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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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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3
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Borgqvist JG, Gerlee P, Lundholm C. Turing pattern formation on the sphere is robust to the removal of a hole. J Math Biol 2024; 88:23. [PMID: 38296874 PMCID: PMC10830788 DOI: 10.1007/s00285-023-02034-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2023] [Revised: 11/17/2023] [Accepted: 12/03/2023] [Indexed: 02/02/2024]
Abstract
The formation of buds on the cell membrane of budding yeast cells is thought to be driven by reactions and diffusion involving the protein Cdc42. These processes can be described by a coupled system of partial differential equations known as the Schnakenberg system. The Schnakenberg system is known to exhibit diffusion-driven pattern formation, thus providing a mechanism for bud formation. However, it is not known how the accumulation of bud scars on the cell membrane affect the ability of the Schnakenberg system to form patterns. We have approached this problem by modelling a bud scar on the cell membrane with a hole on the sphere. We have studied how the spectrum of the Laplace-Beltrami operator, which determines the resulting pattern, is affected by the size of the hole, and by numerically solving the Schnakenberg system on a sphere with a hole using the finite element method. Both theoretical predictions and numerical solutions show that pattern formation is robust to the introduction of a bud scar of considerable size, which lends credence to the hypothesis that bud formation is driven by diffusion-driven instability.
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Affiliation(s)
- Johannes G Borgqvist
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Andrew Wiles Building Radcliffe Observatory Quarter (550) Woodstock Road, Oxford, Oxfordshire, OX2 6GG, UK.
| | - Philip Gerlee
- Mathematical Sciences, University of Gothenburg, Chalmers tvärgata 3, 412 96, Gothenburg, Västra Götaland, Sweden
- Mathematical Sciences, Chalmers University of Technology, Chalmers tvärgata 3, 412 96, Gothenburg, Västra Götaland, Sweden
| | - Carl Lundholm
- Department of Mathematics and Mathematical Statistics, Umeå University, MIT Building, 3rd Floor Linneaus Väg, 907 36, Umeå, Västerbotten, Sweden
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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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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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7
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Frohoff-Hülsmann T, Thiele U, Pismen LM. Non-reciprocity induces resonances in a two-field Cahn-Hilliard model. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2023; 381:20220087. [PMID: 36842986 DOI: 10.1098/rsta.2022.0087] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/29/2022] [Accepted: 01/09/2023] [Indexed: 06/18/2023]
Abstract
We consider a non-reciprocally coupled two-field Cahn-Hilliard system that has been shown to allow for oscillatory behaviour and suppression of coarsening. After introducing the model, we first review the linear stability of steady uniform states and show that all instability thresholds are identical to the ones for a corresponding two-species reaction-diffusion system. Next, we consider a specific interaction of linear modes-a 'Hopf-Turing' resonance-and derive the corresponding amplitude equations using a weakly nonlinear approach. We discuss the weakly nonlinear results and finally compare them with fully nonlinear simulations for a specific conserved amended FitzHugh-Nagumo system. We conclude with a discussion of the limitations of the employed weakly nonlinear approach. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.
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Affiliation(s)
- Tobias Frohoff-Hülsmann
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Str. 9, Münster 48149, Germany
| | - Uwe Thiele
- Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Str. 9, Münster 48149, Germany
- Center for Nonlinear Science (CeNoS), Westfälische Wilhelms-Universität Münster, Corrensstr. 2, Münster 48149, Germany
| | - Len M Pismen
- Department of Chemical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel
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8
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Kuwamura M, Izuhara H, Ei SI. Oscillations and bifurcation structure of reaction-diffusion model for cell polarity formation. J Math Biol 2022; 84:22. [PMID: 35212844 DOI: 10.1007/s00285-022-01723-5] [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: 04/01/2021] [Revised: 01/25/2022] [Accepted: 01/28/2022] [Indexed: 11/28/2022]
Abstract
We investigate the oscillatory dynamics and bifurcation structure of a reaction-diffusion system with bistable nonlinearity and mass conservation, which was proposed by (Otsuji et al., PLoS Comp Biol 3:e108, 2007). The system is a useful model for understanding cell polarity formation. We show that this model exhibits four different spatiotemporal patterns including two types of oscillatory patterns, which can be regarded as cell polarity oscillations with the reversal and non-reversal of polarity, respectively. The trigger causing these patterns is a diffusion-driven (Turing-like) instability. Moreover, we investigate the effects of extracellular signals on the cell polarity oscillations.
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Affiliation(s)
- Masataka Kuwamura
- Graduate School of Human Development and Environment, Kobe University, Kobe, 657-8501, Japan.
| | - Hirofumi Izuhara
- Faculty of Engineering, University of Miyazaki, 1-1 Gakuen Kibanadai-nishi, Miyazaki, 889-2192, Japan
| | - Shin-Ichiro Ei
- Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan
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9
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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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10
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Brauns F, Weyer H, Halatek J, Yoon J, Frey E. Wavelength Selection by Interrupted Coarsening in Reaction-Diffusion Systems. PHYSICAL REVIEW LETTERS 2021; 126:104101. [PMID: 33784126 DOI: 10.1103/physrevlett.126.104101] [Citation(s) in RCA: 16] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/05/2020] [Revised: 10/19/2020] [Accepted: 12/09/2020] [Indexed: 06/12/2023]
Abstract
Wavelength selection in reaction-diffusion systems can be understood as a coarsening process that is interrupted by counteracting processes at certain wavelengths. We first show that coarsening in mass-conserving systems is driven by self-amplifying mass transport between neighboring high-density domains. We derive a general coarsening criterion and show that coarsening is generically uninterrupted in two-component systems that conserve mass. The theory is then generalized to study interrupted coarsening and anticoarsening due to weakly broken mass conservation, providing a general path to analyze wavelength selection in pattern formation far from equilibrium.
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Affiliation(s)
- 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
| | - 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
| | - Jacob Halatek
- Biological Computation Group, Microsoft Research, Cambridge CB1 2FB, United Kingdom
| | - Junghoon Yoon
- 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
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11
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Bressloff PC. Multi-spike solutions of a hybrid reaction–transport model. Proc Math Phys Eng Sci 2021. [DOI: 10.1098/rspa.2020.0829] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Simulations of classical pattern-forming reaction–diffusion systems indicate that they often operate in the strongly nonlinear regime, with the final steady state consisting of a spatially repeating pattern of localized spikes. In activator–inhibitor systems such as the two-component Gierer–Meinhardt (GM) model, one can consider the singular limit
D
a
≪
D
h
, where
D
a
and
D
h
are the diffusivities of the activator and inhibitor, respectively. Asymptotic analysis can then be used to analyse the existence and linear stability of multi-spike solutions. In this paper, we analyse multi-spike solutions in a hybrid reaction–transport model, consisting of a slowly diffusing activator and an actively transported inhibitor that switches at a rate
α
between right-moving and left-moving velocity states. Such a model was recently introduced to account for the formation and homeostatic regulation of synaptic puncta during larval development in
Caenorhabditis elegans
. We exploit the fact that the hybrid model can be mapped onto the classical GM model in the fast switching limit
α
→ ∞, which establishes the existence of multi-spike solutions. Linearization about the multi-spike solution yields a non-local eigenvalue problem that is used to investigate stability of the multi-spike solution by combining analytical results for
α
→ ∞ with a graphical construction for finite
α
.
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Affiliation(s)
- P. C. Bressloff
- Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, UT 84112, USA
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12
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Subramanian S, Murray SM. Pattern selection in reaction diffusion systems. Phys Rev E 2021; 103:012215. [PMID: 33601645 DOI: 10.1103/physreve.103.012215] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2020] [Accepted: 01/07/2021] [Indexed: 11/07/2022]
Abstract
Turing's theory of pattern formation has been used to describe the formation of self-organized periodic patterns in many biological, chemical, and physical systems. However, the use of such models is hindered by our inability to predict, in general, which pattern is obtained from a given set of model parameters. While much is known near the onset of the spatial instability, the mechanisms underlying pattern selection and dynamics away from onset are much less understood. Here, we provide physical insight into the dynamics of these systems. We find that peaks in a Turing pattern behave as point sinks, the dynamics of which is determined by the diffusive fluxes into them. As a result, peaks move toward a periodic steady-state configuration that minimizes the mass of the diffusive species. We also show that the preferred number of peaks at the final steady state is such that this mass is minimized. Our work presents mass minimization as a potential general principle for understanding pattern formation in reaction diffusion systems far from onset.
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Affiliation(s)
| | - Seán M Murray
- Max Planck Institute for Terrestrial Microbiology, 35043 Marburg, Germany
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13
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Goryachev AB, Leda M. Compete or Coexist? Why the Same Mechanisms of Symmetry Breaking Can Yield Distinct Outcomes. Cells 2020; 9:E2011. [PMID: 32882972 PMCID: PMC7563139 DOI: 10.3390/cells9092011] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2020] [Revised: 08/27/2020] [Accepted: 08/28/2020] [Indexed: 12/22/2022] Open
Abstract
Cellular morphogenesis is governed by the prepattern based on the symmetry-breaking emergence of dense protein clusters. Thus, a cluster of active GTPase Cdc42 marks the site of nascent bud in the baker's yeast. An important biological question is which mechanisms control the number of pattern maxima (spots) and, thus, the number of nascent cellular structures. Distinct flavors of theoretical models seem to suggest different predictions. While the classical Turing scenario leads to an array of stably coexisting multiple structures, mass-conserved models predict formation of a single spot that emerges via the greedy competition between the pattern maxima for the common molecular resources. Both the outcome and the kinetics of this competition are of significant biological importance but remained poorly explored. Recent theoretical analyses largely addressed these questions, but their results have not yet been fully appreciated by the broad biological community. Keeping mathematical apparatus and jargon to the minimum, we review the main conclusions of these analyses with their biological implications in mind. Focusing on the specific example of pattern formation by small GTPases, we speculate on the features of the patterning mechanisms that bypass competition and favor formation of multiple coexisting structures and contrast them with those of the mechanisms that harness competition to form unique cellular structures.
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Affiliation(s)
- Andrew B. Goryachev
- SynthSys, Centre for Synthetic and Systems Biology, Institute for Cell Biology, University of Edinburgh, Edinburg EH9 3BD, UK;
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14
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Rusch F, Wosniack ME, Raposo EP, Viswanathan GM, da Luz MGE. Transient dynamics in a nonequilibrium superdiffusive reaction-diffusion process: Nonequilibrium random search as a case study. Phys Rev E 2020; 102:012126. [PMID: 32794983 DOI: 10.1103/physreve.102.012126] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/13/2020] [Accepted: 06/23/2020] [Indexed: 11/07/2022]
Abstract
Transient regimes, often difficult to characterize, can be fundamental in establishing final steady states features of reaction-diffusion phenomena. This is particularly true in ecological problems. Here, through both numerical simulations and an analytic approximation, we analyze the transient of a nonequilibrium superdiffusive random search when the targets are created at a certain rate and annihilated upon encounters (a key dynamics, e.g., in biological foraging). The steady state is achieved when the number of targets stabilizes to a constant value. Our results unveil how key features of the steady state are closely associated to the particularities of the initial evolution. The searching efficiency variation in time is also obtained. It presents a rather surprising universal behavior at the asymptotic limit. These analyses shed some light into the general relevance of transients in reaction-diffusion systems.
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Affiliation(s)
- F Rusch
- Departamento de Física, Universidade Federal do Paraná, Curitiba-PR 81531-980, Brazil
| | - M E Wosniack
- Max Planck Institute for Brain Research, Frankfurt 60438, Germany
| | - E P Raposo
- Laboratório de Física Teórica e Computacional, Departamento de Física, Universidade Federal de Pernambuco, Recife-PE 50670-901, Brazil.,Centre d'Estudis Avançats de Blanes-CEAB-CSIC, Girona 17300, Spain.,Centre for Ecological Research and Forestry Applications-CREAF, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Spain
| | - G M Viswanathan
- National Institute of Science and Technology of Complex Systems and Department of Physics, Universidade Federal do Rio Grande do Norte, Natal-RN 59078-970, Brazil
| | - M G E da Luz
- Departamento de Física, Universidade Federal do Paraná, Curitiba-PR 81531-980, Brazil
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15
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Flow Induced Symmetry Breaking in a Conceptual Polarity Model. Cells 2020; 9:cells9061524. [PMID: 32585819 PMCID: PMC7349905 DOI: 10.3390/cells9061524] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/19/2020] [Revised: 06/18/2020] [Accepted: 06/19/2020] [Indexed: 12/19/2022] Open
Abstract
Important cellular processes, such as cell motility and cell division, are coordinated by cell polarity, which is determined by the non-uniform distribution of certain proteins. Such protein patterns form via an interplay of protein reactions and protein transport. Since Turing’s seminal work, the formation of protein patterns resulting from the interplay between reactions and diffusive transport has been widely studied. Over the last few years, increasing evidence shows that also advective transport, resulting from cytosolic and cortical flows, is present in many cells. However, it remains unclear how and whether these flows contribute to protein-pattern formation. To address this question, we use a minimal model that conserves the total protein mass to characterize the effects of cytosolic flow on pattern formation. Combining a linear stability analysis with numerical simulations, we find that membrane-bound protein patterns propagate against the direction of cytoplasmic flow with a speed that is maximal for intermediate flow speed. We show that the mechanism underlying this pattern propagation relies on a higher protein influx on the upstream side of the pattern compared to the downstream side. Furthermore, we find that cytosolic flow can change the membrane pattern qualitatively from a peak pattern to a mesa pattern. Finally, our study shows that a non-uniform flow profile can induce pattern formation by triggering a regional lateral instability.
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16
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Beta C, Gov NS, Yochelis A. Why a Large-Scale Mode Can Be Essential for Understanding Intracellular Actin Waves. Cells 2020; 9:cells9061533. [PMID: 32585983 PMCID: PMC7349605 DOI: 10.3390/cells9061533] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/16/2020] [Revised: 06/16/2020] [Accepted: 06/18/2020] [Indexed: 01/18/2023] Open
Abstract
During the last decade, intracellular actin waves have attracted much attention due to their essential role in various cellular functions, ranging from motility to cytokinesis. Experimental methods have advanced significantly and can capture the dynamics of actin waves over a large range of spatio-temporal scales. However, the corresponding coarse-grained theory mostly avoids the full complexity of this multi-scale phenomenon. In this perspective, we focus on a minimal continuum model of activator-inhibitor type and highlight the qualitative role of mass conservation, which is typically overlooked. Specifically, our interest is to connect between the mathematical mechanisms of pattern formation in the presence of a large-scale mode, due to mass conservation, and distinct behaviors of actin waves.
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Affiliation(s)
- Carsten Beta
- Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany;
| | - Nir S. Gov
- Department of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot 76100, Israel;
| | - Arik Yochelis
- Department of Solar Energy and Environmental Physics, Blaustein Institutes for Desert Research (BIDR), Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 8499000, Israel
- Department of Physics, Ben-Gurion University of the Negev, Be’er Sheva 8410501, Israel
- Correspondence:
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Wigbers MC, Brauns F, Hermann T, Frey E. Pattern localization to a domain edge. Phys Rev E 2020; 101:022414. [PMID: 32168714 DOI: 10.1103/physreve.101.022414] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/21/2019] [Accepted: 12/03/2019] [Indexed: 06/10/2023]
Abstract
The formation of protein patterns inside cells is generically described by reaction-diffusion models. The study of such systems goes back to Turing, who showed how patterns can emerge from a homogenous steady state when two reactive components have different diffusivities (e.g., membrane-bound and cytosolic states). However, in nature, systems typically develop in a heterogeneous environment, where upstream protein patterns affect the formation of protein patterns downstream. Examples for this are the polarization of Cdc42 adjacent to the previous bud site in budding yeast and the formation of an actin-recruiter ring that forms around a PIP3 domain in macropinocytosis. This suggests that previously established protein patterns can serve as a template for downstream proteins and that these downstream proteins can "sense" the edge of the template. A mechanism for how this edge sensing may work remains elusive. Here we demonstrate and analyze a generic and robust edge-sensing mechanism, based on a two-component mass-conserving reaction-diffusion (McRD) model. Our analysis is rooted in a recently developed theoretical framework for McRD systems, termed local equilibria theory. We extend this framework to capture the spatially heterogeneous reaction kinetics due to the template. This enables us to graphically construct the stationary patterns in the phase space of the reaction kinetics. Furthermore, we show that the protein template can trigger a regional mass-redistribution instability near the template edge, leading to the accumulation of protein mass, which eventually results in a stationary peak at the template edge. We show that simple geometric criteria on the reactive nullcline's shape predict when this edge-sensing mechanism is operational. Thus, our results provide guidance for future studies of biological systems and for the design of synthetic pattern forming systems.
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Affiliation(s)
- Manon C Wigbers
- Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS), Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München, Germany
| | - Fridtjof Brauns
- Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS), Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München, Germany
| | - Tobias Hermann
- Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS), Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München, 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 München, Germany
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18
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Halatek J, Brauns F, Frey E. Self-organization principles of intracellular pattern formation. Philos Trans R Soc Lond B Biol Sci 2019; 373:rstb.2017.0107. [PMID: 29632261 PMCID: PMC5904295 DOI: 10.1098/rstb.2017.0107] [Citation(s) in RCA: 82] [Impact Index Per Article: 16.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 02/09/2018] [Indexed: 11/13/2022] Open
Abstract
Dynamic patterning of specific proteins is essential for the spatio-temporal regulation of many important intracellular processes in prokaryotes, eukaryotes and multicellular organisms. The emergence of patterns generated by interactions of diffusing proteins is a paradigmatic example for self-organization. In this article, we review quantitative models for intracellular Min protein patterns in Escherichia coli, Cdc42 polarization in Saccharomyces cerevisiae and the bipolar PAR protein patterns found in Caenorhabditis elegans. By analysing the molecular processes driving these systems we derive a theoretical perspective on general principles underlying self-organized pattern formation. We argue that intracellular pattern formation is not captured by concepts such as ‘activators’, ‘inhibitors’ or ‘substrate depletion’. Instead, intracellular pattern formation is based on the redistribution of proteins by cytosolic diffusion, and the cycling of proteins between distinct conformational states. Therefore, mass-conserving reaction–diffusion equations provide the most appropriate framework to study intracellular pattern formation. We conclude that directed transport, e.g. cytosolic diffusion along an actively maintained cytosolic gradient, is the key process underlying pattern formation. Thus the basic principle of self-organization is the establishment and maintenance of directed transport by intracellular protein dynamics. This article is part of the theme issue ‘Self-organization in cell biology’.
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Affiliation(s)
- J Halatek
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München, Germany
| | - F Brauns
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München, Germany
| | - E Frey
- Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience, Department of Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, D-80333 München, Germany
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Jacobs B, Molenaar J, Deinum EE. Small GTPase patterning: How to stabilise cluster coexistence. PLoS One 2019; 14:e0213188. [PMID: 30845201 PMCID: PMC6405054 DOI: 10.1371/journal.pone.0213188] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/29/2018] [Accepted: 02/16/2019] [Indexed: 12/28/2022] Open
Abstract
Many biological processes have to occur at specific locations on the cell membrane. These locations are often specified by the localised activity of small GTPase proteins. Some processes require the formation of a single cluster of active GTPase, also called unipolar polarisation (here “polarisation”), whereas others need multiple coexisting clusters. Moreover, sometimes the pattern of GTPase clusters is dynamically regulated after its formation. This raises the question how the same interacting protein components can produce such a rich variety of naturally occurring patterns. Most currently used models for GTPase-based patterning inherently yield polarisation. Such models may at best yield transient coexistence of at most a few clusters, and hence fail to explain several important biological phenomena. These existing models are all based on mass conservation of total GTPase and some form of direct or indirect positive feedback. Here, we show that either of two biologically plausible modifications can yield stable coexistence: including explicit GTPase turnover, i.e., breaking mass conservation, or negative feedback by activation of an inhibitor like a GAP. Since we start from two different polarising models our findings seem independent of the precise self-activation mechanism. By studying the net GTPase flows among clusters, we provide insight into how these mechanisms operate. Our coexistence models also allow for dynamical regulation of the final pattern, which we illustrate with examples of pollen tube growth and the branching of fungal hyphae. Together, these results provide a better understanding of how cells can tune a single system to generate a wide variety of biologically relevant patterns.
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Affiliation(s)
- Bas Jacobs
- Biometris, Department for Mathematical and Statistical Methods, Wageningen University, Wageningen, The Netherlands
| | - Jaap Molenaar
- Biometris, Department for Mathematical and Statistical Methods, Wageningen University, Wageningen, The Netherlands
| | - Eva E Deinum
- Biometris, Department for Mathematical and Statistical Methods, Wageningen University, Wageningen, The Netherlands
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20
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Kuwamura M, Izuhara H. Diffusion-driven destabilization of spatially homogeneous limit cycles in reaction-diffusion systems. CHAOS (WOODBURY, N.Y.) 2017; 27:033112. [PMID: 28364773 DOI: 10.1063/1.4978924] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
We study the diffusion-driven destabilization of a spatially homogeneous limit cycle with large amplitude in a reaction-diffusion system on an interval of finite size under the periodic boundary condition. Numerical bifurcation analysis and simulations show that the spatially homogeneous limit cycle becomes unstable and changes to a stable spatially nonhomogeneous limit cycle for appropriate diffusion coefficients. This is analogous to the diffusion-driven destabilization (Turing instability) of a spatially homogeneous equilibrium. Our approach is based on a reaction-diffusion system with mass conservation and its perturbed system considered as an infinite dimensional slow-fast system (relaxation oscillator).
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Affiliation(s)
- Masataka Kuwamura
- Graduate School of Human Development and Environment, Kobe University, Kobe 657-8501, Japan
| | - Hirofumi Izuhara
- Faculty of Engineering, University of Miyazaki, Miyazaki 889-2192, Japan
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21
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Kuwamura M, Morita Y. Perturbations and dynamics of reaction-diffusion systems with mass conservation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:012908. [PMID: 26274248 DOI: 10.1103/physreve.92.012908] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/12/2014] [Indexed: 06/04/2023]
Abstract
In some reaction-diffusion systems where the total mass of their components is conserved, solutions with initial values near a homogeneous equilibrium converge to a simple localized pattern (spike) after exhibiting Turing-like patterns near the equilibrium for appropriate diffusion coefficients. In this study, we investigate the perturbed reaction-diffusion systems of such conserved systems. We show that a reaction-diffusion model with a globally stable homogeneous equilibrium can exhibit large amplitude Turing-like patterns in the transient dynamics. Moreover, we propose a three-component model, which exhibits an alternating repetition of spatially (almost) homogeneous oscillations and large amplitude Turing-like patterns.
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Affiliation(s)
- Masataka Kuwamura
- Graduate School of Human Development and Environment, Kobe University, Kobe 657-8501, Japan
| | - Yoshihisa Morita
- Department of Applied Mathematics and Informatics, Ryukoku University, Otsu 520-2194, Japan
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22
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Otsuji M, Terashima Y, Ishihara S, Kuroda S, Matsushima K. A conceptual molecular network for chemotactic behaviors characterized by feedback of molecules cycling between the membrane and the cytosol. Sci Signal 2010; 3:ra89. [PMID: 21156936 DOI: 10.1126/scisignal.2001056] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Abstract
Cell chemotaxis has been characterized as the formation of a front-back axis that is triggered by a gradient of chemoattractant; however, chemotaxis is accompanied by more complicated behaviors. These include migration in a straight line with a stable axis [the stable single-axis (SSA) pattern] and repeated splitting of the leading edge of the cell into two regions, followed by the "choice" of one of these as the new leading edge [the split and choice (S&C) pattern]. Indeed, transition between these two behaviors can be observed in individual cells. However, the conceptual framework of the network of signaling molecules that generates these patterns remains to be clarified. We confirmed theoretically that a system that has positive and negative feedback loops involving the reciprocal cycling between the membrane and the cytosol of molecules that promote membrane protrusion or retraction generates SSA and S&C patterns of migratory behavior under similar conditions. We also predicted properties of the instabilities of such a system, which are essential for the generation of these behaviors, and we verified their existence in chemotaxing cells. Our research provides a simple model of network structure for chemotactic behaviors, including cell polarization.
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Affiliation(s)
- Mikiya Otsuji
- Department of Molecular Preventive Medicine, Graduate School of Medicine, University of Tokyo, Tokyo 113-0033, Japan.
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Otsuji M, Ishihara S, Co C, Kaibuchi K, Mochizuki A, Kuroda S. A mass conserved reaction-diffusion system captures properties of cell polarity. PLoS Comput Biol 2007; 3:e108. [PMID: 17559299 PMCID: PMC1892603 DOI: 10.1371/journal.pcbi.0030108] [Citation(s) in RCA: 111] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/08/2006] [Accepted: 04/29/2007] [Indexed: 12/23/2022] Open
Abstract
Cell polarity is a general cellular process that can be seen in various cell types such as migrating neutrophils and Dictyostelium cells. The Rho small GTP(guanosine 5′-tri phosphate)ases have been shown to regulate cell polarity; however, its mechanism of emergence has yet to be clarified. We first developed a reaction–diffusion model of the Rho GTPases, which exhibits switch-like reversible response to a gradient of extracellular signals, exclusive accumulation of Cdc42 and Rac, or RhoA at the maximal or minimal intensity of the signal, respectively, and tracking of changes of a signal gradient by the polarized peak. The previous cell polarity models proposed by Subramanian and Narang show similar behaviors to our Rho GTPase model, despite the difference in molecular networks. This led us to compare these models, and we found that these models commonly share instability and a mass conservation of components. Based on these common properties, we developed conceptual models of a mass conserved reaction–diffusion system with diffusion–driven instability. These conceptual models retained similar behaviors of cell polarity in the Rho GTPase model. Using these models, we numerically and analytically found that multiple polarized peaks are unstable, resulting in a single stable peak (uniqueness of axis), and that sensitivity toward changes of a signal gradient is specifically restricted at the polarized peak (localized sensitivity). Although molecular networks may differ from one cell type to another, the behaviors of cell polarity in migrating cells seem similar, suggesting that there should be a fundamental principle. Thus, we propose that a mass conserved reaction–diffusion system with diffusion-driven instability is one of such principles of cell polarity. Eukaryotic cells such as neutrophils and Dictyostelium cells respond to temporal and spatial gradients of extracellular signals with directional movements. In a migrating cell, specific molecular events take place at the front and back edges. The spatially distinctive molecular accumulation inside cells is known as cell polarity. Despite numerous experimental and theoretical studies, its mechanism of emergence has yet to be clarified. We first developed a mathematical model of the Rho small GTP(guanosine 5′-tri phosphate)ases that cooperatively regulate cell polarity, and showed that the model generates specific spatial accumulation of the molecules. Based on our Rho GTPases model and other models, we further established a conceptual model, a mass conserved reaction–diffusion system, and showed that diffusion-driven instability and a mass conservation of molecules that have active and inactive states are sufficient for polarity formation. We numerically and analytically found that molecular accumulations at multiple sites are unstable, resulting in a single stable front–back axis, and that sensitivity toward changes of a signal gradient is specifically restricted at the front of a polarized cell. We propose that a mass conserved reaction–diffusion system is one of the fundamental principles of cell polarity.
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Affiliation(s)
- Mikiya Otsuji
- Department of Anesthesiology, Faculty of Medicine, University of Tokyo, Tokyo, Japan.
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