Systematic coarse-graining of spectrin-level red blood cell models

Abstract

We present a rigorous procedure to derive coarse-grained red blood cell (RBC) models, which yield accurate mechanical response. Based on a semi-analytic theory the linear and nonlinear elastic properties of healthy and infected RBCs in malaria can be matched with those obtained in optical tweezers stretching experiments. The present analysis predicts correctly the membrane Young's modulus in contrast to about 50% error in predictions by previous models. In addition, we develop a stress-free model which avoids a number of pitfalls of existing RBC models, such as non-smooth or poorly controlled equilibrium shape and dependence of the mechanical properties on the initial triangulation quality. Here we employ dissipative particle dynamics for the implementation but the proposed model is general and suitable for use in many existing continuum and particle-based numerical methods.

Introduction

Recent experiments to probe the mechanical properties of a red blood cell (RBC) include micropipette aspiration [1], [2], RBC deformation by optical tweezers [3], [4], RBC edge flicker microscopy [5] and tracking of fluorescent nanometer beads attached to the RBC [6]. The first two experimental techniques subject the RBC directly to mechanical deformation, while the two latter attempt to extract the mechanical properties from passive observations of thermal fluctuations. The direct deformation techniques report overlapping results for the shear modulus of healthy cells in the range of 4–9 μN/m for micropipette aspiration and 5–12 μN/m in optical tweezers experiments. In contrast, the thermal fluctuations techniques predict the shear modulus to be one to two orders of magnitude smaller than those from the RBC deformation experiments. Recent theoretical developments offer explanations for the discrepancies in experimental results. Li et al. [7] suggest that the erythrocyte cytoskeleton may be subject to a continuous rearrangement due to metabolic activity or large strains. Their numerical model shows that under certain conditions, the RBC membrane consisting of a lipid bilayer with an attached cytoskeleton formed by a spectrin protein network and linked by short actin filaments may experience strain hardening and softening. In addition, the actin cytoskeleton attachments are subject to diffusion within the lipid bilayer, however it is a slow process and hence negligible at short time scales. Gov [8] proposes an active elastic network model, where the metabolic activity controls the stiffness of the cell through the consumption of ATP. The ATP activity could also greatly affect membrane thermal undulations [9] resulting in fluctuations comparable to an effective temperature increase by a factor of three, which would result in a substantial underprediction of the RBC membrane elastic properties. However, recent experiments [10] did not find a strong dependence of RBC elastic properties and fluctuations on ATP.

The experimental findings provide clear evidence that RBCs subject to large deformations are characterized by a complex nonlinear mechanical response. However, it is plausible to assume that a nonlinear elastic model can provide an adequate description of moderate RBC deformations at small strain rates. Thus, the main focus of this paper is to derive consistent coarse-grained nonlinear elastic models, which are able to successfully describe the mechanical deformations of RBCs. Possible membrane strain hardening or softening as well as the effects of metabolic activity can be incorporated into the model, however this is beyond the scope of the present paper.

The healthy human RBC assumes a biconcave shape with an average diameter of 7.8 μm. The lipid bilayer can be considered nearly viscous and area-incompressible [11], while the attached spectrin network is mainly responsible for the membrane elastic response providing RBC integrity as it undergoes severe deformations in narrow capillaries as small as 3 μm in diameter. An RBC model is constructed by a network of springs in combination with a bending energy and constraints for surface-area and volume conservation. Fig. 1 illustrates the difference between network and continuum based models, which are characterized by different parameters.

Atomic force microscopy experiments [12], [13] have shown that the spectrin network of RBCs is highly irregular compared to the regular hexagonal network and has varying lengths of interconnections. The spectrin-level model in this paper corresponds to an effective spectrin network where each spring represents a single spectrin tetramer; the network is regular, i.e. nearly hexagonal. Theoretical analysis of the hexagonal network yields its linear mechanical properties, however the current theoretical results for the spectrin-level model [14] underestimate the effective membrane Young's modulus by about 50%. In this paper, we present the corrected analysis of elastic membrane properties for different spring models and arbitrary levels of coarse-graining. In addition, we propose a stress-free model, which eliminates non-vanishing local artifacts, such as the dependence of mechanical properties on triangulation quality and equilibrium shape stability for realistic membrane bending rigidity; the latter is often compensated with artificially high bending stiffness. In addition, comparison of the spring response at spectrin-level of modeling with the response of a coarse-grained single spectrin tetramer [15] is shown to yield good agreement. This provides additional model validation.

A number of numerical models have recently been developed, which include continuum descriptions [11], [16], [17], and discrete approximations at the spectrin molecular level [18], [19] as well as at the mesoscopic scale [20], [21], [22]. Fully continuum (fluid and solid) modeling often suffers from difficulties in coupling nonlinear solid motions and fluid flow without excessive computational expense. Therefore, “semi-continuum” modeling [16], [17] of deformable particles is developing rapidly and typically employs immersed boundary or front-tracking techniques. Here a membrane is represented by a set of points which move in Lagrangian fashion and are coupled to an Eulerian discretization of the fluid domain. In this work, we focus on the accurate mesoscopic modeling of RBCs. Specifically, we develop a generalized elastic model with major improvements to its mechanical properties.

The paper is organized as follows. In the next section we present the detailed RBC model. Section 3 provides a semi-analytical theory of the RBC membrane elastic properties, and Section 4 compares calculations of the stretching-deformation of healthy and parasitized RBCs in malaria with experimental data. We conclude in Section 5 with suggestions for model development.