1
|
Lutz T, Menzel AM, Daddi-Moussa-Ider A. Internal sites of actuation and activation in thin elastic films and membranes of finite thickness. Phys Rev E 2024; 109:054802. [PMID: 38907440 DOI: 10.1103/physreve.109.054802] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/07/2024] [Accepted: 04/29/2024] [Indexed: 06/24/2024]
Abstract
Functionalized thin elastic films and membranes frequently feature internal sites of net forces or stresses. These are, for instance, active sites of actuation, or rigid inclusions in a strained membrane that induce counterstress upon externally imposed deformations. We theoretically analyze the geometry of isotropic, flat, thin, linearly elastic films or membranes of finite thickness, laterally extended to infinity. At the mathematical core of such characterizations are the fundamental solutions for localized force and stress singularities associated with corresponding Green's functions. We derive such solutions in three dimensions and place them into the context of previous two-dimensional calculations. To this end, we consider both no-slip and stress-free conditions at the top and/or bottom surfaces. We provide an understanding for why the fully free-standing thin elastic membrane leads to diverging solutions in most geometries and compare these situations to the truly two-dimensional case. A no-slip support of at least one of the surfaces stabilizes the solution, which illustrates that the divergences in the fully free-standing case are connected to global deformations. Within the aforementioned framework, our results are important for associated theoretical characterizations of thin elastic films, whether supported or free-standing, and of membranes subject to internal or external forces or stresses.
Collapse
Affiliation(s)
- Tyler Lutz
- Institut für Physik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany and Department of English Language and Literature, University of Chicago, Chicago, Illinois 60637, USA
| | - Andreas M Menzel
- Institut für Physik, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany
| | | |
Collapse
|
2
|
Li C, Nijjer J, Feng L, Zhang Q, Yan J, Zhang S. Agent-based modeling of stress anisotropy driven nematic ordering in growing biofilms. SOFT MATTER 2024; 20:3401-3410. [PMID: 38563244 DOI: 10.1039/d3sm01535a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/04/2024]
Abstract
Living active collectives have evolved with remarkable self-patterning capabilities to adapt to the physical and biological constraints crucial for their growth and survival. However, the intricate process by which complex multicellular patterns emerge from a single founder cell remains elusive. In this study, we utilize an agent-based model, validated through single-cell microscopy imaging, to track the three-dimensional (3D) morphodynamics of cells within growing bacterial biofilms encased by agarose gels. The confined growth conditions give rise to a spatiotemporally heterogeneous stress landscape within the biofilm. In the core of the biofilm, where high hydrostatic and low shear stresses prevail, cell packing appears disordered. In contrast, near the gel-cell interface, a state of high shear stress and low hydrostatic stress emerges, driving nematic ordering, albeit with a time delay inherent to shear stress relaxation. Strikingly, we observe a robust spatiotemporal correlation between stress anisotropy and nematic ordering within these confined biofilms. This correlation suggests a mechanism whereby stress anisotropy plays a pivotal role in governing the spatial organization of cells. The reciprocity between stress anisotropy and cell ordering in confined biofilms opens new avenues for innovative 3D mechanically guided patterning techniques for living active collectives, which hold significant promise for a wide array of environmental and biomedical applications.
Collapse
Affiliation(s)
- Changhao Li
- Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA, USA.
| | - Japinder Nijjer
- Department of Molecular, Cellular and Developmental Biology, Yale University, New Haven, CT, USA.
| | - Luyi Feng
- Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA, USA.
| | - Qiuting Zhang
- Department of Molecular, Cellular and Developmental Biology, Yale University, New Haven, CT, USA.
| | - Jing Yan
- Department of Molecular, Cellular and Developmental Biology, Yale University, New Haven, CT, USA.
- Quantitative Biology Institute, Yale University, New Haven, CT, USA
| | - Sulin Zhang
- Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA, USA.
- Department of Biomedical Engineering, Pennsylvania State University, University Park, PA, USA
- Department of Material Science and Engineering, Pennsylvania State University, University Park, PA, USA
| |
Collapse
|
3
|
Plummer A, Adkins C, Louf JF, Košmrlj A, Datta SS. Obstructed swelling and fracture of hydrogels. SOFT MATTER 2024; 20:1425-1437. [PMID: 38252539 DOI: 10.1039/d3sm01470c] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/24/2024]
Abstract
Obstructions influence the growth and expansion of bodies in a wide range of settings-but isolating and understanding their impact can be difficult in complex environments. Here, we study obstructed growth/expansion in a model system accessible to experiments, simulations, and theory: hydrogels swelling around fixed cylindrical obstacles with varying geometries. When the obstacles are large and widely-spaced, hydrogels swell around them and remain intact. In contrast, our experiments reveal that when the obstacles are narrow and closely-spaced, hydrogels fracture as they swell. We use finite element simulations to map the magnitude and spatial distribution of stresses that build up during swelling at equilibrium in a 2D model, providing a route toward predicting when this phenomenon of self-fracturing is likely to arise. Applying lessons from indentation theory, poroelasticity, and nonlinear continuum mechanics, we also develop a theoretical framework for understanding how the maximum principal tensile and compressive stresses that develop during swelling are controlled by obstacle geometry and material parameters. These results thus help to shed light on the mechanical principles underlying growth/expansion in environments with obstructions.
Collapse
Affiliation(s)
- Abigail Plummer
- Princeton Center for Complex Materials, Princeton University, Princeton, NJ 08540, USA
| | - Caroline Adkins
- Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544, USA
| | - Jean-François Louf
- Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544, USA.
- Department of Chemical Engineering, Auburn University, Auburn, AL 36849, USA
| | - Andrej Košmrlj
- Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA.
- Princeton Materials Institute, Princeton University, Princeton, NJ 08544, USA
| | - Sujit S Datta
- Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544, USA.
| |
Collapse
|
4
|
Nijjer J, Li C, Kothari M, Henzel T, Zhang Q, Tai JSB, Zhou S, Cohen T, Zhang S, Yan J. Biofilms as self-shaping growing nematics. NATURE PHYSICS 2023; 19:1936-1944. [PMID: 39055904 PMCID: PMC11271743 DOI: 10.1038/s41567-023-02221-1] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/07/2022] [Accepted: 08/23/2023] [Indexed: 07/28/2024]
Abstract
Active nematics are the nonequilibrium analogue of passive liquid crystals. They consist of anisotropic units that consume free energy to drive emergent behaviour. Like liquid crystal molecules in displays, ordering and dynamics in active nematics are sensitive to boundary conditions. However, unlike passive liquid crystals, active nematics have the potential to regulate their boundaries through self-generated stresses. Here, we show how a three-dimensional, living nematic can actively shape itself and its boundary to regulate its internal architecture through growth-induced stresses, using bacterial biofilms confined by a hydrogel as a model system. We show that biofilms exhibit a sharp transition in shape from domes to lenses upon changing environmental stiffness or cell-substrate friction, which is explained by a theoretical model that considers the competition between confinement and interfacial forces. The growth mode defines the progression of the boundary, which in turn determines the trajectories and spatial distribution of cell lineages. We further demonstrate that the evolving boundary and corresponding stress anisotropy define the orientational ordering of cells and the emergence of topological defects in the biofilm interior. Our findings may provide strategies for the development of programmed microbial consortia with emergent material properties.
Collapse
Affiliation(s)
- Japinder Nijjer
- Department of Molecular, Cellular and Developmental Biology, Yale University, New Haven, CT, USA
| | - Changhao Li
- Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA, USA
| | - Mrityunjay Kothari
- Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA
- Department of Mechanical Engineering, University of New Hampshire, Durham, NH, USA
| | - Thomas Henzel
- Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA
| | - Qiuting Zhang
- Department of Molecular, Cellular and Developmental Biology, Yale University, New Haven, CT, USA
| | - Jung-Shen B Tai
- Department of Molecular, Cellular and Developmental Biology, Yale University, New Haven, CT, USA
| | - Shuang Zhou
- Department of Physics, University of Massachusetts Amherst, Amherst, MA, USA
| | - Tal Cohen
- Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA
- Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA
| | - Sulin Zhang
- Department of Engineering Science and Mechanics, Pennsylvania State University, University Park, PA, USA
- Department of Biomedical Engineering, Pennsylvania State University, University Park, PA, USA
| | - Jing Yan
- Department of Molecular, Cellular and Developmental Biology, Yale University, New Haven, CT, USA
- Quantitative Biology Institute, Yale University, New Haven, CT, USA
| |
Collapse
|
5
|
Isensee J, Hupe L, Golestanian R, Bittihn P. Stress anisotropy in confined populations of growing rods. J R Soc Interface 2022; 19:20220512. [PMID: 36349447 PMCID: PMC9653230 DOI: 10.1098/rsif.2022.0512] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/15/2022] [Accepted: 10/18/2022] [Indexed: 11/10/2022] Open
Abstract
A central feature of living matter is its ability to grow and multiply. The mechanical activity associated with growth produces both macroscopic flows shaped by confinement, and striking self-organization phenomena, such as orientational order and alignment, which are particularly prominent in populations of rod-shaped bacteria due to their nematic properties. However, how active stresses, passive mechanical interactions and flow-induced effects interact to give rise to the observed global alignment patterns remains elusive. Here, we study in silico colonies of growing rod-shaped particles of different aspect ratios confined in channel-like geometries. A spatially resolved analysis of the stress tensor reveals a strong relationship between near-perfect alignment and an inversion of stress anisotropy for particles with large length-to-width ratios. We show that, in quantitative agreement with an asymptotic theory, strong alignment can lead to a decoupling of active and passive stresses parallel and perpendicular to the direction of growth, respectively. We demonstrate the robustness of these effects in a geometry that provides less restrictive confinement and introduces natural perturbations in alignment. Our results illustrate the complexity arising from the inherent coupling between nematic order and active stresses in growing active matter, which is modulated by geometric and configurational constraints due to confinement.
Collapse
Affiliation(s)
- Jonas Isensee
- Max Planck Institute for Dynamics and Self-Organization, Göttingen 37077, Germany
- Institute for the Dynamics of Complex Systems, Göttingen University, Göttingen 37077, Germany
| | - Lukas Hupe
- Max Planck Institute for Dynamics and Self-Organization, Göttingen 37077, Germany
- Institute for the Dynamics of Complex Systems, Göttingen University, Göttingen 37077, Germany
| | - Ramin Golestanian
- Max Planck Institute for Dynamics and Self-Organization, Göttingen 37077, Germany
- Institute for the Dynamics of Complex Systems, Göttingen University, Göttingen 37077, Germany
- Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU, UK
| | - Philip Bittihn
- Max Planck Institute for Dynamics and Self-Organization, Göttingen 37077, Germany
- Institute for the Dynamics of Complex Systems, Göttingen University, Göttingen 37077, Germany
| |
Collapse
|
6
|
Abstract
The morphogenesis of two-dimensional bacterial colonies has been well studied. However, little is known about the colony morphologies of bacteria growing in three dimensions, despite the prevalence of three-dimensional environments (e.g., soil, inside hosts) as natural bacterial habitats. Using experiments on bacteria in granular hydrogel matrices, we find that dense multicellular colonies growing in three dimensions undergo a common morphological instability and roughen, adopting a characteristic broccoli-like morphology when they exceed a critical size. Analysis of a continuum “active fluid” model of the expanding colony reveals that this behavior originates from an interplay of competition for nutrients with growth-driven colony expansion, both of which vary spatially. These results shed light on the fundamental biophysical principles underlying growth in three dimensions. How do growing bacterial colonies get their shapes? While colony morphogenesis is well studied in two dimensions, many bacteria grow as large colonies in three-dimensional (3D) environments, such as gels and tissues in the body or subsurface soils and sediments. Here, we describe the morphodynamics of large colonies of bacteria growing in three dimensions. Using experiments in transparent 3D granular hydrogel matrices, we show that dense colonies of four different species of bacteria generically become morphologically unstable and roughen as they consume nutrients and grow beyond a critical size—eventually adopting a characteristic branched, broccoli-like morphology independent of variations in the cell type and environmental conditions. This behavior reflects a key difference between two-dimensional (2D) and 3D colonies; while a 2D colony may access the nutrients needed for growth from the third dimension, a 3D colony inevitably becomes nutrient limited in its interior, driving a transition to unstable growth at its surface. We elucidate the onset of the instability using linear stability analysis and numerical simulations of a continuum model that treats the colony as an “active fluid” whose dynamics are driven by nutrient-dependent cellular growth. We find that when all dimensions of the colony substantially exceed the nutrient penetration length, nutrient-limited growth drives a 3D morphological instability that recapitulates essential features of the experimental observations. Our work thus provides a framework to predict and control the organization of growing colonies—as well as other forms of growing active matter, such as tumors and engineered living materials—in 3D environments.
Collapse
|