### Single Particle Motions In Plasmas | Matjale C Mabula - nischengeschaeft.de

Skip to main content. Log In Sign Up. Motion of nanobeads proximate to plasma membranes during single particle tracking. Motion of nanobeads proximate to plasma membranes during single particle tracking Bulletin of mathematical biology, David Broday.

Bulletin of Mathematical Biology 64, — doi: The model considers a bead that translates proximate to a rigid planar interface that separates two distinct Brinkman media. The hydrodynamic resistance is calculated numerically by a modified boundary integral equation for- mulation, where the pertinent boundary conditions result in a hybrid system of Fredholm delta sigma theta chants youtube er of the first and second kinds.

The hydrodynamic resistance on the translating bead catalpa health maroc calculated for different combinations of the Brinkman screening lengths in the two layers, and for different viscosity ratios. Depending on the bead—membrane separation and on the hydrodynamic properties of both the plasma membrane and the pericellular layer, the drag on the bead may be affected by the properties of the plasma membrane.

The Stokes—Einstein relation is applied for calculating the diffusivity of probes colloidal gold nanobeads attached to gly- colipids in the plasma membrane. This approach provides an alternative way for the interpretation of in vitro observations during single particle tracking procedure, and predicts new properties of the plasma membrane structure. Published by Elsevier Science Ltd. All rights reserved. For example, in a single particle tracking SPT pro- cedure the motion of individual nanobeads, presumably attached firmly to mem- brane glycoproteins or glycolipids, is tracked by a nanovid microscopy technique.

Specifically, diffusion coefficients of the complexes are calculated by analyzing video images of trajectories of the beads. This common tool is used to study properties of the plasma membrane Sheetz et al. Broday and Sako,and the dynamics of transmembrane integrins and other adhesion molecules Schmidt et al.

Usually, the bead diameter ranges between 20 and 40 nm, although larger latex beads of diameters as large as and nm are sometimes used as well Sako and Kusumi, ; Sako et al. It is not clear whether the beads are directly attached to the apices of membrane glycoproteins or to glycosaminoglycans and other polysaccharide single particle motion in plasma pdf extending sideways from the core protein as a result of mutual repulsive electrostatic forces.

The polysaccharides and the ectodomains of glycolipids and glycoproteins form the surface glycocalyx, which prevails in most living cells Lee single particle motion in plasma pdf al. The protein—bead and lipid—bead complexes move in two distinct media. The bead and the upper portion of the membrane protein the ectodomain move in the pericellular layer adjacent to the membrane outer face. The thickness of the pericellular layer extends to about — nm Adamson and Clough, ; Lee et al.

The body of the membrane protein moves, however, in the bilayer phospholipid plasma membrane. The thickness of the plasma mem- brane is 4—10 nm, and the ratio of its viscosity to that single particle motion in plasma pdf the pericellular layer is of the order of O 10 Lee et al.

Single particle motion in plasma pdf the plasma membrane and the peri- cellular layer can be described as viscous fluid layers in which obstructions are present Zhang et al. These obstructions may be mobile or immobile, the latter representing integral membrane proteins and glycoproteins, such as adhesion receptors of the integrin family Burridge et al. Unfor- tunately, data on the permeabilities of these layers, required for their representation in terms of a hydrodynamic model, are very limited.

Diffusion coefficients obtained by statistical analysis of time series of positional changes of numerous glycoprotein—bead complexes clearly depend on the overall mobility of the probes. In practice, though, the contribution to the drag coming from the motion of the bead in the presumably less viscous pericellular layer is sel- dom accounted for Sako and Kusumi, ; Sako single particle motion in plasma pdf al.

To our knowledge, the only attempt to include the impeding contribution result- ing from the motion of the bead within the pericellular region in the calculation of the overall hydrodynamic resistance was done by Lee et al. Due to lack of an appropriate theory, drag on glycolipid—bead complexes was calculated by a simple superposition of the drag on an isolated sphere that translates slowly in an unbounded viscous fluid and the drag on the lipid.

The inherent assumption in that calculation was that the coupled motion of the two bodies is separable, and that the bead is not affected by its proximity to the plasma membrane. Nonetheless, the combined hydrodynamic resistance cannot ignore the proximity of the bead to the plasma membrane, which is known to modify the drag in a nonlinear manner.

Motion of Beads Near the Plasma Membrane 1. Problem definition. A new model for the motion of such probes is pro- posed here. In this model the two bodies, which build the probe, translate in two distinct Brinkman effective media.

A possible model for the motion of a protein—bead complex may consist of a sphere attached to a cylinder. The probe performs an in-plane two-dimensional motion in a domain of finite thickness that is composed of two unmixed Brinkman media.

The sphere and a small portion of the cylinder move in the presumably less alsaplayer mp3 layer, whereas the body of the single particle motion in plasma pdf moves in a layer of a higher viscosity.

Electron microscopy suggests that the membrane ahange bogzar ze man az aref arefkia is almost constant, and that single particle motion in plasma pdf ratio of the membrane thickness to its local radius of cur- vature is diminutive; hence the interface between the two fluids may be assumed non-deformable rigid and plane.

Possible boundary conditions for the system of equations relevant to this model include the velocity of the associated sphere— cylinder complex, the stress-free outer boundary of the pericellular layer which is not necessarily flatthe vanishing velocities on the solid wall-like cytoplasmic face of the plasma membrane and normal to the interface separating the two fluid layers, and the matching of tangential velocities and shear stresses on this inter- face.

This model constitutes a very complex system, the solution of which is beyond the scope of this study, since it requires excessive computation time. It is our intention therefore to define a system more practical for numerical solution, which nevertheless is relevant to the biological system under single particle motion in plasma pdf.

The complex nature of the system and the lack of previous studies on the motion of bodies in a Brinkman effective medium that is confined by a general boundary that is neither a solid wall Feng et al. Consequently, it also requires a robust tool for solving the derived equations.

A schematic of the modeling system. Hence the classic manner of matching the stress vector on the common interface Youngren and Acrivos, ; Rallison and Acrivos, is not applicable. To study the convergence and robustness of the present formulation and the numerical technique employed for its solution, we consider the motion of a sphere in close proximity to a rigid interface separating two distinct Brinkman effective media that extend to infinity.

These simplifications result in a system of nine coupled integral equations, the solution of which poses more moderate com- putational demands than those required for solving the eighteen coupled integral equations. In Section 2 we develop and implement a new and efficient boundary integral method that is suitable for solving a three-dimensional flow in an axisymmetric domain with a general flat interface separating two fluids of possibly distinct per- meabilities.

Technical details on the numerical solution of the problem are dis- cussed in Section 3. In Section 4 we test the convergence, accuracy, and robustness of the method by comparing our results for the drag on a sphere translating close to a solid wall or a free interface with various exact and approximated solutions, and with previous numerical calculations.

We then provide in Section 5 new results for the translation of a spherical bead parallel to a proximate interface separating two Brinkman media. In Section 6 we discuss the relevance of our results to the biological system under consideration. Concluding remarks are given in Section 7.

The subscripts in equa- tions 3b — 3d refer to the different media. All together there are nine boundary conditions for the nine unknowns, namely the stress vectors acting on the sphere and on the interface accounting for the different normal stress on each of its facesand the tangential velocities on the interface.

Its single particle motion in plasma pdf implies that no lift i. With respect to the Laplace-like Stokes equation, there are fewer separa- ble solutions in orthogonal coordinate systems for the Helmholtz-like Brinkman equation.

Broday along the boundaries that confine the flow domain. Indeed, the theory and formu- lation of boundary integral equations is well established for steady cf. Ladyzhen- skaya, and unsteady cf. Williams, Stokes flows. Using a boundary integral equation formulation, the velocity field is expressed by integrals of the distributions of the fundamental velocity field and its associated stress tensor over the boundaries of the flow.

The solution to the three-dimensional physical system is therefore reduced to a two-dimensional mathematical problem of calculating the distributions of the Green function kernels that appear single particle motion in plasma pdf the integral equation for- mulation.

Hence boundary integral methods developed previously for Stokes flows may be applied to obtain a solution of 2. The strength of the singularity g is canceled out in 4. The volume integrals are then transformed into surface integrals over the system boundaries by applying the divergence theorem. It follows that since the fundamental solution vanishes at infinity at a sufficiently fast rate, the boundary integrals over the infinite hemispheres do not contribute to the solution.

In general, the pole x0 may be located inside, right on, or outside of the bound- ary 0 enclosing the flow domain. The integrals on the RHS of 7 are called the single-layer single particle motion in plasma pdf the double-layer hydrodynamic potentials, respectively.

For x0 outside of 0 the integrands are regular throughout the flow domain and the integrals vanish. Equation 7 is a general expression for calculating the velocity at any desired location x0 once the single- and double-layer potential distributions are known. To find these distributions the boundary single particle motion in plasma pdf of the flow are used.

Broday known velocity on the boundary, leading to a Fredholm integral equation of the first kind for the surface force f, or to a prescribed surface force, in which case equation 7 resembles a Fredholm integral satisfya imran khan mp3 skull of the second kind for the boundary velocity u.

Some problems involve the matching of shear stresses of two adjacent viscous fluids single particle motion in plasma pdf. This treatment of the discontinuity pertains because the fundamental solution of the singularly forced Stokes equation simply depends on the viscosity of the fluid. Due to different permeabilities of the two layers, which result in distinct Green functions for the different flow domains, the classical treatment of matching the stresses between adjacent layers in Stokes flows cf.

Pozrikidis, is not applicable for Brinkman flows. A possible way to overcome this difficulty is by directly deriving the stress vector as a free term. Upon differentiation of the double-layer potential a non-integrable singularity is encoun- tered. Therefore, to be able to perform the derivation proposed above we choose to represent the flow field in both flow domains in terms of single-layer potentials.

However, formulation with only single-layer potentials does not let one identify the distribution function with the true stress vector f. Sometimes this disadvantage may be of minor importance. For example, at present we are not interested in the force that acts on the interface. However, since our aim is to cal- culate the drag on and the mobility of the translating bead, when considering the sphere we want to eliminate the double-layer potential without losing any infor- mation on f s.

Since equations 9 and 10 are written in terms of only single-layer potentials, they are continuous throughout the domain of the flow as well as across interfaces.

As such, they hold for x0 on the boundaries e. Equation 11 is a Fredholm-type integral equation of the first kind for the modified traction on the single particle motion in plasma pdf, h, and the unknown distributions of the single-layer potential on the interface, q.

Boundary conditions 3b and 3c can be directly used by inserting values into the LHS of equations 12a and 12b or by matching the two equations accounting for the opposite directions of the normals on each side of the interface. Boundary conditions 3don the other hand, call for the differentiation of 12 with respect to x0.

Equation 8 is a general expression for the stress vector. Broday solely by differentiation of the velocity field with respect to x0. This is advanta- geous, since velocity components stand as free terms in the LHS of equations 12a and 12b. The efficiency of the boundary integral method is significantly improved for problems that are characterized by axisymmetric geometry yet the flow field may still be non-axisymmetric, vide infra, Pozrikidis, For Single particle motion in plasma pdf flows, the integration in the azimuthal direction can be expressed in terms of complete elliptic integrals of the first and second kinds cf.

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