Probing short-range protein Brownian motion in the cytoplasm of living cells

Designer peptide-DNA cytoskeletons regulate the function of synthetic cells

The bottom-up engineering of artificial cells requires a reconfigurable cytoskeleton that can organize at distinct locations and dynamically modulate its structural and mechanical properties. Here, inspired by the vast array of actin-binding proteins and their ability to reversibly crosslink or bundle filaments, we have designed a library of peptide-DNA crosslinkers varying in length, valency and geometry. Peptide filaments conjoint through DNA hybridization give rise to tactoid-shaped bundles with tunable aspect ratios and mechanics. When confined in cell-sized water-in-oil droplets, the DNA crosslinker design guides the localization of cytoskeletal structures at the cortex or within the lumen of the synthetic cells. The tunable spatial arrangement regulates the passive diffusion of payloads within the droplets and complementary DNA handles allow for the reversible recruitment and release of payloads on and off the cytoskeleton. Heat-induced reconfiguration of peptide-DNA architectures triggers shape deformations of droplets, regulated by DNA melting temperatures. Altogether, the modular design of peptide-DNA architectures is a powerful strategy towards the bottom-up assembly of synthetic cells.

Segmented fluorescence correlation spectroscopy (FCS) on a commercial laser scanning microscope

Performing accurate Fluorescence Correlation Spectroscopy (FCS) measurements in cells can be challenging due to cellular motion or other intracellular processes. In this respect, it has recently been shown that analysis of FCS data in short temporal segments (segmented FCS) can be very useful to increase the accuracy of FCS measurements inside cells. Here, we demonstrate that segmented FCS can be performed on a commercial laser scanning microscope (LSM), even in the absence of the dedicated FCS module. We show how data can be acquired on a Leica SP8 confocal microscope and then exported and processed with a custom software in MATLAB. The software performs segmentation of the data to extract an average ACF and measure the diffusion coefficient in specific subcellular regions. First of all, we measure the diffusion of fluorophores of different size in solution, to show that good-quality ACFs can be obtained in a commercial LSM. Next, we validate the method by measuring the diffusion coefficient of GFP in the nucleus of HeLa cells, exploiting variations of the intensity to distinguish between nucleoplasm and nucleolus. As expected, the measured diffusion coefficient of GFP is slower in the nucleolus relative to nucleoplasm. Finally, we apply the method to HeLa cells expressing a PARP1 chromobody to measure the diffusion coefficient of PARP1 in different subcellular regions. We find that PARP1 diffusion is slower in the nucleolus compared to the nucleoplasm.

Cytoplasmic fluidization contributes to breaking spore dormancy in fission yeast

The cytoplasm is a complex, crowded environment that influences myriad cellular processes including protein folding and metabolic reactions. Recent studies have suggested that changes in the biophysical properties of the cytoplasm play a key role in cellular homeostasis and adaptation. However, it still remains unclear how cells control their cytoplasmic properties in response to environmental cues. Here, we used fission yeast spores as a model system of dormant cells to elucidate the mechanisms underlying regulation of the cytoplasmic properties. By tracking fluorescent tracer particles, we found that particle mobility decreased in spores compared to vegetative cells and rapidly increased at the onset of dormancy breaking upon glucose addition. This cytoplasmic fluidization depended on glucose-sensing via the cyclic adenosine monophosphate-protein kinase A pathway. PKA activation led to trehalose degradation through trehalase Ntp1, thereby increasing particle mobility as the amount of trehalose decreased. In contrast, the rapid cytoplasmic fluidization did not require de novo protein synthesis, cytoskeletal dynamics, or cell volume increase. Furthermore, the measurement of diffusion coefficients with tracer particles of different sizes suggests that the spore cytoplasm impedes the movement of larger protein complexes (40 to 150 nm) such as ribosomes, while allowing free diffusion of smaller molecules (~3 nm) such as second messengers and signaling proteins. Our experiments have thus uncovered a series of signaling events that enable cells to quickly fluidize the cytoplasm at the onset of dormancy breaking.

Constrained dynamics of DNA oligonucleotides in phase-separated droplets

Understanding the dynamics of biomolecules in complex environments is crucial for elucidating the effect of condensed and heterogeneous environments on their functional properties. A relevant environment-and one that can also be mimicked easily in vitro-is that of phase-separated droplets. While phase-separated droplet systems have been shown to compartmentalize a wide range of functional biomolecules, the effects of internal structuration of droplets on the dynamics and mobility of internalized molecules remain poorly understood. Here, we use fluorescence correlation spectroscopy to measure the dynamics of short oligonucleotides encapsulated within two representative kinds of uncharged and charged phase-separated droplets. We find that the internal structuration controls the oligonucleotide dynamics in these droplets, revealed by measuring physical parameters at high spatiotemporal resolution. By varying oligonucleotide length and salt concentrations (and thereby charge screening), we found that the dynamics are significantly affected in the noncharged droplets compared to the charged system. Our work lays the foundation for unraveling and quantifying the physical parameters governing biomolecular transport in the condensed environment.

Ergodic criterion of a random diffusivity model

The random diffusivity, initially proposed to explain Brownian yet non-Gaussian diffusion, has garnered significant attention due to its capacity not only for elucidating the internal physical mechanism of non-Gaussian diffusion, but also for establishing an analytical framework to characterize particle motion in complex environments. In this paper, based on the correlation function C(t_{1},t_{2})=〈D(t_{1})D(t_{2})〉 of random diffusivity D(t), we quantitatively propose a general criterion of determining the ergodic property of the Langevin equation with the arbitrary random diffusivity D(t). Due to the critical role of correlation function C(t_{1},t_{2}), we derive the criterion for the two cases with stationary diffusivity or nonstationary diffusivity, respectively. By utilizing the quantitative criterion, we can directly judge the ergodic properties of the random diffusivity model based on the correlation function C(t_{1},t_{2}) of random diffusivity D(t). Several typical diffusivities, including the common square of the Brownian motion and of the (fractional) Ornstein-Uhlenbeck process, are found to contribute to different ergodic properties, which validates our proposed criterion built on the correlation function C(t_{1},t_{2}).

Adsorption of active polymers on attractive nanoparticles

The adsorption of active polymers on an attractive nanoparticle (NP) is studied using Langevin dynamics simulations. The active polymers consist of an active Brownian particle (ABP) at the head and a subsequent passive polymer chain. The ABP experiences an active force of magnitude Fa. The interactions between the active polymer and NP are modeled as Lennard-Jones potential with a strength εpn. We find the critical adsorption point εpn* increases with increasing the active force Fa. The increment of εpn*, denoted as Δεpn*, due to Fa can be expressed approximately as Δεpn* ∝ Fa2.5 for the restricted rotating active polymer (RRAP) where the rotation of the head ABP is restricted and Δεpn* ∝ Fa1.7 for the freely rotating active polymer (FRAP) where the ABP rotates freely. Meanwhile, the conformation of the adsorbed polymer, such as adsorbed trains on NP and the tail near the ABP, are also dependent on Fa. When the tail near the ABP is short, the adsorption is significantly affected by the active force. However, when the tail is long, the whole polymer can be viewed as a long tail stretched by the active force and unperturbed adsorption monomers. Simulation results show that the active force has a direct and significant effect on εpn* and the structure of the adsorbed active polymers.

What determines sub-diffusive behavior in crowded protein solutions?

The aqueous environment inside cells is densely packed. A typical cell has a macromolecular concentration in the range 90-450 g/L, with 5%-40% of its volume being occupied by macromolecules, resulting in what is known as macromolecular crowding. The space available for the free diffusion of metabolites and other macromolecules is thus greatly reduced, leading to so-called excluded volume effects. The slow diffusion of macromolecules under crowded conditions has been explained using transient complex formation. However, sub-diffusion noted in earlier works is not well characterized, particularly the role played by transient complex formation and excluded volume effects. We have used Brownian dynamics simulations to characterize the diffusion of chymotrypsin inhibitor 2 in protein solutions of bovine serum albumin and lysozyme at concentrations ranging from 50 to 300 g/L. The predicted changes in diffusion coefficient as a function of crowder concentration are consistent with NMR experiments. The sub-diffusive behavior observed in the sub-microsecond timescale can be explained in terms of a so-called cage effect, arising from rattling motion in a local molecular cage as a consequence of excluded volume effects. By selectively manipulating the nature of interactions between protein molecules, we determined that excluded volume effects induce sub-diffusive dynamics at sub-microsecond timescales. These findings may help to explain the diffusion-mediated effects of protein crowding on cellular processes.

Multiple diffusive behaviors of the random walk in inhomogeneous environments

Anomalous diffusive behaviors are observed in highly inhomogeneous but relatively stable environments such as intracellular media and are increasingly attracting attention. In this paper we develop a coupled continuous-time random walk model in which the waiting time is power-law coupled with the local environmental diffusion coefficient. We provide two forms of the waiting time density, namely, a heavy-tailed density and an exponential density. For different waiting time densities, anomalous diffusions with the diffusion exponent between 0 and 2 and Brownian yet non-Gaussian diffusion can be realized within the present model. The diffusive behaviors are analyzed and discussed by deriving the mean-squared displacement and probability density function. In addition we derive the effective jump length density corresponding to the decoupled form to help distinguish the diffusion types. Our model unifies two kinds of anomalous diffusive behavior with different characteristics in the same inhomogeneous environment into a theoretical framework. The model interprets the random motion of particles in a complex inhomogeneous environment and reproduces the experimental results of different biological and physical systems.

Multilayered noise model for transport in complex environments

Transport in complex fluidic environments often exhibits transient subdiffusive dynamics accompanied by non-Gaussian probability density profiles featuring a nonmonotonic non-Gaussian parameter. Such properties cannot be adequately explained by the original theory of Brownian motion. Based on an extension of kinetic theory, this study introduces a chain of hierarchically coupled random walks approach that effectively captures all these intriguing characteristics. If the environment consists of a series of independent white noise sources, then the problem can be expressed as a system of hierarchically coupled Ornstein-Uhlenbech equations. Due to the linearity of the system, the most essential transport properties have a closed analytical form.

The reproduction of gram-negative protoplasts and the influence of environmental conditions on this process

Bacterial protoplasts are known to reproduce independently of canonical molecular biological processes. Although their reproduction is thought to be influenced by environmental conditions, the growth of protoplasts in their natural habitat has never been empirically studied. Here, we studied the life cycle of protoplasts in their native environment. Contrary to the previous perception that protoplasts reproduce in an erratic manner, cells in our study reproduced in a defined sequence of steps, always leading to viable daughter cells. Their reproduction can be explained by an interplay between intracellular metabolism, the physicochemical properties of cell constituents, and the nature of cations in the growth media. The efficiency of reproduction is determined by the environmental conditions. Under favorable environmental conditions, protoplasts reproduce with nearly similar efficiency to cells that possess a cell wall. In short, here we demonstrate the simplest method of cellular reproduction and the influence of environmental conditions on this process.

Simulation study on the effect of polydisperse nanoparticles on polymer diffusion in crowded environments

The diffusion of polymer chains in a crowded environment with large and small immobile, attractive nanoparticles (NPs) is studied using Langevin dynamics simulations. For orderly distributed NPs on the simple cubic lattice, our results show that the diffusion of polymer chains is dependent on the NP-NP distance or lattice distance d. At low d where NPs are placed closely, subdiffusion occurs at a sufficiently high polydispersity of NPs, PD. Both the apparent diffusion coefficient and subdiffusion exponent of polymer chains decrease with increasing PD, attributed to the adsorption of polymers on NP clusters formed by larger NPs. At large d, normal diffusion is always observed, and the diffusion coefficient increases with increasing PD. The reason is that, at high PD, the difference between single large NP adsorption and double large NP adsorption is reduced, which increases the exchange of a polymer between the two adsorption states. Finally, the impact of size polydispersity of NPs on the diffusion of polymer chains in a crowded environment with randomly distributed NPs is also investigated. The results show that the position disorder of NPs enhances the subdiffusion of the system.

Comparison of MSD analysis from single particle tracking with MSD from images. Getting the best of both worlds

Fluorescence microscopy can provide valuable information about cell interior dynamics. Particularly, mean squared displacement (MSD) analysis is widely used to characterize proteins and sub-cellular structures' mobility providing the laws of molecular diffusion. The MSD curve is traditionally extracted from individual trajectories recorded by single-particle tracking-based techniques. More recently, image correlation methods like iMSD have been shown capable of providing averaged dynamic information directly from images, without the need for isolation and localization of individual particles. iMSD is a powerful technique that has been successfully applied to many different biological problems, over a wide spatial and temporal scales. The aim of this work is to review and compare these two well-established methodologies and their performance in different situations, to give an insight on how to make the most out of their unique characteristics. We show the analysis of the same datasets by the two methods. Regardless of the experimental differences in the input data for MSD or iMSD analysis, our results show that the two approaches can address equivalent questions for free diffusing systems. We focused on studying a range of diffusion coefficients between D = 0.001μm2s-1and D = 0.1μm2s-1, where we verified that the equivalence is maintained even for the case of isolated particles. This opens new opportunities for studying intracellular dynamics using equipment commonly available in any biophysical laboratory.

Simulation Study on the Mechanism of Intermediate Subdiffusion of Diffusive Particles in Crowded Systems

The intermediate subdiffusion of diffusive particles in crowded systems is studied for two model systems: the continuous time random walk (CTRW) model and the obstruction-binding model. For the CTRW model with an arbitrarily given longest waiting time τmax, we find that the diffusive particle exhibits subdiffusion below τmax and recovers normal diffusion above τmax. For the obstruction-binding model with randomly distributed attractive obstacles, the diffusion of the diffusive particle is dependent on the binding energy and the density of obstacles. Interestingly, diffusion curves for different binding strengths can be overlapped by rescaling the simulation time, indicating that the diffusive particle in the obstruction-binding model can change from the intermediate subdiffusion to the normal diffusion at a long-term simulation scale. The results of the two model systems show that the diffusive particles only exhibit intermediate subdiffusion below the longest waiting time. Therefore, long timescale subdiffusion would only be observed in the CTRW model with an infinitely long waiting time and in the obstruction-binding model with an infinitely large binding strength.

Measuring protein-membrane interaction through radial fluorescence correlation in 2 dimensions

The cell membrane has a fundamental role in the cell life cycle but there's still much to be learned about its heterogeneous structure, regulation, and protein interaction. Additionally, the protein-membrane interaction is often overlooked when studying specific protein dynamics. In this work, we present a new tool for a better understanding of protein dynamics and membrane function using live cells and fast non-invasive techniques without the need for individual particle tracking. To this end, we used the 2D-pair correlation function (2D-pCF) to study protein interactions across cellular membranes. We performed numerical simulations and confocal experiments using a GAP-mEGFP fusion construct known to interact with the plasmatic membrane. Our results demonstrate that based on a quantitative correlation analysis as the 2D pair correlation of the signal intensities, is possible to characterize protein-membrane interactions in live systems and real-time. Combining experimental and numerical results this work presents a new powerful approach to the study of the dynamic protein-membrane interaction.

The Material Properties of the Cell Nucleus: A Matter of Scale

Chromatin regulatory processes physically take place in the environment of the cell nucleus, which is filled with the chromosomes and a plethora of smaller biomolecules. The nucleus contains macromolecular assemblies of different sizes, from nanometer-sized protein complexes to micrometer-sized biomolecular condensates, chromosome territories, and nuclear bodies. This multiscale organization impacts the transport processes within the nuclear interior, the global mechanical properties of the nucleus, and the way the nucleus senses and reacts to mechanical stimuli. Here, we discuss recent work on these aspects, including microrheology and micromanipulation experiments assessing the material properties of the nucleus and its subcomponents. We summarize how the properties of multiscale media depend on the time and length scales probed in the experiment, and we reconcile seemingly contradictory observations made on different scales. We also revisit the concept of liquid-like and solid-like material properties for complex media such as the nucleus. We propose that the nucleus can be considered a multiscale viscoelastic medium composed of three major components with distinct properties: the lamina, the chromatin network, and the nucleoplasmic fluid. This multicomponent organization enables the nucleus to serve its different functions as a reaction medium on the nanoscale and as a mechanosensor and structural scaffold on the microscale.

Mathematical model of the cell signaling pathway based on the extended Boolean network model with a stochastic process

BACKGROUND

In cell signaling pathways, proteins interact with each other to determine cell fate in response to either cell-extrinsic (micro-environmental) or intrinsic cues. One of the well-studied pathways, the mitogen-activated protein kinase (MAPK) signaling pathway, regulates cell processes such as differentiation, proliferation, apoptosis, and survival in response to various micro-environmental stimuli in eukaryotes. Upon micro-environmental stimulus, receptors on the cell membrane become activated. Activated receptors initiate a cascade of protein activation in the MAPK pathway. This activation involves protein binding, creating scaffold proteins, which are known to facilitate effective MAPK signaling transduction.

RESULTS

This paper presents a novel mathematical model of a cell signaling pathway coordinated by protein scaffolding. The model is based on the extended Boolean network approach with stochastic processes. Protein production or decay in a cell was modeled considering the stochastic process, whereas the protein-protein interactions were modeled based on the extended Boolean network approach. Our model fills a gap in the binary set applied to previous models. The model simultaneously considers the stochastic process directly. Using the model, we simulated a simplified mitogen-activated protein kinase (MAPK) signaling pathway upon stimulation of both a single receptor at the initial time and multiple receptors at several time points. Our simulations showed that the signal is amplified as it travels down to the pathway from the receptor, generating substantially amplified downstream ERK activity. The noise generated by the stochastic process of protein self-activity in the model was also amplified as the signaling propagated through the pathway.

CONCLUSIONS

The signaling transduction in a simplified MAPK signaling pathway could be explained by a mathematical model based on the extended Boolean network model with a stochastic process. The model simulations demonstrated signaling amplifications when it travels downstream, which was already observed in experimental settings. We also highlight the importance of stochastic activity in regulating protein inactivation.

Protein Condensation, Cellular Organization, and Spatiotemporal Regulation of Cytoplasmic Properties

The cytoplasm is an aqueous, highly crowded solution of active macromolecules. Its properties influence the behavior of proteins, including their folding, motion, and interactions. In particular, proteins in the cytoplasm can interact to form phase-separated assemblies, so-called biomolecular condensates. The interplay between cytoplasmic properties and protein condensation is critical in a number of functional contexts and is the subject of this review. The authors first describe how cytoplasmic properties can affect protein behavior, in particular condensate formation, and then describe the functional implications of this interplay in three cellular contexts, which exemplify how protein self-organization can be adapted to support certain physiological phenotypes. The authors then describe the formation of RNA-protein condensates in highly polarized cells such as neurons, where condensates play a critical role in the regulation of local protein synthesis, and describe how different stressors trigger extensive reorganization of the cytoplasm, both through signaling pathways and through direct stress-induced changes in cytoplasmic properties. Finally, the authors describe changes in protein behavior and cytoplasmic properties that may occur in extremophiles, in particular organisms that have adapted to inhabit environments of extreme temperature, and discuss the implications and functional importance of these changes.

Cell-Size Space Regulates the Behavior of Confined Polymers: From Nano- and Micromaterials Science to Biology

Polymer micromaterials in a liquid or gel phase covered with a surfactant membrane are widely used materials in pharmaceuticals, cosmetics, and foods. In particular, cell-sized micromaterials of biopolymer solutions covered with a lipid membrane have been studied as artificial cells to understand cells from a physicochemical perspective. The characteristics and phase transitions of polymers confined to a microscopic space often differ from those in bulk systems. The effect that causes this difference is referred to as the cell-size space effect (CSE), but the specific physicochemical factors remain unclear. This study introduces the analysis of CSE on molecular diffusion, nanostructure transition, and phase separation and presents their main factors, i.e., short- and long-range interactions with the membrane surface and small volume (finite element nature). This serves as a guide for determining the dominant factors of CSE. Furthermore, we also introduce other factors of CSE such as spatial closure and the relationships among space size, the characteristic length of periodicity, the structure size, and many others produced by biomolecular assemblies through the analysis of protein reaction-diffusion systems and biochemical reactions.

Cytoplasmic organization promotes protein diffusion in Xenopus extracts

The cytoplasm is highly organized. However, the extent to which this organization influences the dynamics of cytoplasmic proteins is not well understood. Here, we use Xenopus laevis egg extracts as a model system to study diffusion dynamics in organized versus disorganized cytoplasm. Such extracts are initially homogenized and disorganized, and self-organize into cell-like units over the course of tens of minutes. Using fluorescence correlation spectroscopy, we observe that as the cytoplasm organizes, protein diffusion speeds up by about a factor of two over a length scale of a few hundred nanometers, eventually approaching the diffusion time measured in organelle-depleted cytosol. Even though the ordered cytoplasm contained organelles and cytoskeletal elements that might interfere with diffusion, the convergence of protein diffusion in the cytoplasm toward that in organelle-depleted cytosol suggests that subcellular organization maximizes protein diffusivity. The effect of organization on diffusion varies with molecular size, with the effects being largest for protein-sized molecules, and with the time scale of the measurement. These results show that cytoplasmic organization promotes the efficient diffusion of protein molecules in a densely packed environment.

Cytoplasmic organization is a hallmark of living cells. Here, the authors make use of self-organizing cell extracts to examine how the emergence of large-scale organizations influences the microscopic diffusion of protein molecules in the cytoplasm.

Rheology and Viscoelasticity of Proteins and Nucleic Acids Condensates

Phase separation is as familiar as watching vinegar separating from oil in vinaigrette. The observation that phase separation of proteins and nucleic acids is widespread in living cells has opened an entire field of research into the biological significance and the biophysical mechanisms of phase separation and protein condensation in biology. Recent evidence indicates that certain proteins and nucleic acids condensates are not simple liquids and instead display both viscous and elastic behaviors, which in turn may have biological significance. The aim of this Perspective is to review the state-of-the-art of this quickly emerging field focusing on the material and rheological properties of protein condensates. Finally, we discuss the different techniques that can be employed to quantify the viscoelasticity of condensates and highlight potential future directions and opportunities for interdisciplinary cross-talk between chemists, physicists, and biologists.

Measuring Molecular Diffusion in Dynamic Subcellular Nanostructures by Fast Raster Image Correlation Spectroscopy and 3D Orbital Tracking

Here we provide demonstration that fast fluorescence fluctuation spectroscopy is a fast and robust approach to extract information on the dynamics of molecules enclosed within subcellular nanostructures (e.g., organelles or vesicles) which are also moving in the complex cellular environment. In more detail, Raster Image Correlation Spectroscopy (RICS) performed at fast timescales (i.e., microseconds) reveals the fast motion of fluorescently labeled molecules within two exemplary dynamic subcellular nanostructures of biomedical interest, the lysosome and the insulin secretory granule (ISG). The measurement of molecular diffusion is then used to extract information on the average properties of subcellular nanostructures, such as macromolecular crowding or molecular aggregation. Concerning the lysosome, fast RICS on a fluorescent tracer allowed us to quantitatively assess the increase in organelle viscosity in the pathological condition of Krabbe disease. In the case of ISGs, fast RICS on two ISG-specific secreting peptides unveiled their differential aggregation propensity depending on intragranular concentration. Finally, a combination of fast RICS and feedback-based 3D orbital tracking was used to subtract the slow movement of subcellular nanostructures from the fast diffusion of molecules contained within them and independently validate the results. Results presented here not only demonstrate the acquired ability to address the dynamic behavior of molecules in moving, nanoscopic reference systems, but prove the relevance of this approach to advance our knowledge on cell function at the subcellular scale.

Perspective: a stirring role for metabolism in cells

Based on recent findings indicating that metabolism might be governed by a limit on the rate at which cells can dissipate Gibbs energy, in this Perspective, we propose a new mechanism of how metabolic activity could globally regulate biomolecular processes in a cell. Specifically, we postulate that Gibbs energy released in metabolic reactions is used to perform work, allowing enzymes to self-propel or to break free from supramolecular structures. This catalysis-induced enzyme movement will result in increased intracellular motion, which in turn can compromise biomolecular functions. Once the increased intracellular motion has a detrimental effect on regulatory mechanisms, this will establish a feedback mechanism on metabolic activity, and result in the observed thermodynamic limit. While this proposed explanation for the identified upper rate limit on cellular Gibbs energy dissipation rate awaits experimental validation, it offers an intriguing perspective of how metabolic activity can globally affect biomolecular functions and will hopefully spark new research.

A simulation study on the subdiffusion of polymer chains in crowded environments containing nanoparticles

Polymer chains in crowded environments often show subdiffusive behavior. We adopt molecular dynamics simulations to study the conditions for the subdiffusion of polymer chains in crowded environments containing randomly distributed, immobile, attractive nanoparticles (NPs). The attraction is strong enough to adsorb polymer chains on NPs. The results show that subdiffusion occurs at a low concentration of polymer chains (c_p). A transition from subdiffusion to normal diffusion is observed when c_p exceeds the transition concentration , which increases with increasing concentration of NPs while decreases with increasing size of NPs. The high concentration and small size of NPs exert a big effect on the subdiffusion of polymer chains. The subdiffusive behavior of polymer chains can be attributed to the strong adsorption of polymer chains on the attractive NPs. For the subdiffusion case, polymer chains are adsorbed strongly on multiple NPs, and they diffuse via the NP-exchange diffusion mechanism. However for the normal diffusion case, polymer chains are either free or weakly adsorbed on one or a few NPs, and they diffuse mainly via the adsorption-and-desorption diffusion mechanism.

Dependence of diffusion in Escherichia coli cytoplasm on protein size, environmental conditions, and cell growth

Inside prokaryotic cells, passive translational diffusion typically limits the rates with which cytoplasmic proteins can reach their locations. Diffusion is thus fundamental to most cellular processes, but the understanding of protein mobility in the highly crowded and non-homogeneous environment of a bacterial cell is still limited. Here, we investigated the mobility of a large set of proteins in the cytoplasm of Escherichia coli, by employing fluorescence correlation spectroscopy (FCS) combined with simulations and theoretical modeling. We conclude that cytoplasmic protein mobility could be well described by Brownian diffusion in the confined geometry of the bacterial cell and at the high viscosity imposed by macromolecular crowding. We observed similar size dependence of protein diffusion for the majority of tested proteins, whether native or foreign to E. coli. For the faster-diffusing proteins, this size dependence is well consistent with the Stokes-Einstein relation once taking into account the specific dumbbell shape of protein fusions. Pronounced subdiffusion and hindered mobility are only observed for proteins with extensive interactions within the cytoplasm. Finally, while protein diffusion becomes markedly faster in actively growing cells, at high temperature, or upon treatment with rifampicin, and slower at high osmolarity, all of these perturbations affect proteins of different sizes in the same proportions, which could thus be described as changes of a well-defined cytoplasmic viscosity.

β-Cell Pathophysiology: A Review of Advanced Optical Microscopy Applications

β-cells convert glucose (input) resulting in the controlled release of insulin (output), which in turn has the role to maintain glucose homeostasis. β-cell function is regulated by a complex interplay between the metabolic processing of the input, its transformation into second-messenger signals, and final mobilization of insulin-containing granules towards secretion of the output. Failure at any level in this process marks β-cell dysfunction in diabetes, thus making β-cells obvious potential targets for therapeutic purposes. Addressing quantitatively β-cell (dys)function at the molecular level in living samples requires probing simultaneously the spatial and temporal dimensions at the proper resolution. To this aim, an increasing amount of research efforts are exploiting the potentiality of biophysical techniques. In particular, using excitation light in the visible/infrared range, a number of optical-microscopy-based approaches have been tailored to the study of β-cell-(dys)function at the molecular level, either in label-free mode (i.e., exploiting intrinsic autofluorescence of cells) or by the use of organic/genetically-encoded fluorescent probes. Here, relevant examples from the literature are reviewed and discussed. Based on this, new potential lines of development in the field are drawn.

High rates of calcium-free diffusion in the cytosol of living cells

Calcium (Ca²⁺) is a universal second messenger that participates in the regulation of innumerous physiological processes. The way in which local elevations of the cytosolic Ca²⁺ concentration spread in space and time is key for the versatility of the signals. Ca²⁺ diffusion in the cytosol is hindered by its interaction with proteins that act as buffers. Depending on the concentrations and the kinetics of the interactions, there is a large range of values at which Ca²⁺ diffusion can proceed. Having reliable estimates of this range, particularly of its highest end, which corresponds to the ions free diffusion, is key to understand how the signals propagate. In this work, we present the first experimental results with which the Ca²⁺-free diffusion coefficient is directly quantified in the cytosol of living cells. By means of fluorescence correlation spectroscopy experiments performed in Xenopus laevis oocytes and in cells of Saccharomyces cerevisiae, we show that the ions can freely diffuse in the cytosol at a higher rate than previously thought.

Effects of Macromolecular Crowding on Nanoparticle Diffusion: New Insights from Mössbauer Spectroscopy

In this work, we used Mössbauer spectroscopy as a new approach for experimental quantification of the self-diffusion coefficient (D_Mössbauer) and hydrodynamic (HD) size of iron-containing nanoparticles (NPs) in complex crowded solutions, mimicking cell cytoplasm. As a probe, we used 9 nm cobalt ferrite NPs (CFNs) dispersed in solutions of bovine serum albumin (BSA) with a volume fraction (φ_BSA) of 0-0.2. Our results show that the broadening of Mössbauer spectra is highly sensitive to the diffusion of CFNs, while when φ_BSA = 0.2, the CFN-normalized diffusivity is reduced by 86% compared to that of a protein-free solution. CFN colloids were also studied by dynamic light scattering (DLS). Comparison of the experimental data shows that DLS significantly underestimates the diffusion coefficient of CFNs and, consequently, overestimates the HD size of CFNs at φ_BSA > 0, which cannot be attributed to the formation of the BSA monolayer on the surface of CFNs.

Single-Molecule Study of Thermomyces lanuginosus Lipase in a Detergency Application System Reveals Diffusion Pattern Remodeling by Surfactants and Calcium

Lipases comprise one of the major enzyme classes in biotechnology with applications within, e.g., baking, brewing, biocatalysis, and the detergent industry. Understanding the mechanisms of lipase function and regulation is therefore important to facilitate the optimization of their function by protein engineering. Advances in single-molecule studies in model systems have provided deep mechanistic insights on lipase function, such as the existence of functional states, their dependence on regulatory cues, and their correlation to activity. However, it is unclear how these observations translate to enzyme behavior in applied settings. Here, single-molecule tracking of individual Thermomyces lanuginosus lipase (TLL) enzymes in a detergency application system allowed real-time direct observation of spatiotemporal localization, and thus diffusional behavior, of TLL enzymes on a lard substrate. Parallelized imaging of thousands of individual enzymes allowed us to observe directly the existence and quantify the abundance and interconversion kinetics between three diffusional states that we recently provided evidence to correlate with function. We observe redistribution of the enzyme's diffusional pattern at the lipid-water interface as well as variations in binding efficiency in response to surfactants and calcium, demonstrating that detergency effectors can drive the sampling of lipase functional states. Our single-molecule results combined with ensemble activity assays and enzyme surface binding efficiency readouts allowed us to deconvolute how application conditions can significantly alter protein functional dynamics and/or surface binding, both of which underpin enzyme performance. We anticipate that our results will inspire further efforts to decipher and integrate the dynamic nature of lipases, and other enzymes, in the design of new biotechnological solutions.

Principles and Applications of Single Particle Tracking in Cell Research

It is a tough challenge for many decades to decipher the complex relationships between cell behaviors and cellular physical properties. Single particle tracking (SPT) with high spatial and temporal resolution has been applied extensively in cell research to understand physicochemical properties of cells and their bio-functions by tracking endogenous or exogenous probes. This review describes the fundamental principles of SPT as well as its applications in intracellular mechanics, membrane dynamics, organelles distribution, and processes of internalization and transport. Finally, challenges and future directions of SPT are also discussed.

Identifiability analysis for stochastic differential equation models in systems biology

Mathematical models are routinely calibrated to experimental data, with goals ranging from building predictive models to quantifying parameters that cannot be measured. Whether or not reliable parameter estimates are obtainable from the available data can easily be overlooked. Such issues of parameter identifiability have important ramifications for both the predictive power of a model, and the mechanistic insight that can be obtained. Identifiability analysis is well-established for deterministic, ordinary differential equation (ODE) models, but there are no commonly adopted methods for analysing identifiability in stochastic models. We provide an accessible introduction to identifiability analysis and demonstrate how existing ideas for analysis of ODE models can be applied to stochastic differential equation (SDE) models through four practical case studies. To assess structural identifiability, we study ODEs that describe the statistical moments of the stochastic process using open-source software tools. Using practically motivated synthetic data and Markov chain Monte Carlo methods, we assess parameter identifiability in the context of available data. Our analysis shows that SDE models can often extract more information about parameters than deterministic descriptions. All code used to perform the analysis is available on Github.

Revealing Plasma Membrane Nano-Domains with Diffusion Analysis Methods

Nano-domains are sub-light-diffraction-sized heterogeneous areas in the plasma membrane of cells, which are involved in cell signalling and membrane trafficking. Throughout the last thirty years, these nano-domains have been researched extensively and have been the subject of multiple theories and models: the lipid raft theory, the fence model, and the protein oligomerization theory. Strong evidence exists for all of these, and consequently they were combined into a hierarchal model. Measurements of protein and lipid diffusion coefficients and patterns have been instrumental in plasma membrane research and by extension in nano-domain research. This has led to the development of multiple methodologies that can measure diffusion and confinement parameters including single particle tracking, fluorescence correlation spectroscopy, image correlation spectroscopy and fluorescence recovery after photobleaching. Here we review the performance and strengths of these methods in the context of their use in identification and characterization of plasma membrane nano-domains.

Axonal mRNA translation in neurological disorders

It is increasingly recognized that local protein synthesis (LPS) contributes to fundamental aspects of axon biology, in both developing and mature neurons. Mutations in RNA-binding proteins (RBPs), as central players in LPS, and other proteins affecting RNA localization and translation are associated with a range of neurological disorders, suggesting disruption of LPS may be of pathological significance. In this review, we substantiate this hypothesis by examining the link between LPS and key axonal processes, and the implicated pathophysiological consequences of dysregulated LPS. First, we describe how the length and autonomy of axons result in an exceptional reliance on LPS. We next discuss the roles of LPS in maintaining axonal structural and functional polarity and axonal trafficking. We then consider how LPS facilitates the establishment of neuronal connectivity through regulation of axonal branching and pruning, how it mediates axonal survival into adulthood and its involvement in neuronal stress responses.

Tracer Diffusion in Tightly-Meshed Homogeneous Polymer Networks: A Brownian Dynamics Simulation Study

We report Brownian dynamics simulations of tracer diffusion in regularly crosslinked polymer networks in order to elucidate the transport of a tracer particle in polymer networks. The average mesh size of homogeneous polymer networks is varied by assuming different degrees of crosslinking or swelling, and the size of a tracer particle is comparable to the average mesh size. Simulation results show subdiffusion of a tracer particle at intermediate time scales and normal diffusion at long times. In particular, the duration of subdiffusion is significantly prolonged as the average mesh size decreases with increasing degree of crosslinking, for which long-time diffusion occurs via the hopping processes of a tracer particle after undergoing rattling motions within a cage of the network mesh for an extended period of time. On the other hand, the cage dynamics and hopping process are less pronounced as the mesh size decreases with increasing polymer volume fractions. The interpretation is provided in terms of fluctuations in network mesh size: at higher polymer volume fractions, the network fluctuations are large enough to allow for collective, structural changes of network meshes, so that a tracer particle can escape from the cage, whereas, at lower volume fractions, the fluctuations are so small that a tracer particle remains trapped within the cage for a significant period of time before making infrequent jumps out of the cage. This work suggests that fluctuation in mesh size, as well as average mesh size itself, plays an important role in determining the dynamics of molecules and nanoparticles that are embedded in tightly meshed polymer networks.

Puromycin reactivity does not accurately localize translation at the subcellular level

Puromycin is a tyrosyl-tRNA mimic that blocks translation by labeling and releasing elongating polypeptide chains from translating ribosomes. Puromycin has been used in molecular biology research for decades as a translation inhibitor. The development of puromycin antibodies and derivatized puromycin analogs has enabled the quantification of active translation in bulk and single-cell assays. More recently, in vivo puromycylation assays have become popular tools for localizing translating ribosomes in cells. These assays often use elongation inhibitors to purportedly inhibit the release of puromycin-labeled nascent peptides from ribosomes. Using in vitro and in vivo experiments in various eukaryotic systems, we demonstrate that, even in the presence of elongation inhibitors, puromycylated peptides are released and diffuse away from ribosomes. Puromycylation assays reveal subcellular sites, such as nuclei, where puromycylated peptides accumulate post-release and which do not necessarily coincide with sites of active translation. Our findings urge caution when interpreting puromycylation assays in vivo.

Barriers to Diffusion in Cells: Visualization of Membraneless Particles in the Nucleus

Transient barriers are fundamental to cell supramolecular organization and assembly. Discontinuities between spaces can be generated by a physical barrier but also by thermodynamic barriers achieved by phase separation of molecules. However, because of the transient nature and the lack of a visible barrier, the existence of phase separation is difficult to demonstrate experimentally. We describe an approach based on the 2-dimensional pair correlation function (2D-pCF) analysis of the spatial connectivity in a cell. The educational aim of the article is to present both a model suitable for explaining diffusion barrier measurements to a broad range of courses and examples of biological situations. If there are no barriers to diffusion, particles could diffuse equally in all directions. In this situation the pair correlation function introduced in this article is independent of the direction and is uniform in all directions. However, in the presence of obstacles, the shape of the 2D-pCF is distorted to reflect how the obstacle position and orientation change the flow of molecules. In the example shown in this article, measurements of diffusion of enhanced green fluorescent protein moving in live cells show the lack of connectivity at the nucleolus surface for shorter distances. We also observe a gradual increase in the connectivity for longer distances or times, presumably because of molecular trajectories around the nucleolus.

Structural compliance: A new metric for protein flexibility

Proteins are the active players in performing essential molecular activities throughout biology, and their dynamics has been broadly demonstrated to relate to their mechanisms. The intrinsic fluctuations have often been used to represent their dynamics and then compared to the experimental B-factors. However, proteins do not move in a vacuum and their motions are modulated by solvent that can impose forces on the structure. In this paper, we introduce a new structural concept, which has been called the structural compliance, for the evaluation of the global and local deformability of the protein structure in response to intramolecular and solvent forces. Based on the application of pairwise pulling forces to a protein elastic network, this structural quantity has been computed and sometimes is even found to yield an improved correlation with the experimental B-factors, meaning that it may serve as a better metric for protein flexibility. The inverse of structural compliance, namely the structural stiffness, has also been defined, which shows a clear anticorrelation with the experimental data. Although the present applications are made to proteins, this approach can also be applied to other biomolecular structures such as RNA. This present study considers only elastic network models, but the approach could be applied further to conventional atomic molecular dynamics. Compliance is found to have a slightly better agreement with the experimental B-factors, perhaps reflecting its bias toward the effects of local perturbations, in contrast to mean square fluctuations. The code for calculating protein compliance and stiffness is freely accessible at https://jerniganlab.github.io/Software/PACKMAN/Tutorials/compliance.

Investigation of the concentration- and temperature-dependent motion of colloidal nanoparticles

Although the motion of a single nanoparticle suspended in a fluid can be easily modeled, things get complicated for non-infinitely diluted systems. Coincidentally, these are the systems of interest in relevant fields such as, nanomedicine, microfluidics and miniaturized energy storage devices. Hence, a better understanding of the dynamics of colloidal nanoparticles is utterly needed. Herein, the motion of colloidal suspension of plasmonic nanoparticles (i.e., gold nanoshells) is investigated via laser speckle imaging. The method relies on the analysis of the speckle pattern generated by colloidal suspensions forced to flow at specific velocities. Temperature-dependent measurements corroborated that the dynamics of non-infinitely diluted nanoparticle suspensions are better described through a diffusive model rather than by the equipartition theorem. Under the tested experimental conditions, an average diffusion velocity between 0.37 and 1.57 mm s-1 was found. Most importantly, these values were largely dependent on the nanoparticle concentration. These results are in agreement with previous reports and indicate the existence of long-range interactions between nanoparticles.

Back to the Future: Genetically Encoded Fluorescent Proteins as Inert Tracers of the Intracellular Environment

Over the past decades, the discovery and development of genetically encoded fluorescent proteins (FPs) has brought a revolution into our ability to study biologic phenomena directly within living matter. First, FPs enabled fluorescence-labeling of a variety of molecules of interest to study their localization, interactions and dynamic behavior at various scales-from cells to whole organisms/animals. Then, rationally engineered FP-based sensors facilitated the measurement of physicochemical parameters of living matter-especially at the intracellular level, such as ion concentration, temperature, viscosity, pressure, etc. In addition, FPs were exploited as inert tracers of the intracellular environment in which they are expressed. This oft-neglected role is made possible by two distinctive features of FPs: (i) the quite null, unspecific interactions of their characteristic β-barrel structure with the molecular components of the cellular environment; and (ii) their compatibility with the use of time-resolved fluorescence-based optical microscopy techniques. This review seeks to highlight the potential of such unique combinations of properties and report on the most significative and original applications (and related advancements of knowledge) produced to date. It is envisioned that the use of FPs as inert tracers of living matter structural organization holds a potential for several lines of further development in the next future, discussed in the last section of the review, which in turn can lead to new breakthroughs in bioimaging.

CRISPR-Cas9 Genome Interrogation: A Facilitated Subdiffusive Target Search Strategy

The functional application of RNA-guided CRISPR-associated Cas9 protein, a bacterial immune system-based protein complex, via which in vivo, highly specific, and well-regulated, gene-editing processes are being monitored at an unprecedented level, has led to remarkable progress in genetic engineering and technology. The complicated in vivo process of genome interrogation followed by gene editing by the Cas9 complex was recently reported by Knight et al. (Science, 2015, 350, 823-826) using an elegant single-particle tracking method, aided by the two-photon fluorescence correlation spectroscopic technique. In contrast to the usually observed fast target-searching and protein-binding events in biophysical systems, an interesting slow genome-interrogation process by the RNA-guided CRISPR-Cas9 system through a crowded chromatin environment of a mammalian cell has been revealed in Knight et al.'s study. Motivated by this experiment, in this paper, we provide a generalized theoretical framework to capture this particular target-searching mechanism of the CRISPR-Cas9 protein complex. We show that an analysis on the basis of 3D subdiffusion under a cylindrical volume, created by several nonspecific off-target interactions from the DNA strands, can capture the essential details of the process. Moreover, on the basis of this model, we quantify the dynamics of this process and estimate the survival probability, first passage time, and the intensity correlation function of the tagged proteins of the experiment. The results from our theoretical predictions are found to be consistent with the experimental observations, and hence, seem to provide a plausible microscopic picture of the process.

Inferring Entropy Production from Short Experiments

We provide a strategy for the exact inference of the average as well as the fluctuations of the entropy production in nonequilibrium systems in the steady state, from the measurements of arbitrary current fluctuations. Our results are built upon the finite-time generalization of the thermodynamic uncertainty relation, and require only very short time series data from experiments. We illustrate our results with exact and numerical solutions for two colloidal heat engines.

Quantitative Analysis of Membrane Surface and Small Confinement Effects on Molecular Diffusion

Molecular behaviors in small liquid droplets (picoliter scale), such as phase transitions and chemical reactions, are essential for the industrial application of small droplets and their use as artificial cells. However, the droplets often differ from those in bulk solutions (milliliter scale). Since the droplet size is much larger than the molecular size, the so-called size effect that draws these differences has attracted attention as a target to be solved. Although the small volume and the membrane surface surrounding the droplet are thought to be the origin of the size effect, there were little attempts to separate and quantify them. To solve the problem, we develop a series of systems for the evaluation. Using these systems, we have evaluated the size effect of concentrated polymer solutions on molecular diffusion by dividing it into small volume and membrane surface contributions. Our results demonstrate that the size effect on the molecular diffusion originates from the long-range interaction with the surface enhanced with decreasing volume. The quantitative size effect revealed by the systems provides novel insights in the biophysical understanding of molecular behaviors in cells and to the regulation and design of micrometer-sized materials.

Structural Basis of Enhanced Facilitated Diffusion of DNA-Binding Protein in Crowded Cellular Milieu

Although the fast association between DNA-binding proteins (DBPs) and DNA is explained by a facilitated diffusion mechanism, in which DBPs adopt a weighted combination of three-dimensional diffusion and one-dimensional (1D) sliding and hopping modes of transportation, the role of cellular environment that contains many nonspecifically interacting proteins and other biomolecules is mostly overlooked. By performing large-scale computational simulations with an appropriately tuned model of protein and DNA in the presence of nonspecifically interacting bulk and DNA-bound crowders (genomic crowders), we demonstrate the structural basis of the enhanced facilitated diffusion of DBPs inside a crowded cellular milieu through, to our knowledge, novel 1D scanning mechanisms. In this one-dimensional scanning mode, the protein can float along the DNA under the influence of nonspecific interactions of bulk crowder molecules. The search mode is distinctly different compared to usual 1D sliding and hopping dynamics in which protein diffusion is regulated by the DNA electrostatics. In contrast, the presence of genomic crowders expedites the target search process by transporting the protein over DNA segments through the formation of a transient protein-crowder bridged complex. By analyzing the ruggedness of the associated potential energy landscape, we underpin the molecular origin of the kinetic advantages of these search modes and show that they successfully explain the experimentally observed acceleration of facilitated diffusion of DBPs by molecular crowding agents and crowder-concentration-dependent enzymatic activity of transcription factors. Our findings provide crucial insights into gene regulation kinetics inside the crowded cellular milieu.

Guided nuclear exploration increases CTCF target search efficiency

The enormous size of mammalian genomes means that for a DNA-binding protein, the number of non-specific, off-target sites vastly exceeds the number of specific, cognate sites. How mammalian DNA-binding proteins overcome this challenge to efficiently locate their target sites is not known. Here through live-cell single-molecule tracking, we show that CCCTC-binding factor, CTCF, is repeatedly trapped in small zones that likely correspond to CTCF clusters, in a manner that is largely dependent on an internal RNA-binding region (RBR_i). We develop a new theoretical model, Anisotropic Diffusion through transient Trapping in Zones (ADTZ), to explain CTCF dynamics. Functionally, transient RBR_i-mediated trapping increases the efficiency of CTCF target search by ~2.5 fold. Overall, our results suggest a “guided” mechanism where CTCF clusters concentrate diffusing CTCF proteins near cognate binding sites, thus increasing the local ON-rate. We suggest that local guiding may allow DNA-binding proteins to more efficiently locate their target sites.

Transport of probe particles in a polymer network: effects of probe size, network rigidity and probe-polymer interaction

Fundamental understanding of the effect of microscopic parameters on the dynamics of probe particles in different complex environments has wide implications. Examples include diffusion of proteins in biological hydrogels, porous media, polymer matrix, etc. Here, we use extensive molecular dynamics simulations to investigate the dynamics of the probe particle in a polymer network on a diamond lattice, which provides substantial crowding to mimic the cellular environment. Our simulations show that the dynamics of the probe increasingly becomes restricted, non-Gaussian and subdiffusive on increasing the network rigidity, binding affinity and probe size. In addition, the velocity autocorrelation functions show negative dips owing to the viscoelasticity and caging due to the surrounding network. These observations go with the general experimental findings. Importantly, for a probe particle of size comparable to the mesh size, unrestricted motion engulfing large length scales has been observed. This happens with a more flexible polymer network, which is easily pushed by the bigger probe. On increasing the rigidity of the network, the bigger probe can not efficiently push the network and as a result the long tail disappears. Our study gives a general qualitative picture of the transport of probes in a gel-like medium, as encountered in different contexts.

Survival and confinement under quenched disorder

We study the survival and confinement of random walkers under quenched disorder characterized by spatially-varying waiting times and decay rates. Spatial heterogeneity and segregation lead to a dynamic coupling between transport and reaction, resulting in history-dependent dynamics exhibiting long survivals and confinement. The survival probability decays as a power law, in contrast to the classical exponential law for decay at a homogeneous rate. The mean squared displacement shows dimension-dependent subdiffusive growth followed by localization, with stronger confinement in higher dimensions.

Direct observation of Thermomyces lanuginosus lipase diffusional states by Single Particle Tracking and their remodeling by mutations and inhibition

Lipases are interfacially activated enzymes that catalyze the hydrolysis of ester bonds and constitute prime candidates for industrial and biotechnological applications ranging from detergent industry, to chiral organic synthesis. As a result, there is an incentive to understand the mechanisms underlying lipase activity at the molecular level, so as to be able to design new lipase variants with tailor-made functionalities. Our understanding of lipase function primarily relies on bulk assay averaging the behavior of a high number of enzymes masking structural dynamics and functional heterogeneities. Recent advances in single molecule techniques based on fluorogenic substrate analogues revealed the existence of lipase functional states, and furthermore so how they are remodeled by regulatory cues. Single particle studies of lipases on the other hand directly observed diffusional heterogeneities and suggested lipases to operate in two different modes. Here to decipher how mutations in the lid region controls Thermomyces lanuginosus lipase (TLL) diffusion and function we employed a Single Particle Tracking (SPT) assay to directly observe the spatiotemporal localization of TLL and rationally designed mutants on native substrate surfaces. Parallel imaging of thousands of individual TLL enzymes and HMM analysis allowed us to observe and quantify the diffusion, abundance and microscopic transition rates between three linearly interconverting diffusional states for each lipase. We proposed a model that correlate diffusion with function that allowed us to predict that lipase regulation, via mutations in lid region or product inhibition, primarily operates via biasing transitions to the active states.