publications

2024

1. 

   Criticality enhances the reinforcement of disordered networks by rigid inclusions

   Jordan L. Shivers, Jingchen Feng, and Fred C. MacKintosh

   arXiv: 2407.19563 (2024)

   Abstract arXiv PDF

   The mechanical properties of biological materials are spatially heterogeneous. Typical tissues are made up of a spanning fibrous extracellular matrix in which various inclusions, such as living cells, are embedded. While the influence of inclusions on the stiffness of common elastic materials such as rubber has been studied for decades and can be understood in terms of the volume fraction and shape of inclusions, the same is not true for disordered filamentous and fibrous networks. Recent work has shown that, in isolation, such networks exhibit unusual viscoelastic behavior indicative of an underlying mechanical phase transition controlled by network connectivity and strain. How this behavior is modified when inclusions are present is unclear. Here, we present a theoretical and computational study of the influence of rigid inclusions on the mechanics of disordered elastic networks near the connectivity-controlled central force rigidity transition. Combining scaling theory and coarse-grained simulations, we predict and confirm an anomalously strong dependence of the composite stiffness on inclusion volume fraction, beyond that seen in ordinary composites. This stiffening exceeds the well-established volume fraction-dependent stiffening expected in conventional composites, e.g., as an elastic analogue of the classic volume fraction dependent increase in the viscosity of liquids first identified by Einstein. We show that this enhancement is a consequence of the interplay between inter-particle spacing and an emergent correlation length, leading to an effective finite-size scaling imposed by the presence of inclusions. We outline the expected scaling of the shear modulus and strain fluctuations with the inclusion volume fraction and network connectivity, confirm these predictions in simulations, and discuss potential experimental tests and implications for our predictions in real systems.

2024

1. 

   Nonlinear Poisson effect in affine semiflexible polymer networks

   Jordan L. Shivers, and Fred C. MacKintosh

   Physical Review E 110, 014502 (2024)

   Abstract arXiv HTML PDF

   Stretching an elastic material along one axis typically induces contraction along the transverse axes, a phenomenon known as the Poisson effect. From these strains, one can compute the specific volume, which generally either increases or, in the incompressible limit, remains constant as the material is stretched. However, in networks of semiflexible or stiff polymers, which are typically highly compressible yet stiffen significantly when stretched, one instead sees a significant reduction in specific volume under finite strains. This volume reduction is accompanied by increasing alignment of filaments along the strain axis and a nonlinear elastic response, with stiffening of the apparent Young’s modulus. For semiflexible networks, in which entropic bending elasticity governs the linear elastic regime, the nonlinear Poisson effect is caused by the nonlinear force-extension relationship of the constituent filaments, which produces a highly asymmetric response of the constituent polymers to stretching and compression. The details of this relationship depend on the geometric and elastic properties of the underlying filaments, which can vary greatly in experimental systems. Here, we provide a comprehensive characterization of the nonlinear Poisson effect in an affine network model and explore the influence of filament properties on essential features of both microscopic and macroscopic response, including strain-driven alignment and volume reduction.

2. 

   Effects of local incompressibility on the rheology of composite biopolymer networks

   Anupama Gannavarapu, Sadjad Arzash, Iain Muntz, Jordan L. Shivers, Anna-Maria Klianeva, Gijsje H. Koenderink, and Fred C. MacKintosh

   The European Physical Journal E 47, 36 (2024)

   Abstract arXiv HTML PDF

   Fibrous networks such as collagen are common in biological systems. Recent theoretical and experimental efforts have shed light on the mechanics of single component networks. Most real biopolymer networks, however, are composites made of elements with different rigidity. For instance, the extracellular matrix in mammalian tissues consists of stiff collagen fibers in a background matrix of flexible polymers such as hyaluronic acid (HA). The interplay between different biopolymer components in such composite networks remains unclear. In this work, we use 2D coarse-grained models to study the nonlinear strainstiffening behavior of composites. We introduce a local volume constraint to model the incompressibility of HA. We also perform rheology experiments on composites of collagen with HA. Theoretically and experimentally, we demonstrate that the linear shear modulus of composite networks can be increased by approximately an order of magnitude above the corresponding moduli of the pure components. Our model shows that this synergistic effect can be understood in terms of the local incompressibility of HA, which acts to suppress density fluctuations of the collagen matrix with which it is entangled.

3. 

   Motor crosslinking augments elasticity in active nematics

   Steven A. Redford, Jonathan Colen, Jordan L. Shivers, Sasha Zemsky, Mehdi Molaei, Carlos Floyd, Paul V. Ruijgrok, Vincenzo Vitelli, Zev Bryant, Aaron R. Dinner, and Margaret L. Gardel

   Soft Matter 20, 2480-2490 (2024)

   Abstract arXiv HTML PDF

   In active materials, uncoordinated internal stresses lead to emergent long-range flows. An understanding of how the behavior of active materials depends on mesoscopic (hydrodynamic) parameters is developing, but there remains a gap in knowledge concerning how hydrodynamic parameters depend on the properties of microscopic elements. In this work, we combine experiments and multiscale modeling to relate the structure and dynamics of active nematics composed of biopolymer filaments and molecular motors to their microscopic properties, in particular motor processivity, speed, and valency. We show that crosslinking of filaments by both motors and passive crosslinkers not only augments the contributions to nematic elasticity from excluded volume effects but dominates them. By altering motor kinetics we show that a competition between motor speed and crosslinking results in a nonmonotonic dependence of nematic flow on motor speed. By modulating passive filament crosslinking we show that energy transfer into nematic flow is in large part dictated by crosslinking. Thus motor proteins both generate activity and contribute to nematic elasticity. Our results provide new insights for rationally engineering active materials.

2023

1. 

   Strain-controlled critical slowing down in the rheology of disordered networks

   Jordan L. Shivers, Abhinav Sharma, and Fred C. MacKintosh

   Physical Review Letters 131, 178201 (2023)

   Abstract arXiv HTML PDF

   Networks and dense suspensions frequently reside near a boundary between soft (or fluidlike) and rigid (or solidlike) regimes. Transitions between these regimes can be driven by changes in structure, density, or applied stress or strain. In general, near the onset or loss of rigidity in these systems, dissipation-limiting heterogeneous nonaffine rearrangements dominate the macroscopic viscoelastic response, giving rise to diverging relaxation times and power-law rheology. Here, we describe a simple quantitative relationship between nonaffinity and the excess viscosity. We test this nonaffinity-viscosity relationship computationally and demonstrate its rheological consequences in simulations of strained filament networks and dense suspensions. We also predict critical signatures in the rheology of semiflexible and stiff biopolymer networks near the strain stiffening transition.

2022

1. 

   Structural features and nonlinear rheology of self-assembled networks of cross-linked semiflexible polymers

   Saamiya Syed, Fred C. MacKintosh, and Jordan L. Shivers

   The Journal of Physical Chemistry B 126, 10741–10749 (2022)

   Abstract arXiv HTML PDF

   Disordered networks of semiflexible filaments are common support structures in biology. Familiar examples include fibrous matrices in blood clots, bacterial biofilms, and essential components of cells and tissues of plants, animals, and fungi. Despite the ubiquity of these networks in biomaterials, we have only a limited understanding of the relationship between their structural features and their highly strain-sensitive mechanical properties. In this work, we perform simulations of three-dimensional networks produced by the irreversible formation of cross-links between linker-decorated semiflexible filaments. We characterize the structure of networks formed by a simple diffusion-dependent assembly process and measure their associated steady-state rheological features at finite temperature over a range of applied prestrains that encompass the strain-stiffening transition. We quantify the dependence of network connectivity on cross-linker availability and detail the associated connectivity dependence of both linear elasticity and nonlinear strain-stiffening behavior, drawing comparisons with prior experimental measurements of the cross-linker concentration-dependent elasticity of actin gels.

2. 

   Signatures of irreversibility in microscopic models of flocking

   Federica Ferretti, Simon Grosse-Holz, Caroline Holmes, Jordan L. Shivers, Irene Giardina, Thierry Mora, and Aleksandra M. Walczak

   Physical Review E 106, 034608 (2022)

   Abstract arXiv HTML PDF

   Flocking in d=2 is a genuine nonequilibrium phenomenon for which irreversibility is an essential ingredient. We study a class of minimal flocking models whose only source of irreversibility is self-propulsion and use the entropy production rate (EPR) to quantify the departure from equilibrium across their phase diagrams. The EPR is maximal in the vicinity of the order-disorder transition, where reshuffling of the interaction network is fast. We show that signatures of irreversibility come in the form of asymmetries in the steady-state distribution of the flock’s microstates. These asymmetries occur as consequences of the time-reversal symmetry breaking in the considered self-propelled systems, independently of the interaction details. In the case of metric pairwise forces, they reduce to local asymmetries in the distribution of pairs of particles. This study suggests a possible use of pair asymmetries both to quantify the departure from equilibrium and to learn relevant information about aligning interaction potentials from data.

3. 

   Unique role of vimentin networks in compression stiffening of cells and protection of nuclei from compressive stress

   Katarzyna Pogoda, Fitzroy Byfield, Piotr Deptuła, Mateusz Cieśluk, Łukasz Suprewicz, Karol Skłodowski, Jordan L. Shivers, Anne van Oosten, Katrina Cruz, Ekaterina Tarasovetc, Ekaterina L. Grishchuk, Fred C. Mackintosh, Robert Bucki, Alison E. Patteson, and Paul A. Janmey

   Nano Letters 22, 4725–4732 (2022)

   Abstract HTML PDF

   In this work, we investigate whether stiffening in compression is a feature of single cells and whether the intracellular polymer networks that comprise the cytoskeleton (all of which stiffen with increasing shear strain) stiffen or soften when subjected to compressive strains. We find that individual cells, such as fibroblasts, stiffen at physiologically relevant compressive strains, but genetic ablation of vimentin diminishes this effect. Further, we show that unlike networks of purified F-actin or microtubules, which soften in compression, vimentin intermediate filament networks stiffen in both compression and extension, and we present a theoretical model to explain this response based on the flexibility of vimentin filaments and their surface charge, which resists volume changes of the network under compression. These results provide a new framework by which to understand the mechanical responses of cells and point to a central role of intermediate filaments in response to compression.

2021

1. 

   Cell-induced confinement effects in soft tissue mechanics

   Dawei Song, Jordan L. Shivers, Fred C. MacKintosh, Alison E. Patteson, and Paul A. Janmey

   Journal of Applied Physics 129, 140901 (2021)

   Abstract HTML PDF

   The mechanical properties of tissues play a critical role in their normal and pathophysiological functions such as tissue development, aging, injury, and disease. Understanding tissue mechanics is important not only for designing realistic biomimetic materials for tissue engineering and drug testing but also for developing novel diagnostic techniques and medical interventions. Tissues are heterogeneous materials consisting of cells confined within extracellular matrices (ECMs), both of which derive their structural integrity, at least in part, from networks of biopolymers. However, the rheology of purified reconstituted biopolymer networks fails to explain many key aspects of tissue mechanics. Notably, purified networks typically soften under applied compression, whereas many soft tissues like liver, fat, and brain instead stiffen when compressed. While continuum models can readily capture this compression-stiffening behavior, the underlying mechanism is not fully understood. In this perspective paper, we discuss several recently proposed microscopic mechanisms that may explain compression stiffening of soft tissues. These mechanisms include (I) interactions between the ECM and volume-preserving inclusions that promote extension-dominated stiffening of fibrous ECMs when subject to uniform compression, (II) ECM interactions with rigid inclusions under non-uniform compression, (III) other internal physical constraints that cause compression stiffening of cells and ECMs, and (IV) propagation of compressive forces through jammed, compression-stiffening cells. We further identify a few of the many open problems in understanding the structure–function relationship of soft-tissue mechanics.

2. 

   Shear-induced phase transition and critical exponents in three-dimensional fiber networks

   Sadjad Arzash, Jordan L. Shivers, and Fred C. MacKintosh

   Physical Review E 104, L022402 (2021)

   Abstract arXiv HTML PDF

   When subject to applied strain, fiber networks exhibit nonlinear elastic stiffening. Recent theory and experiments have shown that this phenomenon is controlled by an underlying mechanical phase transition that is critical in nature. Growing simulation evidence points to non-mean-field behavior for this transition and a hyperscaling relation has been proposed to relate the corresponding critical exponents. Here, we report simulations on two distinct network structures in three dimensions. By performing a finite-size scaling analysis, we test hyperscaling and identify various critical exponents. From the apparent validity of hyperscaling, as well as the non-mean-field exponents we observe, our results suggest that the upper critical dimension for the strain-controlled phase transition is above three, in contrast to the jamming transition that represents another athermal, mechanical phase transition.

2020

1. 

   Compression stiffening of fibrous networks with stiff inclusions

   Jordan L. Shivers, Jingchen Feng, Anne S. G. van Oosten, Herbert Levine, Paul A. Janmey, and Fred C. MacKintosh

   Proceedings of the National Academy of Sciences 117, 21037–21044 (2020)

   Abstract arXiv HTML PDF

   Tissues commonly consist of cells embedded within a fibrous biopolymer network. Whereas cell-free reconstituted biopolymer networks typically soften under applied uniaxial compression, various tissues, including liver, brain, and fat, have been observed to instead stiffen when compressed. The mechanism for this compression-stiffening effect is not yet clear. Here, we demonstrate that when a material composed of stiff inclusions embedded in a fibrous network is compressed, heterogeneous rearrangement of the inclusions can induce tension within the interstitial network, leading to a macroscopic crossover from an initial bending-dominated softening regime to a stretching-dominated stiffening regime, which occurs before and independently of jamming of the inclusions. Using a coarse-grained particle-network model, we first establish a phase diagram for compression-driven, stretching-dominated stress propagation and jamming in uniaxially compressed two- and three-dimensional systems. Then, we demonstrate that a more detailed computational model of stiff inclusions in a subisostatic semiflexible fiber network exhibits quantitative agreement with the predictions of our coarse-grained model as well as qualitative agreement with experiments.

2. 

   Finite size effects in critical fiber networks

   Sadjad Arzash, Jordan L. Shivers, and Fred C. MacKintosh

   Soft Matter 16, 6784–6793 (2020)

   Abstract arXiv HTML PDF

   Fibrous networks such as collagen are common in physiological systems. One important function of these networks is to provide mechanical stability for cells and tissues. At physiological levels of connectivity, such networks would be mechanically unstable with only central-force interactions. While networks can be stabilized by bending interactions, it has also been shown that they exhibit a critical transition from floppy to rigid as a function of applied strain. Beyond a certain strain threshold, it is predicted that underconstrained networks with only central-force interactions exhibit a discontinuity in the shear modulus. We study the finite-size scaling behavior of this transition and identify both the mechanical discontinuity and critical exponents in the thermodynamic limit. We find both non-mean-field behavior and evidence for a hyperscaling relation for the critical exponents, for which the network stiffness is analogous to the heat capacity for thermal phase transitions. Further evidence for this is also found in the self-averaging properties of fiber networks.

3. 

   Nonlinear Poisson effect governed by a mechanical critical transition

   Jordan L. Shivers, Sadjad Arzash, and Fred C. MacKintosh

   Physical Review Letters 124, 038002 (2020)

   Abstract arXiv HTML PDF

   Under extensional strain, fiber networks can exhibit an anomalously large and nonlinear Poisson effect accompanied by a dramatic transverse contraction and volume reduction for applied strains as small as a few percent. We demonstrate that this phenomenon is controlled by a collective mechanical phase transition that occurs at a critical uniaxial strain that depends on network connectivity. This transition is punctuated by an anomalous peak in the apparent Poisson’s ratio and other critical signatures such as diverging nonaffine strain fluctuations.

2019

1. 

   Scaling theory for mechanical critical behavior in fiber networks

   Jordan L. Shivers, Sadjad Arzash, Abhinav Sharma, and Fred C. MacKintosh

   Physical Review Letters 122, 188003 (2019)

   Abstract arXiv HTML PDF

   As a function of connectivity, spring networks exhibit a critical transition between floppy and rigid phases at an isostatic threshold. For connectivity below this threshold, fiber networks were recently shown theoretically to exhibit a rigidity transition with corresponding critical signatures as a function of strain. Experimental collagen networks were also shown to be consistent with these predictions. We develop a scaling theory for this strain-controlled transition. Using a real-space renormalization approach, we determine relations between the critical exponents governing the transition, which we verify for the strain-controlled transition using numerical simulations of both triangular lattice-based and packing-derived fiber networks.

2. 

   Stress-stabilized subisostatic fiber networks in a ropelike limit

   Sadjad Arzash, Jordan L. Shivers, Albert J. Licup, Abhinav Sharma, and Fred C. MacKintosh

   Physical Review E 99, 042412 (2019)

   Abstract arXiv HTML PDF

   The mechanics of disordered fibrous networks such as those that make up the extracellular matrix are strongly dependent on the local connectivity or coordination number. For biopolymer networks this coordination number is typically between 3 and 4. Such networks are sub-isostatic and linearly unstable to deformation with only central force interactions, but exhibit a mechanical phase transition between floppy and rigid states under strain. The introduction of weak bending interactions stabilizes these networks and suppresses the critical signatures of this transition. We show that applying external stress can also stabilize subisostatic networks with only tensile central force interactions, i.e., a ropelike potential. Moreover, we find that the linear shear modulus shows a power-law scaling with the external normal stress, with a non-mean-field exponent. For networks with finite bending rigidity, we find that the critical stain shifts to lower values under prestress.

3. 

   Normal stress anisotropy and marginal stability in athermal elastic networks

   Jordan L. Shivers, Jingchen Feng, Abhinav Sharma, and Fred C. MacKintosh

   Soft Matter 15, 1666–1675 (2019)

   Abstract arXiv HTML PDF

   Hydrogels of semiflexible biopolymers such as collagen have been shown to contract axially under shear strain, in contrast to the axial dilation observed for most elastic materials. Recent work has shown that this behavior can be understood in terms of the porous, two-component nature and consequent time-dependent compressibility of hydrogels. The apparent normal stress measured by a torsional rheometer reflects only the tensile contribution of the axial component σzz on long (compressible) timescales, crossing over to the first normal stress difference, N1 = σxx − σzz at short (incompressible) times. While the behavior of N1 is well understood for isotropic viscoelastic materials undergoing affine shear deformation, biopolymer networks are often anisotropic and deform nonaffinely. Here, we numerically study the normal stresses that arise under shear in subisostatic, athermal semiflexible polymer networks. We show that such systems exhibit strong deviations from affine behavior and that these anomalies are controlled by a rigidity transition as a function of strain.

2017

1. 

   Microfluidic immobilization and subcellular imaging of developing Caenorhabditis elegans

   Jordan Shivers, Sravanti Uppaluri, and Clifford P. Brangwynne

   Microfluidics and Nanofluidics 21, 149 (2017)

   Abstract bioRxiv HTML PDF

   Caenorhabditis elegans has been an essential model organism in the fields of developmental biology, neuroscience, and aging. However, these areas have been limited by our ability to visualize and track individual C. elegans worms, especially at the subcellular scale, over the course of their lifetime. Here we present a microfluidic device to culture individual C. elegans in parallel throughout post-embryonic development. The device allows for periodic mechanical immobilization of the worm, enabling 3D imaging at subcellular precision. The immobilization is sufficient to enable fluorescence recovery after photobleaching (FRAP) measurements on organelles and other substructures within the same specific cells throughout larval development, without the use of chemical anesthetics. Using this device, we measure FRAP recovery of two nucleolar proteins in specific intestinal cells within the same worms during larval development. We show that these proteins exhibit different fluorescence recovery as the worm grows, suggesting differential protein interactions during development. We anticipate that this device will help expand the possible uses of C. elegans as a model organism, enabling its use in addressing fundamental questions at the subcellular scale.

2022

1. Thesis

   Phase transitions in the rheology of biopolymer networks

   Jordan L. Shivers

   Ph.D. Thesis, Rice University, (2022)

   Abstract HTML PDF

   Most biological materials are stabilized internally by disordered networks of long, thin protein assemblies known as biopolymers. These networks occupy a negligible fraction of the space inside cells and tissues, yet are largely responsible for their extraordinary mechanical properties and exceptional resilience against unpredictable physiological loads. An essential contributor to this resilience is their highly strain-dependent stiffness; they easily deform to accommodate small strains, yet resist damage by stiffening significantly in response to larger strains. Recent work has suggested that the phenomenon of strain-induced stiffening in stiff or athermal biopolymer networks, like the collagen-rich extracellular matrix, constitutes a phase transition between distinct mechanical regimes, analogous to connectivity-controlled rigidity transitions observed in networks and other amorphous materials. Simulations have shown that this transition is heralded by classic signatures of continuous phase transitions, including power law scaling of relevant observables and a diverging correlation length. In this thesis, we develop theoretical and computational models to describe the mechanics and dynamics of disordered elastic networks near the onset of rigidity. We develop a real space renormalization-based scaling theory that establishes relationships between the various critical exponents describing the scaling of the elastic moduli and fluctuations in networks near both the strain-controlled and connectivity-controlled rigidity transitions, which we validate using simulations of coarse-grained elastic networks. We then describe the rheology of fluid-immersed networks near the strain-induced stiffening transition and demonstrate that a coupling between diverging strain fluctuations and time-dependent energy dissipation leads to emergent power law rheology at a critical prestrain. Next, we explore the effects of criticality on a phenomenon in biopolymer networks known as the nonlinear Poisson effect, which describes their tendency to shrink dramatically and strongly align when stretched, a behavior with potentially major consequences for matrix-embedded cells. We show that this effect coincides with an analogous extension-controlled rigidity transition and describe the influence of this transition on network rearrangement and the scaling of the apparent Young’s modulus. We further propose a physical mechanism for the unusual compression-driven stiffening effect observed in tissues. Considering a simplified model tissue consisting of a disordered network with embedded stiff particles, we construct a phase diagram describing a unique regime of compression-driven, tension-dominated mechanical stability that arises in these systems before conventional jamming, which we validate in simulations.