1
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Ebata H, Inagaki S. Self-replicating segregation patterns in horizontally vibrated binary mixture of granules. Sci Rep 2024; 14:5329. [PMID: 38438466 PMCID: PMC10912327 DOI: 10.1038/s41598-024-55876-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/26/2023] [Accepted: 02/28/2024] [Indexed: 03/06/2024] Open
Abstract
Fluidized granular mixtures of various particle sizes exhibit intriguing patterns as different species segregate and condense. However, understanding the segregation dynamics is hindered by the inability to directly observe the time evolution of the internal structure. We discover self-replicating bands within a quasi-2D container subjected to horizontal agitation, resulting in steady surface waves. Through direct observation of surface flow and evolving internal structures, we reveal the crucial role of coupling among segregation, surface flow, and hysteresis in granular fluidity. We develop Bonhoeffer-van der Pol type equations grounded in experimental observations, reproducing complex band dynamics, such as replication, oscillation, and breathing. It suggests the similarity between pattern formation in granular segregation and that in reaction-diffusion systems.
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Affiliation(s)
- Hiroyuki Ebata
- Department of Physics, Kyushu University, Fukuoka, 819-0395, Japan.
| | - Shio Inagaki
- Department of Physics, Kyushu University, Fukuoka, 819-0395, Japan
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2
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de Wit XM, Fruchart M, Khain T, Toschi F, Vitelli V. Pattern formation by turbulent cascades. Nature 2024; 627:515-521. [PMID: 38509279 PMCID: PMC10954557 DOI: 10.1038/s41586-024-07074-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/04/2023] [Accepted: 01/15/2024] [Indexed: 03/22/2024]
Abstract
Fully developed turbulence is a universal and scale-invariant chaotic state characterized by an energy cascade from large to small scales at which the cascade is eventually arrested by dissipation1-6. Here we show how to harness these seemingly structureless turbulent cascades to generate patterns. Pattern formation entails a process of wavelength selection, which can usually be traced to the linear instability of a homogeneous state7. By contrast, the mechanism we propose here is fully nonlinear. It is triggered by the non-dissipative arrest of turbulent cascades: energy piles up at an intermediate scale, which is neither the system size nor the smallest scales at which energy is usually dissipated. Using a combination of theory and large-scale simulations, we show that the tunable wavelength of these cascade-induced patterns can be set by a non-dissipative transport coefficient called odd viscosity, ubiquitous in chiral fluids ranging from bioactive to quantum systems8-12. Odd viscosity, which acts as a scale-dependent Coriolis-like force, leads to a two-dimensionalization of the flow at small scales, in contrast with rotating fluids in which a two-dimensionalization occurs at large scales4. Apart from odd viscosity fluids, we discuss how cascade-induced patterns can arise in natural systems, including atmospheric flows13-19, stellar plasma such as the solar wind20-22, or the pulverization and coagulation of objects or droplets in which mass rather than energy cascades23-25.
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Affiliation(s)
- Xander M de Wit
- Department of Applied Physics and Science Education, Eindhoven University of Technology, Eindhoven, The Netherlands
| | - Michel Fruchart
- Gulliver, ESPCI Paris, Université PSL, CNRS, Paris, France
- James Franck Institute, The University of Chicago, Chicago, IL, USA
| | - Tali Khain
- James Franck Institute, The University of Chicago, Chicago, IL, USA
| | - Federico Toschi
- Department of Applied Physics and Science Education, Eindhoven University of Technology, Eindhoven, The Netherlands.
- CNR-IAC, Rome, Italy.
| | - Vincenzo Vitelli
- James Franck Institute, The University of Chicago, Chicago, IL, USA.
- Kadanoff Center for Theoretical Physics, The University of Chicago, Chicago, IL, USA.
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3
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Deinum EE, Jacobs B. Rho of Plants patterning: linking mathematical models and molecular diversity. JOURNAL OF EXPERIMENTAL BOTANY 2024; 75:1274-1288. [PMID: 37962515 PMCID: PMC10901209 DOI: 10.1093/jxb/erad447] [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: 04/25/2023] [Accepted: 11/08/2023] [Indexed: 11/15/2023]
Abstract
ROPs (Rho of Plants) are plant specific small GTPases involved in many membrane patterning processes and play important roles in the establishment and communication of cell polarity. These small GTPases can produce a wide variety of patterns, ranging from a single cluster in tip-growing root hairs and pollen tubes to an oriented stripe pattern controlling protoxylem cell wall deposition. For an understanding of what controls these various patterns, models are indispensable. Consequently, many modelling studies on small GTPase patterning exist, often focusing on yeast or animal cells. Multiple patterns occurring in plants, however, require the stable co-existence of multiple active ROP clusters, which does not occur with the most common yeast/animal models. The possibility of such patterns critically depends on the precise model formulation. Additionally, different small GTPases are usually treated interchangeably in models, even though plants possess two types of ROPs with distinct molecular properties, one of which is unique to plants. Furthermore, the shape and even the type of ROP patterns may be affected by the cortical cytoskeleton, and cortex composition and anisotropy differ dramatically between plants and animals. Here, we review insights into ROP patterning from modelling efforts across kingdoms, as well as some outstanding questions arising from these models and recent experimental findings.
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Affiliation(s)
- Eva E Deinum
- Mathematical and Statistical Methods (Biometris), Plant Science Group, Wageningen University, 6708 PB Wageningen, The Netherlands
| | - Bas Jacobs
- Mathematical and Statistical Methods (Biometris), Plant Science Group, Wageningen University, 6708 PB Wageningen, The Netherlands
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4
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Weyer H, Brauns F, Frey E. Coarsening and wavelength selection far from equilibrium: A unifying framework based on singular perturbation theory. Phys Rev E 2023; 108:064202. [PMID: 38243507 DOI: 10.1103/physreve.108.064202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/24/2023] [Accepted: 08/29/2023] [Indexed: 01/21/2024]
Abstract
Intracellular protein patterns are described by (nearly) mass-conserving reaction-diffusion systems. While these patterns initially form out of a homogeneous steady state due to the well-understood Turing instability, no general theory exists for the dynamics of fully nonlinear patterns. We develop a unifying theory for nonlinear wavelength-selection dynamics in (nearly) mass-conserving two-component reaction-diffusion systems independent of the specific mathematical model chosen. Previous work has shown that these systems support an extremely broad band of stable wavelengths, but the mechanism by which a specific wavelength is selected has remained unclear. We show that an interrupted coarsening process selects the wavelength at the threshold to stability. Based on the physical intuition that coarsening is driven by competition for mass and interrupted by weak source terms that break strict mass conservation, we develop a singular perturbation theory for the stability of stationary patterns. The resulting closed-form analytical expressions enable us to quantitatively predict the coarsening dynamics and the final pattern wavelength. We find excellent agreement with numerical results throughout the diffusion- and reaction-limited regimes of the dynamics, including the crossover region. Further, we show how, in these limits, the two-component reaction-diffusion systems map to generalized Cahn-Hilliard and conserved Allen-Cahn dynamics, therefore providing a link to these two fundamental scalar field theories. The systematic understanding of the length-scale dynamics of fully nonlinear patterns in two-component systems provided here builds the basis to reveal the mechanisms underlying wavelength selection in multicomponent systems with potentially several conservation laws.
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Affiliation(s)
- Henrik Weyer
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München, Germany
| | - Fridtjof Brauns
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München, Germany
| | - Erwin Frey
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München, Germany
- Max Planck School Matter to Life, Hofgartenstraße 8, D-80539 Munich, Germany
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5
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Frohoff-Hülsmann T, Thiele U. Nonreciprocal Cahn-Hilliard Model Emerges as a Universal Amplitude Equation. PHYSICAL REVIEW LETTERS 2023; 131:107201. [PMID: 37739387 DOI: 10.1103/physrevlett.131.107201] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/29/2022] [Accepted: 08/08/2023] [Indexed: 09/24/2023]
Abstract
Oscillatory behavior is ubiquitous in out-of-equilibrium systems showing spatiotemporal pattern formation. Starting from a linear large-scale oscillatory instability-a conserved-Hopf instability-that naturally occurs in many active systems with two conservation laws, we derive a corresponding amplitude equation. It belongs to a hierarchy of such universal equations for the eight types of instabilities in homogeneous isotropic systems resulting from the combination of three features: large-scale vs small-scale instability, stationary vs oscillatory instability, and instability without and with conservation law(s). The derived universal equation generalizes a phenomenological model of considerable recent interest, namely, the nonreciprocal Cahn-Hilliard model, and may be of a similar relevance for the classification of pattern forming systems as the complex Ginzburg-Landau equation.
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Affiliation(s)
- Tobias Frohoff-Hülsmann
- Institute of Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany
| | - Uwe Thiele
- Institute of Theoretical Physics, University of Münster, Wilhelm-Klemm-Strasse 9, 48149 Münster, Germany
- Center for Nonlinear Science (CeNoS), University of Münster, Corrensstrasse 2, 48149 Münster, Germany
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6
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Ouazan-Reboul V, Agudo-Canalejo J, Golestanian R. Self-organization of primitive metabolic cycles due to non-reciprocal interactions. Nat Commun 2023; 14:4496. [PMID: 37495589 PMCID: PMC10372013 DOI: 10.1038/s41467-023-40241-w] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/23/2023] [Accepted: 07/13/2023] [Indexed: 07/28/2023] Open
Abstract
One of the greatest mysteries concerning the origin of life is how it has emerged so quickly after the formation of the earth. In particular, it is not understood how metabolic cycles, which power the non-equilibrium activity of cells, have come into existence in the first instances. While it is generally expected that non-equilibrium conditions would have been necessary for the formation of primitive metabolic structures, the focus has so far been on externally imposed non-equilibrium conditions, such as temperature or proton gradients. Here, we propose an alternative paradigm in which naturally occurring non-reciprocal interactions between catalysts that can partner together in a cyclic reaction lead to their recruitment into self-organized functional structures. We uncover different classes of self-organized cycles that form through exponentially rapid coarsening processes, depending on the parity of the cycle and the nature of the interaction motifs, which are all generic but have readily tuneable features.
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Affiliation(s)
- Vincent Ouazan-Reboul
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077, Göttingen, Germany
| | - Jaime Agudo-Canalejo
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077, Göttingen, Germany
| | - Ramin Golestanian
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077, Göttingen, Germany.
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, OX1 3PU, Oxford, UK.
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7
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Würthner L, Goychuk A, Frey E. Geometry-induced patterns through mechanochemical coupling. Phys Rev E 2023; 108:014404. [PMID: 37583206 DOI: 10.1103/physreve.108.014404] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/07/2022] [Accepted: 05/15/2023] [Indexed: 08/17/2023]
Abstract
Intracellular protein patterns regulate a variety of vital cellular processes such as cell division and motility, which often involve dynamic cell-shape changes. These changes in cell shape may in turn affect the dynamics of pattern-forming proteins, hence leading to an intricate feedback loop between cell shape and chemical dynamics. While several computational studies have examined the rich resulting dynamics, the underlying mechanisms are not yet fully understood. To elucidate some of these mechanisms, we explore a conceptual model for cell polarity on a dynamic one-dimensional manifold. Using concepts from differential geometry, we derive the equations governing mass-conserving reaction-diffusion systems on time-evolving manifolds. Analyzing these equations mathematically, we show that dynamic shape changes of the membrane can induce pattern-forming instabilities in parts of the membrane, which we refer to as regional instabilities. Deformations of the local membrane geometry can also (regionally) suppress pattern formation and spatially shift already existing patterns. We explain our findings by applying and generalizing the local equilibria theory of mass-conserving reaction-diffusion systems. This allows us to determine a simple onset criterion for geometry-induced pattern-forming instabilities, which is linked to the phase-space structure of the reaction-diffusion system. The feedback loop between membrane shape deformations and reaction-diffusion dynamics then leads to a surprisingly rich phenomenology of patterns, including oscillations, traveling waves, and standing waves, even if these patterns do not occur in systems with a fixed membrane shape. Our paper reveals that the local conformation of the membrane geometry acts as an important dynamical control parameter for pattern formation in mass-conserving reaction-diffusion systems.
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Affiliation(s)
- Laeschkir Würthner
- Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS), Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 Munich, Germany
| | - Andriy Goychuk
- Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS), Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 Munich, Germany
| | - Erwin Frey
- Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS), Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 Munich, Germany
- Max Planck School Matter to Life, Hofgartenstraße 8, D-80539 Munich, Germany
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8
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Demarchi L, Goychuk A, Maryshev I, Frey E. Enzyme-Enriched Condensates Show Self-Propulsion, Positioning, and Coexistence. PHYSICAL REVIEW LETTERS 2023; 130:128401. [PMID: 37027840 DOI: 10.1103/physrevlett.130.128401] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/29/2022] [Accepted: 02/03/2023] [Indexed: 06/19/2023]
Abstract
Enzyme-enriched condensates can organize the spatial distribution of their substrates by catalyzing nonequilibrium reactions. Conversely, an inhomogeneous substrate distribution induces enzyme fluxes through substrate-enzyme interactions. We find that condensates move toward the center of a confining domain when this feedback is weak. Above a feedback threshold, they exhibit self-propulsion, leading to oscillatory dynamics. Moreover, catalysis-driven enzyme fluxes can lead to interrupted coarsening, resulting in equidistant condensate positioning, and to condensate division.
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Affiliation(s)
- Leonardo Demarchi
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München, Germany
| | - Andriy Goychuk
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München, Germany
| | - Ivan Maryshev
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München, Germany
| | - Erwin Frey
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München, Germany
- Max Planck School Matter to Life, Hofgartenstraße 8, D-80539 München, Germany
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9
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Cusseddu D, Madzvamuse A. Numerical investigations of the bulk-surface wave pinning model. Math Biosci 2022; 354:108925. [PMID: 36397641 DOI: 10.1016/j.mbs.2022.108925] [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: 05/25/2022] [Revised: 10/12/2022] [Accepted: 10/15/2022] [Indexed: 11/15/2022]
Abstract
The bulk-surface wave pinning model is a reaction-diffusion system for studying cell polarisation. It is constituted by a surface reaction-diffusion equation, coupled to a bulk diffusion equation with a non-linear boundary condition. Cell polarisation arises as the surface component develops specific patterns. Since proteins diffuse much faster in the cell interior than on the membrane, in the literature, the bulk component is often assumed to be spatially homogeneous. Therefore, the model can be reduced to a single surface equation. However, in real applications a spatially non-uniform bulk component might be an important player to take into account. In this paper, we study, through numerical computations, the role of the bulk component and, more specifically, how different bulk diffusion rates might affect the polarisation response. We find that the bulk component is indeed a key factor in determining the surface polarisation response. Moreover, for certain geometries, it is the spatial heterogeneity of the bulk component that triggers the polarisation response, which might not be possible in a reduced model. Understanding how polarisation depends on bulk diffusivity might be crucial when studying models of migrating cells, which are naturally subject to domain deformation.
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Affiliation(s)
- Davide Cusseddu
- Grupo de Física-Matematica, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Lisboa, 1749-016, Portugal; Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Sussex House, Falmer, Brighton, BN1 9RH, United Kingdom.
| | - Anotida Madzvamuse
- Faculty of Science, Vancouver Campus, Mathematics Department, 121 - 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2; Department of Mathematics, School of Mathematical and Physical Sciences, University of Sussex, Sussex House, Falmer, Brighton, BN1 9RH, United Kingdom.
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10
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Cotton MW, Golestanian R, Agudo-Canalejo J. Catalysis-Induced Phase Separation and Autoregulation of Enzymatic Activity. PHYSICAL REVIEW LETTERS 2022; 129:158101. [PMID: 36269959 DOI: 10.1103/physrevlett.129.158101] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2022] [Accepted: 09/09/2022] [Indexed: 06/16/2023]
Abstract
We present a thermodynamically consistent model describing the dynamics of a multicomponent mixture where one enzyme component catalyzes a reaction between other components. We find that the catalytic activity alone can induce phase separation for sufficiently active systems and large enzymes, without any equilibrium interactions between components. In the limit of fast reaction rates, binodal lines can be calculated using a mapping to an effective free energy. We also explain how this catalysis-induced phase separation can act to autoregulate the enzymatic activity, which points at the biological relevance of this phenomenon.
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Affiliation(s)
- Matthew W Cotton
- Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG United Kingdom
- Department of Living Matter Physics, Max Planck Institute for Dynamics and Self-Organization, D-37077 Göttingen, Germany
| | - Ramin Golestanian
- Department of Living Matter Physics, Max Planck Institute for Dynamics and Self-Organization, D-37077 Göttingen, Germany
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, United Kingdom
| | - Jaime Agudo-Canalejo
- Department of Living Matter Physics, Max Planck Institute for Dynamics and Self-Organization, D-37077 Göttingen, Germany
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11
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Würthner L, Brauns F, Pawlik G, Halatek J, Kerssemakers J, Dekker C, Frey E. Bridging scales in a multiscale pattern-forming system. Proc Natl Acad Sci U S A 2022; 119:e2206888119. [PMID: 35960842 PMCID: PMC9388104 DOI: 10.1073/pnas.2206888119] [Citation(s) in RCA: 10] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/28/2022] [Accepted: 07/13/2022] [Indexed: 01/08/2023] Open
Abstract
Self-organized pattern formation is vital for many biological processes. Reaction-diffusion models have advanced our understanding of how biological systems develop spatial structures, starting from homogeneity. However, biological processes inherently involve multiple spatial and temporal scales and transition from one pattern to another over time, rather than progressing from homogeneity to a pattern. To deal with such multiscale systems, coarse-graining methods are needed that allow the dynamics to be reduced to the relevant degrees of freedom at large scales, but without losing information about the patterns at small scales. Here, we present a semiphenomenological approach which exploits mass conservation in pattern formation, and enables reconstruction of information about patterns from the large-scale dynamics. The basic idea is to partition the domain into distinct regions (coarse grain) and determine instantaneous dispersion relations in each region, which ultimately inform about local pattern-forming instabilities. We illustrate our approach by studying the Min system, a paradigmatic model for protein pattern formation. By performing simulations, we first show that the Min system produces multiscale patterns in a spatially heterogeneous geometry. This prediction is confirmed experimentally by in vitro reconstitution of the Min system. Using a recently developed theoretical framework for mass-conserving reaction-diffusion systems, we show that the spatiotemporal evolution of the total protein densities on large scales reliably predicts the pattern-forming dynamics. Our approach provides an alternative and versatile theoretical framework for complex systems where analytical coarse-graining methods are not applicable, and can, in principle, be applied to a wide range of systems with an underlying conservation law.
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Affiliation(s)
- Laeschkir Würthner
- Arnold Sommerfeld Center for Theoretical Physics, Department of Physics, Ludwig-Maximilians-Universität München, D-80333 München, Germany
- Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, D-80333 München, Germany
| | - Fridtjof Brauns
- Arnold Sommerfeld Center for Theoretical Physics, Department of Physics, Ludwig-Maximilians-Universität München, D-80333 München, Germany
- Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, D-80333 München, Germany
| | - Grzegorz Pawlik
- Department of Bionanoscience, Kavli Institute of Nanoscience Delft, Delft University of Technology, 2629 HZ Delft, The Netherlands
| | - Jacob Halatek
- Arnold Sommerfeld Center for Theoretical Physics, Department of Physics, Ludwig-Maximilians-Universität München, D-80333 München, Germany
- Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, D-80333 München, Germany
- Research Department, Oxford BioMedica Ltd., Oxford OX4 6LT, United Kingdom
| | - Jacob Kerssemakers
- Department of Bionanoscience, Kavli Institute of Nanoscience Delft, Delft University of Technology, 2629 HZ Delft, The Netherlands
| | - Cees Dekker
- Department of Bionanoscience, Kavli Institute of Nanoscience Delft, Delft University of Technology, 2629 HZ Delft, The Netherlands
| | - Erwin Frey
- Arnold Sommerfeld Center for Theoretical Physics, Department of Physics, Ludwig-Maximilians-Universität München, D-80333 München, Germany
- Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, D-80333 München, Germany
- Max Planck School Matter to Life, D-80539 Munich, Germany
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12
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Miller PW, Fortunato D, Muratov C, Greengard L, Shvartsman S. Forced and spontaneous symmetry breaking in cell polarization. NATURE COMPUTATIONAL SCIENCE 2022; 2:504-511. [PMID: 37309402 PMCID: PMC10260237 DOI: 10.1038/s43588-022-00295-0] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/20/2022] [Accepted: 07/12/2022] [Indexed: 06/14/2023]
Abstract
How does breaking the symmetry of an equation alter the symmetry of its solutions? Here, we systematically examine how reducing underlying symmetries from spherical to axisymmetric influences the dynamics of an archetypal model of cell polarization, a key process of biological spatial self-organization. Cell polarization is characterized by nonlinear and non-local dynamics, but we overcome the theory challenges these traits pose by introducing a broadly applicable numerical scheme allowing us to efficiently study continuum models in a wide range of geometries. Guided by numerical results, we discover a dynamical hierarchy of timescales that allows us to reduce relaxation to a purely geometric problem of area-preserving geodesic curvature flow. Through application of variational results, we analytically construct steady states on a number of biologically relevant shapes. In doing so, we reveal non-trivial solutions for symmetry breaking.
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Affiliation(s)
- Pearson W. Miller
- Center for Computational Biology, Flatiron Institute, New York, NY, USA
- These authors contributed equally: Pearson W. Miller, Daniel Fortunato
| | - Daniel Fortunato
- Center for Computational Mathematics, Flatiron Institute, New York, NY, USA
- These authors contributed equally: Pearson W. Miller, Daniel Fortunato
| | - Cyrill Muratov
- Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, USA
- Dipartimento di Matematica, Università di Pisa, Pisa, Italy
| | - Leslie Greengard
- Center for Computational Mathematics, Flatiron Institute, New York, NY, USA
- Courant Institute, New York University, New York, NY, USA
| | - Stanislav Shvartsman
- Center for Computational Biology, Flatiron Institute, New York, NY, USA
- Courant Institute, New York University, New York, NY, USA
- Department of Molecular Biology, Princeton University, Princeton, NJ, USA
- Lewis-Sigler Institute for Integrative Genomics, Princeton University, Princeton, NJ, USA
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13
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Escárcega-Bobadilla MV, Maldonado-Domínguez M, Romero-Ávila M, Zelada-Guillén GA. Turing patterns by supramolecular self-assembly of a single salphen building block. iScience 2022; 25:104545. [PMID: 35747384 PMCID: PMC9209723 DOI: 10.1016/j.isci.2022.104545] [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] [Received: 01/07/2022] [Revised: 05/15/2022] [Accepted: 06/02/2022] [Indexed: 11/02/2022] Open
Abstract
In the 1950s, Alan Turing showed that concerted reactions and diffusion of activating and inhibiting chemical species can autonomously generate patterns without previous positional information, thus providing a chemical basis for morphogenesis in Nature. However, access to these patterns from only one molecular component that contained all the necessary information to execute agonistic and antagonistic signaling is so far an elusive goal, since two or more participants with different diffusivities are a must. Here, we report on a single-molecule system that generates Turing patterns arrested in the solid state, where supramolecular interactions are used instead of chemical reactions, whereas diffusional differences arise from heterogeneously populated self-assembled products. We employ a family of hydroxylated organic salphen building blocks based on a bis-Schiff-base scaffold with portions responsible for either activation or inhibition of assemblies at different hierarchies through purely supramolecular reactions, only depending upon the solvent dielectric constant and evaporation as fuel.
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Affiliation(s)
- Martha V Escárcega-Bobadilla
- School of Chemistry, National Autonomous University of Mexico (UNAM), Circuito Escolar s/n, Ciudad Universitaria, 04510 Mexico City, Mexico
| | - Mauricio Maldonado-Domínguez
- School of Chemistry, National Autonomous University of Mexico (UNAM), Circuito Escolar s/n, Ciudad Universitaria, 04510 Mexico City, Mexico.,Department of Computational Chemistry, J. Heyrovský Institute of Physical Chemistry, The Czech Academy of Sciences, Dolejškova 3, 18223 Prague 8, Czech Republic
| | - Margarita Romero-Ávila
- School of Chemistry, National Autonomous University of Mexico (UNAM), Circuito Escolar s/n, Ciudad Universitaria, 04510 Mexico City, Mexico
| | - Gustavo A Zelada-Guillén
- School of Chemistry, National Autonomous University of Mexico (UNAM), Circuito Escolar s/n, Ciudad Universitaria, 04510 Mexico City, Mexico
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14
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Diffusion-mediated HEI10 coarsening can explain meiotic crossover positioning in Arabidopsis. Nat Commun 2021; 12:4674. [PMID: 34344879 PMCID: PMC8333306 DOI: 10.1038/s41467-021-24827-w] [Citation(s) in RCA: 51] [Impact Index Per Article: 17.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2020] [Accepted: 07/12/2021] [Indexed: 11/08/2022] Open
Abstract
In most organisms, the number and distribution of crossovers that occur during meiosis are tightly controlled. All chromosomes must receive at least one ‘obligatory crossover’ and crossovers are prevented from occurring near one another by ‘crossover interference’. However, the mechanistic basis of this phenomenon of crossover interference has remained mostly mysterious. Using quantitative super-resolution cytogenetics and mathematical modelling, we investigate crossover positioning in the Arabidopsis thaliana wild-type, an over-expressor of the conserved E3 ligase HEI10, and a hei10 heterozygous line. We show that crossover positions can be explained by a predictive, diffusion-mediated coarsening model, in which large, approximately evenly-spaced HEI10 foci grow at the expense of smaller, closely-spaced clusters. We propose this coarsening process explains many aspects of Arabidopsis crossover positioning, including crossover interference. Consistent with this model, we also demonstrate that crossover positioning can be predictably modified in vivo simply by altering HEI10 dosage, with higher and lower dosage leading to weaker and stronger crossover interference, respectively. As HEI10 is a conserved member of the RING finger protein family that functions in the interference-sensitive pathway for crossover formation, we anticipate that similar mechanisms may regulate crossover positioning in diverse eukaryotes. Crossover numbers and positions are tightly controlled but the mechanism involved is still obscure. Here, the authors, using quantitative super-resolution cytogenetics and mathematical modelling, show that diffusion mediated coarsening of HEI10, an E3-ligase domain containing protein, may explain meiotic crossover positioning in Arabidopsis.
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Frohoff-Hülsmann T, Wrembel J, Thiele U. Suppression of coarsening and emergence of oscillatory behavior in a Cahn-Hilliard model with nonvariational coupling. Phys Rev E 2021; 103:042602. [PMID: 34006003 DOI: 10.1103/physreve.103.042602] [Citation(s) in RCA: 10] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/02/2020] [Accepted: 03/04/2021] [Indexed: 12/27/2022]
Abstract
We investigate a generic two-field Cahn-Hilliard model with variational and nonvariational coupling. It describes, for instance, passive and active ternary mixtures, respectively. Already a linear stability analysis of the homogeneous mixed state shows that activity not only allows for the usual large-scale stationary (Cahn-Hilliard) instability of the well-known passive case but also for small-scale stationary (Turing) and large-scale oscillatory (Hopf) instabilities. In consequence of the Turing instability, activity may completely suppress the usual coarsening dynamics. In a fully nonlinear analysis, we first briefly discuss the passive case before focusing on the active case. Bifurcation diagrams and selected direct time simulations are presented that allow us to establish that nonvariational coupling (i) can partially or completely suppress coarsening and (ii) may lead to the emergence of drifting and oscillatory states. Throughout, we emphasize the relevance of conservation laws and related symmetries for the encountered intricate bifurcation behavior.
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Affiliation(s)
- Tobias Frohoff-Hülsmann
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Str. 9, 48149 Münster, Germany
| | - Jana Wrembel
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Str. 9, 48149 Münster, Germany
| | - Uwe Thiele
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Str. 9, 48149 Münster, Germany.,Center for Nonlinear Science (CeNoS), Westfälische Wilhelms-Universität Münster, Corrensstr. 2, 48149 Münster, Germany.,Center for Multiscale Theory and Computation (CMTC), Westfälische Wilhelms-Universität, Corrensstr. 40, 48149 Münster, Germany
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Chiou JG, Moran KD, Lew DJ. How cells determine the number of polarity sites. eLife 2021; 10:e58768. [PMID: 33899733 PMCID: PMC8116050 DOI: 10.7554/elife.58768] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2020] [Accepted: 04/23/2021] [Indexed: 12/15/2022] Open
Abstract
The diversity of cell morphologies arises, in part, through regulation of cell polarity by Rho-family GTPases. A poorly understood but fundamental question concerns the regulatory mechanisms by which different cells generate different numbers of polarity sites. Mass-conserved activator-substrate (MCAS) models that describe polarity circuits develop multiple initial polarity sites, but then those sites engage in competition, leaving a single winner. Theoretical analyses predicted that competition would slow dramatically as GTPase concentrations at different polarity sites increase toward a 'saturation point', allowing polarity sites to coexist. Here, we test this prediction using budding yeast cells, and confirm that increasing the amount of key polarity proteins results in multiple polarity sites and simultaneous budding. Further, we elucidate a novel design principle whereby cells can switch from competition to equalization among polarity sites. These findings provide insight into how cells with diverse morphologies may determine the number of polarity sites.
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Affiliation(s)
- Jian-geng Chiou
- Department of Pharmacology and Cancer Biology, Duke University Medical CenterDurhamUnited States
| | - Kyle D Moran
- Department of Pharmacology and Cancer Biology, Duke University Medical CenterDurhamUnited States
| | - Daniel J Lew
- Department of Pharmacology and Cancer Biology, Duke University Medical CenterDurhamUnited States
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