Generic rigidity percolation: The pebble game

Free volume theory explains the unusual behavior of viscosity in a non-confluent tissue during morphogenesis

A recent experiment on zebrafish blastoderm morphogenesis showed that the viscosity (η) of a non-confluent embryonic tissue grows sharply until a critical cell packing fraction (ϕS). The increase in η up to ϕS is similar to the behavior observed in several glass-forming materials, which suggests that the cell dynamics is sluggish or glass-like. Surprisingly, η is a constant above ϕS. To determine the mechanism of this unusual dependence of η on ϕ, we performed extensive simulations using an agent-based model of a dense non-confluent two-dimensional tissue. We show that polydispersity in the cell size, and the propensity of the cells to deform, results in the saturation of the available free area per cell beyond a critical packing fraction. Saturation in the free space not only explains the viscosity plateau above ϕS but also provides a relationship between equilibrium geometrical packing to the dramatic increase in the relaxation dynamics.

Schramm-Loewner Evolution in 2D Rigidity Percolation

Amorphous solids may resist external deformation such as shear or compression, while they do not present any long-range translational order or symmetry at the microscopic scale. Yet, it was recently discovered that, when they become rigid, such materials acquire a high degree of symmetry hidden in the disorder fluctuations: their microstructure becomes statistically conformally invariant. In this Letter, we exploit this finding to characterize the universality class of central-force rigidity percolation (RP), using Schramm-Loewner evolution (SLE) theory. We provide numerical evidence that the interfaces of the mechanically stable structures (rigid clusters), at the rigidification transition, are consistently described by SLE_{κ}, showing that this powerful framework can be applied to a mechanical percolation transition. Using well-known relations between different SLE observables and the universal diffusion constant κ, we obtain the estimation κ∼2.9 for central-force RP. This value is consistent, through relations coming from conformal field theory, with previously measured values for the clusters' fractal dimension D_{f} and correlation length exponent ν, providing new, nontrivial relations between critical exponents for RP. These findings open the way to a fine understanding of the microstructure in other important classes of rigidity and jamming transitions.

Rigidity percolation in a random tensegrity via analytic graph theory

Functional structures from across the engineered and biological world combine rigid elements such as bones and columns with flexible ones such as cables, fibers, and membranes. These structures are known loosely as tensegrities, since these cable-like elements have the highly nonlinear property of supporting only extensile tension. Marginally rigid systems are of particular interest because the number of structural constraints permits both flexible deformation and the support of external loads. We present a model system in which tensegrity elements are added at random to a regular backbone. This system can be solved analytically via a directed graph theory, revealing a mechanical critical point generalizing that of Maxwell. We show that even the addition of a few cable-like elements fundamentally modifies the nature of this transition point, as well as the later transition to a fully rigid structure. Moreover, the tensegrity network displays a collective avalanche behavior, in which the addition of a single cable leads to the elimination of multiple floppy modes, a phenomenon that becomes dominant at the transition point. These phenomena have implications for systems with nonlinear mechanical constraints, from biopolymer networks to soft robots to jammed packings to origami sheets.

Emergence of rigidity percolation in flowing granular systems

Jammed granular media and glasses exhibit spatial long-range correlations as a result of mechanical equilibrium. However, the existence of such correlations in the flowing matter, where the mechanical equilibrium is unattainable, has remained elusive. Here, we investigate this problem in the context of the percolation of interparticle forces in flowing granular media. We find that the flow rate introduces an effective long-range correlation, which plays the role of a relevant perturbation giving rise to a spectrum of varying exponents on a critical line as a function of the flow rate. Our numerical simulations along with analytical arguments predict a crossover flow rate [Formula: see text] below which the effect of induced disorder is weak and the universality of the force chain structure is shown to be given by the standard rigidity percolation. We also find a power-law behavior for the critical exponents with the flow rate [Formula: see text].

Heterogeneous Dynamics of Sheared Particle-Laden Fluid Interfaces with Janus Particle Doping

The formation of particle clusters can substantially modify the dynamics and mechanical properties of suspensions in both two and three dimensions. While it has been well established that large network-spanning clusters increase the rigidity of particle systems, it is still unclear how the presence of localized nonpercolating clusters affects the dynamics and mechanical properties of particle suspensions. Here, we introduce self-assembled localized particle clusters at a fluid-fluid interface by mixing a fraction of Janus particles in a monolayer of homogeneous colloids. Each Janus particle binds to a few nearby homogeneous colloids, resulting in numerous small clusters uniformly distributed across the interface. Using a custom magnetic rod interfacial stress rheometer, we apply linear oscillatory shear to the particle-laden fluid interface. By analyzing the local affine deformation of particles from optical microscopy, we show that particles in localized clusters experience substantially lower shear-induced stretching than their neighbors outside clusters. We hypothesize that such heterogeneous dynamics induced by particle clusters increase the effective surface coverage of particles, which in turn enhances the shear moduli of the interface, as confirmed by direct interfacial rheological measurements. Our study illustrates the microscopic dynamics of small clusters in a shear flow and reveals their profound effects on the macroscopic rheology of particle-laden fluid interfaces. Our findings open an avenue for designing interfacial materials with improved mechanical properties via the control of formation of localized particle clusters.

Range and strength of mechanical interactions of force dipoles in elastic fiber networks

Mechanical forces generated by myosin II molecular motors drive diverse cellular processes, most notably shape change, division and locomotion. These forces may be transmitted over long range through the cytoskeletal medium - a disordered, viscoelastic network of biopolymers. The resulting cell size scale force chains can in principle mediate mechanical interactions between distant actomyosin units, leading to self-organized structural order in the cell cytoskeleton. Inspired by such force transmission through elastic structures in the cytoskeleton, we consider a percolated fiber lattice network, where fibers are represented as linear elastic elements that can both bend and stretch, and the contractile activity of myosin motors is represented by force dipoles. Then, by using a variety of metrics, we show how two such contractile force dipoles interact with each other through their mutual mechanical deformations of the elastic fiber network. As a prelude to two-dipole interactions, we quantify how forces propagate through the network from a single anisotropic force dipole by analyzing clusters of nodes connected by highly strained bonds, as well as through the decay rate of strain energy with distance from a force dipole. We show that predominant fiber bending screens out force propagation, resulting in reduced and strongly network configuration-dependent dipole interactions. On the other hand, stretching-dominated networks support longer-ranged inter-dipole interactions that recapitulate the predictions of linear elasticity theory. By characterizing the differences between tensile and compressive force propagation in the fiber network, we show how inter-dipole interaction depends on the dipoles' mutual separation and orientation. The resulting elastic interaction energy may mediate a force between multiple distant dipoles, leading to their self-organization into ordered configurations. This provides a potential pathway for active mechanical force-driven structural order in elastic biopolymer networks.

Evidences of Conformal Invariance in 2D Rigidity Percolation

The rigidity transition occurs when, as the density of microscopic components is increased, a disordered medium becomes able to transmit and ensure macroscopic mechanical stability, owing to the appearance of a space-spanning rigid connected component, or cluster. As a second-order phase transition it exhibits a scale invariant critical point, at which the rigid clusters are random fractals. We show, using numerical analysis, that these clusters are also conformally invariant, and we use conformal field theory to predict the form of universal finite-size effects. Furthermore, although connectivity and rigidity percolation are usually thought to be of fundamentally different natures, we provide evidence of unexpected similarities between the statistical properties of their random clusters at criticality. Our work opens a new research avenue through the application of the powerful 2D conformal field theory tools to understand the critical behavior of a wide range of physical and biological materials exhibiting such a mechanical transition.

Explosive Percolation Obeys Standard Finite-Size Scaling in an Event-Based Ensemble

Explosive percolation in the Achlioptas process, which has attracted much research attention, is known to exhibit a rich variety of critical phenomena that are anomalous from the perspective of continuous phase transitions. Hereby, we show that, in an event-based ensemble, the critical behaviors in explosive percolation are rather clean and obey the standard finite-size scaling theory, except for the large fluctuation of pseudo-critical points. In the fluctuation window, multiple fractal structures emerge and the values can be derived from a crossover scaling theory. Further, their mixing effects account well for the previously observed anomalous phenomena. Making use of the clean scaling in the event-based ensemble, we determine with a high precision the critical points and exponents for a number of bond-insertion rules and clarify ambiguities about their universalities. Our findings hold true for any spatial dimensions.

Molecular simulation unravels the amyloidogenic misfolding of nascent ApoA1 protein, driven by deleterious point mutations occurring in between 170-178 hotspot region

Protein ApoA1 is extensively studied for its role in lipid metabolism. Its seedy dark side of amyloid formulation remains relatively understudied yet. Due to genetic mutations, the protein pathologically misshapes into its amyloid form that gets accumulated in various organs, including the heart. To contrive effective therapeutics against this debilitating congenital disorder, it is imperative to comprehend the structural ramifications induced by mutations in APoA1's dynamic conformation. Till now, several point mutations have been implicated in ApoA1's amyloidosis, although only a handful has been examined considerably. Especially, the single nucleotide polymorphisms (SNPs) that occur in-between 170-178 mutation hotspot site of APoA1 needs to be investigated, since most of them are culpable of amyloid deposition in the heart. To that effect, in the present study, we have computationally quantified and studied the ApoA1's biomolecular modifications fostered by SNPs in the 170-178 mutation hotspot. Findings from discrete molecular dynamics simulation studies indicate that the SNPs have noticeably steered the ApoA1's behaviour from its native structural dynamics. Analysis of protein's secondary structural changes exhibits a considerable change upon mutations. Further, subjecting the protein structures to simulated thermal denaturation shows increased resistance to denaturation among mutants when compared to native. Further, normal mode analysis of protein's dynamic motion also shows discrepancy in its dynamic structural change upon SNP. These structural digressions induced by SNPs can very well be the biomolecular incendiary that drives ApoA1 into its amyloidogenesis. And, understanding these structural modifications initiates a better understanding of SNP's amyloidogenic pathology on APoA1.Communicated by Ramaswamy H. Sarma.

Universal scaling for disordered viscoelastic matter near the onset of rigidity

The onset of rigidity in interacting liquids, as they undergo a transition to a disordered solid, is associated with a rearrangement of the low-frequency vibrational spectrum. In this Letter, we derive scaling forms for the singular dynamical response of disordered viscoelastic networks near both jamming and rigidity percolation. Using effective-medium theory, we extract critical exponents, invariant scaling combinations, and analytical formulas for universal scaling functions near these transitions. Our scaling forms describe the behavior in space and time near the various onsets of rigidity, for rigid and floppy phases and the crossover region, including diverging length scales and timescales at the transitions.

Omnimodal topological polarization of bilayer networks: Analysis in the Maxwell limit and experiments on a 3D-printed prototype

Periodic networks on the verge of mechanical instability, called Maxwell lattices, are known to exhibit zero-frequency modes localized to their boundaries. Topologically polarized Maxwell lattices, in particular, focus these zero modes to one of their boundaries in a manner that is protected against disorder by the reciprocal-space topology of the lattice's band structure. Here, we introduce a class of mechanical bilayers as a model system for designing topologically protected edge modes that couple in-plane dilational and shearing modes to out-of-plane flexural modes, a paradigm that we refer to as "omnimodal polarization." While these structures exhibit a high-dimensional design space that makes it difficult to predict the topological polarization of generic geometries, we are able to identify a family of mirror-symmetric bilayers that inherit the in-plane modal localization of their constitutive monolayers, whose topological polarization can be determined analytically. Importantly, the coupling between the layers results in the emergence of omnimodal polarization, whereby in-plane and out-of-plane edge modes localize on the same edge. We demonstrate these theoretical results by fabricating a mirror-symmetric, topologically polarized kagome bilayer consisting of a network of elastic beams via additive manufacturing and confirm this finite-frequency polarization via finite element analysis and laser-vibrometry experiments.

Emergence of zero modes in disordered solids under periodic tiling

In computational models of particle packings with periodic boundary conditions, it is assumed that the packing is attached to exact copies of itself in all possible directions. The periodicity of the boundary then requires that all of the particles' images move together. An infinitely repeated structure, on the other hand, does not necessarily have this constraint. As a consequence, a jammed packing (or a rigid elastic network) under periodic boundary conditions may have a corresponding infinitely repeated lattice representation that is not rigid or indeed may not even be at a local energy minimum. In this manuscript, we prove this claim and discuss ways in which periodic boundary conditions succeed in capturing the physics of repeated structures and where they fall short.

Scaling to zero of compressive modulus in disordered isostatic cubic networks

Networks with as many mechanical constraints as degrees of freedom and no redundant constraints are minimally rigid or isostatic. Isostatic networks are relevant in the study of network glasses, soft matter, and sphere packings. Because of being at the verge of mechanical collapse, they have anomalous elastic and dynamical properties not found in the more commonly occurring hyperstatic networks. In particular, while hyperstatic networks are only slightly affected by geometric disorder, the elastic properties of isostatic networks are dramatically altered by it. In this paper, we show how disorder and system size strongly affect the ability of isostatic networks to sustain a compressive load. We develop an analytic method to calculate the bulk compressive modulus B for various boundary conditions as a function of disorder strength and system size. For simplicity, we consider square and cubic lattices with L^{d} sites, each having d mechanical degrees of freedom, and dL^{d} rotatable springs in the presence of hot-solid disorder of magnitude ε. Additionally, ∼L^{θ} sites may be fixed, thus introducing a nonextensive number of redundancies, either in the bulk or on the boundaries of the system. In all cases, B is analytically and numerically shown to decay as L^{-μ} with μ_{large}=d-θ for large disorder and μ_{small}=max{(d-θ-1),0} for small disorder. Furthermore B(L,ε)L^{μ_{small}}=g(λ) with λ=L^{(μ_{large}-μ_{small})}ε^{2} a scaling variable such that λ<<1 is small disorder and λ>1 is large disorder. The faster decay to zero of B in the large disorder regime results from a broad distribution of spring tensions, including tensions of both signs in equal proportions, which is remarkable since the system is under a purely compressive load. Notably, the bulk modulus is discontinuous at ε=0, a consequence of the fact that the regular network sits at an unstable degenerate configuration.

Uncoupling of Voltage- and Ligand-Induced Activation in HCN2 Channels by Glycine Inserts

Hyperpolarization-activated cyclic nucleotide-modulated (HCN) channels are tetramers that generate electrical rhythmicity in special brain neurons and cardiomyocytes. The channels are activated by membrane hyperpolarization. The binding of cAMP to the four available cyclic nucleotide-binding domains (CNBD) enhances channel activation. We analyzed in the present study the mechanism of how the effect of cAMP binding is transmitted to the pore domain. Our strategy was to uncouple the C-linker (CL) from the channel core by inserting one to five glycine residues between the S6 gate and the A′-helix (constructs 1G to 5G). We quantified in full-length HCN2 channels the resulting functional effects of the inserted glycines by current activation as well as the structural dynamics and statics using molecular dynamics simulations and Constraint Network Analysis. We show functionally that already in 1G the cAMP effect on activation is lost and that with the exception of 3G and 5G the concentration-activation relationships are shifted to depolarized voltages with respect to HCN2. The strongest effect was found for 4G. Accordingly, the activation kinetics were accelerated by all constructs, again with the strongest effect in 4G. The simulations reveal that the average residue mobility of the CL and CNBD domains is increased in all constructs and that the junction between the S6 and A′-helix is turned into a flexible hinge, resulting in a destabilized gate in all constructs. Moreover, for 3G and 4G, there is a stronger downward displacement of the CL-CNBD than in HCN2 and the other constructs, resulting in an increased kink angle between S6 and A′-helix, which in turn loosens contacts between the S4-helix and the CL. This is suggested to promote a downward movement of the S4-helix, similar to the effect of hyperpolarization. In addition, exclusively in 4G, the selectivity filter in the upper pore region and parts of the S4-helix are destabilized. The results provide new insights into the intricate activation of HCN2 channels.

Modular representation and control of floppy networks

Geometric graph models of systems as diverse as proteins, DNA assemblies, architected materials and robot swarms are useful abstract representations of these objects that also unify ways to study their properties and control them in space and time. While much work has been done in the context of characterizing the behaviour of these networks close to critical points associated with bond and rigidity percolation, isostaticity, etc., much less is known about floppy, underconstrained networks that are far more common in nature and technology. Here, we combine geometric rigidity and algebraic sparsity to provide a framework for identifying the zero energy floppy modes via a representation that illuminates the underlying hierarchy and modularity of the network and thence the control of its nestedness and locality. Our framework allows us to demonstrate a range of applications of this approach that include robotic reaching tasks with motion primitives, and predicting the linear and nonlinear response of elastic networks based solely on infinitesimal rigidity and sparsity, which we test using physical experiments. Our approach is thus likely to be of use broadly in dissecting the geometrical properties of floppy networks using algebraic sparsity to optimize their function and performance.

Viscoelastic Scaling Regimes for Marginally Rigid Fractal Spring Networks

A family of marginally rigid (isostatic) spring networks with fractal structure up to a controllable length was devised, and the viscoelastic spectra G^{*}(ω) calculated. Two nontrivial scaling regimes were observed, (i) G^{'}≈G^{''}∝ω^{Δ} at low frequencies, consistent with Δ=1/2, and (ii) G^{'}∝G^{''}∝ω^{Δ^{'}} for intermediate frequencies corresponding to fractal structure, consistent with a theoretical prediction Δ^{'}=(ln3-ln2)/(ln3+ln2). The crossover between these two regimes occurred at lower frequencies for larger fractals in a manner suggesting diffusivelike dispersion. Solid gels generated by introducing internal stresses exhibited similar behavior above a low-frequency cutoff, indicating the relevance of these findings to real-world applications.

Structure and Function of Redox-Sensitive Superfolder Green Fluorescent Protein Variant

Aims: Genetically encoded green fluorescent protein (GFP)-based redox biosensors are widely used to monitor specific and dynamic redox processes in living cells. Over the last few years, various biosensors for a variety of applications were engineered and enhanced to match the organism and cellular environments, which should be investigated. In this context, the unicellular intraerythrocytic parasite Plasmodium, the causative agent of malaria, represents a challenge, as the small size of the organism results in weak fluorescence signals that complicate precise measurements, especially for cell compartment-specific observations. To address this, we have functionally and structurally characterized an enhanced redox biosensor superfolder roGFP2 (sfroGFP2). Results: SfroGFP2 retains roGFP2-like behavior, yet with improved fluorescence intensity (FI) in cellulo. SfroGFP2-based redox biosensors are pH insensitive in a physiological pH range and show midpoint potentials comparable with roGFP2-based redox biosensors. Using crystallography and rigidity theory, we identified the superfolding mutations as being responsible for improved structural stability of the biosensor in a redox-sensitive environment, thus explaining the improved FI in cellulo. Innovation: This work provides insight into the structure and function of GFP-based redox biosensors. It describes an improved redox biosensor (sfroGFP2) suitable for measuring oxidizing effects within small cells where applicability of other redox sensor variants is limited. Conclusion: Improved structural stability of sfroGFP2 gives rise to increased FI in cellulo. Fusion to hGrx1 (human glutaredoxin-1) provides the hitherto most suitable biosensor for measuring oxidizing effects in Plasmodium. This sensor is of major interest for studying glutathione redox changes in small cells, as well as subcellular compartments in general. Antioxid. Redox Signal. 37, 1-18.

A generalized Flory-Stockmayer kinetic theory of connectivity percolation and rigidity percolation of cytoskeletal networks

Actin networks are essential for living cells to move, reproduce, and sense their environments. The dynamic and rheological behavior of actin networks is modulated by actin-binding proteins such as α-actinin, Arp2/3, and myosin. There is experimental evidence that actin-binding proteins modulate the cooperation of myosin motors by connecting the actin network. In this work, we present an analytical mean field model, using the Flory-Stockmayer theory of gelation, to understand how different actin-binding proteins change the connectivity of the actin filaments as the networks are formed. We follow the kinetics of the networks and estimate the concentrations of actin-binding proteins that are needed to reach connectivity percolation as well as to reach rigidity percolation. We find that Arp2/3 increases the actomyosin connectivity in the network in a non-monotonic way. We also describe how changing the connectivity of actomyosin networks modulates the ability of motors to exert forces, leading to three possible phases of the networks with distinctive dynamical characteristics: a sol phase, a gel phase, and an active phase. Thus, changes in the concentration and activity of actin-binding proteins in cells lead to a phase transition of the actin network, allowing the cells to perform active contraction and change their rheological properties.

The actin cytoskeleton is a complex dynamic system, regulated by multiple proteins that bind to actin filaments. Some actin-binding proteins are crosslinkers, which can bind pairs of actin filaments, forming actin networks. Actin crosslinkers can be passive linkers, providing only structural integrity, or can be active linkers such as myosin motors, which exert forces on the network. Experiments have shown that crosslinked actin networks can behave viscously when the number of passive crosslinkers is low, but become elastic, when there are many crosslinkers. Motors can only lead to contraction of the network when there is an intermediate concentration of passive crosslinkers. The behavior of networks in the cell depends on the concentration and activity of several distinct crosslinkers, which have different binding sites, geometries, affinities, and concentrations. In this work we propose a simple analytical model based on chemical kinetics and the Flory-Stockmayer theory that gives us insight into how different crosslinkers interact with the actin filaments so as to give rise to the emergent mechanical behavior. This theory also allows us to compute analytically several crucial aspects of the development of the mechanical properties during network assembly.

Stretchy and disordered: Toward understanding fracture in soft network materials via mesoscopic computer simulations

Soft network materials exist in numerous forms ranging from polymer networks, such as elastomers, to fiber networks, such as collagen. In addition, in colloidal gels, an underlying network structure can be identified, and several metamaterials and textiles can be considered network materials as well. Many of these materials share a highly disordered microstructure and can undergo large deformations before damage becomes visible at the macroscopic level. Despite their widespread presence, we still lack a clear picture of how the network structure controls the fracture processes of these soft materials. In this Perspective, we will focus on progress and open questions concerning fracture at the mesoscopic scale, in which the network architecture is clearly resolved, but neither the material-specific atomistic features nor the macroscopic sample geometries are considered. We will describe concepts regarding the network elastic response that have been established in recent years and turn out to be pre-requisites to understand the fracture response. We will mostly consider simulation studies, where the influence of specific network features on the material mechanics can be cleanly assessed. Rather than focusing on specific systems, we will discuss future challenges that should be addressed to gain new fundamental insights that would be relevant across several examples of soft network materials.

CGRAP: A Web Server for Coarse-Grained Rigidity Analysis of Proteins

Elucidating protein rigidity offers insights about protein conformational changes. An understanding of protein motion can help speed drug development, and provide general insights into the dynamic behaviors of biomolecules. Existing rigidity analysis techniques employ fine-grained, all-atom modeling, which has a costly run-time, particularly for proteins made up of more than 500 residues. In this work, we introduce coarse-grained rigidity analysis, and showcase that it provides flexibility information about a protein that is similar in accuracy to an all-atom modeling approach. We assess the accuracy of the coarse-grained method relative to an all-atom approach via a comparison metric that reasons about the largest rigid clusters of the two methods. The apparent symmetry between the all-atom and coarse-grained methods yields very similar results, but the coarse-grained method routinely exhibits 40% reduced run-times. The CGRAP web server outputs rigid cluster information, and provides data visualization capabilities, including a interactive protein visualizer.

Designing self-assembling kinetics with differentiable statistical physics models

The inverse problem of designing component interactions to target emergent structure is fundamental to numerous applications in biotechnology, materials science, and statistical physics. Equally important is the inverse problem of designing emergent kinetics, but this has received considerably less attention. Using recent advances in automatic differentiation, we show how kinetic pathways can be precisely designed by directly differentiating through statistical physics models, namely free energy calculations and molecular dynamics simulations. We consider two systems that are crucial to our understanding of structural self-assembly: bulk crystallization and small nanoclusters. In each case, we are able to assemble precise dynamical features. Using gradient information, we manipulate interactions among constituent particles to tune the rate at which these systems yield specific structures of interest. Moreover, we use this approach to learn nontrivial features about the high-dimensional design space, allowing us to accurately predict when multiple kinetic features can be simultaneously and independently controlled. These results provide a concrete and generalizable foundation for studying nonstructural self-assembly, including kinetic properties as well as other complex emergent properties, in a vast array of systems.

F/G Region Rigidity is Inversely Correlated to Substrate Promiscuity of Human CYP Isoforms Involved in Metabolism

Of 57 human cytochrome P450 (CYP) enzymes, 12 metabolize 90% of xenobiotics. To our knowledge, no study has addressed the relation between enzyme dynamics and substrate promiscuity for more than three CYPs. Here, we show by constraint dilution simulations with the Constraint Network Analysis for the 12 isoforms that structural rigidity of the F/G region is significantly inversely correlated to the enzymes' substrate promiscuity. This highlights the functional importance of structural dynamics of the substrate tunnel.

Elasticity of colloidal gels: structural heterogeneity, floppy modes, and rigidity

Rheological measurements of model colloidal gels reveal that large variations in the shear moduli as colloidal volume-fraction changes are not reflected by simple structural parameters such as the coordination number, which remains almost a constant. We resolve this apparent contradiction by conducting a normal-mode analysis of experimentally measured bond networks of gels of colloidal particles with short-ranged attraction. We find that structural heterogeneity of the gels, which leads to floppy modes and a nonaffine-affine crossover as frequency increases, evolves as a function of the volume fraction and is key to understanding the frequency-dependent elasticity. Without any free parameters, we achieve good qualitative agreement with the measured mechanical response. Furthermore, we achieve universal collapse of the shear moduli through a phenomenological spring-dashpot model that accounts for the interplay between fluid viscosity, particle dissipation, and contributions from the affine and non-affine network deformation.

Aqueous ionic liquids redistribute local enzyme stability via long-range perturbation pathways

Ionic liquids (IL) and aqueous ionic liquids (aIL) are attractive (co-)solvents for biocatalysis due to their unique properties. On the other hand, the incubation of enzymes in IL or aIL often reduces enzyme activity. Recent studies proposed various aIL-induced effects to explain the reduction, classified as direct effects, e.g., local dehydration or competitive inhibition, and indirect effects, e.g., structural perturbations or disturbed catalytic site integrity. However, the molecular origin of indirect effects has largely remained elusive. Here we show by multi-μs long molecular dynamics simulations, free energy computations, and rigidity analyses that aIL favorably interact with specific residues of Bacillus subtilis Lipase A (BsLipA) and modify the local structural stability of this model enzyme by inducing long-range perturbations of noncovalent interactions. The perturbations percolate over neighboring residues and eventually affect the catalytic site and the buried protein core. Validation against a complete experimental site saturation mutagenesis library of BsLipA (3620 variants) reveals that the residues of the perturbation pathways are distinguished sequence positions where substitutions highly likely yield significantly improved residual activity. Our results demonstrate that identifying these perturbation pathways and specific IL ion-residue interactions there effectively predicts focused variant libraries with improved aIL tolerance.

Elastic moduli of body-centered cubic lattice near rigidity percolation threshold: Finite-size effects and evidence for first-order phase transition

Extensive numerical simulations of rigidity percolation with only central forces in large three-dimensional lattices have indicated that many of their topological properties undergo a first-order phase transition at the rigidity percolation threshold p_{ce}. In contrast with such properties, past numerical calculations of the elastic moduli of the same lattices had provided evidence for a second-order phase transition. In this paper we present the results of extensive simulation of rigidity percolation in large body-centered cubic (bcc) lattices, and show that as the linear size L of the lattice increases, the elastic moduli close to p_{ce} decrease in a stepwise, discontinuous manner, a feature that is absent in lattices with L<30. The number and size of such steps increase with L. As p_{ce} is approached, long-range, nondecaying orientational correlations are built up, giving rise to compact, nonfractal clusters. As a result, we find that the backbone of the lattice at p_{ce} is compact with a fractal dimension D_{bb}≈3. The absence of fractal, scale-invariant clusters, the hallmark of second-order phase transitions, together with the stairwise behavior of the elastic moduli, provide strong evidence that, at least in bcc lattices, many of the topological properties of rigidity percolation as well as its elastic moduli may undergo a first-order phase transition at p_{ce}. In relatively small lattices, however, the boundary effects interfere with the nonlocal nature of the rigidity percolation. As a result, only when such effects diminish in large lattices does the true nature of the phase transition emerge.

Promiscuous Esterases Counterintuitively Are Less Flexible than Specific Ones

Understanding mechanisms of promiscuity is increasingly important from a fundamental and application point of view. As to enzyme structural dynamics, more promiscuous enzymes generally have been recognized to also be more flexible. However, examples for the opposite received much less attention. Here, we exploit comprehensive experimental information on the substrate promiscuity of 147 esterases tested against 96 esters together with computationally efficient rigidity analyses to understand the molecular origin of the observed promiscuity range. Unexpectedly, our data reveal that promiscuous esterases are significantly less flexible than specific ones, are significantly more thermostable, and have a significantly increased specific activity. These results may be reconciled with a model according to which structural flexibility in the case of specific esterases serves for conformational proofreading. Our results signify that an esterase sequence space can be screened by rigidity analyses for promiscuous esterases as starting points for further exploration in biotechnology and synthetic chemistry.

Rigidity percolation uncovers a structural basis for embryonic tissue phase transitions

Embryo morphogenesis is impacted by dynamic changes in tissue material properties, which have been proposed to occur via processes akin to phase transitions (PTs). Here, we show that rigidity percolation provides a simple and robust theoretical framework to predict material/structural PTs of embryonic tissues from local cell connectivity. By using percolation theory, combined with directly monitoring dynamic changes in tissue rheology and cell contact mechanics, we demonstrate that the zebrafish blastoderm undergoes a genuine rigidity PT, brought about by a small reduction in adhesion-dependent cell connectivity below a critical value. We quantitatively predict and experimentally verify hallmarks of PTs, including power-law exponents and associated discontinuities of macroscopic observables. Finally, we show that this uniform PT depends on blastoderm cells undergoing meta-synchronous divisions causing random and, consequently, uniform changes in cell connectivity. Collectively, our theoretical and experimental findings reveal the structural basis of material PTs in an organismal context.

Spongelike Rigid Structures in Frictional Granular Packings

We show how rigidity emerges in experiments on sheared two-dimensional frictional granular materials by using generalizations of two methods for identifying rigid structures. Both approaches, the force-based dynamical matrix and the topology-based rigidity percolation, agree with each other and identify similar rigid structures. As the system becomes jammed, at a critical contact number z_{c}=2.4±0.1, a rigid backbone interspersed with floppy, particle-filled holes of a broad range of sizes emerges, creating a spongelike morphology. While the pressure within rigid structures always exceeds the pressure outside the rigid structures, they are not identified with the force chains of shear jamming. These findings highlight the need to focus on mechanical stability arising through arch structures and hinges at the mesoscale.

Correlated rigidity percolation in fractal lattices

Rigidity percolation (RP) is the emergence of mechanical stability in networks. Motivated by the experimentally observed fractal nature of materials like colloidal gels and disordered fiber networks, we study RP in a fractal network where intrinsic correlations in particle positions is controlled by the fractal iteration. Specifically, we calculate the critical packing fractions of site-diluted lattices of Sierpiński gaskets (SG's) with varying degrees of fractal iteration. Our results suggest that although the correlation length exponent and fractal dimension of the RP of these lattices are identical to that of the regular triangular lattice, the critical volume fraction is dramatically lower due to the fractal nature of the network. Furthermore, we develop a simplified model for an SG lattice based on the fragility analysis of a single SG. This simplified model provides an upper bound for the critical packing fractions of the full fractal lattice, and this upper bound is strictly obeyed by the disorder averaged RP threshold of the fractal lattices. Our results characterize rigidity in ultralow-density fractal networks.

Can constraint network analysis guide the identification phase of KnowVolution? A case study on improved thermostability of an endo-β-glucanase

Cellulases are industrially important enzymes, e.g., in the production of bioethanol, in pulp and paper industry, feedstock, and textile. Thermostability is often a prerequisite for high process stability and improving thermostability without affecting specific activities at lower temperatures is challenging and often time-consuming. Protein engineering strategies that combine experimental and computational are emerging in order to reduce experimental screening efforts and speed up enzyme engineering campaigns. Constraint Network Analysis (CNA) is a promising computational method that identifies beneficial positions in enzymes to improve thermostability. In this study, we compare CNA and directed evolution in the identification of beneficial positions in order to evaluate the potential of CNA in protein engineering campaigns (e.g., in the identification phase of KnowVolution). We engineered the industrially relevant endoglucanase EGLII from Penicillium verruculosum towards increased thermostability. From the CNA approach, six variants were obtained with an up to 2-fold improvement in thermostability. The overall experimental burden was reduced to 40% utilizing the CNA method in comparison to directed evolution. On a variant level, the success rate was similar for both strategies, with 0.27% and 0.18% improved variants in the epPCR and CNA-guided library, respectively. In essence, CNA is an effective method for identification of positions that improve thermostability.

Control of connectivity and rigidity in prismatic assemblies

How can we manipulate the topological connectivity of a three-dimensional prismatic assembly to control the number of internal degrees of freedom and the number of connected components in it? To answer this question in a deterministic setting, we use ideas from elementary number theory to provide a hierarchical deterministic protocol for the control of rigidity and connectivity. We then show that it is possible to also use a stochastic protocol to achieve the same results via a percolation transition. Together, these approaches provide scale-independent algorithms for the cutting or gluing of three-dimensional prismatic assemblies to control their overall connectivity and rigidity.

Emergent solidity of amorphous materials as a consequence of mechanical self-organisation

Amorphous solids have peculiar properties distinct from crystals. One of the most fundamental mysteries is the emergence of solidity in such nonequilibrium, disordered state without the protection by long-range translational order. A jammed system at zero temperature, although marginally stable, has solidity stemming from the space-spanning force network, which gives rise to the long-range stress correlation. Here, we show that such nonlocal correlation already appears at the nonequilibrium glass transition upon cooling. This is surprising since we also find that the system suffers from giant anharmonic fluctuations originated from the fractal-like potential energy landscape. We reveal that it is the percolation of the force-bearing network that allows long-range stress transmission even under such circumstance. Thus, the emergent solidity of amorphous materials is a consequence of nontrivial self-organisation of the disordered mechanical architecture. Our findings point to the significance of understanding amorphous solids and nonequilibrium glass transition from a mechanical perspective.

Finite size effects in critical fiber networks

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.

Multifunctional twisted kagome lattices: Tuning by pruning mechanical metamaterials

This article investigates phonons and elastic response in randomly diluted lattices constructed by combining (via the addition of next-nearest bonds) a twisted kagome lattice, with bulk modulus B=0 and shear modulus G>0, with either a generalized untwisted kagome lattice with B>0 and G>0 or with a honeycomb lattice with B>0 and G=0. These lattices exhibit jamming-like critical endpoints at which B, G, or both B and G jump discontinuously from zero while the remaining moduli (if any) begin to grow continuously from zero. Pairs of these jamming points are joined by lines of continuous rigidity percolation transitions at which both B and G begin to grow continuously from zero. The Poisson ratio and G/B can be continuously tuned throughout their physical range via random dilution in a manner analogous to "tuning by pruning" in random jammed lattices. These lattices can be produced with modern techniques, such as three-dimensional printing, for constructing metamaterials.

Two-Scale Scenario of Rigidity Percolation of Sticky Particles

In the presence of attraction, the jamming transition of packings of frictionless particles corresponds to the rigidity percolation. When the range of attraction is long, the distribution of the size of rigid clusters, P(s), is continuous and shows a power-law decay. For systems with short-range attractions, however, P(s) appears discontinuous. There is a power-law decay for small cluster sizes, followed by a low probability gap and a peak near the system size. We find that this appearing "discontinuity" does not mean that the transition is discontinuous. In fact, it signifies the coexistence of two distinct length scales, associated with the largest cluster and smaller ones, respectively. The comparison between the largest and second largest clusters indicates that their growth rates with system size are rather different. However, both cluster sizes tend to diverge in the large system size limit, suggesting that the jamming transition of systems with short-range attractions is still continuous. In the framework of the two-scale scenario, we also derive a generalized hyperscaling relation. With robust evidence, our work challenges the former single-scale view of the rigidity percolation.

Continuum Theory for Topological Edge Soft Modes

Topological edge zero modes and states of self stress have been intensively studied in discrete lattices at the Maxwell point, offering robust properties concerning surface and interface stiffness and stress focusing. In this Letter, we present a topological elasticity theory for general continuous media where a gauge-invariant bulk topological index independent of microscopic details is defined. This index directly predicts the number of zero modes on edges at long length scales, and it naturally extends to media that deviate from the Maxwell point, depicting how topological zero modes turn into topological soft modes.

Deterministic and stochastic control of kirigami topology

Kirigami, the creative art of paper cutting, is a promising paradigm for mechanical metamaterials. However, to make kirigami-inspired structures a reality requires controlling the topology of kirigami to achieve connectivity and rigidity. We address this question by deriving the maximum number of cuts (minimum number of links) that still allow us to preserve global rigidity and connectivity of the kirigami. A deterministic hierarchical construction method yields an efficient topological way to control both the number of connected pieces and the total degrees of freedom. A statistical approach to the control of rigidity and connectivity in kirigami with random cuts complements the deterministic pathway, and shows that both the number of connected pieces and the degrees of freedom show percolation transitions as a function of the density of cuts (links). Together, this provides a general framework for the control of rigidity and connectivity in planar kirigami.

Dynamic arrest of adhesive hard rod dispersions

The phenomenon of dynamic arrest, more commonly referred to as gel and glass formation, originates as particle motion slows significantly. Current understanding of gels and glasses stems primarily from dispersions of spherical particles, but much less is known about how particle shape affects dynamic arrest transitions. To better understand the effects of particle shape anisotropy on gel and glass formation, we systematically measure the rheology, particle dynamics, and static microstructure of thermoreversible colloidal dispersions of adhesive hard rods (AHR). First, the dynamic arrest transitions are mapped as a function of temperature T, aspect ratio L/D≈ 3 to 7, and volume fraction φ≈ 0.1 to 0.5. The critical gel temperature T_gel and glass volume fraction φ_g are determined from the particle dynamics and rheology. Second, an effective orientation-averaged, short-range attraction between rods is quantified from small-angle scattering measurements and characterized by a reduced temperature τ. Similar τ is found at low rod concentrations, indicating that rod gelation occurs at similar effective attraction strength independent of L/D. Monte Carlo simulations reveal a similar convergence in τ when rods cluster and percolate with an average bond coordination number 〈n_c〉≈ 2.4, supporting the link between physical gelation and rigidity percolation. Lastly, AHR results are mapped onto a dimensionless state diagram to compare with previous predictions of attraction-driven gels, repulsion-driven glasses, and liquid crystal phases.

Systematically Scrutinizing the Impact of Substitution Sites on Thermostability and Detergent Tolerance for Bacillus subtilis Lipase A

Improving an enzyme's (thermo-)stability or tolerance against solvents and detergents is highly relevant in protein engineering and biotechnology. Recent developments have tended toward data-driven approaches, where available knowledge about the protein is used to identify substitution sites with high potential to yield protein variants with improved stability, and subsequently, substitutions are engineered by site-directed or site-saturation (SSM) mutagenesis. However, the development and validation of algorithms for data-driven approaches have been hampered by the lack of availability of large-scale data measured in a uniform way and being unbiased with respect to substitution types and locations. Here, we extend our knowledge on guidelines for protein engineering following a data-driven approach by scrutinizing the impact of substitution sites on thermostability or/and detergent tolerance for Bacillus subtilis lipase A (BsLipA) at very large scale. We systematically analyze a complete experimental SSM library of BsLipA containing all 3439 possible single variants, which was evaluated as to thermostability and tolerances against four detergents under respectively uniform conditions. Our results provide systematic and unbiased reference data at unprecedented scale for a biotechnologically important protein, identify consistently defined hot spot types for evaluating the performance of data-driven protein-engineering approaches, and show that the rigidity theory and ensemble-based approach Constraint Network Analysis yields hot spot predictions with an up to ninefold gain in precision over random classification.

Salt bridge impact on global rigidity and thermostability in thermophilic citrate synthase

It has been suggested that structural rigidity is connected to thermostability, e.g. in enzymes from thermophilic microorganisms. We examine the importance of correctly handling salt bridges, and interactions which we term 'strong polars', when constructing the constraint network for global rigidity analysis in these systems. Through a comparison of rigidity in citrate synthases, we clarify the relationship between rigidity and thermostability. In particular, with our corrected handling of strong polar interactions, the difference in rigidity between mesophilic and thermophilic structures is detected more clearly than in previous studies. The increase in rigidity did not detract from the functional flexibility of the active site in all systems once their respective temperature range had been reached. We then examine the distribution of salt bridges in thermophiles that were previously unaccounted for in flexibility studies. We show that in hyperthermophiles these have stabilising roles in the active site; occuring in close proximity to key residues involved in catalysis and binding of the protein.

Characterizing nonaffinity upon decompression of soft-sphere packings

Athermal elastic moduli of soft-sphere packings are known to exhibit universal scaling properties near the unjamming point, most notably the vanishing of the shear-to-bulk moduli ratio G/B upon decompression. Interestingly, the smallness of G/B stems from the large nonaffinity of deformation-induced displacements under shear strains, compared to insignificant nonaffinity of displacements under compressive strains. In this work, we show using numerical simulations that the relative weights of the affine and nonaffine contributions to the bulk modulus, and their dependence on the proximity to the unjamming point, can differ qualitatively between different models that feature the same generic unjamming phenomenology. In canonical models of unjamming, we observe that the ratio of the nonaffine to total bulk moduli B_{na}/B approaches a constant upon decompression, while in other, less well-studied models, it vanishes. We show that the vanishing of B_{na}/B in noncanonical models stems from the emergence of an invariance of net (zero) forces on the constituent particles to compressive strains at the onset of unjamming. We provide a theoretical scaling analysis that fully explains our numerical observations, and allows us to predict the scaling behavior of B_{na}/B upon unjamming, given the functional form of the pairwise interaction potential.

Mechanical Heterogeneity in Tissues Promotes Rigidity and Controls Cellular Invasion

We study the influence of cell-level mechanical heterogeneity in epithelial tissues using a vertex-based model. Heterogeneity is introduced into the cell shape index (p_{0}) that tunes the stiffness at a single-cell level. The addition of heterogeneity can always enhance the mechanical rigidity of the epithelial layer by increasing its shear modulus, hence making it more rigid. There is an excellent scaling collapse of our data as a function of a single scaling variable f_{r}, which accounts for the overall fraction of rigid cells. We identify a universal threshold f_{r}^{*} that demarcates fluid versus solid tissues. Furthermore, this rigidity onset is far below the contact percolation threshold of rigid cells. These results give rise to a separation of rigidity and contact percolation processes that leads to distinct types of solid states. We also investigate the influence of heterogeneity on tumor invasion dynamics. There is an overall impedance of invasion as the tissue becomes more rigid. Invasion can also occur in an intermediate heterogeneous solid state that is characterized by significant spatial-temporal intermittency.

Correlated Rigidity Percolation and Colloidal Gels

Rigidity percolation (RP) occurs when mechanical stability emerges in disordered networks as constraints or components are added. Here we discuss RP with structural correlations, an effect ignored in classical theories albeit relevant to many liquid-to-amorphous-solid transitions, such as colloidal gelation, which are due to attractive interactions and aggregation. Using a lattice model, we show that structural correlations shift RP to lower volume fractions. Through molecular dynamics simulations, we show that increasing attraction in colloidal gelation increases structural correlation and thus lowers the RP transition, agreeing with experiments. Hence, the emergence of rigidity at colloidal gelation can be understood as a RP transition, but occurs at volume fractions far below values predicted by the classical RP, due to attractive interactions which induce structural correlation.

Excess floppy modes and multibranched mechanisms in metamaterials with symmetries

Floppy modes-deformations that cost zero energy-are central to the mechanics of a wide class of systems. For disordered systems, such as random networks and particle packings, it is well-understood how the number of floppy modes is controlled by the topology of the connections. Here we uncover that symmetric geometries, present in, e.g., mechanical metamaterials, can feature an unlimited number of excess floppy modes that are absent in generic geometries, and in addition can support floppy modes that are multibranched. We study the number Δ of excess floppy modes by comparing generic and symmetric geometries with identical topologies, and show that Δ is extensive, peaks at intermediate connection densities, and exhibits mean-field scaling. We then develop an approximate yet accurate cluster counting algorithm that captures these findings. Finally, we leverage our insights to design metamaterials with multiple folding mechanisms.

TCR-induced alteration of primary MHC peptide anchor residue

The HLA-A*02:01-restricted decapeptide EAAGIGILTV, derived from melanoma antigen recognized by T-cells-1 (MART-1) protein, represents one of the best-studied tumor associated T-cell epitopes, but clinical results targeting this peptide have been disappointing. This limitation may reflect the dominance of the nonapeptide, AAGIGILTV, at the melanoma cell surface. The decapeptide and nonapeptide are presented in distinct conformations by HLA-A*02:01 and TCRs from clinically relevant T-cell clones recognize the nonapeptide poorly. Here, we studied the MEL5 TCR that potently recognizes the nonapeptide. The structure of the MEL5-HLA-A*02:01-AAGIGILTV complex revealed an induced fit mechanism of antigen recognition involving altered peptide-MHC anchoring. This "flexing" at the TCR-peptide-MHC interface to accommodate the peptide antigen explains previously observed incongruences in this well-studied system and has important implications for future therapeutic approaches. Finally, this study expands upon the mechanisms by which molecular plasticity can influence antigen recognition by T cells.

Emergence and percolation of rigid domains during the colloidal glass transition

Using video microscopy, we measure local spatial constraints in disordered binary colloidal samples, ranging from dilute fluids to jammed glasses, and probe their spatial and temporal correlations to local dynamics during the glass transition. We observe the emergence of significant correlations between constraints and local dynamics within the Lindemann criterion, which coincides with the onset of glassy dynamics in supercooled liquids. Rigid domains in fluids are identified based on local constraints and demonstrate a percolation transition near the glass transition, accompanied by the emergence of dynamical heterogeneities. Our results show that spatial constraint instead of the geometry of amorphous structures is the key that connects the complex spatial-temporal correlations in disordered materials.

Scaling Theory for Mechanical Critical Behavior in Fiber Networks

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.

Direct link between mechanical stability in gels and percolation of isostatic particles

Colloidal gels have unique mechanical and transport properties that stem from their bicontinuous nature, in which a colloidal network is intertwined with a viscous solvent, and have found numerous applications in foods, cosmetics, and construction materials and for medical applications, such as cartilage replacements. So far, our understanding of the process of colloidal gelation is limited to long-time dynamical effects, where gelation is viewed as a phase separation process interrupted by the glass transition. However, this purely out-of-equilibrium thermodynamic picture does not address the emergence of mechanical stability. With confocal microscopy experiments, we reveal that mechanical metastability is reached only after isotropic percolation of locally isostatic environments, establishing a direct link between the load-bearing ability of gels and the isostaticity condition. Our work suggests an operative description of gels based on mechanical equilibrium and isostaticity, providing the physical basis for the stability and rheology of these materials.

A broader view on jamming: from spring networks to circle packings

Jamming occurs when objects like grains are packed tightly together (e.g. grain silos). It is highly cooperative and can lead to phenomena like earthquakes, traffic jams, etc. In this paper we point out the paramount importance of the underlying contact network for jammed systems; the network must have one contact in excess of isostaticity and a finite bulk modulus. Isostatic means that the number of degrees of freedom is exactly balanced by the number of constraints. This defines a large class of networks that can be constructed without the necessity of packing particles together compressively (either in the lab or computationally). One such construction, which we explore here, involves setting up the Delaunay triangulation of a Poisson disk sampling and then removing edges to maximize the bulk modulus until the isostatic plus one edge is reached. This construction works in any dimensions and here we give results in 2D where we also show how such networks can be transformed into disk packs.