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Smith CA, Yates CA. Incorporating domain growth into hybrid methods for reaction-diffusion systems. J R Soc Interface 2021; 18:20201047. [PMID: 33849339 DOI: 10.1098/rsif.2020.1047] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Reaction-diffusion mechanisms are a robust paradigm that can be used to represent many biological and physical phenomena over multiple spatial scales. Applications include intracellular dynamics, the migration of cells and the patterns formed by vegetation in semi-arid landscapes. Moreover, domain growth is an important process for embryonic growth and wound healing. There are many numerical modelling frameworks capable of simulating such systems on growing domains; however, each of these may be well suited to different spatial scales and particle numbers. Recently, spatially extended hybrid methods on static domains have been produced to bridge the gap between these different modelling paradigms in order to represent multi-scale phenomena. However, such methods have not been developed with domain growth in mind. In this paper, we develop three hybrid methods on growing domains, extending three of the prominent static-domain hybrid methods. We also provide detailed algorithms to allow others to employ them. We demonstrate that the methods are able to accurately model three representative reaction-diffusion systems accurately and without bias.
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Affiliation(s)
- Cameron A Smith
- Centre for Mathematical Biology, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
| | - Christian A Yates
- Centre for Mathematical Biology, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
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Yates CA, George A, Jordana A, Smith CA, Duncan AB, Zygalakis KC. The blending region hybrid framework for the simulation of stochastic reaction-diffusion processes. J R Soc Interface 2020; 17:20200563. [PMID: 33081647 PMCID: PMC7653393 DOI: 10.1098/rsif.2020.0563] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022] Open
Abstract
The simulation of stochastic reaction–diffusion systems using fine-grained representations can become computationally prohibitive when particle numbers become large. If particle numbers are sufficiently high then it may be possible to ignore stochastic fluctuations and use a more efficient coarse-grained simulation approach. Nevertheless, for multiscale systems which exhibit significant spatial variation in concentration, a coarse-grained approach may not be appropriate throughout the simulation domain. Such scenarios suggest a hybrid paradigm in which a computationally cheap, coarse-grained model is coupled to a more expensive, but more detailed fine-grained model, enabling the accurate simulation of the fine-scale dynamics at a reasonable computational cost. In this paper, in order to couple two representations of reaction–diffusion at distinct spatial scales, we allow them to overlap in a ‘blending region’. Both modelling paradigms provide a valid representation of the particle density in this region. From one end of the blending region to the other, control of the implementation of diffusion is passed from one modelling paradigm to another through the use of complementary ‘blending functions’ which scale up or down the contribution of each model to the overall diffusion. We establish the reliability of our novel hybrid paradigm by demonstrating its simulation on four exemplar reaction–diffusion scenarios.
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Affiliation(s)
- Christian A Yates
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
| | - Adam George
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
| | - Armand Jordana
- Centre de Mathématiques et de Leurs Applications, CNRS, ENS Paris-Saclay, Université Paris-Saclay, 94235 Cachan cedex, France
| | - Cameron A Smith
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
| | - Andrew B Duncan
- Department of Mathematics, Imperial College London, London SW7 2AZ, UK
| | - Konstantinos C Zygalakis
- School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK
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Morris B, Curtin L, Hawkins-Daarud A, Hubbard ME, Rahman R, Smith SJ, Auer D, Tran NL, Hu LS, Eschbacher JM, Smith KA, Stokes A, Swanson KR, Owen MR. Identifying the spatial and temporal dynamics of molecularly-distinct glioblastoma sub-populations. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2020; 17:4905-4941. [PMID: 33120534 PMCID: PMC8382158 DOI: 10.3934/mbe.2020267] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Glioblastomas (GBMs) are the most aggressive primary brain tumours and have no known cure. Each individual tumour comprises multiple sub-populations of genetically-distinct cells that may respond differently to targeted therapies and may contribute to disappointing clinical trial results. Image-localized biopsy techniques allow multiple biopsies to be taken during surgery and provide information that identifies regions where particular sub-populations occur within an individual GBM, thus providing insight into their regional genetic variability. These sub-populations may also interact with one another in a competitive or cooperative manner; it is important to ascertain the nature of these interactions, as they may have implications for responses to targeted therapies. We combine genetic information from biopsies with a mechanistic model of interacting GBM sub-populations to characterise the nature of interactions between two commonly occurring GBM sub-populations, those with EGFR and PDGFRA genes amplified. We study population levels found across image-localized biopsy data from a cohort of 25 patients and compare this to model outputs under competitive, cooperative and neutral interaction assumptions. We explore other factors affecting the observed simulated sub-populations, such as selection advantages and phylogenetic ordering of mutations, which may also contribute to the levels of EGFR and PDGFRA amplified populations observed in biopsy data.
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Affiliation(s)
- Bethan Morris
- School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK
| | - Lee Curtin
- Mathematical NeuroOncology Lab, Mayo Clinic, Phoenix, Arizona, 85054, USA
| | | | - Matthew E. Hubbard
- School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK
| | - Ruman Rahman
- School of Medicine and Health Sciences, University of Nottingham, Nottingham, NG7 2UH, UK
| | - Stuart J. Smith
- School of Medicine and Health Sciences, University of Nottingham, Nottingham, NG7 2UH, UK
| | - Dorothee Auer
- School of Medicine and Health Sciences, University of Nottingham, Nottingham, NG7 2UH, UK
| | - Nhan L. Tran
- Mathematical NeuroOncology Lab, Mayo Clinic, Phoenix, Arizona, 85054, USA
- Department of Cancer Biology, Mayo Clinic, Phoenix, Arizona 85054, USA
| | - Leland S. Hu
- Mathematical NeuroOncology Lab, Mayo Clinic, Phoenix, Arizona, 85054, USA
- Department of Radiology, Mayo Clinic, Phoenix, Arizona 85054, USA
| | - Jennifer M. Eschbacher
- Department of Pathology, Barrow Neurological Institute - St. Joseph’s Hospital and Medical Center, Phoenix, Arizona 85013, USA
| | - Kris A. Smith
- Department of Neurosurgery, Barrow Neurological Institute - St. Joseph’s Hospital and Medical Center, Phoenix, Arizona 85013, USA
| | - Ashley Stokes
- Department of Imaging Research, Barrow Neurological Institute - St. Joseph’s Hospital and Medical Center, Phoenix, Arizona 85013, USA
| | - Kristin R. Swanson
- Mathematical NeuroOncology Lab, Mayo Clinic, Phoenix, Arizona, 85054, USA
- Department of Neurosurgery, Mayo Clinic, Phoenix, Arizona 85054, USA
| | - Markus R. Owen
- School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK
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Abstract
Domain growth is a key process in many areas of biology, including embryonic development, the growth of tissue, and limb regeneration. As a result, mechanisms for incorporating it into traditional models for cell movement, interaction, and proliferation are of great importance. A previously well-used method to incorporate domain growth into on-lattice reaction-diffusion models causes a buildup of particles on the boundaries of the domain, which is particularly evident when diffusion is low in comparison to the rate of domain growth. Here we present an alternative method which addresses this unphysical buildup of particles at the boundaries and demonstrate that it is accurate for scenarios in which the previous method fails. Further, we discuss for which parameter regimes it is feasible to continue using the original method due to diffusion dominating the domain growth mechanism.
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Affiliation(s)
- Cameron A Smith
- Centre for Mathematical Biology, Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
| | - Cécile Mailler
- Probability Laboratory, Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
| | - Christian A Yates
- Centre for Mathematical Biology, Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom
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Fu Y, Yogurtcu ON, Kothari R, Thorkelsdottir G, Sodt AJ, Johnson ME. An implicit lipid model for efficient reaction-diffusion simulations of protein binding to surfaces of arbitrary topology. J Chem Phys 2019; 151:124115. [PMID: 31575182 DOI: 10.1063/1.5120516] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022] Open
Abstract
Localization of proteins to a membrane is an essential step in a broad range of biological processes such as signaling, virion formation, and clathrin-mediated endocytosis. The strength and specificity of proteins binding to a membrane depend on the lipid composition. Single-particle reaction-diffusion methods offer a powerful tool for capturing lipid-specific binding to membrane surfaces by treating lipids explicitly as individual diffusible binding sites. However, modeling lipid particle populations is expensive. Here, we present an algorithm for reversible binding of proteins to continuum surfaces with implicit lipids, providing dramatic speed-ups to many body simulations. Our algorithm can be readily integrated into most reaction-diffusion software packages. We characterize changes to kinetics that emerge from explicit vs implicit lipids as well as surface adsorption models, showing excellent agreement between our method and the full explicit lipid model. Compared to models of surface adsorption, which couple together binding affinity and lipid concentration, our implicit lipid model decouples them to provide more flexibility for controlling surface binding properties and lipid inhomogeneity, thus reproducing binding kinetics and equilibria. Crucially, we demonstrate our method's application to membranes of arbitrary curvature and topology, modeled via a subdivision limit surface, again showing excellent agreement with explicit lipid simulations. Unlike adsorption models, our method retains the ability to bind lipids after proteins are localized to the surface (through, e.g., a protein-protein interaction), which can greatly increase the stability of multiprotein complexes on the surface. Our method will enable efficient cell-scale simulations involving proteins localizing to realistic membrane models, which is a critical step for predictive modeling and quantification of in vitro and in vivo dynamics.
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Affiliation(s)
- Yiben Fu
- T. C. Jenkins Department of Biophysics, Johns Hopkins University, 3400 N. Charles St., Baltimore, Maryland 21218, USA
| | - Osman N Yogurtcu
- T. C. Jenkins Department of Biophysics, Johns Hopkins University, 3400 N. Charles St., Baltimore, Maryland 21218, USA
| | - Ruchita Kothari
- Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health, 9000 Rockville Pike, Bethesda, Maryland 20892, USA
| | - Gudrun Thorkelsdottir
- Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health, 9000 Rockville Pike, Bethesda, Maryland 20892, USA
| | - Alexander J Sodt
- Eunice Kennedy Shriver National Institute of Child Health and Human Development, National Institutes of Health, 9000 Rockville Pike, Bethesda, Maryland 20892, USA
| | - Margaret E Johnson
- T. C. Jenkins Department of Biophysics, Johns Hopkins University, 3400 N. Charles St., Baltimore, Maryland 21218, USA
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Smith CA, Yates CA. Spatially extended hybrid methods: a review. J R Soc Interface 2018; 15:20170931. [PMID: 29491179 PMCID: PMC5832735 DOI: 10.1098/rsif.2017.0931] [Citation(s) in RCA: 19] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2017] [Accepted: 02/08/2018] [Indexed: 12/13/2022] Open
Abstract
Many biological and physical systems exhibit behaviour at multiple spatial, temporal or population scales. Multiscale processes provide challenges when they are to be simulated using numerical techniques. While coarser methods such as partial differential equations are typically fast to simulate, they lack the individual-level detail that may be required in regions of low concentration or small spatial scale. However, to simulate at such an individual level throughout a domain and in regions where concentrations are high can be computationally expensive. Spatially coupled hybrid methods provide a bridge, allowing for multiple representations of the same species in one spatial domain by partitioning space into distinct modelling subdomains. Over the past 20 years, such hybrid methods have risen to prominence, leading to what is now a very active research area across multiple disciplines including chemistry, physics and mathematics. There are three main motivations for undertaking this review. Firstly, we have collated a large number of spatially extended hybrid methods and presented them in a single coherent document, while comparing and contrasting them, so that anyone who requires a multiscale hybrid method will be able to find the most appropriate one for their need. Secondly, we have provided canonical examples with algorithms and accompanying code, serving to demonstrate how these types of methods work in practice. Finally, we have presented papers that employ these methods on real biological and physical problems, demonstrating their utility. We also consider some open research questions in the area of hybrid method development and the future directions for the field.
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Affiliation(s)
- Cameron A Smith
- Centre for Mathematical Biology, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
| | - Christian A Yates
- Centre for Mathematical Biology, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
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