Numerical study of clathrin-mediated endocytosis of nanoparticles by cells under tension

In this study, a three-dimensional mathematical model was used to study the contribution of clathrins during the process of cellular uptake of spherical nanoparticles under different membrane tensions. The clathrin-coated pit (CCP) that forms around the inward budding of the cell membrane was modeled as a vesicle with bending rigidity. An optimization algorithm was proposed for minimizing the total energy of the system, which comprises the deforming nanoparticle, receptor-ligand bonds, cell membrane, and CCP, in which way, the profile of the system is acquired. The results showed that the CCP enable full wrapping of the nanoparticles at various membrane tensions. When the cell membrane tension increases, the total deformation energy also increases, but the ratio of CCP bending to the minimum value of the total energy of the system decreases. The results also showed that the diameter of the endocytic vesicles determined by the competition between the stretching of the cell membrane and confinement of the coated pits are much larger than the nanoparticles, which is quit different as the results in passive endocytosis that is not facilitated by the CCPs. The present results indicate that variations of tension on cell membranes constitutes a biophysical marker for understanding the size distribution of CCPs observed in experiments. The present results also suggest that the early abortion of endocytosis is related to that the receptor-ligand bonds cannot generate adequate force to wrap the nanoparticles into the cell membrane before the clathrins respond to support the endocytic vesicles. Correspondingly, late abortion may relate to the inability of CCPs to confine the nanoparticles until the occurrence of the necking stage of endocytosis.

A Numerical Study of Passive Receptor-Mediated Endocytosis of Nanoparticles:The Effect of Mechanical Properties

In this work,a three-dimensional axisymmetric model with nanoparticle,receptor-ligand bonds and cell membrane as a system was used to study the quasi-static receptor-mediated endocytosis process of spherical nanoparticles in drug delivery.The minimization of the system energy function was carried out numerically,and the deformations of nanoparticle,receptor-ligand bonds and cell membrane were predicted.Results show that passive endocytosis may fail due to the rupture of receptor-ligand bonds during the wrapping process,and the size and rigidity of nanoparticles affect the total deformation energy and the terminal wrapping stage.Our results suggest that,in addition to the energy requirement,the success of passive endocytosis also depends on the maximum strength of the receptor-ligand bonds.

2008 Stability Analysis of the Immersed Boundary Method for a Two-Dimensional Membrane with Bending Rigidity

In this paper,we analyze the stability of the Immersed Boundary Methodapplied to a membrane-fluid system with a plasma membrane immersed in an incompressibleviscous fluid.We show that for small deformations,the planar rest state isstable for a membrane with bending rigidity.The smoothed version,using a standardregularization technique for the singular force,is also shown to be stable.Furthermore,we show that the coupled fluid-membrane system is stiff and smoothing helpsto reduce the stiffness.Compared to the system of elastic fibers immersed in an incompressiblefluid,membrane with bending rigidity consist of a wider range of decayrates.Therefore numerical instability could occur more easily for an explicit methodwhen the time step size is not sufficiently small,even though the continuous problemis stable.

A Full Eulerian Fluid-Membrane Coupling Method with a Smoothed Volume-of-Fluid Approach

A novel full Eulerian fluid-elastic membrane coupling method on the fixed Cartesian coordinate mesh is proposed within the framework of the volume-of-fluid approach.The present method is based on a full Eulerian fluid-(bulk)structure coupling solver(Sugiyama et al.,J.Comput.Phys.,230(2011)596–627),with the bulk structure replaced by elastic membranes.In this study,a closed membrane is considered,and it is described by a volume-of-fluid or volume-fraction information generally called VOF function.A smoothed indicator(or characteristic)function is introduced as a phase indicator which results in a smoothed VOF function.This smoothed VOF function uses a smoothed delta function,and it enables a membrane singular force to be incorporated into a mixture momentum equation.In order to deal with a membrane deformation on the Eulerian mesh,a deformation tensor is introduced and updated within a compactly supported region near the interface.Both the neo-Hookean and the Skalak models are employed in the numerical simulations.A smoothed(and less dissipative)interface capturing method is employed for the advection of the VOF function and the quantities defined on the membrane.The stability restriction due to membrane stiffness is relaxed by using a quasi-implicit approach.The present method is validated by using the spherical membrane deformation problems,and is applied to a pressure-driven flow with the biconcave membrane capsules(red blood cells).

A dynamic model of calcium signaling in mast cells and LTC_4 release induced by mechanical stimuli

Mast cells(MCs) play an important role in the immune system. It is known that mechanical stimuli can induce intracellular Ca2+signal and release a variety of mediators, including leukotriene C4(LTC4), leading to other cellular and physiological changes. In this paper, we present a mathematical model to explore signalling pathways in MCs, by including cellular mechanisms for intracellular Ca2t increase and LTC4release in response to mechanical stimuli, thapsigargin(TG, SERCA pump inhibitor), and LTC4 stimuli. We show that(i) mechanical stimuli activate mechano-sensitive ion channels and induce inward ion fluxes and Ca2?entry which increases intracellular Ca2+concentration and releases LTC4;(ii) TG inhibits SERCA pumps, empties the internal Ca2+ stores,which activates Ca2+release-activated Ca2+channels and results in sustained intracellular Ca2+increase; and(iii)LTC4activates receptors on MCs surface and increases intracellular Ca2+concentration. Our results are consistent with experimental observations, and furthermore, they also reveal that mechanical stimuli can increase intracellular Ca2+even when LTC4release is blocked, which suggests a feed forward loop involved in LTC4production. This study may facilitate our understanding of the mechanotransduction process in MCs and provide a useful modeling tool for quantitatively analyzing immune mechanisms involving MCs.

Numerical Simulation of Moving Contact Lines with Surfactant by Immersed Boundary Method

In this paper,we present an immersed boundary method for simulating moving contact lines with surfactant.The governing equations are the incompressible Navier-Stokes equations with the usual mixture of Eulerian fluid variables and Lagrangian interfacial markers.The immersed boundary force has two components:one from the nonhomogeneous surface tension determined by the distribution of surfactant along the fluid interface,and the other from unbalanced Young’s force at the moving contact lines.An artificial tangential velocity has been added to the Lagrangian markers to ensure that the markers are uniformly distributed at all times.The corresponding modified surfactant equation is solved in a way such that the total surfactant mass is conserved.Numerical experiments including convergence analysis are carefully conducted.The effect of the surfactant on the motion of hydrophilic and hydrophobic drops are investigated in detail.

Numerical Simulation for a Droplet Fission Process of Electrowetting on Dielectric Device

Electrowetting has been proposed as a technique for manipulating dropletssurrounded by air or oil. In this paper, we discuss the modeling and simulation of thedroplet fission process between two parallel plates inside an electrowetting on dielectric (EWOD) device. Since the gap between the plates is small, we use the two-phaseHele-Shaw flow as a model. While there are several high order methods around, suchas the immersed interface methods [1, 2], we decide to use two first-order methods forsimplicity. A ghost-fluid (GF) method is employed to solve the governing equationsand a local level set method is used to track the drop interface. For comparison purposes, the same set of two-phase Hele-Shaw equations are also solved directly usingthe immersed boundary (IB) method. Numerical results are consistent with experimental observations reported in the literature.

Coarse Grained Molecular Dynamics Simulation of Interaction between Hemagglutinin Fusion Peptides and Lipid Bilayer Membranes

Microscopic level interaction between fusion-peptides and lipid bilayer membranes plays a crucial role in membrane fusion,a key step of viral infection.In this paper,we use coarse-grained molecular dynamics(CGMD)simulations to study the interaction between hemagglutinin fusion-peptides and phospholipid bilayer membranes.With CGMD,we are able to simulate the interaction of fusion peptides with a relatively large piece of membrane for a sufficiently long time period,which is necessary for a detailed understanding of the fusion process.A conformation of the peptide with a kink at the level of phosphate group is obtained,consistent with NMR and EPR studies.Our results show that the N-terminal segment of the peptide inserts more deeply into the membrane bilayer compared to the C-terminal segment,as observed in previous experiments.Our simulations also show that the presence of fusion peptides inside the membrane may cause bilayer thinning and lipid molecule disorder.Finally,our results reveal that peptides tend to aggregate,indicating cluster formation as seen in many experiments.

A Front-Tracking Method for Motion by Mean Curvature with Surfactant

In this paper,we present a finite difference method to track a network of curves whose motion is determined by mean curvature.To study the effect of inhomogeneous surface tension on the evolution of the network of curves,we include surfactant which can diffuse along the curves.The governing equations consist of one parabolic equation for the curve motion coupled with a convection-diffusion equation for the surfactant concentration along each curve.Our numerical method is based on a direct discretization of the governing equations which conserves the total surfactant mass in the curve network.Numerical experiments are carried out to examine the effects of inhomogeneous surface tension on the motion of the network,including the von Neumann law for cell growth in two space dimensions.

ASimplified Neuronal Model for the Instigation and Propagation of Cortical Spreading Depression

In this paper,we construct a simplified neuronal model that is capable of simulating the instigation of cortical spreading depression(CSD)and propagation of a CSD wave.Our model is a simplification and extension of a single neuron model proposed in the literature for studying the instigation of CSD.Using the simplified neuronal model,we construct a network of these simplified neurons.This network model shows that the propagation of a CSD wave occurs naturally after it is instigated electrically or chemically.Although the model is simple,the speed of the CSD wave predicted by our model is consistent with experimentally observed values.Finally,our model allows us to investigate the effects of specific ion channels on the spread of a CSD wave.

Effect of Ocean Iron Fertilization on the Phytoplankton Biological Carbon Pump

It has been proposed that photosynthetic plankton can be used as a biological carbon pump tp absorb and sequester carbon dioxide in the ocean.In this paper,plankton population dynamics are simulated in a single stratified water column to predict carbon dioxide sequestering due to surface iron fertilization in deep ocean.Using a predator-prey model and realistic parameter values,iron fertilization was found to only cause temporary blooms up to 5 months in duration,and relatively small increases in adsorption of atmospheric CO_(2).

Influenza Viral Membrane Deformation Due to Refolding of HA-Protein:Two-dimensional Model and Analysis

In this paper we study influenza viral membrane deformation related to the refolding of Hemagglutinin(HA)protein.The focus of the paper is to understand membrane deformation and budding due to experimentally observed linear HA-protein clusters,which have not been mathematically studied before.The viral membrane is modeled as a two dimensional incompressible lipid bilayer with bending rigidity.For tensionless membranes,we derive an analytical solution while for membrane under tension we solve the problem numerically.Our solution for tensionless membranes shows that the height of membrane deformation increases monotonically with the bending moment exerted by HA-proteins and attains its maximum when the size of the protein cluster reaches a critical value.Our results also show that the hypothesis of dimple formation proposed in the literature is valid in the two dimensional setting.Our comparative study of axisymmetric HA-clusters and linear HA-clusters reveals that the linear HA-clusters are not favorable to provide a sufficient energy required to overcome an energy barrier for a successful fusion,despite their capability to cause membrane deformation and budding.

A Constrained Level Set Method for Simulating the Formation of Liquid Bridges

In this paper,we investigate the dynamic process of liquid bridge formation between two parallel hydrophobic plates with hydrophilic patches,previously studied in[1].We propose a dynamic Hele-Shaw model to take advantage of the small aspect ratio between the gap width and the plate size.A constrained level set method is applied to solve the model equations numerically,where a global constraint is imposed in the evolution[2]stage together with local constraints in the reinitialization[3]stage of level set function in order to limit numerical mass loss.In contrast to the finite element method used in[2],we use a finite difference method with a 5th order HJWENO scheme for spatial discretization.To illustrate the effectiveness of the constrained method,we have compared the results obtained by the standard level set method with those from the constrained version.Our results show that the constrained level set method produces physically reasonable results while that of the standard method is less reliable.Our numerical results also show that the dynamic nature of the flow plays an important role in the process of liquid bridge formation and criteria based on static energy minimization approach has limited applicability.

Hydrophobic Effect in a Continuum Model of the Lipid Bilayer

We study a continuum paradigm of the lipid bilayer based on minimizing the free energy of a mixture of water and lipid molecules.This paper extends previous work of Blom and Peletier[European J.Appl.Math.,15(2004),pp.487-508]in the following ways.(a)It formulates a more general model of the hydrophobic effect to facilitate connections with microscale simulations and first-principles analysis.(b)It clarifies the meaning and role of the model parameters.(c)It outlines a method for determining parameter values so that physically-realistic bilayer density profiles can be obtained,for example for use in macroscale simulations.Points(a)-(c)suggest that the model has potential to robustly connect some micro-and macroscale levels of multiscale blood flow simulations.The mathematical modelling in point(a)is based upon a consideration of the underlying physics of inter-molecular forces.The governing equations thus obtained are minimized by gradient flows via a novel numerical approach;this enables point(b).The numerical results are shown to behave physically in terms of the effect of background concentration,in contrast to the earlier model which is shown here to not display the expected behaviour.A“short-tail”approximation of the lipid molecules also gives an analytical tool which yields critical values of some parameters under certain conditions.Point(c)involves the first quantitative comparison of the numerical data with physical experimental results.