Quantitative theory for the diffusive dynamics of liquid condensates.

Key processes of biological condensates are diffusion and material exchange with their environment. Experimentally, diffusive dynamics are typically probed via fluorescent labels. However, to date, a physics-based, quantitative framework for the dynamics of labeled condensate components is lacking. Here, we derive the corresponding dynamic equations, building on the physics of phase separation, and quantitatively validate the related framework via experiments. We show that by using our framework, we can precisely determine diffusion coefficients inside liquid condensates via a spatio-temporal analysis of fluorescence recovery after photobleaching (FRAP) experiments. We showcase the accuracy and precision of our approach by considering space- and time-resolved data of protein condensates and two different polyelectrolyte-coacervate systems. Interestingly, our theory can also be used to determine a relationship between the diffusion coefficient in the dilute phase and the partition coefficient, without relying on fluorescence measurements in the dilute phase. This enables us to investigate the effect of salt addition on partitioning and bypasses recently described quenching artifacts in the dense phase. Our approach opens new avenues for theoretically describing molecule dynamics in condensates, measuring concentrations based on the dynamics of fluorescence intensities, and quantifying rates of biochemical reactions in liquid condensates.

Phosphofructokinase relocalizes into subcellular compartments with liquid-like properties in vivo.

Although much is known about the biochemical regulation of glycolytic enzymes, less is understood about how they are organized inside cells. We systematically examine the dynamic subcellular localization of glycolytic protein phosphofructokinase-1/PFK-1.1 in Caenorhabditis elegans. We determine that endogenous PFK-1.1 localizes to subcellular compartments in vivo. In neurons, PFK-1.1 forms phase-separated condensates near synapses in response to energy stress from transient hypoxia. Restoring animals to normoxic conditions results in cytosolic dispersion of PFK-1.1. PFK-1.1 condensates exhibit liquid-like properties, including spheroid shapes due to surface tension, fluidity due to deformations, and fast internal molecular rearrangements. Heterologous self-association domain cryptochrome 2 promotes formation of PFK-1.1 condensates and recruitment of aldolase/ALDO-1. PFK-1.1 condensates do not correspond to stress granules and might represent novel metabolic subcompartments. Our studies indicate that glycolytic protein PFK-1.1 can dynamically form condensates in vivo.

Erratum: Salt-Dependent Rheology and Surface Tension of Protein Condensates Using Optical Traps [Phys. Rev. Lett. 121, 258101 (2018)].

This corrects the article DOI: 10.1103/PhysRevLett.121.258101.

Salt-Dependent Rheology and Surface Tension of Protein Condensates Using Optical Traps.

An increasing number of proteins with intrinsically disordered domains have been shown to phase separate in buffer to form liquidlike phases. These protein condensates serve as simple models for the investigation of the more complex membraneless organelles in cells. To understand the function of such proteins in cells, the material properties of the condensates they form are important. However, these material properties are not well understood. Here, we develop a novel method based on optical traps to study the frequency-dependent rheology and the surface tension of P-granule protein PGL-3 condensates as a function of salt concentration. We find that PGL-3 droplets are predominantly viscous but also exhibit elastic properties. As the salt concentration is reduced, their elastic modulus, viscosity, and surface tension increase. Our findings show that salt concentration has a strong influence on the rheology and dynamics of protein condensates suggesting an important role of electrostatic interactions for their material properties.

Physical limits to biomechanical sensing in disordered fibre networks.

Cells actively probe and respond to the stiffness of their surroundings. Since mechanosensory cells in connective tissue are surrounded by a disordered network of biopolymers, their in vivo mechanical environment can be extremely heterogeneous. Here we investigate how this heterogeneity impacts mechanosensing by modelling the cell as an idealized local stiffness sensor inside a disordered fibre network. For all types of networks we study, including experimentally-imaged collagen and fibrin architectures, we find that measurements applied at different points yield a strikingly broad range of local stiffnesses, spanning roughly two decades. We verify via simulations and scaling arguments that this broad range of local stiffnesses is a generic property of disordered fibre networks. Finally, we show that to obtain optimal, reliable estimates of global tissue stiffness, a cell must adjust its size, shape, and position to integrate multiple stiffness measurements over extended regions of space.

Stress controls the mechanics of collagen networks.

Collagen is the main structural and load-bearing element of various connective tissues, where it forms the extracellular matrix that supports cells. It has long been known that collagenous tissues exhibit a highly nonlinear stress-strain relationship, although the origins of this nonlinearity remain unknown. Here, we show that the nonlinear stiffening of reconstituted type I collagen networks is controlled by the applied stress and that the network stiffness becomes surprisingly insensitive to network concentration. We demonstrate how a simple model for networks of elastic fibers can quantitatively account for the mechanics of reconstituted collagen networks. Our model points to the important role of normal stresses in determining the nonlinear shear elastic response, which can explain the approximate exponential relationship between stress and strain reported for collagenous tissues. This further suggests principles for the design of synthetic fiber networks with collagen-like properties, as well as a mechanism for the control of the mechanics of such networks.

Strain history dependence of the nonlinear stress response of fibrin and collagen networks.

We show that the nonlinear mechanical response of networks formed from un-cross-linked fibrin or collagen type I continually changes in response to repeated large-strain loading. We demonstrate that this dynamic evolution of the mechanical response arises from a shift of a characteristic nonlinear stress-strain relationship to higher strains. Therefore, the imposed loading does not weaken the underlying matrices but instead delays the occurrence of the strain stiffening. Using confocal microscopy, we present direct evidence that this behavior results from persistent lengthening of individual fibers caused by an interplay between fiber stretching and fiber buckling when the networks are repeatedly strained. Moreover, we show that covalent cross-linking of fibrin or collagen inhibits the shift of the nonlinear material response, suggesting that the molecular origin of individual fiber lengthening may be slip of monomers within the fibers. Thus, a fibrous architecture in combination with constituents that exhibit internal plasticity creates a material whose mechanical response adapts to external loading conditions. This design principle may be useful to engineer novel materials with this capability.

A blind spot in confocal reflection microscopy: the dependence of fiber brightness on fiber orientation in imaging biopolymer networks.

We investigate the dependence of fiber brightness on three-dimensional fiber orientation when imaging biopolymer networks with confocal reflection microscopy (CRM) and confocal fluorescence microscopy (CFM). We compare image data of fluorescently labeled type I collagen networks concurrently acquired using each imaging modality. For CRM, fiber brightness decreases for more vertically oriented fibers, leaving fibers above approximately 50 degrees from the imaging plane entirely undetected. As a result, the three-dimensional network structure appears aligned with the imaging plane. In contrast, CFM data exhibit little variation of fiber brightness with fiber angle, thus revealing an isotropic collagen network. Consequently, we find that CFM detects almost twice as many fibers as are visible with CRM, thereby yielding more complete structural information for three-dimensional fiber networks. We offer a simple explanation that predicts the detected fiber brightness as a function of fiber orientation in CRM.

An algorithm for extracting the network geometry of three-dimensional collagen gels.

The geometric structure of a biopolymer network impacts its mechanical and biological properties. In this paper, we develop an algorithm for extracting the network architecture of three-dimensional (3d) fluorescently labeled collagen gels, building on the initial work of Wu et al., (2003). Using artificially generated images, the network extraction algorithm is then validated for its ability to reconstruct the correct bulk properties of the network, including fiber length, persistence length, cross-link density, and shear modulus.

Robust pore size analysis of filamentous networks from three-dimensional confocal microscopy.

We describe a robust method for determining morphological properties of filamentous biopolymer networks, such as collagen or other connective tissue matrices, from confocal microscopy image stacks. Morphological properties including pore size distributions and percolation thresholds are important for transport processes, e.g., particle diffusion or cell migration through the extracellular matrix. The method is applied to fluorescently labeled fiber networks prepared from rat-tail tendon and calf-skin collagen, at concentrations of 1.2, 1.6, and 2.4 mg/ml. The collagen fibers form an entangled and branched network. The medial axes, or skeletons, representing the collagen fibers are extracted from the image stack by threshold intensity segmentation and distance-ordered homotopic thinning. The size of the fluid pores as defined by the radii of largest spheres that fit into the cavities between the collagen fibers is derived from Euclidean distance maps and maximal covering radius transforms of the fluid phase. The size of the largest sphere that can traverse the fluid phase between the collagen fibers across the entire probe, called the percolation threshold, was computed for both horizontal and vertical directions. We demonstrate that by representing the fibers as the medial axis the derived morphological network properties are both robust against changes of the value of the segmentation threshold intensity and robust to problems associated with the point-spread function of the imaging system. We also provide empirical support for a recent claim that the percolation threshold of a fiber network is close to the fiber diameter for which the Euler index of the networks becomes zero.