Spontaneous buckling of contractile poroelastic actomyosin sheets.

Shape transitions in developing organisms can be driven by active stresses, notably, active contractility generated by myosin motors. The mechanisms generating tissue folding are typically studied in epithelia. There, the interaction between cells is also coupled to an elastic substrate, presenting a major difficulty for studying contraction induced folding. Here we study the contraction and buckling of active, initially homogeneous, thin elastic actomyosin networks isolated from bounding surfaces. The network behaves as a poroelastic material, where a flow of fluid is generated during contraction. Contraction starts at the system boundaries, proceeds into the bulk, and eventually leads to spontaneous buckling of the sheet at the periphery. The buckling instability resulted from system self-organization and from the spontaneous emergence of density gradients driven by the active contractility. The buckling wavelength increases linearly with sheet thickness. Our system offers a well-controlled way to study mechanically induced, spontaneous shape transitions in active matter.

Mechanobiological induction of long-range contractility by diffusing biomolecules and size scaling in cell assemblies.

Mechanobiological studies of cell assemblies have generally focused on cells that are, in principle, identical. Here we predict theoretically the effect on cells in culture of locally introduced biochemical signals that diffuse and locally induce cytoskeletal contractility which is initially small. In steady-state, both the concentration profile of the signaling molecule as well as the contractility profile of the cell assembly are inhomogeneous, with a characteristic length that can be of the order of the system size. The long-range nature of this state originates in the elastic interactions of contractile cells (similar to long-range "macroscopic modes" in non-living elastic inclusions) and the non-linear diffusion of the signaling molecules, here termed mechanogens. We suggest model experiments on cell assemblies on substrates that can test the theory as a prelude to its applicability in embryo development where spatial gradients of morphogens initiate cellular development.

Transcription rates in DNA brushes.

We theoretically predict the rate of transcription (TX) in DNA brushes by introducing the concept of TX dipoles that takes into account the unidirectional motion of enzymes (RNAP) along DNA during transcription as correlated pairs of sources and sinks in the relevant diffusion equation. Our theory predicts that the TX rates dramatically change upon the inversion of the orientation of the TX dipoles relative to the substrate because TX dipoles modulate the concentrations of RNAP in the solution. Comparing our theory with experiments suggests that, in some cases, DNA chain segments are relatively uniformly distributed in the brush, in contrast to the parabolic profile expected for flexible polymer brushes.

Effect of charge inhomogeneity and mobility on colloid aggregation.

The aggregation of inhomogeneously charged colloids with the same average charge is analyzed using Monte Carlo simulations. We find aggregation of colloids for sizes in the range 10-200 nm, which is similar to the range in which aggregation is observed in several experiments. The attraction arises from the strongly correlated electrostatic interactions associated with the increase in the counterion density in the region between the particles; this effect is enhanced by the discreteness and mobility of the surface charges. Larger colloids attract more strongly when their surface charges are discrete. We study the aggregation as functions of the surface charge density, counterion valence, and volume fraction.

Long-range interaction between heterogeneously charged membranes.

Despite their neutrality, surfaces or membranes with equal amounts of positive and negative charge can exhibit long-range electrostatic interactions if the surface charge is heterogeneous; this can happen when the surface charges form finite-size domain structures. These domains can be formed in lipid membranes where the balance of the different ranges of strong but short-ranged hydrophobic interactions and longer-ranged electrostatic repulsion result in a finite, stable domain size. If the domain size is large enough, oppositely charged domains in two opposing surfaces or membranes can be strongly correlated by the electrostatic interactions; these correlations give rise to an attractive interaction of the two membranes or surfaces over separations on the order of the domain size. We use numerical simulations to demonstrate the existence of strong attractions at separations of tens of nanometers. Large line tensions result in larger domains but also increase the charge density within the domain. This promotes correlations and, as a result, increases the intermembrane attraction. On the other hand, increasing the salt concentration increases both the domain size and degree of domain anticorrelation, but the interactions are ultimately reduced due to increased screening. The result is a decrease in the net attraction as salt concentration is increased.

Optimal matrix rigidity for stress fiber polarization in stem cells.

The shape and differentiation of human mesenchymal stem cells is especially sensitive to the rigidity of their environment; the physical mechanisms involved are unknown. A theoretical model and experiments demonstrate here that the polarization/alignment of stress-fibers within stem cells is a non-monotonic function of matrix rigidity. We treat the cell as an active elastic inclusion in a surrounding matrix whose polarizability, unlike dead matter, depends on the feedback of cellular forces that develop in response to matrix stresses. The theory correctly predicts the monotonic increase of the cellular forces with the matrix rigidity and the alignment of stress-fibers parallel to the long axis of cells. We show that the anisotropy of this alignment depends non-monotonically on matrix rigidity and demonstrate it experimentally by quantifying the orientational distribution of stress-fibers in stem cells. These findings offer a first physical insight for the dependence of stem cell differentiation on tissue elasticity.

Cell shape, spreading symmetry and the polarization of stress-fibers in cells.

The active regulation of cellular forces during cell adhesion plays an important role in the determination of cell size, shape and internal structure. While on flat, homogeneous and isotropic substrates some cells spread isotropically, others spread anisotropically and assume elongated structures. In addition, in their native environment as well as in vitro experiments, the cell shape and spreading asymmetry can be modulated by the local distribution of adhesive molecules and topography of the environment. We present a simple elastic model, and experiments on stem cells to explain the variation of cell size with the matrix rigidity. In addition, we predict the experimental consequences of two mechanisms of acto-myosin polarization and focus here on the effect of the cell spreading asymmetry on the regulation of the stress-fiber alignment in the cytoskeleton. We show that when cell spreading is sufficiently asymmetric the alignment of acto-myosin forces in the cell increases monotonically with the matrix rigidity; however, in general this alignment is non-monotonic as shown previously. These results highlight the importance of the symmetry characteristics of cell spreading in the regulation of cytoskeleton structure and suggest a mechanism by which different cell types may acquire different morphologies and internal structures in different mechanical environments.

Lamellar phase coexistence induced by electrostatic interactions.

Membranes containing highly charged biomolecules can have a minimal free-energy state at small separations that originates in the strongly correlated electrostatic interactions mediated by counterions. This phenomenon can lead to a condensed, lamellar phase of charged membranes that coexists in thermodynamic equilibrium with a very dilute membrane phase. Although the dilute phase is mostly water, entropy dictates that this phase must contain some membranes and counterions. Thus, electrostatics alone can give rise to the coexistence of a condensed and an unbound lamellar phase. We use numerical simulations to predict the nature of this coexistence when the charge density of the membrane is large, for the case of multivalent counterions and for a membrane charge that is characteristic of biomolecules. We also investigate the effects of counterion size and salt on the two coexisting phases. With increasing salt concentration, we predict that electrostatic screening by salt can destroy the phase separation.

Theory of the mechanical response of focal adhesions to shear flow.

The response of cells to shear flow is primarily determined by the asymmetry of the external forces and moments that are sensed by each member of a focal adhesion pair connected by a contractile stress fiber. In the theory presented here, we suggest a physical model in which each member of such a pair of focal adhesions is treated as an elastic body subject to both a myosin-activated contractile force and the shear stress induced by the external flow. The elastic response of a focal adhesion complex is much faster than the active cellular processes that determine the size of the associated focal adhesions and the direction of the complex relative to the imposed flow. Therefore, the complex attains its mechanical equilibrium configuration which may change because of the cellular activity. Our theory is based on the experimental observation that focal adhesions modulate their cross-sectional area in order to attain an optimal shear. Using this assumption, our elastic model shows that such a complex can passively change its orientation to align parallel to the direction of the flow.

Direct measurement of sub-Debye-length attraction between oppositely charged surfaces.

Using a surface force balance with fast video analysis, we have measured directly the attractive forces between oppositely charged solid surfaces (charge densities sigma(+), sigma(-)) across water over the entire range of interaction, in particular, at surface separations D below the Debye screening length lambda(S). At very low salt concentration we find a long-ranged attraction between the surfaces (onset ca. 100 nm), whose variation at D

The cellular response to curvature-induced stress.

We present a theoretical model to explain recent observations of the orientational response of cells to unidirectional curvature. Experiments show that some cell types when plated on a rigid cylindrical surface tend to reorient their shape and stress fibers along the axis of the cylinder, while others align their stress fibers perpendicular to that axis. Our model focuses on the competition of the shear stress--that results from cell adhesion and active contractility--and the anisotropic bending stiffness of the stress fibers. We predict the cell orientation angle that results from the balance of these two forces in a mechanical equilibrium. The conditions under which the different experimental observations can be obtained are discussed in terms of the theory.

Hybrid lipids as a biological surface-active component.

Cell membranes contain small domains (on the order of nanometers in size, sometimes called rafts) of lipids whose hydrocarbon chains are more ordered than those of the surrounding bulk-phase lipids. Whether these domains are fluctuations, metastable, or thermodynamically stable, is still unclear. Here, we show theoretically how a lipid with one saturated hydrocarbon chain that prefers the ordered environment and one partially unsaturated chain that prefers the less ordered phase, can act as a line-active component. We present a unified model that treats the lipids in both the bulk and at the interface and show how they lower the line tension between domains, eventually driving it to zero at sufficiently large interaction strengths or at sufficiently low temperatures. In this limit, finite-sized domains stabilized by the packing of these hybrid lipids can form as equilibrium structures.

Nonlinear dynamics of cell orientation.

The nonlinear dependence of cellular orientation on an external, time-varying stress field determines the distribution of orientations in the presence of noise and the characteristic time, tauc, for the cell to reach its steady-state orientation. The short, local cytoskeletal relaxation time distinguishes between high-frequency (nearly perpendicular) and low-frequency (random or parallel) orientations. However, tauc is determined by the much longer, orientational relaxation time. This behavior is related to experiments for which we predict the angle and characteristic time as a function of frequency.

Self assembly modulated by interactions of two heterogeneously charged surfaces.

Recent experiments have measured attractive interactions between two surfaces that each bear two molecular species with opposite charge. Such surfaces form charged domains of finite size. We present a theoretical model that predicts the dependence of the domain size, phase behavior and the interlayer forces as a function of spacing and salt concentration for two such interacting surfaces. A strong correlation between two length scales, the screening length and the surface separation, at the spinodal is shown. Remarkably, the first-order phase transition to infinite sized domains depends logarithmically on the ratio of the domain size to the molecular size. Finally, we fit the predicted pressure with experiments.

Scattering form factors for self-assembled network junctions.

The equilibrium microstructures in microemulsions and other self-assembled systems show complex, connected shapes such as symmetric bicontinuous spongelike structures and asymmetric bicontinuous networks formed by cylinders interconnected at junctions. In microemulsions, these cylinder network microstructures may mediate the structural transition from a spherical or globular phase to the bicontinuous microstructure. To understand the structural and statistical properties of such cylinder network microstructures as measured by scattering experiments, models are needed to extract the real-space structure from the scattering data. In this paper, we calculate the scattering functions appropriate for cylinder network microstructures. We focus on such networks that contain a high density of network junctions that connect the cylindrical elements. In this limit, the network microstructure can be regarded as an assembly of randomly oriented, closed packed network junctions (i.e., the cylinder scattering contributions are neglected). Accordingly, the scattering spectrum of the network microstructure can be calculated as the product of the junction number density, the junction form factor, which describes the scattering from the surface of a single junction, and a structure factor, which describes the local correlations of different junctions due to junction interactions (including their excluded volume). This approach is applied to analyze the scattering data from a bicontinuous microemulsion with equal volumes of water and oil. In a second approach, we included the cylinder scattering contribution in the junction form factor by calculating the scattering intensity of Y junctions to which three rods with spherical cross section are attached. The respective theoretical predictions are compared with results of neutron scattering measurements on a water-in-oil microemulsion with a connected microstructure.

Active self-polarization of contractile cells in asymmetrically shaped domains.

Mechanical forces generated by contractile cells allow the cells to sense their environment and to interact with other cells. By locally pulling on their environment, cells can sense and respond to mechanical features such as the local stress (or strain), the shape of a cellular domain, and the surrounding rigidity; at the same time, they also modify the mechanical state of the system. This creates a mechanical feedback loop that can result in self-polarization of cells. In this paper, we present a quantitative mechanical model that predicts the self-polarization of cells in spheroidally shaped domains, comprising contractile cells and an elastic matrix, that are embedded in a three-dimensional, cell-free gel. The theory is based on a generalization of the known results for passive inclusions in solids to include the effects of cell activity. We use the active cellular susceptibility tensor presented by Zemel [Phys. Rev. Lett. 97, 128103 (2006)] to calculate the polarization response and hence the elastic stress field developed by the cells in the cellular domain. The cell polarization is analyzed as a function of the shape and the elastic moduli of the cellular domain compared with the cell-free surrounding material. Consistent with experiment, our theory predicts the development of a stronger contractile force for cells in a gel that is surrounded by a large, cell-free material whose elastic modulus is stiffer than that of the gel that contains the cells. This provides a quantitative explanation of the differences in the development of cellular forces as observed in free and fixed gels. In the case of an asymmetrically shaped (spheroidal) domain of cells, we show that the anisotropic elastic field within the domain leads to a spontaneous self-polarization of the cells along the long axis of the domain.

Equilibrium domains on heterogeneously charged surfaces.

Recent experiments have shown that salt solutions containing surfaces with two oppositely charged species show stable, possibly equilibrium, structures with finite domain sizes. The short-range interactions between the two species would normally result in phase separation that is driven by the line tension with macroscopically large domains of each species. In this paper, we show that, when at least one of the charged species is mobile, finite domains can occur in equilibrium. The domain size is determined by a competition of the electrostatic free energy that promotes charge mixing and small domains, with the line tension that promotes macroscopic phase separation. We calculate the equilibrium patch size as a function of the surface charge and the concentration of dissolved monovalent salts in the bulk phase. An important finding is the prediction of a first-order transition from finite patches to macroscopic phase separation of the two charge species as the salt concentration is increased.

Fluctuations of coupled fluid and solid membranes with application to red blood cells.

The fluctuation spectra and the intermembrane interaction of two membranes at a fixed average distance are investigated. Each membrane can either be a fluid or a solid membrane, and in isolation, its fluctuations are described by a bare or a wave-vector-dependent bending modulus, respectively. The membranes interact via their excluded-volume interaction; the average distance is maintained by an external, homogeneous pressure. For strong coupling, the fluctuations can be described by a single, effective membrane that combines the elastic properties. For weak coupling, the fluctuations of the individual, noninteracting membranes are recovered. The case of a composite membrane consisting of one fluid and one solid membrane can serve as a microscopic model for the plasma membrane and cytoskeleton of the red blood cell. We find that, despite the complex microstructure of bilayers and cytoskeletons in a real cell, the fluctuations with wavelengths lambda greater, similar 400 nm are well described by the fluctuations of a single, polymerized membrane (provided that there are no inhomogeneities of the microstructure). The model is applied to the fluctuation data of discocytes ("normal" red blood cells), a stomatocyte, and an echinocyte. The elastic parameters of the membrane and an effective temperature that quantifies active, metabolically driven fluctuations are extracted from the experiments.

Active elasticity of gels with contractile cells.

Cells play an active role in the maintenance of mechanical homeostasis within tissues and their response to elastic forces is important for tissue engineering. We predict the collective response of an ensemble of contractile cells in a three-dimensional elastic medium to externally applied strain fields. Motivated by experiment, we model the cells as polarizable force dipoles that change their orientation in response to the local elastic strain. The analogy between the mechanical response of these systems and the dielectric response of polar molecules is used to calculate the elastic response function. We use this analogy to evaluate the average cell orientation, the mean polarization stress, and the effective elastic constants of the material, as a function of the cell concentration and matrix properties.

Junctions and end-caps in self-assembled non-ionic cylindrical micelles.

Cylindrical micelles are known to exhibit two types of morphologies: branched networks and linear, worm-like (or thread-like) micelles. These structures correspond to two types of topological defects: end-caps and junction points. Although either type of defect increases the micelle energy (when compared to the cylindrical sections), they are stabilized by an increase in the translational (end-caps) or configurational (junctions) entropy. End-caps reduce the length of the cylindrical micelles, resulting in a suspension of linear, worm-like micelles. Y-junction branch points cause the formation of a network structure that may percolate and coexist thermodynamically with a "sol" of finite cylinders with end-caps. In this paper, we review current experimental and theoretical studies of non-ionic cylindrical micelles in aqueous solutions. We focus on single and multicomponent amphiphiles, and consider both small molecules and macromolecules (polymers), in order to identify the driving forces that determine the type of topological 'defect' and the resulting system morphology.