Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
| | - Aleš Buček
- Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
| | - Thomas Bourguignon
- Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
- Faculty of Tropical AgriSciences, Czech University of Life Sciences, Kamýcká 129, CZ-165 00 Prague, Czech Republic
| | - Simone Pigolotti
- Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
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8
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Turbulent coherent structures and early life below the Kolmogorov scale. Nat Commun 2020; 11:2192. [PMID: 32366844 PMCID: PMC7198613 DOI: 10.1038/s41467-020-15780-1] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/16/2019] [Accepted: 03/27/2020] [Indexed: 01/28/2023] Open
Abstract
Major evolutionary transitions, including the emergence of life, likely occurred in aqueous environments. While the role of water’s chemistry in early life is well studied, the effects of water’s ability to manipulate population structure are less clear. Population structure is known to be critical, as effective replicators must be insulated from parasites. Here, we propose that turbulent coherent structures, long-lasting flow patterns which trap particles, may serve many of the properties associated with compartments — collocalization, division, and merging — which are commonly thought to play a key role in the origins of life and other evolutionary transitions. We substantiate this idea by simulating multiple proposed metabolisms for early life in a simple model of a turbulent flow, and find that balancing the turnover times of biological particles and coherent structures can indeed enhance the likelihood of these metabolisms overcoming extinction either via parasitism or via a lack of metabolic support. Our results suggest that group selection models may be applicable with fewer physical and chemical constraints than previously thought, and apply much more widely in aqueous environments. Models of the origin of life generally require a mechanism to structure emerging populations. Here, Krieger et al. develop spatial models showing that coherent structures arising in turbulent flows in aquatic environments could have provided compartments that facilitated the origin of life.
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9
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Guccione G, Benzi R, Plummer A, Toschi F. Discrete Eulerian model for population genetics and dynamics under flow. Phys Rev E 2020; 100:062105. [PMID: 31962443 DOI: 10.1103/physreve.100.062105] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/20/2019] [Indexed: 11/07/2022]
Abstract
Marine species reproduce and compete while being advected by turbulent flows. It is largely unknown, both theoretically and experimentally, how population dynamics and genetics are changed by the presence of fluid flows. Discrete agent-based simulations in continuous space allow for accurate treatment of advection and number fluctuations, but can be computationally expensive for even modest organism densities. In this report, we propose an algorithm to overcome some of these challenges. We first provide a thorough validation of the algorithm in one and two dimensions without flow. Next, we focus on the case of weakly compressible flows in two dimensions. This models organisms such as phytoplankton living at a specific depth in the three-dimensional, incompressible ocean experiencing upwelling and/or downwelling events. We show that organisms born at sources in a two-dimensional time-independent flow experience an increase in fixation probability.
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Affiliation(s)
- Giorgia Guccione
- Department of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands and Department of Physics and INFN, University of Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Rome, Italy
| | - Roberto Benzi
- Department of Physics and INFN, University of Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Rome, Italy
| | - Abigail Plummer
- Department of Physics, Harvard University, 17 Oxford Street, Cambridge, Massachusetts 02138, USA
| | - Federico Toschi
- Department of Applied Physics, Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands and CNR-IAC, Via dei Taurini 19, I-00185 Rome, Italy
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10
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Motion, fixation probability and the choice of an evolutionary process. PLoS Comput Biol 2019; 15:e1007238. [PMID: 31381556 PMCID: PMC6746388 DOI: 10.1371/journal.pcbi.1007238] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/15/2019] [Revised: 09/16/2019] [Accepted: 07/02/2019] [Indexed: 11/21/2022] Open
Abstract
Seemingly minor details of mathematical and computational models of evolution are known to change the effect of population structure on the outcome of evolutionary processes. For example, birth-death dynamics often result in amplification of selection, while death-birth processes have been associated with suppression. In many biological populations the interaction structure is not static. Instead, members of the population are in motion and can interact with different individuals at different times. In this work we study populations embedded in a flowing medium; the interaction network is then time dependent. We use computer simulations to investigate how this dynamic structure affects the success of invading mutants, and compare these effects for different coupled birth and death processes. Specifically, we show how the speed of the motion impacts the fixation probability of an invading mutant. Flows of different speeds interpolate between evolutionary dynamics on fixed heterogeneous graphs and well-stirred populations; this allows us to systematically compare against known results for static structured populations. We find that motion has an active role in amplifying or suppressing selection by fragmenting and reconnecting the interaction graph. While increasing flow speeds suppress selection for most evolutionary models, we identify characteristic responses to flow for the different update rules we test. In particular we find that selection can be maximally enhanced or suppressed at intermediate flow speeds. Whether a mutation spreads in a population or not is one of the most important questions in biology. The evolution of cancer and antibiotic resistance, for example, are mediated by invading mutants. Recent work has shown that population structure can have important consequences for the outcome of evolution. For instance, a mutant can have a higher or a lower chance of invasion than in unstructured populations. These effects can depend on seemingly minor details of the evolutionary model, such as the order of birth and death events. Many biological populations are in motion, for example due to external stirring. Experimentally this is known to be important; the performance of mutants in E. coli populations, for example, depends on the rate of mixing. Here, we focus on simulations of populations in a flowing medium, and compare the success of a mutant for different flow speeds. We contrast different evolutionary models, and identify what features of the evolutionary model affect mutant success for different speeds of the flow. We find that the chance of mutant invasion can be at its highest (or lowest) at intermediate flow speeds, depending on the order in which birth and death events occur in the evolutionary process.
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11
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Horowitz JM, Kardar M. Bacterial range expansions on a growing front: Roughness, fixation, and directed percolation. Phys Rev E 2019; 99:042134. [PMID: 31108639 DOI: 10.1103/physreve.99.042134] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2018] [Indexed: 06/09/2023]
Abstract
Directed percolation (DP) is a classic model for nonequilibrium phase transitions into a single absorbing state (fixation). It has been extensively studied by analytical and numerical techniques in diverse contexts. Recently, DP has appeared as a generic model for the evolutionary and ecological dynamics of competing bacterial populations. Range expansion-the stochastic reproduction of bacteria competing for space to be occupied by their progeny-leads to a fluctuating and rough growth front, which is known from experiment and simulation to affect the underlying critical behavior of the DP transition. In this work, we employ symmetry arguments to construct a pair of nonlinear stochastic partial differential equations describing the coevolution of surface roughness with the composition field of DP. Macroscopic manifestations (phenomenology) of these equations on growth patterns and genealogical tracks of range expansion are discussed; followed by a renormalization group analysis of possible scaling behaviors at the DP transition.
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Affiliation(s)
- Jordan M Horowitz
- Physics of Living Systems Group, Department of Physics, Massachusetts Institute of Technology, 400 Technology Square, Cambridge, Massachusetts 02139, USA
- Department of Biophysics, University of Michigan, Ann Arbor, Michigan 48109, USA
- Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48104, USA
| | - Mehran Kardar
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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