Nonequilibrium statistical mechanical models for cytoskeletal assembly: towards understanding tensegrity in cells

Hydrogels with prestressed tensegrity structures

Tensegrity structures are isolated rigid compression components held in place by a continuous network of tensile components, and are central to natural systems such as the extracellular matrix and the cell cytoskeleton. These structures enable the nonreciprocal mechanical properties essential for dynamic biological functions. Here, we introduce a synthetic approach to engineer hydrogels with tensegrity architectures, drawing inspiration from the mechanochemical principles underlying biological systems. By employing in-situ enzyme-induced amino acid crystal growth within preformed polymeric networks, we achieve a hierarchical integration of micro crystal sticks randomly interlocked in the prestressed polymer matrice. This design mirrors natural tensegrity structures, balancing mechanical forces to maintain high stiffness (tensile moduli up to 30 MPa), fracture toughness (2600 J m⁻²), and water content (exceeding 80%). The resultant hydrogels exhibit bimodulus behavior due to their tensegrity structure, featuring a tensile-to-compressive modulus ratio of 13. This biomimetic approach provides a strategy for creating robust, adaptive materials for applications in tissue engineering and beyond.

Chromatin folding through nonuniform motorization by responsive motor proteins

Chromatin is partially structured through the effects of biological motors. "Swimming motors" such as RNA polymerases and chromatin remodelers are thought to act differentially on the active parts of the genome and the stored inactive part. By systematically expanding the many-body master equation for chromosomes driven by swimming motors, we show that this nonuniform aspect of motorization leads to heterogeneously folded conformations, thereby contributing to chromosome compartmentalization.

Motorized chain models of the ideal chromosome

An array of motor proteins consumes chemical energy in setting up the architectures of chromosomes. Here, we explore how the structure of ideal polymer chains is influenced by two classes of motors. The first class which we call "swimming motors" acts to propel the chromatin fiber through three-dimensional space. They represent a caricature of motors such as RNA polymerases. Previously, they have often been described by adding a persistent flow onto Brownian diffusion of the chain. The second class of motors, which we call "grappling motors" caricatures the loop extrusion processes in which segments of chromatin fibers some distance apart are brought together. We analyze these models using a self-consistent variational phonon approximation to a many-body Master equation incorporating motor activities. We show that whether the swimming motors lead to contraction or expansion depends on the susceptibility of the motors, that is, how their activity depends on the forces they must exert. Grappling motors in contrast to swimming motors lead to long-ranged correlations that resemble those first suggested for fractal globules and that are consistent with the effective interactions inferred by energy landscape analyses of Hi-C data on the interphase chromosome.

Mechano-immunology in microgravity

Life on Earth has evolved to thrive in the Earth's natural gravitational field; however, as space technology advances, we must revisit and investigate the effects of unnatural conditions on human health, such as gravitational change. Studies have shown that microgravity has a negative impact on various systemic parts of humans, with the effects being more severe in the human immune system. Increasing costs, limited experimental time, and sample handling issues hampered our understanding of this field. To address the existing knowledge gap and provide confidence in modelling the phenomena, in this review, we highlight experimental works in mechano-immunology under microgravity and different computational modelling approaches that can be used to address the existing problems.

Theory of Active Chromatin Remodeling

Nucleosome positioning controls the accessible regions of chromatin and plays essential roles in DNA-templated processes. ATP driven remodeling enzymes are known to be crucial for its establishment in vivo, but their nonequilibrium nature has hindered the development of a unified theoretical framework for nucleosome positioning. Using a perturbation theory, we show that the effect of these enzymes can be well approximated by effective equilibrium models with rescaled temperatures and interactions. Numerical simulations support the accuracy of the theory in predicting both kinetic and steady-state quantities, including the effective temperature and the radial distribution function, in biologically relevant regimes. The energy landscape view emerging from our study provides an intuitive understanding for the impact of remodeling enzymes in either reinforcing or overwriting intrinsic signals for nucleosome positioning, and may help improve the accuracy of computational models for its prediction in silico.

Rotational and translational diffusion in an interacting active dumbbell system

We study the dynamical properties of a two-dimensional ensemble of self-propelled dumbbells with only repulsive interactions. This model undergoes a phase transition between a homogeneous and a segregated phase and we focus on the former. We analyze the translational and rotational mean-square displacements in terms of the Péclet number, describing the relative role of active forces and thermal fluctuations, and of particle density. We find that the four distinct regimes of the translational mean-square displacement of the single active dumbbell survive at finite density for parameters that lead to a separation of time scales. We establish the Péclet number and density dependence of the diffusion constant in the last diffusive regime. We prove that the ratio between the diffusion constant and its value for the single dumbbell depends on temperature and active force only through the Péclet number at all densities explored. We also study the rotational mean-square displacement proving the existence of a rich behavior with intermediate regimes only appearing at finite density. The ratio of the rotational late-time diffusion constant and its vanishing density limit depends on the Péclet number and density only. At low Péclet number it is a monotonically decreasing function of density. At high Péclet number it first increases to reach a maximum and then decreases as a function of density. We interpret the latter result advocating the presence of large-scale fluctuations close to the transition, at large-enough density, that favor coherent rotation inhibiting, however, rotational motion for even larger packing fractions.

Dynamics of a homogeneous active dumbbell system

We analyze the dynamics of a two-dimensional system of interacting active dumbbells. We characterize the mean-square displacement, linear response function, and deviation from the equilibrium fluctuation-dissipation theorem as a function of activity strength, packing fraction, and temperature for parameters such that the system is in its homogeneous phase. While the diffusion constant in the last diffusive regime naturally increases with activity and decreases with packing fraction, we exhibit an intriguing nonmonotonic dependence on the activity of the ratio between the finite-density and the single-particle diffusion constants. At fixed packing fraction, the time-integrated linear response function depends nonmonotonically on activity strength. The effective temperature extracted from the ratio between the integrated linear response and the mean-square displacement in the last diffusive regime is always higher than the ambient temperature, increases with increasing activity, and, for small active force, monotonically increases with density while for sufficiently high activity it first increases and next decreases with the packing fraction. We ascribe this peculiar effect to the existence of finite-size clusters for sufficiently high activity and density at the fixed (low) temperatures at which we worked. The crossover occurs at lower activity or density the lower the external temperature. The finite-density effective temperature is higher (lower) than the single dumbbell one below (above) a crossover value of the Péclet number.

Tensegrity, cellular biophysics, and the mechanics of living systems

The recent convergence between physics and biology has led many physicists to enter the fields of cell and developmental biology. One of the most exciting areas of interest has been the emerging field of mechanobiology that centers on how cells control their mechanical properties, and how physical forces regulate cellular biochemical responses, a process that is known as mechanotransduction. In this article, we review the central role that tensegrity (tensional integrity) architecture, which depends on tensile prestress for its mechanical stability, plays in biology. We describe how tensional prestress is a critical governor of cell mechanics and function, and how use of tensegrity by cells contributes to mechanotransduction. Theoretical tensegrity models are also described that predict both quantitative and qualitative behaviors of living cells, and these theoretical descriptions are placed in context of other physical models of the cell. In addition, we describe how tensegrity is used at multiple size scales in the hierarchy of life—from individual molecules to whole living organisms—to both stabilize three-dimensional form and to channel forces from the macroscale to the nanoscale, thereby facilitating mechanochemical conversion at the molecular level.

Molecular transport modulates the adaptive response of branched actin networks to an external force

Actin networks are an integral part of the cytoskeleton of eukaryotic cells and play an essential role in determining cellular shape and movement. Understanding the underlying mechanism of actin network assembly is of fundamental importance. We developed in this work a minimal motility model and performed stochastic simulations to study mechanical regulation of the growth dynamics of lamellipodia-like branched actin networks, characterized by various force-velocity relations. In such networks, the treadmilling process leads to a concentration gradient of G-actin, and thus G-actin transport is essential to effective actin network assembly. We first explore how capping protein modulates force-velocity relations and then discuss how actin transport due to diffusion and facilitated transport such as advective flow tunes the growth dynamics of the branched actin network. Our work demonstrates the important role of molecular transport in determining the adaptive response of the actin network to an external force.

Glassy dynamics, cell mechanics, and endothelial permeability

A key feature of all inflammatory processes is disruption of the vascular endothelial barrier. Such disruption is initiated in part through active contraction of the cytoskeleton of the endothelial cell (EC). Because contractile forces are propagated from cell to cell across a great many cell-cell junctions, this contractile process is strongly cooperative and highly nonlocal. We show here that the characteristic length scale of propagation is modulated by agonists and antagonists that impact permeability of the endothelial barrier. In the presence of agonists including thrombin, histamine, and H2O2, force correlation length increases, whereas in the presence of antagonists including sphingosine-1-phosphate, hepatocyte growth factor, and the rho kinase inhibitor, Y27632, force correlation length decreases. Intercellular force chains and force clusters are also evident, both of which are reminiscent of soft glassy materials approaching a glass transition.

Swimming motility plays a key role in the stochastic dynamics of cell clumping

Dynamic cell-to-cell interactions are a prerequisite to many biological processes, including development and biofilm formation. Flagellum induced motility has been shown to modulate the initial cell-cell or cell-surface interaction and to contribute to the emergence of macroscopic patterns. While the role of swimming motility in surface colonization has been analyzed in some detail, a quantitative physical analysis of transient interactions between motile cells is lacking. We examined the Brownian dynamics of swimming cells in a crowded environment using a model of motorized adhesive tandem particles. Focusing on the motility and geometry of an exemplary motile bacterium Azospirillum brasilense, which is capable of transient cell-cell association (clumping), we constructed a physical model with proper parameters for the computer simulation of the clumping dynamics. By modulating mechanical interaction ('stickiness') between cells and swimming speed, we investigated how equilibrium and active features affect the clumping dynamics. We found that the modulation of active motion is required for the initial aggregation of cells to occur at a realistic time scale. Slowing down the rotation of flagellar motors (and thus swimming speeds) is correlated to the degree of clumping, which is consistent with the experimental results obtained for A. brasilense.

Communication: Effective temperature and glassy dynamics of active matter

A systematic expansion of the many-body master equation for active matter, in which motors power configurational changes as in the cytoskeleton, is shown to yield a description of the steady state and responses in terms of an effective temperature. The effective temperature depends on the susceptibility of the motors and a Peclet number which measures their strength relative to thermal Brownian diffusion. The analytic prediction is shown to agree with previous numerical simulations and experiments. The mapping also establishes a description of aging in active matter that is also kinetically jammed.

On the spontaneous collective motion of active matter

Spontaneous directed motion, a hallmark of cell biology, is unusual in classical statistical physics. Here we study, using both numerical and analytical methods, organized motion in models of the cytoskeleton in which constituents are driven by energy-consuming motors. Although systems driven by small-step motors are described by an effective temperature and are thus quiescent, at higher order in step size, both homogeneous and inhomogeneous, flowing and oscillating behavior emerges. Motors that respond with a negative susceptibility to imposed forces lead to an apparent negative-temperature system in which beautiful structures form resembling the asters seen in cell division.

Active biological materials

Cells make use of dynamic internal structures to control shape and create movement. By consuming energy to assemble into highly organized systems of interacting parts, these structures can generate force and resist compression, as well as adaptively change in response to their environment. Recent progress in reconstituting cytoskeletal structures in vitro has provided an opportunity to characterize the mechanics and dynamics of filament networks formed from purified proteins. Results indicate that a complex interplay between length scales and timescales underlies the mechanical responses of these systems and that energy consumption, as manifested in molecular motor activity and cytoskeletal filament growth, can drive transitions between distinct material states. This review discusses the basic characteristics of these active biological materials that set them apart from conventional materials and that create a rich array of unique behaviors.

Universal behavior of the osmotically compressed cell and its analogy to the colloidal glass transition

Mechanical robustness of the cell under different modes of stress and deformation is essential to its survival and function. Under tension, mechanical rigidity is provided by the cytoskeletal network; with increasing stress, this network stiffens, providing increased resistance to deformation. However, a cell must also resist compression, which will inevitably occur whenever cell volume is decreased during such biologically important processes as anhydrobiosis and apoptosis. Under compression, individual filaments can buckle, thereby reducing the stiffness and weakening the cytoskeletal network. However, the intracellular space is crowded with macromolecules and organelles that can resist compression. A simple picture describing their behavior is that of colloidal particles; colloids exhibit a sharp increase in viscosity with increasing volume fraction, ultimately undergoing a glass transition and becoming a solid. We investigate the consequences of these 2 competing effects and show that as a cell is compressed by hyperosmotic stress it becomes progressively more rigid. Although this stiffening behavior depends somewhat on cell type, starting conditions, molecular motors, and cytoskeletal contributions, its dependence on solid volume fraction is exponential in every instance. This universal behavior suggests that compression-induced weakening of the network is overwhelmed by crowding-induced stiffening of the cytoplasm. We also show that compression dramatically slows intracellular relaxation processes. The increase in stiffness, combined with the slowing of relaxation processes, is reminiscent of a glass transition of colloidal suspensions, but only when comprised of deformable particles. Our work provides a means to probe the physical nature of the cytoplasm under compression, and leads to results that are universal across cell type.

Biomechanics: cell research and applications for the next decade

With the recent revolution in Molecular Biology and the deciphering of the Human Genome, understanding of the building blocks that comprise living systems has advanced rapidly. We have yet to understand, however, how the physical forces that animate life affect the synthesis, folding, assembly, and function of these molecular building blocks. We are equally uncertain as to how these building blocks interact dynamically to create coupled regulatory networks from which integrative biological behaviors emerge. Here we review recent advances in the field of biomechanics at the cellular and molecular levels, and set forth challenges confronting the field. Living systems work and move as multi-molecular collectives, and in order to understand key aspects of health and disease we must first be able to explain how physical forces and mechanical structures contribute to the active material properties of living cells and tissues, as well as how these forces impact information processing and cellular decision making. Such insights will no doubt inform basic biology and rational engineering of effective new approaches to clinical therapy.

Effective temperature of active matter

We follow the dynamics of an ensemble of interacting self-propelled motorized particles in contact with an equilibrated thermal bath. We find that the fluctuation-dissipation relation allows for the definition of an effective temperature that is compatible with the results obtained using a tracer particle as a thermometer. The effective temperature takes a value which is higher than the temperature of the bath, and it is continuously controlled by the motor intensity.

Cellular mechanotransduction: putting all the pieces together again

Analysis of cellular mechanotransduction, the mechanism by which cells convert mechanical signals into biochemical responses, has focused on identification of critical mechanosensitive molecules and cellular components. Stretch-activated ion channels, caveolae, integrins, cadherins, growth factor receptors, myosin motors, cytoskeletal filaments, nuclei, extracellular matrix, and numerous other structures and signaling molecules have all been shown to contribute to the mechanotransduction response. However, little is known about how these different molecules function within the structural context of living cells, tissues, and organs to produce the orchestrated cellular behaviors required for mechanosensation, embryogenesis, and physiological control. Recent work from a wide range of fields reveals that organ, tissue, and cell anatomy are as important for mechanotransduction as individual mechanosensitive proteins and that our bodies use structural hierarchies (systems within systems) composed of interconnected networks that span from the macroscale to the nanoscale in order to focus stresses on specific mechanotransducer molecules. The presence of isometric tension (prestress) at all levels of these multiscale networks ensures that various molecular scale mechanochemical transduction mechanisms proceed simultaneously and produce a concerted response. Future research in this area will therefore require analysis, understanding, and modeling of tensionally integrated (tensegrity) systems of mechanochemical control.