The deformation of spherical vesicles with permeable, constant-area membranes: application to the red blood cell

In vitro investigation of the mechanics of fixed red blood cells based on optical trap micromanipulation and image analysis

Erythrocyte deformability correlates with various diseases. Single-cell measurements via optical tweezers (OTs) enable quantitative exploration but may encounter inaccuracies due to erythrocyte life cycle mixing. We present a three-step methodology to address these challenges. Firstly, density gradient centrifugation minimizes erythrocyte variations. Secondly, OTs measure membrane shear force across layers. Thirdly, MATLAB analyzes dynamic cell areas. Results combined with membrane shear force data reveal erythrocyte deformational capacity. To further characterize the deformability of diseased erythrocytes, the experiments used glutaraldehyde-fixed erythrocytes to simulate diseased cells. OTs detect increased shear modulus, while image recognition indicates decreased deformation. The integration of OTs and image recognition presents a comprehensive approach to deformation analysis, introducing novel ideas and methodologies for investigating erythrocytic lesions.

Non-Invasive Dynamic Reperfusion of Microvessels In Vivo Controlled by Optical Tweezers

Distributive shock is considered to be a condition of microvascular hypoperfusion, which can be fatal in severe cases. However, traditional therapeutic methods to restore the macro blood flow are difficult to accurately control the blood perfusion of microvessels, and the currently developed manipulation techniques are inevitably incompatible with biological systems. In our approach, infrared optical tweezers are used to dynamically control the microvascular reperfusion within subdermal capillaries in the pinna of mice. Furthermore, we estimate the effect of different optical trap positions on reperfusion at branch and investigate the effect of the laser power on reperfusion. The results demonstrate the ability of optical tweezers to control microvascular reperfusion. This strategy allows near-noninvasive reperfusion of the microvascular hypoperfusion in vivo. Hence, our work is expected to provide unprecedented insights into the treatment of distributive shock.

Inertial migration of a deformable capsule in an oscillatory flow in a microchannel

Dynamics of a deformable capsule in an oscillatory flow of a Newtonian fluid in a microchannel has been studied numerically. The effects of oscillation frequency, capsule deformability, and channel flow rate have been explored by simulating the capsule within a microchannel. In addition, the simulation captures the effect of the type of imposed pressure oscillations on the migration pattern of the capsule. An oscillatory channel flow enables the focusing of extremely small biological particles by eliminating the need to design impractically long channels. The presented results show that the equilibrium position of the capsule changes not only by the addition of an oscillatory component to the pressure gradient but it also is influenced by the capsule deformability and channel flow rate. Furthermore, it has been shown that the amplitude of oscillation of capsules decreases as the channel flow rate and the rigidity of the capsule increases.

Probing force in living cells with optical tweezers: from single-molecule mechanics to cell mechanotransduction

The invention of optical tweezers more than three decades ago has opened new avenues in the study of the mechanical properties of biological molecules and cells. Quantitative force measurements still represent a challenging task in living cells due to the complexity of the cellular environment. Here, we review different methodologies to quantitatively measure the mechanical properties of living cells, the strength of adhesion/receptor bonds, and the active force produced during intracellular transport, cell adhesion, and migration. We discuss experimental strategies to attain proper calibration of optical tweezers and molecular resolution in living cells. Finally, we show recent studies on the transduction of mechanical stimuli into biomolecular and genetic signals that play a critical role in cell health and disease.

Study of in vitro RBCs membrane elasticity with AOD scanning optical tweezers

The elasticity of red cell membrane is a critical physiological index for the activity of RBC. Study of the inherent mechanism for RBCs membrane elasticity transformation is attention-getting all along. This paper proposes an optimized measurement method of erythrocytes membrane shear modulus incorporating acousto-optic deflector (AOD) scanning optical tweezers system. By use of this method, both membrane shear moduli and sizes of RBCs with different in vitro times were determined. The experimental results reveal that the RBCs membrane elasticity and size decline with in vitro time extension. In addition, semi quantitative measurements of S-nitrosothiol content in blood using fluorescent spectrometry during in vitro storage show that RBCs membrane elasticity change is positively associated with the S-nitrosylation level of blood. The analysis considered that the diminished activity of the nitric oxide synthase makes the S-nitrosylation of in vitro blood weaker gradually. The main reason for worse elasticity of the in vitro RBCs is that S-nitrosylation effect of spectrin fades. These results will provide a guideline for further study of in vitro cells activity and other clinical applications.

Cell mechanics in biomedical cavitation

Studies on the deformation behaviours of cellular entities, such as coated microbubbles and liposomes subject to a cavitation flow, become increasingly important for the advancement of ultrasonic imaging and drug delivery. Numerical simulations for bubble dynamics of ultrasound contrast agents based on the boundary integral method are presented in this work. The effects of the encapsulating shell are estimated by adapting Hoff's model used for thin-shell contrast agents. The viscosity effects are estimated by including the normal viscous stress in the boundary condition. In parallel, mechanical models of cell membranes and liposomes as well as state-of-the-art techniques for quantitative measurement of viscoelasticity for a single cell or coated microbubbles are reviewed. The future developments regarding modelling and measurement of the material properties of the cellular entities for cutting-edge biomedical applications are also discussed.

Microcapsule mechanics: from stability to function

Microcapsules are reviewed with special emphasis on the relevance of controlled mechanical properties for functional aspects. At first, assembly strategies are presented that allow control over the decisive geometrical parameters, diameter and wall thickness, which both influence the capsule's mechanical performance. As one of the most powerful approaches the layer-by-layer technique is identified. Subsequently, ensemble and, in particular, single-capsule deformation techniques are discussed. The latter generally provide more in-depth information and cover the complete range of applicable forces from smaller than pN to N. In a theory chapter, we illustrate the physics of capsule deformation. The main focus is on thin shell theory, which provides a useful approximation for many deformation scenarios. Finally, we give an overview of applications and future perspectives where the specific design of mechanical properties turns microcapsules into (multi-)functional devices, enriching especially life sciences and material sciences.

Buckling of a growing tissue and the emergence of two-dimensional patterns

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We model the growth of gut epithelial cells cultured upon a deformable substrate.

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Growth generates buckling instabilities, contributing to crypt formation in vivo.

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Variations in mechanical properties have little effect on resulting configurations.

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Configurations are controlled by growth patterns & interactions with strata below.

The process of biological growth and the associated generation of residual stress has previously been considered as a driving mechanism for tissue buckling and pattern selection in numerous areas of biology. Here, we develop a two-dimensional thin plate theory to simulate the growth of cultured intestinal epithelial cells on a deformable substrate, with the goal of elucidating how a tissue engineer might best recreate the regular array of invaginations (crypts of Lieberkühn) found in the wall of the mammalian intestine. We extend the standard von Kármán equations to incorporate inhomogeneity in the plate’s mechanical properties and surface stresses applied to the substrate by cell proliferation. We determine numerically the configurations of a homogeneous plate under uniform cell growth, and show how tethering to an underlying elastic foundation can be used to promote higher-order buckled configurations. We then examine the independent effects of localised softening of the substrate and spatial patterning of cellular growth, demonstrating that (within a two-dimensional framework, and contrary to the predictions of one-dimensional models) growth patterning constitutes a more viable mechanism for control of crypt distribution than does material inhomogeneity.

Axisymmetric Optical-Trap Measurement of Red Blood Cell Membrane Elasticity

The elastic properties of the cell membrane play a crucial role in determining the equilibrium shape of the cell, as well as its response to the external forces it experiences in its physiological environment. Red blood cells are a favored system for studying membrane properties because of their simple structure: a lipid bilayer coupled to a membrane cytoskeleton and no cytoplasmic cytoskeleton. An optical trap is used to stretch a red blood cell, fixed to a glass surface, along its symmetry axis by pulling on a micron-sized latex bead that is bound at the center of the exposed cell dimple. The system, at equilibrium, shows Hookean behavior with a spring constant of 1.5×10−6 N/m over a 1–2 μm range of extension. This choice of simple experimental geometry preserves the axial symmetry of the native cell throughout the stretch, probes membrane deformations in the small-extension regime, and facilitates theoretical analysis. The axisymmetry makes the experiment amenable to simulation using a simple model that makes no a priori assumption on the relative importance of shear and bending in membrane deformations. We use an iterative relaxation algorithm to solve for the geometrical configuration of the membrane at mechanical equilibrium for a range of applied forces. We obtain estimates for the out-of-plane membrane bending modulus B≈1×10−19 Nm and an upper limit to the in-plane shear modulus H<2×10−6 N/m. The partial agreement of these results with other published values may serve to highlight the dependence of the cell’s resistance to deformation on the scale and geometry of the deformation.

An optical-manipulation technique for cells in physiological flows

We have developed a technique to manipulate human red blood cells (RBCs) in hydrodynamic flows. This method applies optical tweezers to trap and move microbead-attached RBCs in a liquid medium at various speeds, while it significantly minimizes laser heating and photon-induced stress for normal operation with laser-trapped cells. Computational fluid dynamics is applied to simulate flow-induced shear stress over the cell membrane and to correlate quantitatively the forces with the cell deformations. RBCs can be manipulated under physiological conditions by this approach, which may open an avenue to design principles for the next generation of cell sorting and delivery.

The nonlinear mechanical response of the red blood cell

We measure the dynamical mechanical properties of human red blood cells. A single cell response is measured with optical tweezers. We investigate both the stress relaxation following a fast deformation and the effect of varying the strain rate. We find a power-law decay of the stress as a function of time, down to a plateau stress, and a power-law increase of the cell's elasticity as a function of the strain rate. Interestingly, the exponents of these quantities violate the linear superposition principle, indicating a nonlinear response. We propose that this is due to the breaking of a fraction of the crosslinks during the deformation process. The soft glassy rheology model accounts for the relation between the exponents we observe experimentally. This picture is consistent with recent models of bond remodeling in the red blood cell's molecular structure. Our results imply that the blood cell's mechanical behavior depends critically on the deformation process.

Nanomechanical characterization of red blood cells using optical tweezers

Deformation behaviours of red blood cells (RBCs) have been studied by applying stretching forces via optical tweezers. Combined with finite-element analyses (FEA), the RBCs' mechanical properties are determined quantitatively based on a best fitting between the experimental deformed geometries and the simulated counterparts. Experimentally, a silica beads attached erythrocyte is optical-mechanically stretched to different lengths. On the theoretical front, a large deformation model with Mooney-Rivlin constitutive equations has been simulated by using FEA to predict the cell deformation geometries. The numerically simulated transverse and longitudinal strains which are in a good agreement with the experimental measurements facilitate the determination of elastic constants of the cells.

The Deformation of an Erythrocyte Under the Radiation Pressure by Optical Stretch

In this paper, the mechanical properties of erythrocytes were studied numerically based upon the mechanical model originally developed by Pamplona and Calladine (ASME J. Biomech. Eng., 115, p. 149, 1993) for liposomes. The case under study is the erythrocyte stretched by a pair of laser beams in opposite directions within buffer solutions. The study aims to elucidate the effect of radiation pressure from the optical laser because up to now little is known about its influence on the cell deformation. Following an earlier study by Guck et al. (Phys. Rev. Lett., 84, p. 5451, 2000; Biophys. J., 81, p. 767, 2001), the empirical results of the radiation pressure were introduced and imposed on the cell surface to simulate the real experimental situation. In addition, an algorithm is specially designed to implement the simulation. For better understanding of the radiation pressure on the cell deformation, a large number of simulations were conducted for different properties of cell membrane. Results are first discussed parametrically and then evaluated by comparing with the experimental data reported by Guck et al. An optimization approach through minimizing the errors between experimental and numerical data is used to determine the optimal values of membrane properties. The results showed that an average shear stiffness around 4.611×10-6Nm−1, when the nondimensional ratio of shear modulus to bending modulus ranges from 10 to 300. These values are in a good agreement with those reported in literature.

Coupling bending and shear effects on liposome deformation

Cell membrane deformation induced by external mechanical stimuli has been studied extensively over the past three decades. The present study focuses on the coupling of in-plane shear H and out-of-plane bending B of liposome membrane and its influences on the deformation of a single vesicle subjected to (i) external compressive load via two parallel platens and (ii) contact forces caused by a rigid substrate. Our results show that the increase of membrane resultant stress in both loading configurations causes the liposome to become more rigid and the degree of vesicle deformation decreases when the in-plane shearing effect is dominant. A theoretical approach is developed to facilitate cell membrane characterization under different biomechanical stimuli.

Mechanics of tether formation in liposomes

It is well-known that a "tether" may be drawn out from a pressurized liposome by means of a suitably applied radial-outward force applied locally to the lipid bilayer. The tether is a narrow, uniform cylindrical tube, which joins the main vesicle in a short "transition region." A first-order energy analysis establishes the broad relationship between the force F needed to draw the tether, the radius R0 of the tether, the bending-stiffness constant B for the lipid bilayer and the membrane tension T in the pressurized liposome. The aim of the present paper is to study in detail the "transition region" between the tether and the main vesicle, by means of a careful application of the engineering theory of axisymmetric shell structures. It turns out that the well-known textbook "thin-shell" theory is inadequate for this purpose, because the tether is evidently an example of a thick-walled shell; and a novel ingredient of the present study is the introduction of elastic constitutive relations that are appropriate to the thick-shell situation. The governing equations are set up in dimensionless form, and are solved by means of a "shooting" technique, starting with a single disposable parameter at a point on the meridian in the tether, which can be adjusted until the boundary conditions at the far "equator" of the main vessel are satisfied. It turns out that the "transition region" between the tether and the main vessel is well characterized by only a few parameters, while the tether and main vessel themselves are described by very simple equations. Introduction of the thick-shell constitutive relation makes little difference to the conformation of and stress-resultants in, the main vessel; but it makes a great deal of difference in the tether itself Indeed, a kind of phase-change appears to take place in the "transition region" between these two zones of the liposome.

Theoretical predictions of electromechanical deformation of cells subjected to high voltages for membrane electroporation

An electromechanical analysis based on thin-shell theory is presented to analyze cell shape changes in response to external electric fields. This approach can be extended to include osmotic-pressure changes. Our calculations demonstrate that at large fields, the spherical cell geometry can be significantly modified, and even ellipsoidal forms would be inappropriate to account for the deformation. Values of the surface forces obtained from our calculations are in very good agreement with the 1--10 mN/m range for membrane rupture reported in the literature. The results, in keeping with reports in the literature, demonstrate that the final shape depends on membrane thickness. This has direct implications for tissues in which significant molecular restructuring can occur. It is also shown that, at least for the smaller electric fields, both the cellular surface area and volume change roughly in a quadratic manner with the electric field. Finally, it is shown that the bending moments are generally quite small and can be neglected for a simpler analysis.

Theoretical analysis of shape transformation of V-79 cells after treatment with cytochalasin B

We observed that after treatment of V-79 fibroblasts with cytochalasin B the area of cell contact with the substrate is essentially reduced, the microtubules are organized into rodlike structures and the actin filaments are disintegrated. Remnants of the actin cortex become concentrated in the form of discrete patches under the plasma membrane. The described changes in the organization of the cytoskeleton and of the cortical shell are accompanied by the formation of a cell shape resembling the Greek letter phi. We calculated that the phi shape corresponds to the minimum of the stretching energy of the cortical shell at relevant geometrical constraints. In line with this result, if cytochalasin B treatment was followed by colchicine application which disrupted the microtubular rod, the characteristic phi shape completely disappeared. This study suggests that the effect of the microtubular rod on the cell shape can be theoretically well described by taking into account some basic conditions for the mechanical equilibrium of the cell cortical shell and the appropriate geometrical constraints.

Elasticity of the red cell membrane and its relation to hemolytic disorders: an optical tweezers study

We have used optical tweezers to study the elasticity of red cell membranes; force was applied to a bead attached to a permeabilized spherical ghost and the force-extension relation was obtained from the response of a second bead bound at a diametrically opposite position. Interruption of the skeletal network by dissociation of spectrin tetramers or extraction of the actin junctions engendered a fourfold reduction in stiffness at low applied force, but only a twofold change at larger extensions. Proteolytic scission of the ankyrin, which links the membrane skeleton to the integral membrane protein, band 3, induced a similar effect. The modified, unlike the native membranes, showed plastic relaxation under a prolonged stretch. Flaccid giant liposomes showed no measurable elasticity. Our observations indicate that the elastic character is at least as much a consequence of the attachment of spectrin as of a continuous membrane-bound network, and they offer a rationale for formation of elliptocytes in genetic conditions associated with membrane-skeletal perturbations. The theory of Parker and Winlove for elastic deformation of axisymmetric shells (accompanying paper) allows us to determine the function BH(2) for the spherical saponin-permeabilized ghost membranes (where B is the bending modulus and H the shear modulus); taking the literature value of 2 x 10(-19) Nm for B, H then emerges as 2 x 10(-6) Nm(-1). This is an order of magnitude higher than the value reported for intact cells from micropipette aspiration. Reasons for the difference are discussed.