Passive endocytosis in model protocells.

Semipermeable membranes are a key feature of all living organisms. While specialized membrane transporters in cells can import otherwise impermeable nutrients, the earliest cells would have lacked a mechanism to import nutrients rapidly under nutrient-rich circumstances. Using both experiments and simulations, we find that a process akin to passive endocytosis can be recreated in model primitive cells. Molecules that are too impermeable to be absorbed can be taken up in a matter of seconds in an endocytic vesicle. The internalized cargo can then be slowly released over hours, into the main lumen or putative cytoplasm. This work demonstrates a way by which primitive life could have broken the symmetry of passive permeation prior to the evolution of protein transporters.

Influence of the vessel wall geometry on the wall-induced migration of red blood cells.

The geometry of the blood vessel wall plays a regulatory role on the motion of red blood cells (RBCs). The overall topography of the vessel wall depends on many features, among which the endothelial lining of the endothelial surface layer (ESL) is an important one. The endothelial lining of vessel walls presents a large surface area for exchanging materials between blood and tissues. The ESL plays a critical role in regulating vascular permeability, hindering leukocyte adhesion as well as inhibiting coagulation during inflammation. Changes in the ESL structure are believed to cause vascular hyperpermeability and entrap immune cells during sepsis, which could significantly alter the vessel wall geometry and disturb interactions between RBCs and the vessel wall, including the wall-induced migration of RBCs and the thickening of a cell-free layer. To investigate the influence of the vessel wall geometry particularly changed by the ESL under various pathological conditions, such as sepsis, on the motion of RBCs, we developed two models to represent the ESL using the immersed boundary method in two dimensions. In particular, we used simulations to study how the lift force and drag force on a RBC near the vessel wall vary with different wall thickness, spatial variation, and permeability associated with changes in the vessel wall geometry. We find that the spatial variation of the wall has a significant effect on the wall-induced migration of the RBC for a high permeability, and that the wall-induced migration is significantly inhibited as the vessel diameter is increased.

Modeling the Transition between Localized and Extended Deposition in Flow Networks through Packings of Glass Beads.

We use a theoretical model to explore how fluid dynamics, in particular, the pressure gradient and wall shear stress in a channel, affect the deposition of particles flowing in a microfluidic network. Experiments on transport of colloidal particles in pressure-driven systems of packed beads have shown that at lower pressure drop, particles deposit locally at the inlet, while at higher pressure drop, they deposit uniformly along the direction of flow. We develop a mathematical model and use agent-based simulations to capture these essential qualitative features observed in experiments. We explore the deposition profile over a two-dimensional phase diagram defined in terms of the pressure and shear stress threshold, and show that two distinct phases exist. We explain this apparent phase transition by drawing an analogy to simple one-dimensional mass-aggregation models in which the phase transition is calculated analytically.

Leaky cell model of hard spheres.

We study packings of hard spheres on lattices. The partition function, and therefore the pressure, may be written solely in terms of the accessible free volume, i.e., the volume of space that a sphere can explore without touching another sphere. We compute these free volumes using a leaky cell model, in which the accessible space accounts for the possibility that spheres may escape from the local cage of lattice neighbors. We describe how elementary geometry may be used to calculate the free volume exactly for this leaky cell model in two- and three-dimensional lattice packings and compare the results to the well-known Carnahan-Starling and Percus-Yevick liquid models. We provide formulas for the free volumes of various lattices and use the common tangent construction to identify several phase transitions between them in the leaky cell regime, indicating the possibility of coexistence in crystalline materials.

Global asymptotic stability of the active disassembly model of flagellar length control.

Organelle size control is a fundamental question in biology that demonstrates the fascinating ability of cells to maintain homeostasis within their highly variable environments. Theoretical models describing cellular dynamics have the potential to help elucidate the principles underlying size control. Here, we perform a detailed study of the active disassembly model proposed in Fai et al. (elife 8:e42599, 2019). We construct a hybrid system which is shown to be well-behaved throughout the domain. We rule out the possibility of oscillations arising in the model and prove global asymptotic stability in the case of two flagella by the construction of a suitable Lyapunov function. Finally, we generalize the model to the case of arbitrary flagellar number in order to study olfactory sensory neurons, which have up to twenty cilia per cell. We show that our theoretical results may be extended to this case and explore the implications of this universal mechanism of size control.

Dynamics of Vesicles Driven Into Closed Constrictions by Molecular Motors.

We study the dynamics of a model of membrane vesicle transport into dendritic spines, which are bulbous intracellular compartments in neurons driven by molecular motors. We reduce the lubrication model proposed in Fai et al. (Phys Rev Fluids 2:113601, 2017) to a fast-slow system, yielding an analytically and numerically tractable equation equivalent to the original model in the overdamped limit. The model's key parameters include: (1) the ratio of motors that prefer to push toward the head of the dendritic spine to the motors that prefer to push in the opposite direction, and (2) the viscous drag exerted on the vesicle by the spine constriction. We perform a numerical bifurcation analysis in these parameters and find that steady-state vesicle velocities appear and disappear through several saddle-node bifurcations. This process allows us to identify the region of parameter space in which multiple stable velocities exist. We show by direct calculations that there can only be unidirectional motion for sufficiently close vesicle-to-spine diameter ratios. Our analysis predicts the critical vesicle-to-spine diameter ratio, at which there is a transition from unidirectional to bidirectional motion, consistent with experimental observations of vesicle trajectories in the literature.

Dynamics of Growth and Form in Prebiotic Vesicles.

The growth, form, and division of prebiotic vesicles, membraneous bags of fluid of varying components and shapes is hypothesized to have served as the substrate for the origin of life. The dynamics of these out-of-equilibrium structures is controlled by physicochemical processes that include the intercalation of amphiphiles into the membrane, fluid flow across the membrane, and elastic deformations of the membrane. To understand prebiotic vesicular forms and their dynamics, we construct a minimal model that couples membrane growth, deformation, and fluid permeation, ultimately couched in terms of two dimensionless parameters that characterize the relative rate of membrane growth and the membrane permeability. Numerical simulations show that our model captures the morphological diversity seen in extant precursor mimics of cellular life, and might provide simple guidelines for the synthesis of these complex shapes from simple ingredients.

Image-based model of the spectrin cytoskeleton for red blood cell simulation.

We simulate deformable red blood cells in the microcirculation using the immersed boundary method with a cytoskeletal model that incorporates structural details revealed by tomographic images. The elasticity of red blood cells is known to be supplied by both their lipid bilayer membranes, which resist bending and local changes in area, and their cytoskeletons, which resist in-plane shear. The cytoskeleton consists of spectrin tetramers that are tethered to the lipid bilayer by ankyrin and by actin-based junctional complexes. We model the cytoskeleton as a random geometric graph, with nodes corresponding to junctional complexes and with edges corresponding to spectrin tetramers such that the edge lengths are given by the end-to-end distances between nodes. The statistical properties of this graph are based on distributions gathered from three-dimensional tomographic images of the cytoskeleton by a segmentation algorithm. We show that the elastic response of our model cytoskeleton, in which the spectrin polymers are treated as entropic springs, is in good agreement with the experimentally measured shear modulus. By simulating red blood cells in flow with the immersed boundary method, we compare this discrete cytoskeletal model to an existing continuum model and predict the extent to which dynamic spectrin network connectivity can protect against failure in the case of a red cell subjected to an applied strain. The methods presented here could form the basis of disease- and patient-specific computational studies of hereditary diseases affecting the red cell cytoskeleton.

FAST SOLVER FOR DIFFUSIVE TRANSPORT TIMES ON DYNAMIC INTRACELLULAR NETWORKS.

The transport of particles in cells is influenced by the properties of intracellular networks they traverse while searching for localized target regions or reaction partners. Moreover, given the rapid turnover in many intracellular structures, it is crucial to understand how temporal changes in the network structure affect diffusive transport. In this work, we use network theory to characterize complex intracellular biological environments across scales. We develop an efficient computational method to compute the mean first passage times for simulating a particle diffusing along two-dimensional planar networks extracted from fluorescence microscopy imaging. We first benchmark this methodology in the context of synthetic networks, and subsequently apply it to live-cell data from endoplasmic reticulum tubular networks.

An approach for quantitative mapping of synaptic periactive zone architecture and organization.

Following exocytosis at active zones, synaptic vesicle membranes and membrane-bound proteins must be recycled. The endocytic machinery that drives this recycling accumulates in the periactive zone (PAZ), a region of the synapse adjacent to active zones, but the organization of this machinery within the PAZ, and how PAZ composition relates to active zone release properties, remains unknown. The PAZ is also enriched for cell adhesion proteins, but their function at these sites is poorly understood. Here, using Airyscan and stimulated emission depletion imaging of Drosophila synapses, we develop a quantitative framework describing the organization and ultrastructure of the PAZ. Different endocytic proteins localize to distinct regions of the PAZ, suggesting that subdomains are specialized for distinct biochemical activities, stages of membrane remodeling, or synaptic functions. We find that the accumulation and distribution of endocytic but not adhesion PAZ proteins correlate with the abundance of the scaffolding protein Bruchpilot at active zones-a structural correlate of release probability. These data suggest that endocytic and exocytic activities are spatially correlated. Taken together, our results identify novel relationships between the exocytic and endocytic apparatus at the synapse and provide a new conceptual framework to quantify synaptic architecture.

Passive endocytosis in model protocells.

Semipermeable membranes are a key feature of all living organisms. While specialized membrane transporters in cells can import otherwise impermeable nutrients, the earliest cells would have lacked a mechanism to import nutrients rapidly under nutrient-rich circumstances. Using both experiments and simulations, we find that a process akin to passive endocytosis can be recreated in model primitive cells. Molecules that are too impermeable to be absorbed can be taken up in a matter of seconds in an endocytic vesicle. The internalized cargo can then be slowly released over hours, into the main lumen or putative cytoplasm. This work demonstrates a way by which primitive life could have broken the symmetry of passive permeation prior to the evolution of protein transporters.

Coarse-grained Stochastic Model of Myosin-Driven Vesicles into Dendritic Spines.

We study the dynamics of membrane vesicle motor transport into dendritic spines, which are bulbous intracellular compartments in neurons that play a key role in transmitting signals between neurons. We consider the stochastic analog of the vesicle transport model in [Park and Fai, The Dynamics of Vesicles Driven Into Closed Constrictions by Molecular Motors. Bull. Math. Biol. 82, 141 (2020)]. The stochastic version, which may be considered as an agent-based model, relies mostly on the action of individual myosin motors to produce vesicle motion. To aid in our analysis, we coarse-grain this agent-based model using a master equation combined with a partial differential equation describing the probability of local motor positions. We confirm through convergence studies that the coarse-graining captures the essential features of bistability in velocity (observed in experiments) and waiting-time distributions to switch between steady-state velocities. Interestingly, these results allow us to reformulate the translocation problem in terms of conditional mean first passage times for a run-and-tumble particle moving on a finite domain with absorbing boundaries at the two ends. We conclude by presenting numerical and analytical calculations of vesicle translocation.

Control of nuclear size by osmotic forces in Schizosaccharomyces pombe.

The size of the nucleus scales robustly with cell size so that the nuclear-to-cell volume ratio (N/C ratio) is maintained during cell growth in many cell types. The mechanism responsible for this scaling remains mysterious. Previous studies have established that the N/C ratio is not determined by DNA amount but is instead influenced by factors such as nuclear envelope mechanics and nuclear transport. Here, we developed a quantitative model for nuclear size control based upon colloid osmotic pressure and tested key predictions in the fission yeast Schizosaccharomyces pombe. This model posits that the N/C ratio is determined by the numbers of macromolecules in the nucleoplasm and cytoplasm. Osmotic shift experiments showed that the fission yeast nucleus behaves as an ideal osmometer whose volume is primarily dictated by osmotic forces. Inhibition of nuclear export caused accumulation of macromolecules in the nucleoplasm, leading to nuclear swelling. We further demonstrated that the N/C ratio is maintained by a homeostasis mechanism based upon synthesis of macromolecules during growth. These studies demonstrate the functions of colloid osmotic pressure in intracellular organization and size control.

An autoinhibitory clamp of actin assembly constrains and directs synaptic endocytosis.

Synaptic membrane-remodeling events such as endocytosis require force-generating actin assembly. The endocytic machinery that regulates these actin and membrane dynamics localizes at high concentrations to large areas of the presynaptic membrane, but actin assembly and productive endocytosis are far more restricted in space and time. Here we describe a mechanism whereby autoinhibition clamps the presynaptic endocytic machinery to limit actin assembly to discrete functional events. We found that collective interactions between the Drosophila endocytic proteins Nwk/FCHSD2, Dap160/intersectin, and WASp relieve Nwk autoinhibition and promote robust membrane-coupled actin assembly in vitro. Using automated particle tracking to quantify synaptic actin dynamics in vivo, we discovered that Nwk-Dap160 interactions constrain spurious assembly of WASp-dependent actin structures. These interactions also promote synaptic endocytosis, suggesting that autoinhibition both clamps and primes the synaptic endocytic machinery, thereby constraining actin assembly to drive productive membrane remodeling in response to physiological cues.

Neurons constantly talk to each other by sending chemical signals across the tiny gap, or 'synapse', that separates two cells. While inside the emitting cell, these molecules are safely packaged into small, membrane-bound vessels. Upon the right signal, the vesicles fuse with the external membrane of the neuron and spill their contents outside, for the receiving cell to take up and decode. The emitting cell must then replenish its vesicle supply at the synapse through a recycling mechanism known as endocytosis. To do so, it uses dynamically assembling rod-like 'actin' filaments, which work in concert with many other proteins to pull in patches of membrane as new vesicles. The proteins that control endocytosis and actin assembly abound at neuronal synapses, and, when mutated, are linked to many neurological diseases. Unlike other cell types, neurons appear to 'pre-deploy' these actin-assembly proteins to synaptic membranes, but to keep them inactive under normal conditions. How neurons control the way this machinery is recruited and activated remains unknown. To investigate this question, Del Signore et al. conducted two sets of studies. First, they exposed actin to several different purified proteins in initial 'test tube' experiments. This revealed that, depending on the conditions, a group of endocytosis proteins could prevent or promote actin assembly: assembly occurred only if the proteins were associated with membranes. Next, Del Signore et al. mutated these proteins in fruit fly larvae, and performed live cell microscopy to determine their impact on actin assembly and endocytosis. Consistent with the test tube findings, endocytosis mutants had more actin assembly overall, implying that the proteins were required to prevent random actin assembly. However, the same mutants had reduced levels of endocytosis, suggesting that the proteins were also necessary for productive actin assembly. Together, these experiments suggest that, much like a mousetrap holds itself poised ready to spring, some endocytic proteins play a dual role to restrain actin assembly when and where it is not needed, and to promote it at sites of endocytosis. These results shed new light on how neurons might build and maintain effective, working synapses. Del Signore et al. hope that this knowledge may help to better understand and combat neurological diseases, such as Alzheimer's, which are linked to impaired membrane traffic and cell signalling.

Length regulation of multiple flagella that self-assemble from a shared pool of components.

The single-celled green algae Chlamydomonas reinhardtii with its two flagella-microtubule-based structures of equal and constant lengths-is the canonical model organism for studying size control of organelles. Experiments have identified motor-driven transport of tubulin to the flagella tips as a key component of their length control. Here we consider a class of models whose key assumption is that proteins responsible for the intraflagellar transport (IFT) of tubulin are present in limiting amounts. We show that the limiting-pool assumption is insufficient to describe the results of severing experiments, in which a flagellum is regenerated after it has been severed. Next, we consider an extension of the limiting-pool model that incorporates proteins that depolymerize microtubules. We show that this 'active disassembly' model of flagellar length control explains in quantitative detail the results of severing experiments and use it to make predictions that can be tested in experiments.

Quantitative biology: where modern biology meets physical sciences.

Quantitative methods and approaches have been playing an increasingly important role in cell biology in recent years. They involve making accurate measurements to test a predefined hypothesis in order to compare experimental data with predictions generated by theoretical models, an approach that has benefited physicists for decades. Building quantitative models in experimental biology not only has led to discoveries of counterintuitive phenomena but has also opened up novel research directions. To make the biological sciences more quantitative, we believe a two-pronged approach needs to be taken. First, graduate training needs to be revamped to ensure biology students are adequately trained in physical and mathematical sciences and vice versa. Second, students of both the biological and the physical sciences need to be provided adequate opportunities for hands-on engagement with the methods and approaches necessary to be able to work at the intersection of the biological and physical sciences. We present the annual Physiology Course organized at the Marine Biological Laboratory (Woods Hole, MA) as a case study for a hands-on training program that gives young scientists the opportunity not only to acquire the tools of quantitative biology but also to develop the necessary thought processes that will enable them to bridge the gap between these disciplines.