Phase plane dynamics of ERK phosphorylation.

The extracellular signal-regulated kinase (ERK) controls multiple critical processes in the cell and is deregulated in human cancers, congenital abnormalities, immune diseases, and neurodevelopmental syndromes. Catalytic activity of ERK requires dual phosphorylation by an upstream kinase, in a mechanism that can be described by two sequential Michaelis-Menten steps. The estimation of individual reaction rate constants from kinetic data in the full mechanism has proved challenging. Here, we present an analytically tractable approach to parameter estimation that is based on the phase plane representation of ERK activation and yields two combinations of six reaction rate constants in the detailed mechanism. These combinations correspond to the ratio of the specificities of two consecutive phosphorylations and the probability that monophosphorylated substrate does not dissociate from the enzyme before the second phosphorylation. The presented approach offers a language for comparing the effects of mutations that disrupt ERK activation and function in vivo. As an illustration, we use phase plane representation to analyze dual phosphorylation under heterozygous conditions, when two enzyme variants compete for the same substrate.

Quartz Crystal Microbalance Frequency Response to Discrete Adsorbates in Liquids.

Quartz crystal microbalance with dissipation monitoring (QCM-D) has become a major tool enabling accurate investigation of the adsorption kinetics of nanometric objects such as DNA fragments, polypeptides, proteins, viruses, liposomes, polymer, and metal nanoparticles. However, in liquids, a quantitative analysis of the experimental results is often intricate because of the complex interplay of hydrodynamic and adhesion forces varying with the physicochemical properties of adsorbates and functionalized QCM-D sensors. In the present paper, we dissect the role of hydrodynamics for the analytically tractable case of stiff contact, whereas the adsorbed rigid particles oscillate with the resonator without rotation. Under the assumption of the low surface coverage, we theoretically study the excess shear force exerted on the resonator, which has two contributions: (i) the fluid-mediated force due to flow disturbance created by the particle and (ii) the force exerted on the particle by the fluid and transmitted to the sensor via contact. The theoretical analysis enables an accurate interpretation of the QCM-D impedance measurements. It is demonstrated inter alia that for particles of the size comparable with protein molecules, the hydrodynamic force dominates over the inertial force and that the apparent mass derived from QCM independently of the overtone is about 10 times the Sauerbrey (inertial) mass. The theoretical results show excellent agreement with the results of experiments and advanced numerical simulations for a wide range of particle sizes and oscillation frequencies.

Protrusion membrane pearling emerges during 3D cell division.

Cell division is accompanied by dramatic changes in shape that ultimately lead to the physical separation of one cell into two. In 2D microenvironments, cells round up and remain adhered onto the substrate by thin retraction fibers during division. In contrast, in 3D environments, cells divide exhibiting long protrusions that guide the orientation of the division axis. However, the mechanism of cell division in three dimensions still remains poorly understood. Here we report the spontaneous formation of transient quasiperiodic membrane pearling on extended mitotic protrusions during 3D cell division. Protrusion membrane pearling may be initiated by the non-uniform distribution of focal adhesions and consequent stationary instability of the protrusive membrane. Overall, membrane pearling emergence may provide insights into a novel modality of 3D cell division with potential physiological relevance.

Myosin Clusters of Finite Size Develop Contractile Stress in 1D Random Actin Arrays.

Myosin-powered force generation and contraction in nonmuscle cells underlies many cell biological processes and is based on contractility of random actin arrays. This contractility must rely on a microscopic asymmetry, the precise mechanism of which is not completely clear. A number of models of mechanical and structural asymmetries in actomyosin contraction have been posited. Here, we examine a contraction mechanism based on a finite size of myosin clusters and anisotropy of force generation by myosin heads at the ends of the myosin clusters. We use agent-based numerical simulations to demonstrate that if average lengths of actin filaments and myosin clusters are similar, then the proposed microscopic asymmetry leads to effective contraction of random 1D actomyosin arrays. We discuss the model's implication for mechanics of contractile rings and stress fibers.

A Combination of Actin Treadmilling and Cross-Linking Drives Contraction of Random Actomyosin Arrays.

We investigate computationally the self-organization and contraction of an initially random actomyosin ring. In the framework of a detailed physical model for a ring of cross-linked actin filaments and myosin-II clusters, we derive the force balance equations and solve them numerically. We find that to contract, actin filaments have to treadmill and to be sufficiently cross linked, and myosin has to be processive. The simulations reveal how contraction scales with mechanochemical parameters. For example, they show that the ring made of longer filaments generates greater force but contracts slower. The model predicts that the ring contracts with a constant rate proportional to the initial ring radius if either myosin is released from the ring during contraction and actin filaments shorten, or if myosin is retained in the ring, while the actin filament number decreases. We demonstrate that a balance of actin nucleation and compression-dependent disassembly can also sustain contraction. Finally, the model demonstrates that with time pattern formation takes place in the ring, worsening the contractile process. The more random the actin dynamics are, the higher the contractility will be.

On the origins of the mitotic shift in proliferating cell layers.

BACKGROUND: During plant and animal development, monolayer cell sheets display a stereotyped distribution of polygonal cell shapes. In interphase cells these shapes range from quadrilaterals to decagons, with a robust average of six sides per cell. In contrast, the subset of cells in mitosis exhibits a distinct distribution with an average of seven sides. It remains unclear whether this 'mitotic shift' reflects a causal relationship between increased polygonal sidedness and increased division likelihood, or alternatively, a passive effect of local proliferation on cell shape. METHODS: We use a combination of probabilistic analysis and mathematical modeling to predict the geometry of mitotic polygonal cells in a proliferating cell layer. To test these predictions experimentally, we use Flp-Out stochastic labeling in the Drosophila wing disc to induce single cell clones, and confocal imaging to quantify the polygonal topologies of these clones as a function of cellular age. For a more generic test in an idealized cell layer, we model epithelial sheet proliferation in a finite element framework, which yields a computationally robust, emergent prediction of the mitotic cell shape distribution. RESULTS: Using both mathematical and experimental approaches, we show that the mitotic shift derives primarily from passive, non-autonomous effects of mitoses in neighboring cells on each cell's geometry over the course of the cell cycle. Computationally, we predict that interphase cells should passively gain sides over time, such that cells at more advanced stages of the cell cycle will tend to have a larger number of neighbors than those at earlier stages. Validating this prediction, experimental analysis of randomly labeled epithelial cells in the Drosophila wing disc demonstrates that labeled cells exhibit an age-dependent increase in polygonal sidedness. Reinforcing these data, finite element simulations of epithelial sheet proliferation demonstrate in a generic framework that passive side-gaining is sufficient to generate a mitotic shift. CONCLUSIONS: Taken together, our results strongly suggest that the mitotic shift reflects a time-dependent accumulation of shared cellular interfaces over the course of the cell cycle. These results uncover fundamental constraints on the relationship between cell shape and cell division that should be general in adherent, polarized cell layers.

Modeling the Evolution of S. pombe Populations with Multiple Killer Meiotic Drivers.

Meiotic drivers are selfish genetic loci that can be transmitted to more than half of the viable gametes produced by a heterozygote. This biased transmission gives meiotic drivers an evolutionary advantage that can allow them to spread over generations until all members of a population carry the driver. This evolutionary power can also be exploited to modify natural populations using synthetic drivers known as 'gene drives.' Recently, it has become clear that natural drivers can spread within genomes to birth multicopy gene families. To understand intragenomic spread of drivers, we model the evolution of two or more distinct meiotic drivers in a population. We employ the wtf killer meiotic drivers from Schizosaccharomyces pombe, which are multicopy in all sequenced isolates, as models. We find that a duplicate wtf driver identical to the parent gene can spread in a population unless, or until, the original driver is fixed. When the duplicate driver diverges to be distinct from the parent gene, we find that both drivers spread to fixation under most conditions, but both drivers can be lost under some conditions. Finally, we show that stronger drivers make weaker drivers go extinct in most, but not all, polymorphic populations with absolutely linked drivers. These results reveal the strong potential for natural meiotic drive loci to duplicate and diverge within genomes. Our findings also highlight duplication potential as a factor to consider in the design of synthetic gene drives.

Gas-assisted microfluidic step-emulsification for generating micron- and submicron-sized droplets.

Micron- and submicron-sized droplets have extensive applications in biomedical diagnosis and drug delivery. Moreover, accurate high-throughput analysis requires a uniform droplet size distribution and high production rates. Although the previously reported microfluidic coflow step-emulsification method can be used to generate highly monodispersed droplets, the droplet diameter (d) is constrained by the microchannel height (b), d≳3b, while the production rate is limited by the maximum capillary number of the step-emulsification regime, impeding emulsification of highly viscous liquids. In this paper, we report a novel, gas-assisted coflow step-emulsification method, where air serves as the innermost phase of a precursor hollow-core air/oil/water emulsion. Air gradually diffuses out, producing oil droplets. The size of the hollow-core droplets and the ultrathin oil layer thickness both follow the scaling laws of triphasic step-emulsification. The minimal droplet size attains d≈1.7b, inaccessible in standard all-liquid biphasic step-emulsification. The production rate per single channel is an order-of-magnitude higher than that in the standard all-liquid biphasic step-emulsification and is also superior to alternative emulsification methods. Due to low gas viscosity, the method can also be used to generate micron- and submicron-sized droplets of high-viscosity fluids, while the inert nature of the auxiliary gas offers high versatility.

Nature of the Poynting Vector Field Singularities in Resonant Light Scattering by Nanoparticles.

Singularities of the Poynting vector field subwavelength patterns in resonant light scattering by nanoparticles are discussed and classified. There are two generic types of the singularities, namely, (i) the singularities related to the vanishing of the magnetic (and/or electric) field at the singular points and (ii) the singularities related to the formation of standing waves in proximity to the singular points. The connection of these types of singularities to the topology of the singular points, space dimension (3D vs. 2D), and energy conservation law are revealed. In particular, it is shown that in 2D cases in non-dissipative media, the energy conservation reduces the possible types of generic singular points to saddles and centers only. In 3D cases, a universal expression connecting different components of the Poynting vector and valid for any generic singularities is derived and numerically checked for various types of singular points.

Correction: Shape-controlled anisotropy of superparamagnetic micro-/nanohelices.

Correction for 'Shape-controlled anisotropy of superparamagnetic micro-/nanohelices' by Alexander M. Leshansky et al., Nanoscale, 2016, 8, 14127-14138, https://doi.org/10.1039/C6NR01803C.

The Poynting Vector Field Generic Singularities in Resonant Scattering of Plane Linearly Polarized Electromagnetic Waves by Subwavelength Particles.

We present the results of a study of the Poynting vector field generic singularities at the resonant light scattering of a plane monochromatic linearly polarized electromagnetic wave by a subwavelength particle. We reveal the impact of the problem symmetry, the spatial dimension, and the energy conservation law on the properties of the singularities. We show that, in the cases when the problem symmetry results in the existence of an invariant plane for the Poynting vector field lines, a formation of a standing wave in the immediate vicinity of a singularity gives rise to a saddle-type singular point. All other types of singularities are associated with vanishing at the singular points, either (i) magnetic field, for the polarization plane parallel to the invariant plane, or (ii) electric field, at the perpendicular orientation of the polarization plane. We also show that in the case of two-dimensional problems (scattering by a cylinder), the energy conservation law restricts the types of possible singularities only to saddles and centers in the non-dissipative media and to saddles, foci, and nodes in dissipative. Finally, we show that dissipation affects the (i)-type singularities much stronger than the (ii)-type. The same conclusions are valid for the imaginary part of the Poynting vector in problems where the latter is regarded as a complex quantity. The singular points associated with the formation of standing waves are different for real and imaginary parts of this complex vector field, while all other singularities are common. We illustrate the general discussion by analyzing singularities at light scattering by a subwavelength Germanium cylinder with the actual dispersion of its refractive index.

Double emulsions with ultrathin shell by microfluidic step-emulsification.

Double emulsions with ultrathin shells are important in some biomedical applications, such as controlled drug release. However, the existing production techniques require two or more manipulation steps, or more complicated channel geometry, to form thin-shell double emulsions. This work presents a novel microfluidic tri-phasic step-emulsification device, with an easily fabricated double-layer PDMS channel, for production of oil-in-oil-in-water and water-in-water-in-oil double emulsions in a single step. The shell thickness is controlled by the flow rates and can reach 1.4% of the µm-size droplet diameter. Four distinct emulsification regimes are observed depending on the experimental conditions. A theoretical model for the tri-phasic step-emulsification is proposed to predict the boundaries separating the four regimes of emulsification in plane of two dimensionless capillary numbers, Ca. The theory yields two coupled nonlinear differential equations that can be solved numerically to find the approximate shape of the free interfaces in the shallow (Hele-Shaw) microfluidic channel. This approximation is then used as the initial guess for the more accurate finite element method solution, showing very good agreement with the experimental findings. This study demonstrates the feasibility of co-flow step-emulsification as a promising method to production of double (and multiple) emulsions and micro-capsules with ultrathin shells of controllable thickness.

Modeling Propulsion of Soft Magnetic Nanowires.

The emergent interest in artificial nanostructures that can be remotely navigated a specific location in a fluidic environment is motivated by the enormous potential this technology offers to biomedical applications. Originally, bio-inspired micro-/nanohelices driven by a rotating magnetic field were proposed. However, fabrication of 3D helical nanostructures is complicated. One idea to circumvent complex microfabrication is to use 1D soft magnetic nanowires that acquire chiral shape when actuated by a rotating field. The paper describes the comprehensive numerical approach for modeling propulsion of externally actuated soft magnetic nanowires. The proposed bead-spring model allows for arbitrary filament geometry and flexibility and takes rigorous account of intra-filament hydrodynamic interactions. The comparison of the numerical predictions with the previous experimental results on propulsion of composite two-segment (Ni-Ag) nanowires shows an excellent agreement. Using our model we could substantiate and rationalize important and previously unexplained details, such as bidirectional propulsion of three-segment (Ni-Ag-Au) nanowires.

The membrane periodic skeleton is an actomyosin network that regulates axonal diameter and conduction.

Neurons have a membrane periodic skeleton (MPS) composed of actin rings interconnected by spectrin. Here, combining chemical and genetic gain- and loss-of-function assays, we show that in rat hippocampal neurons the MPS is an actomyosin network that controls axonal expansion and contraction. Using super-resolution microscopy, we analyzed the localization of axonal non-muscle myosin II (NMII). We show that active NMII light chains are colocalized with actin rings and organized in a circular periodic manner throughout the axon shaft. In contrast, NMII heavy chains are mostly positioned along the longitudinal axonal axis, being able to crosslink adjacent rings. NMII filaments can play contractile or scaffolding roles determined by their position relative to actin rings and activation state. We also show that MPS destabilization through NMII inactivation affects axonal electrophysiology, increasing action potential conduction velocity. In summary, our findings open new perspectives on axon diameter regulation, with important implications in neuronal biology.

Shape-controlled anisotropy of superparamagnetic micro-/nanohelices.

Micro-/nanopropellers can be actuated remotely by a rotating magnetic field and steered at high precision through various fluidic environments. Recent progress comprises microfabrication of superparamagnetic microhelices not possessing remanent magnetization, but rather magnetized by an applied magnetic field. In this article we present a numerical approach for computing, from first principles, the effective susceptibility of polarizable helical micro-/nanopropellers. We show that nanopropeller geometry, in particular, filament cross-section elongation and orientation, play a central role in determining its magnetic anisotropy and polarizability. The numerical predictions are in qualitative agreement with the previously reported experiments, showing that tight polarizable helices are propulsive. The numerical results are also supported by the approximate slender-body theory. Finally, we propose a semi-quantitative energy criterion to rank polarizable helices with different geometries of the filament by their propulsive capacity and also estimate their maximal propulsion speed.

Long-term dynamics of multisite phosphorylation.

Multisite phosphorylation cycles are ubiquitous in cell regulation systems and are studied at multiple levels of complexity, from molecules to organisms, with the ultimate goal of establishing predictive understanding of the effects of genetic and pharmacological perturbations of protein phosphorylation in vivo. Achieving this goal is essentially impossible without mathematical models, which provide a systematic framework for exploring dynamic interactions of multiple network components. Most of the models studied to date do not discriminate between the distinct partially phosphorylated forms and focus on two limiting reaction regimes, distributive and processive, which differ in the number of enzyme-substrate binding events needed for complete phosphorylation or dephosphorylation. Here we use a minimal model of extracellular signal-related kinase regulation to explore the dynamics of a reaction network that includes all essential phosphorylation forms and arbitrary levels of reaction processivity. In addition to bistability, which has been studied extensively in distributive mechanisms, this network can generate periodic oscillations. Both bistability and oscillations can be realized at high levels of reaction processivity. Our work provides a general framework for systematic analysis of dynamics in multisite phosphorylation systems.

Theory of axisymmetric pendular rings.

We present the theory of liquid bridges between two solids, sphere and plane, with prescribed contact angles. We give explicit expressions for curvature, volume and surface area of pendular ring as functions of the filling angle ψ for all available types of menisci: catenoid, sphere, cylinder, nodoid and unduloid (the meridional profile of the latter may have inflection points). There exists a rich set of solutions of the Young-Laplace equation for the shape of an axisymmetric meniscus of constant mean curvature. In case when the solids do not contact each other, these solutions extend Plateau's sequence of meniscus evolution observed with increase of the liquid volume to include the unduloids at small filling angle, unduloids with multiple inflection points and multiple catenoids. The Young-Laplace equation with boundary conditions can be viewed as a nonlinear eigenvalue problem. Its unduloid solutions, menisci shapes and curvatures H(n)(s)(ψ), exhibit a discrete spectrum and are enumerated by two indices: the number n of inflection points on the meniscus meridional profile M and the convexity index s=±1 determined by the shape of a segment of M contacting the solid sphere: the shape is either convex, s=1, or concave, s=-1. For the fixed contact angles the set of the functions H(n)(s)(ψ) behaves in such a way that in the plane {ψ,H} there exists a bounded domain where H(n)(s)(ψ) do not exist for any distance between solids. The curves H(n)(s)(ψ) may be tangent to the boundary of domain which is a smooth closed curve. This topological representation allows to classify possible curves and introduce a saddle point notion. We observe several types of saddle points, and give their classification.