Selection for Size in Molecular Self-Assembly Drives the De Novo Evolution of a Molecular Machine.

The functioning of machines typically requires a concerted action of their parts. This requirement also holds for molecular motors that drive vital cellular processes and imposes constraints on their conformational changes as well as the rates at which they occur. It remains unclear whether, during evolution, features required for functional molecular machines can emerge simultaneously or require sequential adaptation to different selection pressures. We address this question by theoretically analyzing the evolution of filament treadmilling. This process refers to the self-assembly of linear polymers that grow and shrink at equal rates at their opposite ends. It constitutes a simple biological molecular machine that is involved in bacterial cell division and requires that several conditions are met. In our simulation framework, treadmilling emerges as a consequence of selecting for a target average polymer length. We discuss why other forms of assembly dynamics, which also reach the imposed target length, do not emerge in our simulations. Our work shows that complex molecular functions can evolve de novo under selection for a single physical feature.

Localized States in Active Fluids.

Biological active matter is typically tightly coupled to chemical reaction networks affecting its assembly-disassembly dynamics and stress generation. We show that localized states can emerge spontaneously if assembly of active matter is regulated by chemical species that are advected with flows resulting from gradients in the active stress. The mechanochemical localized patterns form via a subcritical bifurcation and for parameter values for which patterns do not exist in absence of the advective coupling. Our work identifies a generic mechanism underlying localized cellular patterns.

Calcium storage in multivesicular endo-lysosome.

It is now established that endo-lysosomes, also referred to as late endosomes, serve as intracellular calcium store, in addition to the endoplasmic reticulum. While abundant calcium-binding proteins provide the latter compartment with its calcium storage capacity, essentially nothing is known about the mechanism responsible for calcium storage in endo-lysosomes. In this paper, we propose that the structural organization of endo-lysosomal membranes drives the calcium storage capacity of the compartment. Indeed, endo-lysosomes exhibit a characteristic multivesicular ultrastructure, with intralumenal membranes providing a large amount of additional bilayer surface. We used a theoretical approach to investigate the calcium storage capacity of endosomes, using known calcium binding affinities for bilayers and morphological data on endo-lysosome membrane organization. Finally, we tested our predictions experimentally after Sorting Nexin 3 depletion to decrease the intralumenal membrane content. We conclude that the major negatively-charge lipids and proteins of endo-lysosomes serve as calcium-binding molecules in the acidic calcium stores of mammalian cells, while the large surface area of intralumenal membranes provide the necessary storage capacity.

Phase separation on surfaces in the presence of matter exchange.

We present a field theory to describe the composition of a surface spontaneously exchanging matter with its bulk environment. By only assuming matter conservation in the system, we show with extensive numerical simulations that, depending on the matter exchange rates, a complex patterned composition distribution emerges on the surface. For one-dimensional systems we show analytically and numerically that coarsening is arrested and as a consequence domains have a characteristic length scale. Our results show that the causes of heterogeneous lipid composition in cellular membranes may be justified in simple physical terms.

The effect of motor-induced shaft dynamics on microtubule stability and length.

Control of microtubule abundance, stability, and length is crucial to regulate intracellular transport as well as cell polarity and division. How microtubule stability depends on tubulin addition or removal at the dynamic ends is well studied. However, microtubule rescue, the event when a microtubule switches from shrinking to growing, occurs at tubulin exchange sites along the shaft. Molecular motors have recently been shown to promote such exchanges. Using a stochastic theoretical description, we study how microtubule stability and length depend on motor-induced tubulin exchange and thus rescue. Our theoretical description matches our in vitro experiments on microtubule dynamics in the presence of kinesin-1 molecular motors. Although the overall dynamics of a population of microtubules can be captured by an effective rescue rate, by assigning rescue to exchange sites, we reveal that the dynamics of individual microtubules within the population differ dramatically. Furthermore, we study in detail a transition from bounded to unbounded microtubule growth. Our results provide novel insights into how molecular motors imprint information of microtubule stability on the microtubule network.

Density-Polarity Coupling in Confined Active Polar Films: Asters, Spirals, and Biphasic Orientational Phases.

Topological defects in active polar fluids can organize spontaneous flows and influence macroscopic density patterns. Both of them play an important role during animal development. Yet the influence of density on active flows is poorly understood. Motivated by experiments on cell monolayers confined to disks, we study the coupling between density and polar order for a compressible active polar fluid in the presence of a +1 topological defect. As in the experiments, we find a density-controlled spiral-to-aster transition. In addition, biphasic orientational phases emerge as a generic outcome of such coupling. Our results highlight the importance of density gradients as a potential mechanism for controlling flow and orientational patterns in biological systems.

Integer topological defects organize stresses driving tissue morphogenesis.

Tissues acquire function and shape via differentiation and morphogenesis. Both processes are driven by coordinating cellular forces and shapes at the tissue scale, but general principles governing this interplay remain to be discovered. Here we report that self-organization of myoblasts around integer topological defects, namely spirals and asters, suffices to establish complex multicellular architectures. In particular, these arrangements can trigger localized cell differentiation or, alternatively, when differentiation is inhibited, they can drive the growth of swirling protrusions. Both localized differentiation and growth of cellular vortices require specific stress patterns. By analysing the experimental velocity and orientational fields through active gel theory, we show that integer topological defects can generate force gradients that concentrate compressive stresses. We reveal these gradients by assessing spatial changes in nuclear volume and deformations of elastic pillars. We propose integer topological defects as mechanical organizing centres controlling differentiation and morphogenesis.

Motor usage imprints microtubule stability along the shaft.

Tubulin dimers assemble into dynamic microtubules, which are used by molecular motors as tracks for intracellular transport. Organization and dynamics of the microtubule network are commonly thought to be regulated at the polymer ends, where tubulin dimers can be added or removed. Here, we show that molecular motors running on microtubules cause exchange of dimers along the shaft in vitro and in cells. These sites of dimer exchange act as rescue sites where depolymerizing microtubules stop shrinking and start re-growing. Consequently, the average length of microtubules increases depending on how frequently they are used as motor tracks. An increase of motor activity densifies the cellular microtubule network and enhances cell polarity. Running motors leave marks in the shaft, serving as traces of microtubule usage to organize the polarity landscape of the cell.

Pinching the cortex of live cells reveals thickness instabilities caused by myosin II motors.

The cell cortex is a contractile actin meshwork, which determines cell shape and is essential for cell mechanics, migration, and division. Because its thickness is below optical resolution, there is a tendency to consider the cortex as a thin uniform two-dimensional layer. Using two mutually attracted magnetic beads, one inside the cell and the other in the extracellular medium, we pinch the cortex of dendritic cells and provide an accurate and time-resolved measure of its thickness. Our observations draw a new picture of the cell cortex as a highly dynamic layer, harboring large fluctuations in its third dimension because of actomyosin contractility. We propose that the cortex dynamics might be responsible for the fast shape-changing capacity of highly contractile cells that use amoeboid-like migration.

Integer topological defects of cell monolayers: Mechanics and flows.

Monolayers of anisotropic cells exhibit long-ranged orientational order and topological defects. During the development of organisms, orientational order often influences morphogenetic events. However, the linkage between the mechanics of cell monolayers and topological defects remains largely unexplored. This holds specifically at the timescales relevant for tissue morphogenesis. Here, we build on the physics of liquid crystals to determine material parameters of cell monolayers. In particular, we use a hydrodynamical description of an active polar fluid to study the steady-state mechanical patterns at integer topological defects. Our description includes three distinct sources of activity: traction forces accounting for cell-substrate interactions as well as anisotropic and isotropic active nematic stresses accounting for cell-cell interactions. We apply our approach to C2C12 cell monolayers in small circular confinements, which form isolated aster or spiral topological defects. By analyzing the velocity and orientational order fields in spirals as well as the forces and cell number density fields in asters, we determine mechanical parameters of C2C12 cell monolayers. Our work shows how topological defects can be used to fully characterize the mechanical properties of biological active matter.

Excitable actin dynamics and amoeboid cell migration.

Amoeboid cell migration is characterized by frequent changes of the direction of motion and resembles a persistent random walk on long time scales. Although it is well known that cell migration is typically driven by the actin cytoskeleton, the cause of this migratory behavior remains poorly understood. We analyze the spontaneous dynamics of actin assembly due to nucleation promoting factors, where actin filaments lead to an inactivation of these factors. We show that this system exhibits excitable dynamics and can spontaneously generate waves, which we analyze in detail. By using a phase-field approach, we show that these waves can generate cellular random walks. We explore how the characteristics of these persistent random walks depend on the parameters governing the actin-nucleator dynamics. In particular, we find that the effective diffusion constant and the persistence time depend strongly on the speed of filament assembly and the rate of nucleator inactivation. Our findings point to a deterministic origin of the random walk behavior and suggest that cells could adapt their migration pattern by modifying the pool of available actin.

Quantifying Material Properties of Cell Monolayers by Analyzing Integer Topological Defects.

In developing organisms, internal cellular processes generate mechanical stresses at the tissue scale. The resulting deformations depend on the material properties of the tissue, which can exhibit long-ranged orientational order and topological defects. It remains a challenge to determine these properties on the time scales relevant for developmental processes. Here, we build on the physics of liquid crystals to determine material parameters of cell monolayers. Specifically, we use a hydrodynamic description to characterize the stationary states of compressible active polar fluids around defects. We illustrate our approach by analyzing monolayers of C2C12 cells in small circular confinements, where they form a single topological defect with integer charge. We find that such monolayers exert compressive stresses at the defect centers, where localized cell differentiation and formation of three-dimensional shapes is observed.

Buckling of an Epithelium Growing under Spherical Confinement.

Many organs are formed through folding of an epithelium. This change in shape is usually attributed to tissue heterogeneities, for example, local apical contraction. In contrast, compressive stresses have been proposed to fold a homogeneous epithelium by buckling. While buckling is an appealing mechanism, demonstrating that it underlies folding requires measurement of the stress field and the material properties of the tissue, which are currently inaccessible in vivo. Here, we show that monolayers of identical cells proliferating on the inner surface of elastic spherical shells can spontaneously fold. By measuring the elastic deformation of the shell, we infer the forces acting within the monolayer and its elastic modulus. Using analytical and numerical theories linking forces to shape, we find that buckling quantitatively accounts for the shape changes of our monolayers. Our study shows that forces arising from epithelial growth in three-dimensional confinement are sufficient to drive folding by buckling.

Deterministic actin waves as generators of cell polarization cues.

Dendritic cells "patrol" the human body to detect pathogens. In their search, dendritic cells perform a random walk by amoeboid migration. The efficiency of pathogen detection depends on the properties of the random walk. It is not known how the dendritic cells control these properties. Here, we quantify dendritic cell migration under well-defined 2-dimensional confinement and in a 3-dimensional collagen matrix through recording their long-term trajectories. We find 2 different migration states: persistent migration, during which the dendritic cells move along curved paths, and diffusive migration, which is characterized by successive sharp turns. These states exhibit differences in the actin distributions. Our theoretical and experimental analyses indicate that this kind of motion can be generated by spontaneous actin polymerization waves that contribute to dendritic cell polarization and migration. The relative distributions of persistent and diffusive migration can be changed by modification of the molecular actin filament nucleation and assembly rates. Thus, dendritic cells can control their migration patterns and adapt to specific environments. Our study offers an additional perspective on how dendritic cells tune their searches for pathogens.

Accuracy of position determination in Ca

A living cell senses its environment and responds to external signals. In this paper, we study theoretically the precision at which cells can determine the position of a spatially localized transient extracellular signal. To this end, we focus on the case where the stimulus is converted into the release of a small molecule that acts as a second messenger, for example, Ca^{2+}, and activates kinases that change the activity of enzymes by phosphorylating them. We analyze the spatial distribution of phosphorylation events using stochastic simulations as well as a mean-field approach. Kinases that need to bind to the cell membrane for getting activated provide more accurate estimates than cytosolic kinases. Our results could explain why the rate of Ca^{2+} detachment from the membrane-binding conventional protein kinase Cα is larger than its phosphorylation rate.

Positional Information Readout in Ca

Living cells respond to spatially confined signals. Intracellular signal transmission often involves the release of second messengers like Ca^{2+}. They eventually trigger a physiological response, for example, by activating kinases that in turn activate target proteins through phosphorylation. Here, we investigate theoretically how positional information can be accurately read out by protein phosphorylation in spite of rapid second messenger diffusion. We find that accuracy is increased by binding of kinases to the cell membrane prior to phosphorylation and by increasing the rate of Ca^{2+} loss from the cell interior. These findings could explain some salient features of the conventional protein kinase Cα.

Sound of an axon's growth.

Axons are linear structures of nerve cells that can range from a few tens of micrometers up to meters in length. In addition to external cues, the length of an axon is also regulated by unknown internal mechanisms. Molecular motors have been suggested to generate oscillations with an axon-length-dependent frequency that could be used to measure an axon's extension. Here, we present a mechanism for determining the axon length that couples the mechanical properties of an axon to the spectral decomposition of the oscillatory signal.

Aurora A depletion reveals centrosome-independent polarization mechanism in Caenorhabditis elegans.

How living systems break symmetry in an organized manner is a fundamental question in biology. In wild-type Caenorhabditis elegans zygotes, symmetry breaking during anterior-posterior axis specification is guided by centrosomes, resulting in anterior-directed cortical flows and a single posterior PAR-2 domain. We uncover that C. elegans zygotes depleted of the Aurora A kinase AIR-1 or lacking centrosomes entirely usually establish two posterior PAR-2 domains, one at each pole. We demonstrate that AIR-1 prevents symmetry breaking early in the cell cycle, whereas centrosomal AIR-1 instructs polarity initiation thereafter. Using triangular microfabricated chambers, we establish that bipolarity of air-1(RNAi) embryos occurs effectively in a cell-shape and curvature-dependent manner. Furthermore, we develop an integrated physical description of symmetry breaking, wherein local PAR-2-dependent weakening of the actin cortex, together with mutual inhibition of anterior and posterior PAR proteins, provides a mechanism for spontaneous symmetry breaking without centrosomes.

FtsZ filaments have the opposite kinetic polarity of microtubules.

FtsZ is the ancestral homolog of tubulin and assembles into the Z ring that organizes the division machinery to drive cell division in most bacteria. In contrast to tubulin that assembles into 13 stranded microtubules that undergo dynamic instability, FtsZ assembles into single-stranded filaments that treadmill to distribute the peptidoglycan synthetic machinery at the septum. Here, using longitudinal interface mutants of FtsZ, we demonstrate that the kinetic polarity of FtsZ filaments is opposite to that of microtubules. A conformational switch accompanying the assembly of FtsZ generates the kinetic polarity of FtsZ filaments, which explains the toxicity of interface mutants that function as a capper and reveals the mechanism of cooperative assembly. This approach can also be employed to determine the kinetic polarity of other filament-forming proteins.

Effects of geometry and topography on Min-protein dynamics.

In the rod-shaped bacterium Escherichia coli, the center is selected by the Min-proteins as the site of cell division. To this end, the proteins periodically translocate between the two cell poles, where they suppress assembly of the cell division machinery. Ample evidence notably obtained from in vitro reconstitution experiments suggests that the oscillatory pattern results from self-organization of the proteins MinD and MinE in presence of a membrane. A mechanism built on cooperative membrane attachment of MinD and persistent MinD removal from the membrane induced by MinE has been shown to be able to reproduce the observed Min-protein patterns in rod-shaped E. coli and on flat supported lipid bilayers. Here, we report our results of a numerical investigation of patterns generated by this mechanism in various geoemtries. Notably, we consider the dynamics on membrane patches of different forms, on topographically structured lipid bilayers, and in closed geometries of various shapes. We find that all previously described patterns can be reproduced by the mechanism. However, it requires different parameter sets for reproducing the patterns in closed and in open geometries.