A multiscale model for red blood cell mechanics

Scale-invariance in miniature coarse-grained red blood cells by fluctuation analysis

To accurately represent the morphological and elastic properties of a human red blood cell, Fu et al. [Fu et al., Lennard-Jones type pair-potential method for coarse-grained lipid bilayer membrane simulations in LAMMPS, 2017, 210, 193-203] recently developed a coarse-grained molecular dynamics model with particular detail in the membrane. However, such a model accrues an extremely high computational cost for whole-cell simulation when assuming an appropriate length scaling - that of the bilayer thickness. To date, the model has only simulated "miniature" cells in order to circumvent this, with the a priori assumption that these miniaturised cells correctly represent their full-sized counterparts. The present work assesses the validity of this approach, by testing the scale invariance of the model through simulating cells of various diameters; first qualitatively in their shape evolution, then quantitatively by measuring their bending rigidity through fluctuation analysis. Cells of diameter of at least 0.5 μm were able to form the characteristic biconcave shape of human red blood cells, though smaller cells instead equilibrated to bowl-shaped stomatocytes. Thermal fluctuation analysis showed the bending rigidity to be constant over all cell sizes tested, and consistent between measurements on the whole-cell and on a planar section of bilayer. This is as expected from the theory on both counts. Therefore, we confirm that the evaluated model is a good representation of a full-size RBC when the model diameter is ≥0.5 μm, in terms of the morphological and mechanical properties investigated.

Dissipative particle dynamics for modeling micro-objects in microfluidics: application to dielectrophoresis

The dissipative particle dynamics (DPD) technique is employed to model the trajectories of micro-objects in a practical microfluidic device. The simulation approach is first developed using an in-house Fortran code to model Stokes flow at Reynolds number of 0.01. The extremely low Reynolds number is achieved by adjusting the DPD parameters, such as force coefficients, thermal energies of the particles, and time steps. After matching the numerical flow profile with the analytical results, the technique is developed further to simulate the deflection of micro-objects under the effect of a deflecting external force in a rectangular microchannel. A mapping algorithm is introduced to establish the scaling relationship for the deflecting force between the physical device and the DPD domain. Dielectrophoresis is studied as a case study for the deflecting force, and the trajectory of a single red blood cell under the influence of the dielectrophoretic force is simulated. The device is fabricated using standard microfabrication techniques, and the experiments involving a dilute sample of red blood cells are performed at two different cases of the actuation voltage. Good agreement between the numerical and experimental results is achieved.

Relative contributions of chain density and topology to the elasticity of two-dimensional polymer networks

Understanding the relationships between the structure of polymer networks and their mechanical properties is important for the design of advanced soft materials with optimal properties. However, classical rubber elasticity theories often fall short in their description of the network structure, while simulation techniques at molecular scale remain impractical at that length scale. Here we develop a computational approach based on random discrete networks, in which the polymer network is represented as an assembly of non-linear springs connected at crosslinking points. The density of elastically-effective chains, average network coordination and chain contour lengths are varied independently in order to identify their respective contributions to the network elasticity. Numerical results suggest scaling relations between network parameters and elastic properties that are markedly different from the predictions of classical rubber elasticity theories. In particular, the elastic modulus of 2D random networks is found to be independent of density at constant topology, and proportional to the average coordination at constant density. The discrepancy is due to the pre-straining of the chains in the discrete network, which is not accounted for in classical models of rubber elasticity. Our results have implications for the interpretation of experimental data for ideal network gels that are formed by the cross-coupling of macromolecular building blocks in solution.

Continuum modeling of deformation and aggregation of red blood cells

In order to gain better understanding for rheology of an isolated red blood cell (RBC) and a group of multiple RBCs, new continuum models for describing mechanical properties of cellular structures of an RBC and inter-cellular interactions among multiple RBCs are developed. The viscous property of an RBC membrane, which characterizes dynamic behaviors of an RBC under stress loading and unloading processes, is determined using a generalized Maxwell model. The present model is capable of predicting stress relaxation and stress-strain hysteresis, of which prediction is not possible using the commonly used Kelvin-Voigt model. Nonlinear elasticity of an RBC is determined using the Yeoh hyperelastic material model in a framework of continuum mechanics using finite-element approximation. A novel method to model inter-cellular interactions among multiple adjacent RBCs is also developed. Unlike the previous modeling approaches for aggregation of RBCs, where interaction energy for aggregation is curve-fitted using a Morse-type potential function, the interaction energy is analytically determined. The present aggregation model, therefore, allows us to predict various effects of physical parameters such as the osmotic pressure, the thickness of a glycocalyx layer, the penetration depth, and the permittivity, on the depletion and electrostatic energy among RBCs. Simulations for elongation and recovery deformation of an RBC and for aggregation of multiple RBCs are conducted to evaluate the efficacy of the present continuum modeling methods.

A novel strain energy relationship for red blood cell membrane skeleton based on spectrin stiffness and its application to micropipette deformation

Red blood cell (RBC) membrane skeleton is a closed two-dimensional elastic network of spectrin tetramers with nodes formed by short actin filaments. Its three-dimensional shape conforms to the shape of the bilayer, to which it is connected through vertical linkages to integral membrane proteins. Numerous methods have been devised over the years to predict the response of the RBC membrane to applied forces and determine the corresponding increase in the skeleton elastic energy arising either directly from continuum descriptions of its deformation, or seeking to relate the macroscopic behavior of the membrane to its molecular constituents. In the current work, we present a novel continuum formulation rooted in the molecular structure of the membrane and apply it to analyze model deformations similar to those that occur during aspiration of RBCs into micropipettes. The microscopic elastic properties of the skeleton are derived by treating spectrin tetramers as simple linear springs. For a given local deformation of the skeleton, we determine the average bond energy and define the corresponding strain energy function and stress-strain relationships. The lateral redistribution of the skeleton is determined variationally to correspond to the minimum of its total energy. The predicted dependence of the length of the aspirated tongue on the aspiration pressure is shown to describe the experimentally observed system behavior in a quantitative manner by taking into account in addition to the skeleton energy an energy of attraction between RBC membrane and the micropipette surface.

A mechanochemical model for embryonic pattern formation: coupling tissue mechanics and morphogen expression

Motivated by recent experimental findings, we propose a novel mechanism of embryonic pattern formation based on coupling of tissue curvature with diffusive signaling by a chemical factor. We derive a new mathematical model using energy minimization approach and show that the model generates a variety of morphogen and curvature patterns agreeing with experimentally observed structures. The mechanism proposed transcends the classical Turing concept which requires interactions between two morphogens with a significantly different diffusivity. Our studies show how biomechanical forces may replace the elusive long-range inhibitor and lead to formation of stable spatially heterogeneous structures without existence of chemical prepatterns. We propose new experimental approaches to decisively test our central hypothesis that tissue curvature and morphogen expression are coupled in a positive feedback loop.

Analysis of the variation in the determination of the shear modulus of the erythrocyte membrane: Effects of the constitutive law and membrane modeling

Despite research spanning several decades, the exact value of the shear modulus Gs of the erythrocyte membrane is still ambiguous, and a wealth of studies, using measurements based on micropipette aspirations, ektacytometry systems and other flow chambers, and optical tweezers, as well as application of several models, have found different average values in the range 2-10μN/m. Our study shows that different methodologies have predicted the correct shear modulus for the specific membrane modeling employed, i.e., the variation in the shear modulus determination results from the specific membrane modeling. Available experimental findings from ektacytometry systems and optical tweezers suggest that the dynamics of the erythrocyte membrane is strain hardening at both moderate and large deformations. Thus the erythrocyte shear modulus cannot be determined accurately using strain-softening models (such as the neo-Hookean and Evans laws) or strain-softening/strain-hardening models (such as the Yeoh law), which overestimate the erythrocyte shear modulus. According to our analysis, the only available strain-hardening constitutive law, the Skalak et al. law, is able to match well both deformation-shear rate data from ektacytometry and force-extension data from optical tweezers at moderate and large strains, using an average value of the shear modulus of Gs=2.4-2.75μN/m, i.e., very close to that found in the linear regime of deformations via force-extension data from optical tweezers, Gs=2.5±0.4μN/m. In addition, our analysis suggests that a standard deviation in Gs of 0.4-0.5μN/m (owing to the inherent differences between erythrocytes within a large population) describes well the findings from optical tweezers at small and large strains as well as from micropipette aspirations.

A multiscale approach to the elastic moduli of biomembrane networks

We develop equilibrium fluctuation formulae for the isothermal elastic moduli of discrete biomembrane models at different scales. We account for the coupling of large stretching and bending strains of triangulated network models endowed with harmonic and dihedral angle potentials, on the basis of the discrete-continuum approach presented in Schmidt and Fraternali (J Mech Phys Solids 60:172-180, 2012). We test the proposed equilibrium fluctuation formulae with reference to a coarse-grained molecular dynamics model of the red blood cell (RBC) membrane (Marcelli et al. in Biophys J 89:2473-2480, 2005; Hale et al. in Soft Matter 5:3603-3606, 2009), employing a local maximum-entropy regularization of the fluctuating configurations (Fraternali et al. in J Comput Phys 231:528-540, 2012). We obtain information about membrane stiffening/softening due to stretching, curvature, and microscopic undulations of the RBC model. We detect local dependence of the elastic moduli over the RBC membrane, establishing comparisons between the present theory and different approaches available in the literature.

Thermodynamic concepts in the study of microbial populations: age structure in Plasmodium falciparum infected red blood cells

Variability is a hallmark of microbial systems. On the one hand, microbes are subject to environmental heterogeneity and undergo changeable conditions in their immediate surroundings. On the other hand, microbial populations exhibit high cellular diversity. The relation between microbial diversity and variability of population dynamics is difficult to assess. This connection can be quantitatively studied from a perspective that combines in silico models and thermodynamic methods and interpretations. The infection process of Plasmodium falciparum parasitizing human red blood cells under laboratory cultivation conditions is used to illustrate the potential of Individual-based models in the context of predictive microbiology and parasitology. Experimental data from several in vitro cultures are compared to the outcome of an individual-based model and analysed from a thermodynamic perspective. This approach allows distinguishing between intrinsic and external constraints that give rise to the diversity in the infection forms, and it provides a criterion to quantitatively define transient and stationary regimes in the culture. Increasing the ability of models to discriminate between different states of microbial populations enhances their predictive capability which finally leads to a better the control over culture systems. The strategy here presented is of general application and it can substantially improve modelling of other types of microbial communities.

Equivalent mechanical properties of biological membranes from lattice homogenization

The goal of this manuscript is to set up a novel methodology for the calculation of the effective mechanical properties of biological membranes viewed as repetitive networks of elastic filaments, based on the discrete asymptotic homogenization method. We will show that for some lattice configurations, flexional effects due to internal structure mechanisms at the unit cell scale lead to additional flexional effects at the continuum scale, accounted for by an internal length associated to a micropolar behavior. Thereby, a systematic methodology is established, allowing the prediction of the overall mechanical properties of biological membranes for a given network topology, as closed form expressions of the geometrical and mechanical micro-parameters. The peptidoglycan and the erythrocyte have been analyzed using this methodology, and their effective moduli are calculated and recorded versus the geometrical and mechanical lattice parameters. A classification of lattices with respect to the choice of the equivalent continuum model is proposed: The Cauchy continuum and a micropolar continuum are adopted as two possible effective medium, for a given beam model. The relative ratio of the characteristic length of the micropolar continuum to the unit cell size determines the relevant choice of the equivalent medium. In most cases, the Cauchy continuum is sufficient to model membranes in most of their configurations. The peptidoglycan network may exhibit a re-entrant hexagonal lattice, for which micropolar effects become important. This is attested by the characteristic length becoming larger than the beam length for such configurations. The homogenized moduli give accurate results for both membranes, as revealed by comparison with experimental measurements or simulation results from the literature at the network scale. A first insight into the nonlinear mechanical behavior of the hexagonal and triangular networks is lastly investigated using a perturbative method.