Heterogeneous nucleation and growth of sessile chemically active droplets.

Droplets are essential for spatially controlling biomolecules in cells. To work properly, cells need to control the emergence and morphology of droplets. On the one hand, driven chemical reactions can affect droplets profoundly. For instance, reactions can control how droplets nucleate and how large they grow. On the other hand, droplets coexist with various organelles and other structures inside cells, which could affect their nucleation and morphology. To understand the interplay of these two aspects, we study a continuous field theory of active phase separation. Our numerical simulations reveal that reactions suppress nucleation while attractive walls enhance it. Intriguingly, these two effects are coupled, leading to shapes that deviate substantially from the spherical caps predicted for passive systems. These distortions result from anisotropic fluxes responding to the boundary conditions dictated by the Young-Dupré equation. Interestingly, an electrostatic analogy of chemical reactions confirms these effects. We thus demonstrate how driven chemical reactions affect the emergence and morphology of droplets, which could be crucial for understanding biological cells and improving technical applications, e.g., in chemical engineering.

Optimized PAR-2 RING dimerization mediates cooperative and selective membrane binding for robust cell polarity.

Cell polarity networks are defined by quantitative features of their constituent feedback circuits, which must be tuned to enable robust and stable polarization, while also ensuring that networks remain responsive to dynamically changing cellular states and/or spatial cues during development. Using the PAR polarity network as a model, we demonstrate that these features are enabled by the dimerization of the polarity protein PAR-2 via its N-terminal RING domain. Combining theory and experiment, we show that dimer affinity is optimized to achieve dynamic, selective, and cooperative binding of PAR-2 to the plasma membrane during polarization. Reducing dimerization compromises positive feedback and robustness of polarization. Conversely, enhanced dimerization renders the network less responsive due to kinetic trapping of PAR-2 on internal membranes and reduced sensitivity of PAR-2 to the anterior polarity kinase, aPKC/PKC-3. Thus, our data reveal a key role for a dynamically oligomeric RING domain in optimizing interaction affinities to support a robust and responsive cell polarity network, and highlight how optimization of oligomerization kinetics can serve as a strategy for dynamic and cooperative intracellular targeting.

Vesicle condensation induced by synapsin: condensate size, geometry, and vesicle shape deformations.

We study the formation of vesicle condensates induced by the protein synapsin, as a cell-free model system mimicking vesicle pool formation in the synapse. The system can be considered as an example of liquid-liquid phase separation (LLPS) in biomolecular fluids, where one phase is a complex fluid itself consisting of vesicles and a protein network. We address the pertinent question why the LLPS is self-limiting and stops at a certain size, i.e., why macroscopic phase separation is prevented. Using fluorescence light microscopy, we observe different morphologies of the condensates (aggregates) depending on the protein-to-lipid ratio. Cryogenic electron microscopy then allows us to resolve individual vesicle positions and shapes in a condensate and notably the size and geometry of adhesion zones between vesicles. We hypothesize that the membrane tension induced by already formed adhesion zones then in turn limits the capability of vesicles to bind additional vesicles, resulting in a finite condensate size. In a simple numerical toy model we show that this effect can be accounted for by redistribution of effective binding particles on the vesicle surface, accounting for the synapsin-induced adhesion zone.

Power-law growth models explain incidences and sizes of pancreatic cancer precursor lesions and confirm spatial genomic findings.

Pancreatic ductal adenocarcinoma is a rare but lethal cancer. Recent evidence reveals that pancreatic intraepithelial neoplasms (PanINs), the microscopic precursor lesions in the pancreatic ducts that can give rise to invasive pancreatic cancer, are significantly larger and more prevalent than previously believed. Better understanding of the growth law dynamics of PanINs may improve our ability to understand how a miniscule fraction of these lesions makes the transition to invasive cancer. Here, using artificial intelligence (AI)-based three-dimensional (3D) tissue mapping method, we measured the volumes of >1,000 PanIN and found that lesion size is distributed according to a power law with a fitted exponent of -1.7 over > 3 orders of magnitude. Our data also suggest that PanIN growth is not very sensitive to the pancreatic microenvironment or an individual's age, family history, and lifestyle, and is rather shaped by general growth behavior. We analyze several models of PanIN growth and fit the predicted size distributions to the observed data. The best fitting models suggest that both intraductal spread of PanIN lesions and fusing of multiple lesions into large, highly branched structures drive PanIN growth patterns. This work lays the groundwork for future mathematical modeling efforts integrating PanIN incidence, morphology, genomic, and transcriptomic features to understand pancreas tumorigenesis, and demonstrates the utility of combining experimental measurement of human tissues with dynamic modeling for understanding cancer tumorigenesis.

Influence of physical interactions on spatiotemporal patterns.

Spatiotemporal patterns are often modeled using reaction-diffusion equations, which combine complex reactions between constituents with ideal diffusive motion. Such descriptions neglect physical interactions between constituents, which might affect resulting patterns. To overcome this, we study how physical interactions affect cyclic dominant reactions, like the seminal rock-paper-scissors game, which exhibits spiral waves for ideal diffusion. Generalizing diffusion to incorporate physical interactions, we find that weak interactions change the length- and time scales of spiral waves, consistent with a mapping to the complex Ginzburg-Landau equation. In contrast, strong repulsive interactions typically generate oscillating lattices, and strong attraction leads to an interplay of phase separation and chemical oscillations, like droplets co-locating with cores of spiral waves. Our work suggests that physical interactions are relevant for forming spatiotemporal patterns in nature, and it might shed light on how biodiversity is maintained in ecological settings.

Physical interactions in non-ideal fluids promote Turing patterns.

Turing's mechanism is often invoked to explain periodic patterns in nature, although direct experimental support is scarce. Turing patterns form in reaction-diffusion systems when the activating species diffuse much slower than the inhibiting species, and the involved reactions are highly nonlinear. Such reactions can originate from cooperativity, whose physical interactions should also affect diffusion. We here take direct interactions into account and show that they strongly affect Turing patterns. We find that weak repulsion between the activator and inhibitor can substantially lower the required differential diffusivity and reaction nonlinearity. By contrast, strong interactions can induce phase separation, but the resulting length scale is still typically governed by the fundamental reaction-diffusion length scale. Taken together, our theory connects traditional Turing patterns with chemically active phase separation, thus describing a wider range of systems. Moreover, we demonstrate that even weak interactions affect patterns substantially, so they should be incorporated when modelling realistic systems.

Nucleation of Chemically Active Droplets.

Driven chemical reactions can control the macroscopic properties of droplets, like their size. Such active droplets are critical in structuring the interior of biological cells. Cells also need to control where and when droplets appear, so they need to control droplet nucleation. Our numerical simulations demonstrate that reactions generally suppress nucleation if they stabilize the homogeneous state. An equilibrium surrogate model reveals that reactions increase the effective energy barrier of nucleation, enabling quantitative predictions of the increased nucleation times. Moreover, the surrogate model allows us to construct a phase diagram, which summarizes how reactions affect the stability of the homogeneous phase and the droplet state. This simple picture provides accurate predictions of how driven reactions delay nucleation, which is relevant for understanding droplets in biological cells and chemical engineering.

The regulation of meiotic crossover distribution: a coarse solution to a century-old mystery?

Meiotic crossovers, which are exchanges of genetic material between homologous chromosomes, are more evenly and distantly spaced along chromosomes than expected by chance. This is because the occurrence of one crossover reduces the likelihood of nearby crossover events - a conserved and intriguing phenomenon called crossover interference. Although crossover interference was first described over a century ago, the mechanism allowing coordination of the fate of potential crossover sites half a chromosome away remains elusive. In this review, we discuss the recently published evidence supporting a new model for crossover patterning, coined the coarsening model, and point out the missing pieces that are still needed to complete this fascinating puzzle.

Memory capacity of adaptive flow networks.

Biological flow networks adapt their network morphology to optimize flow while being exposed to external stimuli from different spatial locations in their environment. These adaptive flow networks retain a memory of the stimulus location in the network morphology. Yet, what limits this memory and how many stimuli can be stored are unknown. Here, we study a numerical model of adaptive flow networks by applying multiple stimuli subsequently. We find strong memory signals for stimuli imprinted for a long time into young networks. Consequently, networks can store many stimuli for intermediate stimulus duration, which balance imprinting and aging.

Effective simulations of interacting active droplets.

Droplets form a cornerstone of the spatiotemporal organization of biomolecules in cells. These droplets are controlled using physical processes like chemical reactions and imposed gradients, which are costly to simulate using traditional approaches, like solving the Cahn-Hilliard equation. To overcome this challenge, we here present an alternative, efficient method. The main idea is to focus on the relevant degrees of freedom, like droplet positions and sizes. We derive dynamical equations for these quantities using approximate analytical solutions obtained from a sharp interface limit and linearized equations in the bulk phases. We verify our method against fully-resolved simulations and show that it can describe interacting droplets under the influence of chemical reactions and external gradients using only a fraction of the computational costs of traditional methods. Our method can be extended to include other processes in the future and will thus serve as a relevant platform for understanding the dynamics of droplets in cells.

Joint control of meiotic crossover patterning by the synaptonemal complex and HEI10 dosage.

Meiotic crossovers are limited in number and are prevented from occurring close to each other by crossover interference. In many species, crossover number is subject to sexual dimorphism, and a lower crossover number is associated with shorter chromosome axes lengths. How this patterning is imposed remains poorly understood. Here, we show that overexpression of the Arabidopsis pro-crossover protein HEI10 increases crossovers but maintains some interference and sexual dimorphism. Disrupting the synaptonemal complex by mutating ZYP1 also leads to an increase in crossovers but, in contrast, abolishes interference and disrupts the link between chromosome axis length and crossovers. Crucially, combining HEI10 overexpression and zyp1 mutation leads to a massive and unprecedented increase in crossovers. These observations support and can be predicted by, a recently proposed model in which HEI10 diffusion along the synaptonemal complex drives a coarsening process leading to well-spaced crossover-promoting foci, providing a mechanism for crossover patterning.

Memory Formation in Adaptive Networks.

The continuous adaptation of networks like our vasculature ensures optimal network performance when challenged with changing loads. Here, we show that adaptation dynamics allow a network to memorize the position of an applied load within its network morphology. We identify that the irreversible dynamics of vanishing network links encode memory. Our analytical theory successfully predicts the role of all system parameters during memory formation, including parameter values which prevent memory formation. We thus provide analytical insight on the theory of memory formation in disordered systems.

Evolved interactions stabilize many coexisting phases in multicomponent liquids.

Phase separation has emerged as an essential concept for the spatial organization inside biological cells. However, despite the clear relevance to virtually all physiological functions, we understand surprisingly little about what phases form in a system of many interacting components, like in cells. Here we introduce a numerical method based on physical relaxation dynamics to study the coexisting phases in such systems. We use our approach to optimize interactions between components, similar to how evolution might have optimized the interactions of proteins. These evolved interactions robustly lead to a defined number of phases, despite substantial uncertainties in the initial composition, while random or designed interactions perform much worse. Moreover, the optimized interactions are robust to perturbations, and they allow fast adaption to new target phase counts. We thus show that genetically encoded interactions of proteins provide versatile control of phase behavior. The phases forming in our system are also a concrete example of a robust emergent property that does not rely on fine-tuning the parameters of individual constituents.

Self-generated oxygen gradients control collective aggregation of photosynthetic microbes.

For billions of years, photosynthetic microbes have evolved under the variable exposure to sunlight in diverse ecosystems and microhabitats all over our planet. Their abilities to dynamically respond to alterations of the luminous intensity, including phototaxis, surface association and diurnal cell cycles, are pivotal for their survival. If these strategies fail in the absence of light, the microbes can still sustain essential metabolic functionalities and motility by switching their energy production from photosynthesis to oxygen respiration. For suspensions of motile C. reinhardtii cells above a critical density, we demonstrate that this switch reversibly controls collective microbial aggregation. Aerobic respiration dominates over photosynthesis in conditions of low light, which causes the microbial motility to sensitively depend on the local availability of oxygen. For dense microbial populations in self-generated oxygen gradients, microfluidic experiments and continuum theory based on a reaction-diffusion mechanism show that oxygen-regulated motility enables the collective emergence of highly localized regions of high and low cell densities.

Computational Fluid Dynamics Modeling of Nasal Obstruction and Associations with Patient-Reported Outcomes.

BACKGROUND: Nasal obstruction is a common problem, with significant impact on quality of life. Accurate diagnosis may be challenging because of the complex and dynamic nature of the involved anatomy. Computational fluid dynamics modeling has the ability to identify specific anatomical defects, allowing for a targeted surgical approach. The goal of the current study is to better understand nasal obstruction as it pertains to disease-specific quality of life by way of a novel computational fluid dynamics model of nasal airflow. METHODS: Fifty-three patients with nasal obstruction underwent computational fluid dynamics modeling based on computed tomographic imaging. Nasal resistance was compared to demographic data and baseline subjective nasal patency based on Nasal Obstructive Symptom Evaluation scores. RESULTS: Mean Nasal Obstructive Symptom Evaluation score among all patients was 72.6. Nasal Obstructive Symptom Evaluation score demonstrated a significant association with nasal resistance in patients with static obstruction (p = 0.03). There was a positive correlation between Nasal Obstructive Symptom Evaluation score and nasal resistance in patients with static bilateral nasal obstruction (R2 = 0.32) and poor correlation in patients with dynamic bilateral obstruction caused by nasal valve collapse (R2 = 0.02). Patients with moderate and severe bilateral symptoms had significantly higher nasal resistance compared to those with unilateral symptoms (p = 0.048). CONCLUSIONS: Nasal obstruction is a multifactorial condition in most patients. This study shows correlation between simulated nasal resistance and Nasal Obstructive Symptom Evaluation score in a select group of patients. There is currently no standardized diagnostic algorithm or gold standard objective measure of nasal airflow; however, computational fluid dynamics may better inform treatment planning and surgical techniques on an individual basis. CLINICAL QUESTION/LEVEL OF EVIDENCE: Risk, V.

Cavitation controls droplet sizes in elastic media.

Biological cells use droplets to separate components and spatially control their interior. Experiments demonstrate that the complex, crowded cellular environment affects the droplet arrangement and their sizes. To understand this behavior, we here construct a theoretical description of droplets growing in an elastic matrix, which is motivated by experiments in synthetic systems where monodisperse emulsions form during a temperature decrease. We show that large droplets only form when they break the surrounding matrix in a cavitation event. The energy barrier associated with cavitation stabilizes small droplets on the order of the mesh size and diminishes the stochastic effects of nucleation. Consequently, the cavitated droplets have similar sizes and highly correlated positions. In particular, we predict the density of cavitated droplets, which increases with faster cooling, as in the experiments. Our model also suggests how adjusting the cooling protocol and the density of nucleation sites affects the droplet size distribution. In summary, our theory explains how elastic matrices affect droplets in the synthetic system, and it provides a framework for understanding the biological case.

Controlling biomolecular condensates via chemical reactions.

Biomolecular condensates are small droplets forming spontaneously in biological cells through phase separation. They play a role in many cellular processes, but it is unclear how cells control them. Cellular regulation often relies on post-translational modifications of proteins. For biomolecular condensates, such chemical modifications could alter the molecular interaction of key condensate components. Here, we test this idea using a theoretical model based on non-equilibrium thermodynamics. In particular, we describe the chemical reactions using transition-state theory, which accounts for the non-ideality of phase separation. We identify that fast control, as in cell signalling, is only possible when external energy input drives the reaction out of equilibrium. If this reaction differs inside and outside the droplet, it is even possible to control droplet sizes. Such an imbalance in the reaction could be created by enzymes localizing to the droplet. Since this situation is typical inside cells, we speculate that our proposed mechanism is used to stabilize multiple droplets with independently controlled size and count. Our model provides a novel and thermodynamically consistent framework for describing droplets subject to non-equilibrium chemical reactions.

Elastic stresses reverse Ostwald ripening.

When liquid droplets nucleate and grow in a polymer network, compressive stresses can significantly increase their internal pressure, reaching values that far exceed the Laplace pressure. When droplets have grown in a polymer network with a stiffness gradient, droplets in relatively stiff regions of the network tend to dissolve, favoring growth of droplets in softer regions. Here, we show that this elastic ripening can be strong enough to reverse the direction of Ostwald ripening: large droplets can shrink to feed the growth of smaller ones. To numerically model these experiments, we generalize the theory of elastic ripening to account for gradients in solubility alongside gradients in mechanical stiffness.

Theory of droplet ripening in stiffness gradients.

Liquid droplets embedded in soft solids are a new composite material whose properties are not very well explored. In particular, it is unclear how the elastic properties of the matrix affect the dynamics of the droplets. Here, we study theoretically how stiffness gradients influence droplet growth and arrangement. We show that stiffness gradients imply concentration gradients in the dilute phase, which transport droplet material from stiff to soft regions. Consequently, droplets dissolve in the stiff region, creating a dissolution front. Using a mean-field theory, we predict that the front emerges where the curvature of the elasticity profile is large and that it propagates diffusively. This elastic ripening can occur at much higher rates than classical Ostwald ripening, thus driving the dynamics. Our work shows how gradients in elastic properties control the arrangement of droplets, which has potential applications in soft matter physics and biological cells.

Elastic ripening and inhibition of liquid-liquid phase separation.

Phase separation is a central concept of materials physics [1-3] and has recently emerged as an important route to compartmentalization within living cells [4-6]. Biological phase separation features activity [7], complex compositions [8], and elasticity [9], which reveal important gaps in our understanding of this universal physical phenomenon. Here, we explore the impact of elasticity on phase separation in synthetic polymer networks. We show that compressive stresses in a polymer network can suppress phase separation of the solvent that swells it, stabilizing mixtures well beyond the liquid-liquid phase separation boundary. Network stresses also drive a new form of ripening, driven by transport of solute down stiffness gradients. This elastic ripening can be much faster than conventional surface tension driven Ostwald ripening.