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Vo HK, Dawes JHP, Kelsh RN. Oscillatory differentiation dynamics fundamentally restricts the resolution of pseudotime reconstruction algorithms. J R Soc Interface 2024; 21:20230537. [PMID: 38503342 PMCID: PMC10950464 DOI: 10.1098/rsif.2023.0537] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/15/2023] [Accepted: 02/20/2024] [Indexed: 03/21/2024] Open
Abstract
The challenge to understand differentiation and cell lineages in development has resulted in many bioinformatics software tools, notably those working with gene expression data obtained via single-cell RNA sequencing obtained at snapshots in time. Reconstruction methods for trajectories often proceed by dimension reduction, data clustering and then computation of a tree graph in which edges indicate closely related clusters. Cell lineages can then be deduced by following paths through the tree. In the case of multi-potent cells undergoing differentiation, this trajectory reconstruction involves the reconstruction of multiple distinct lineages corresponding to commitment to each of a set of distinct fates. Recent work suggests that there may be cases in which the cell differentiation process involves trajectories that explore, in a dynamic and oscillatory fashion, propensity to differentiate into a number of possible cell fates before commitment finally occurs. Here, we show theoretically that the presence of such oscillations provides intrinsic constraints on the quality and resolution of the trajectory reconstruction process, even for idealized noise-free data. These constraints point to inherent common limitations of current methodologies and serve both to provide additional challenge in the development of software tools and also may help to understand features observed in recent experiments.
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Affiliation(s)
- Huy K. Vo
- Department of Mathematical Sciences, University of Bath, BA2 7AY Bath, UK
| | | | - Robert N. Kelsh
- Department of Life Sciences, University of Bath, BA2 7AY Bath, UK
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2
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Waters FR, Yates CA, Dawes JHP. Minimal reaction schemes for pattern formation. J R Soc Interface 2024; 21:20230490. [PMID: 38412962 PMCID: PMC10898969 DOI: 10.1098/rsif.2023.0490] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/22/2023] [Accepted: 01/30/2024] [Indexed: 02/29/2024] Open
Abstract
We link continuum models of reaction-diffusion systems that exhibit diffusion-driven instability to constraints on the particle-scale interactions underpinning this instability. While innumerable biological, chemical and physical patterns have been studied through the lens of Alan Turing's reaction-diffusion pattern-forming mechanism, the connections between models of pattern formation and the nature of the particle interactions generating them have been relatively understudied in comparison with the substantial efforts that have been focused on understanding proposed continuum systems. To derive the necessary reactant combinations for the most parsimonious reaction schemes, we analyse the emergent continuum models in terms of possible generating elementary reaction schemes. This analysis results in the complete list of such schemes containing the fewest reactions; these are the simplest possible hypothetical mass-action models for a pattern-forming system of two interacting species.
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Affiliation(s)
- Fraser R. Waters
- Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
- Centre for Mathematical Biology, University of Bath, Bath BA2 7AY, UK
| | - Christian A. Yates
- Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
- Centre for Mathematical Biology, University of Bath, Bath BA2 7AY, UK
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3
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Subkhankulova T, Camargo Sosa K, Uroshlev LA, Nikaido M, Shriever N, Kasianov AS, Yang X, Rodrigues FSLM, Carney TJ, Bavister G, Schwetlick H, Dawes JHP, Rocco A, Makeev VJ, Kelsh RN. Zebrafish pigment cells develop directly from persistent highly multipotent progenitors. Nat Commun 2023; 14:1258. [PMID: 36878908 PMCID: PMC9988989 DOI: 10.1038/s41467-023-36876-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2021] [Accepted: 02/17/2023] [Indexed: 03/08/2023] Open
Abstract
Neural crest cells are highly multipotent stem cells, but it remains unclear how their fate restriction to specific fates occurs. The direct fate restriction model hypothesises that migrating cells maintain full multipotency, whilst progressive fate restriction envisages fully multipotent cells transitioning to partially-restricted intermediates before committing to individual fates. Using zebrafish pigment cell development as a model, we show applying NanoString hybridization single cell transcriptional profiling and RNAscope in situ hybridization that neural crest cells retain broad multipotency throughout migration and even in post-migratory cells in vivo, with no evidence for partially-restricted intermediates. We find that leukocyte tyrosine kinase early expression marks a multipotent stage, with signalling driving iridophore differentiation through repression of fate-specific transcription factors for other fates. We reconcile the direct and progressive fate restriction models by proposing that pigment cell development occurs directly, but dynamically, from a highly multipotent state, consistent with our recently-proposed Cyclical Fate Restriction model.
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Affiliation(s)
| | - Karen Camargo Sosa
- Department of Life Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK
| | - Leonid A Uroshlev
- Vavilov Institute of General Genetics, Russian Academy of Sciences, Ul. Gubkina 3, Moscow, 119991, Russia
| | - Masataka Nikaido
- Department of Life Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK
- Graduate School of Science, University of Hyogo, Ako-gun, Hyogo Pref., 678-1297, Japan
| | - Noah Shriever
- Department of Life Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK
| | - Artem S Kasianov
- Vavilov Institute of General Genetics, Russian Academy of Sciences, Ul. Gubkina 3, Moscow, 119991, Russia
- Department of Medical and Biological Physics, Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russia
- A.A. Kharkevich Institute for Information Transmission Problems (IITP), Russian Academy of Sciences, Bolshoy Karetny per. 19, build.1, Moscow, 127051, Russia
| | - Xueyan Yang
- Department of Life Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK
- The MOE Key Laboratory of Contemporary Anthropology, School of Life Sciences, Fudan University, Shanghai, 200438, PR China
| | | | - Thomas J Carney
- Department of Life Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK
- Lee Kong Chian School of Medicine, Experimental Medicine Building, Yunnan Garden Campus, Nanyang Technological University, 59 Nanyang Drive, Yunnan Garden, 636921, Singapore
| | - Gemma Bavister
- Department of Life Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK
| | - Hartmut Schwetlick
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK
| | - Jonathan H P Dawes
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK
| | - Andrea Rocco
- Department of Microbial Sciences, FHMS, University of Surrey, GU2 7XH, Guildford, UK
- Department of Physics, FEPS, University of Surrey, GU2 7XH, Guildford, UK
| | - Vsevolod J Makeev
- Vavilov Institute of General Genetics, Russian Academy of Sciences, Ul. Gubkina 3, Moscow, 119991, Russia
- Department of Medical and Biological Physics, Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russia
- Laboratory 'Regulatory Genomics', Institute of Fundamental Medicine and Biology, Kazan Federal University, 18 Kremlyovskaya street, Kazan, 420008, Russia
| | - Robert N Kelsh
- Department of Life Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK.
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4
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Dawes JHP, Zhou X, Moinuddin M. System-level consequences of synergies and trade-offs between SDGs: quantitative analysis of interlinkage networks at country level. Sustain Sci 2022; 17:1435-1457. [PMID: 35251357 PMCID: PMC8882233 DOI: 10.1007/s11625-022-01109-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/31/2021] [Accepted: 02/03/2022] [Indexed: 06/14/2023]
Abstract
UNLABELLED The Sustainable Development Goals (SDGs) present a complex system of 17 goals and 169 individual targets whose interactions can be described in terms of co-benefits and trade-offs between policy actions. We analyse in detail target-by-target interlinkage networks established by the Institute for Global Environmental Strategies (IGES) SDG Interlinkages Tool. We discuss two quantitative measures of network structure; the leading eigenvector of the interlinkage networks ('eigencentrality') and a notion of hierarchy within the network motivated by the concept of trophic levels for species in food webs. We use three interlinkage matrices generated by IGES: the framework matrix which provides a generic network model of the interlinkages at the target level, and two country-specific matrices for Bangladesh and Indonesia that combine SDG indicator data with the generic framework matrix. Our results echo, and are confirmed by, similar work at the level of whole SDGs that has shown that SDGs 1-3 (ending poverty, and providing food security and healthcare) are much more likely to be achieved than the environmentally- related SDGs 13-15 concerned with climate action, life on land and life below water. Our results here provide a refinement in terms of specific targets within each of these SDGs. We find that not all targets within SDGs 1-3 are equally well-supported, and not all targets within SDGs 13-15 are equally at risk of not being achieved. Finally, we point to the recurring issue of data gaps that hinders our quantitative analysis, in particular for SDGs 5 (gender equality) and 13 (climate action) where the huge gaps in indicator data that mean the true nature of the interlinkages and importance of these two SDGs are not fully recognised. SUPPLEMENTARY INFORMATION The online version contains supplementary material available at 10.1007/s11625-022-01109-y.
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Affiliation(s)
- Jonathan H. P. Dawes
- Centre for Networks and Collective Behaviour, University of Bath, Bath, BA2 7AY UK
- Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY UK
| | - Xin Zhou
- Institute for Global Environmental Strategies (IGES), 2108-11 Kamiyamaguchi, Hayama, Kanagawa 240-0115 Japan
| | - Mustafa Moinuddin
- Institute for Global Environmental Strategies (IGES), 2108-11 Kamiyamaguchi, Hayama, Kanagawa 240-0115 Japan
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Dawes JHP, Kelsh RN. Cell Fate Decisions in the Neural Crest, from Pigment Cell to Neural Development. Int J Mol Sci 2021; 22:13531. [PMID: 34948326 PMCID: PMC8706606 DOI: 10.3390/ijms222413531] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2021] [Revised: 12/14/2021] [Accepted: 12/15/2021] [Indexed: 11/17/2022] Open
Abstract
The neural crest shows an astonishing multipotency, generating multiple neural derivatives, but also pigment cells, skeletogenic and other cell types. The question of how this process is controlled has been the subject of an ongoing debate for more than 35 years. Based upon new observations of zebrafish pigment cell development, we have recently proposed a novel, dynamic model that we believe goes some way to resolving the controversy. Here, we will firstly summarize the traditional models and the conflicts between them, before outlining our novel model. We will also examine our recent dynamic modelling studies, looking at how these reveal behaviors compatible with the biology proposed. We will then outline some of the implications of our model, looking at how it might modify our views of the processes of fate specification, differentiation, and commitment.
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Affiliation(s)
- Jonathan H. P. Dawes
- Centre for Networks and Collective Behaviour, University of Bath, Bath BA2 7AY, UK;
- Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
| | - Robert N. Kelsh
- Centre for Mathematical Biology, University of Bath, Bath BA2 7AY, UK
- Department of Biology & Biochemistry, University of Bath, Bath BA2 7AY, UK
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6
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Kelsh RN, Camargo Sosa K, Farjami S, Makeev V, Dawes JHP, Rocco A. Cyclical fate restriction: a new view of neural crest cell fate specification. Development 2021; 148:273451. [PMID: 35020872 DOI: 10.1242/dev.176057] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022]
Abstract
Neural crest cells are crucial in development, not least because of their remarkable multipotency. Early findings stimulated two hypotheses for how fate specification and commitment from fully multipotent neural crest cells might occur, progressive fate restriction (PFR) and direct fate restriction, differing in whether partially restricted intermediates were involved. Initially hotly debated, they remain unreconciled, although PFR has become favoured. However, testing of a PFR hypothesis of zebrafish pigment cell development refutes this view. We propose a novel 'cyclical fate restriction' hypothesis, based upon a more dynamic view of transcriptional states, reconciling the experimental evidence underpinning the traditional hypotheses.
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Affiliation(s)
- Robert N Kelsh
- Department of Biology & Biochemistry, University of Bath, Bath, BA2 7AY, UK
| | - Karen Camargo Sosa
- Department of Biology & Biochemistry, University of Bath, Bath, BA2 7AY, UK
| | - Saeed Farjami
- Department of Microbial Sciences, FHMS, University of Surrey, Guildford, GU2 7XH, UK
| | - Vsevolod Makeev
- Department of Computational Systems Biology, Vavilov Institute of General Genetics, Russian Academy of Sciences, Ul. Gubkina 3, Moscow, 119991, Russian Federation.,Department of Biological and Medical Physics, Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, 141701, Russian Federation
| | - Jonathan H P Dawes
- Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK
| | - Andrea Rocco
- Department of Microbial Sciences, FHMS, University of Surrey, Guildford, GU2 7XH, UK.,Department of Physics, FEPS, University of Surrey, Guildford, GU2 7XH, UK
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7
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Farjami S, Camargo Sosa K, Dawes JHP, Kelsh RN, Rocco A. Novel generic models for differentiating stem cells reveal oscillatory mechanisms. J R Soc Interface 2021; 18:20210442. [PMID: 34610261 PMCID: PMC8492175 DOI: 10.1098/rsif.2021.0442] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/27/2021] [Accepted: 09/09/2021] [Indexed: 12/03/2022] Open
Abstract
Understanding cell fate selection remains a central challenge in developmental biology. We present a class of simple yet biologically motivated mathematical models for cell differentiation that generically generate oscillations and hence suggest alternatives to the standard framework based on Waddington's epigenetic landscape. The models allow us to suggest two generic dynamical scenarios that describe the differentiation process. In the first scenario, gradual variation of a single control parameter is responsible for both entering and exiting the oscillatory regime. In the second scenario, two control parameters vary: one responsible for entering, and the other for exiting the oscillatory regime. We analyse the standard repressilator and four variants of it and show the dynamical behaviours associated with each scenario. We present a thorough analysis of the associated bifurcations and argue that gene regulatory networks with these repressilator-like characteristics are promising candidates to describe cell fate selection through an oscillatory process.
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Affiliation(s)
- Saeed Farjami
- Department of Microbial Sciences, University of Surrey, Guildford GU2 7XH, UK
| | - Karen Camargo Sosa
- Department of Biology and Biochemistry, University of Bath, Bath BA2 7AY, UK
| | | | - Robert N. Kelsh
- Department of Biology and Biochemistry, University of Bath, Bath BA2 7AY, UK
| | - Andrea Rocco
- Department of Microbial Sciences, University of Surrey, Guildford GU2 7XH, UK
- Department of Physics, University of Surrey, Guildford GU2 7XH, UK
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8
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Adamson MW, Dawes JHP, Hastings A, Hilker FM. Forecasting resilience profiles of the run-up to regime shifts in nearly-one-dimensional systems. J R Soc Interface 2020; 17:20200566. [PMID: 32933374 DOI: 10.1098/rsif.2020.0566] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
The forecasting of sudden, irreversible shifts in natural systems is a challenge of great importance, whose realization could allow pre-emptive action to be taken to avoid or mitigate catastrophic transitions, or to help systems adapt to them. In recent years, there have been many advances in the development of such early warning signals. However, much of the current toolbox is based around the tracking of statistical trends and therefore does not aim to estimate the future time scale of transitions or resilience loss. Metric-based indicators are also difficult to implement when systems have inherent oscillations which can dominate the indicator statistics. To resolve these gaps in the toolbox, we use additional system properties to fit parsimonious models to dynamics in order to predict transitions. Here, we consider nearly-one-dimensional systems-higher dimensional systems whose dynamics can be accurately captured by one-dimensional discrete time maps. We show how the nearly one-dimensional dynamics can be used to produce model-based indicators for critical transitions which produce forecasts of the resilience and the time of transitions in the system. A particularly promising feature of this approach is that it allows us to construct early warning signals even for critical transitions of chaotic systems. We demonstrate this approach on two model systems: of phosphorous recycling in a shallow lake, and of an overcompensatory fish population.
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Affiliation(s)
- Matthew W Adamson
- Institute for Environmental Systems Research and Institute of Mathematics, University of Osnabrück, Barbarastraße 12, 49076 Osnabrück, Germany
| | - Jonathan H P Dawes
- Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
| | - Alan Hastings
- Department of Environmental Science and Policy, University of California, Davis, CA 95616, USA.,Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
| | - Frank M Hilker
- Institute for Environmental Systems Research and Institute of Mathematics, University of Osnabrück, Barbarastraße 12, 49076 Osnabrück, Germany
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9
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Abstract
Sliding frictional interfaces at a range of length scales are observed to generate travelling waves; these are considered relevant, for example, to both earthquake ground surface movements and the performance of mechanical brakes and dampers. We propose an explanation of the origins of these waves through the study of an idealized mechanical model: a thin elastic plate subject to uniform shear stress held in frictional contact with a rigid flat surface. We construct a nonlinear wave equation for the deformation of the plate, and couple it to a spinodal rate-and-state friction law which leads to a mathematically well-posed problem that is capable of capturing many effects not accessible in a Coulomb friction model. Our model sustains a rich variety of solutions, including periodic stick–slip wave trains, isolated slip and stick pulses, and detachment and attachment fronts. Analytical and numerical bifurcation analysis is used to show how these states are organized in a two-parameter state diagram. We discuss briefly the possible physical interpretation of each of these states, and remark also that our spinodal friction law, though more complicated than other classical rate-and-state laws, is required in order to capture the full richness of wave types.
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Affiliation(s)
- T Putelat
- Department of Engineering Mathematics, University of Bristol, Bristol BS8 1UB, UK
| | - J H P Dawes
- Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
| | - A R Champneys
- Department of Engineering Mathematics, University of Bristol, Bristol BS8 1UB, UK
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10
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Abstract
We analyse the model for vegetation growth in a semi-arid landscape proposed by von Hardenberg et al. (Phys. Rev. Lett. 87:198101, 2001), which consists of two parabolic partial differential equations that describe the evolution in space and time of the water content of the soil and the level of vegetation. This model is a generalisation of one proposed by Klausmeier but it contains additional terms that capture additional physical effects. By considering the limit in which the diffusion of water in the soil is much faster than the spread of vegetation, we reduce the system to an asymptotically simpler parabolic-elliptic system of equations that describes small amplitude instabilities of the uniform vegetated state. We carry out a thorough weakly nonlinear analysis to investigate bifurcations and pattern formation in the reduced model. We find that the pattern forming instabilities are subcritical except in a small region of parameter space. In the original model at large amplitude there are localised solutions, organised by homoclinic snaking curves. The resulting bifurcation structure is well known from other models for pattern forming systems. Taken together our results describe how the von Hardenberg model displays a sequence of (often hysteretic) transitions from a non-vegetated state, to localised patches of vegetation that exist with uniform low-level vegetation, to periodic patterns, to higher-level uniform vegetation as the precipitation parameter increases.
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Affiliation(s)
- J H P Dawes
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK.
| | - J L M Williams
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK
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11
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Dawes JHP, Susanto H. Variational approximation and the use of collective coordinates. Phys Rev E Stat Nonlin Soft Matter Phys 2013; 87:063202. [PMID: 23848797 DOI: 10.1103/physreve.87.063202] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/26/2012] [Revised: 04/02/2013] [Indexed: 06/02/2023]
Abstract
We consider propagating, spatially localized waves in a class of equations that contain variational and nonvariational terms. The dynamics of the waves is analyzed through a collective coordinate approach. Motivated by the variational approximation, we show that there is a natural choice of projection onto collective variables for reducing the governing (nonlinear) partial differential equation (PDE) to coupled ordinary differential equations (ODEs). This projection produces ODEs whose solutions are exactly the stationary states of the effective Lagrangian that would be considered in applying the variational approximation method. We illustrate our approach by applying it to a modified Fisher equation for a traveling front, containing a non-constant-coefficient nonlinear term. We present numerical results that show that our proposed projection captures both the equilibria and the dynamics of the PDE much more closely than previously proposed projections.
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Affiliation(s)
- J H P Dawes
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom
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12
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Dawes JHP. The emergence of a coherent structure for coherent structures: localized states in nonlinear systems. Philos Trans A Math Phys Eng Sci 2010; 368:3519-3534. [PMID: 20603365 DOI: 10.1098/rsta.2010.0057] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/29/2023]
Abstract
Coherent structures emerge from the dynamics of many kinds of dissipative, externally driven, nonlinear systems, and continue to provoke new questions that challenge our physical and mathematical understanding. In one specific subclass of such problems, in which a pattern-forming, or 'Turing', instability occurs, rapid progress has been made recently in our understanding of the formation of localized states: patches of regular pattern surrounded by the unpatterned homogeneous background state. This short review article surveys the progress that has been made for localized states and proposes three areas of application for these ideas that would take the theory in new directions and ultimately be of substantial benefit to areas of applied science. Finally, I offer speculations for future work, based on localized states, that may help researchers to understand coherent structures more generally.
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Affiliation(s)
- J H P Dawes
- Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK.
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13
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Morris SW, Dawes JHP, Ribe NM, Lister JR. Meandering instability of a viscous thread. Phys Rev E Stat Nonlin Soft Matter Phys 2008; 77:066218. [PMID: 18643361 DOI: 10.1103/physreve.77.066218] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/24/2007] [Indexed: 05/05/2023]
Abstract
A viscous thread falling from a nozzle onto a surface exhibits the famous rope-coiling effect, in which the thread buckles to form loops. If the surface is replaced by a belt moving with speed U , the rotational symmetry of the buckling instability is broken and a wealth of interesting states are observed [see S. Chiu-Webster and J. R. Lister, J. Fluid Mech. 569, 89 (2006)]. We experimentally studied this "fluid-mechanical sewing machine" in a more precise apparatus. As U is reduced, the steady catenary thread bifurcates into a meandering state in which the thread displacements are only transverse to the motion of the belt. We measured the amplitude and frequency omega of the meandering close to the bifurcation. For smaller U , single-frequency meandering bifurcates to a two-frequency "figure-8" state, which contains a significant 2omega component and parallel as well as transverse displacements. This eventually reverts to single-frequency coiling at still smaller U . More complex, highly hysteretic states with additional frequencies are observed for larger nozzle heights. We propose to understand this zoology in terms of the generic amplitude equations appropriate for resonant interactions between two oscillatory modes with frequencies omega and 2omega . The form of the amplitude equations captures both the axisymmetry of the U=0 coiling state and the symmetry-breaking effects induced by the moving belt.
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Affiliation(s)
- Stephen W Morris
- Department of Physics, University of Toronto, 60 St. George Street, Toronto, Ontario, Canada M5S 1A7
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14
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Dawes JHP, Tsai TL. Frequency locking and complex dynamics near a periodically forced robust heteroclinic cycle. Phys Rev E Stat Nonlin Soft Matter Phys 2006; 74:055201. [PMID: 17279960 DOI: 10.1103/physreve.74.055201] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2006] [Indexed: 05/13/2023]
Abstract
Robust heteroclinic cycles occur naturally in many classes of nonlinear differential equations with invariant hyperplanes. In particular they occur frequently in models for ecological dynamics and fluid mechanical instabilities. We consider the effect of small-amplitude time-periodic forcing and describe how to reduce the dynamics to a two-dimensional map. In the limit where the heteroclinic cycle loses asymptotic stability, intervals of frequency locking appear. In the opposite limit, where the heteroclinic cycle becomes strongly stable, the dynamics remains chaotic and no frequency locking is observed.
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Affiliation(s)
- J H P Dawes
- DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom.
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15
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Abstract
We examine a generalised SIR model for the infection dynamics of four competing disease strains. This model contains four previously-studied models as special cases. The different strains interact indirectly by the mechanism of cross-immunity; individuals in the host population may become immune to infection by a particular strain even if they have only been infected with different but closely related strains. Several different models of cross-immunity are compared in the limit where the death rate is much smaller than the rate of recovery from infection. In this limit an asymptotic analysis of the dynamics of the models is possible, and we are able to compute the location and nature of the Takens-Bogdanov bifurcation associated with the presence of oscillatory dynamics observed by previous authors.
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Affiliation(s)
- J H P Dawes
- DAMTP, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK.
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