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

Abstract

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.

Appendices

Appendix 1: Description of the numerical procedure

Equation 18 is a differential equation of second order, and we translate it to a system of two differential equations of first order:

$$\begin{aligned}&\frac{\hbox {d}s_0 }{\hbox {d}s}=\frac{1}{\lambda _\mathrm{m} }\nonumber \\&\begin{aligned} \frac{d\lambda _\mathrm{m} }{\hbox {d}s}&=\left( {r_0 \frac{\partial ^{2}u}{\partial \lambda _\mathrm{m}^2 }} \right) ^{-1}\left[ \frac{\partial ^{2}u}{\partial \lambda _\mathrm{m} \partial \lambda _\mathrm{p}}\left( {\frac{1}{\lambda _\mathrm{m} }\frac{r}{r_0 }\frac{\hbox {d}r_0 }{\hbox {d}s_0 }-\frac{\hbox {d}r}{\hbox {d}s}} \right) \right. \\&\quad \left. +\,\,\frac{1}{\lambda _\mathrm{m} }\left( {\frac{\partial u}{\partial \lambda _\mathrm{p}}\frac{\hbox {d}r}{\hbox {d}s}-\frac{\partial u}{\partial \lambda _\mathrm{m} }\frac{\hbox {d}r_0 }{\hbox {d}s_0 }} \right) \right] \end{aligned} \end{aligned}$$

(30)

The undeformed RBC area is a disk with radius $R_{\mathrm{III}}= (A_{0}/4\pi )^{1/2}$ and is divided into three sections, each corresponding to a different region of the deformed membrane: cap, cylinder and annulus. Because each section has a different contour function s(r), the corresponding deformations are obtained from different versions of Eq. 30 subject to the constraints that s and $s_{0}$ must be continuous across boundaries from the beginning of Sect. 1 to the end of Sect. 3 (Fig. 6a).

The contour functions in case of deformation from a flat disk to a spherical cap are:

$$\begin{aligned}&r = R\sin \frac{s}{R} \nonumber \\&r_0 =s_0 \end{aligned}$$

(31)

where R is the radius of the spherical cap and can be determined from the meniscus height h and the pipette radius $R_{p}$

$$\begin{aligned} R=\frac{h^{2}+R_\mathrm{p}^2 }{2h} \end{aligned}$$

(32)

For the cylindrical section, the contour functions are:

$$\begin{aligned}&r = R_\mathrm{p}\nonumber \\&r_0 = s_0 \end{aligned}$$

(33)

and for deformation from a larger to a smaller annulus, the relationships are:

$$\begin{aligned}&r = s-s_\mathrm{p}+R_\mathrm{p}\nonumber \\&r_0 =s_0 \end{aligned}$$

(34)

where $s_\mathrm{p}$ is the length of the contour within the micropipette. The mapping function between the shapes $s_0 \left( s \right)$ is determined by requiring area preservation and applying the condition that the extension ratios at the pole are the same $\lambda _\mathrm{m} =\lambda _\mathrm{p}$. Numerically, we set the value of $\lambda _\mathrm{m}$ at the pole as $\lambda _\mathrm{m} =1/\gamma$ and making the first step of integration $s_0 =\gamma s$. The value of $\upgamma$ is obtained by iteration to satisfy the requirement that the value of $s_0$ at the end of the third section matches the initial disk radius $R_{\mathrm{III}}$.

Appendix 2

For our present model, it is straightforward to calculate the dependence of the continuum moduli $\kappa$ and $\mu$ on extension ratios:

$$\begin{aligned} \mu= & {} \frac{Kn_0 X_0^2 }{2}\left\{ \lambda _1 \lambda _2 -\frac{4}{\pi }\frac{\lambda _1^2 \lambda _2^3 }{\left( {\lambda _1^2 -\lambda _2^2 } \right) ^{2}}\right. \nonumber \\&\times \left. \left[ {\left( {\frac{\lambda _1^2 }{\lambda _2^2 }+1} \right) E\left( {1-\frac{\lambda _2^2 }{\lambda _1^2 }} \right) -2K\left( {1-\frac{\lambda _2^2 }{\lambda _1^2 }} \right) } \right] \right\} \end{aligned}$$

(35)

$$\begin{aligned} \kappa= & {} \frac{Kn_0 X_0^2 }{2\pi }\frac{1}{\lambda _1 \lambda _2^2 }E\left( {1-\frac{\lambda _2^2 }{\lambda _1^2 }} \right) \end{aligned}$$

(36)

for $\lambda _{1} > \lambda _{2}$, and with indices interchanged when the inequality is not satisfied. At first, the second term in the expression for $\mu$ appears to be singular for $\lambda _{1}=\lambda _{2}$, but a careful analysis of this term reveals the following limit:

$$\begin{aligned} \mathop {\lim }\limits _{\lambda _1 \rightarrow \lambda _2 } \left( \mu \right) =\frac{Kn_0 X_0^2 }{2}\left( {\lambda _1 \lambda _2 -\frac{3\lambda _2 }{4}} \right) \end{aligned}$$

(37)

From Fig. 9, it is evident that these two coefficients for a material made up of randomly oriented springs are strongly dependent on deformation, with the shear modulus increasing dramatically with extension and the area modulus decreasing with expansion. This indicates that such a material would “prefer” to decrease its local density than stretch when membrane deformations are large. As indicated in the discussion, experiments involving aspiration of small portions of red cell membrane into a micropipette, by themselves, do not enable us to determine whether this behavior is exhibited by RBC membrane, but future experiments using fluorescence to image changes in local skeletal density could provide a test of this prediction.