Elastic properties of the red blood cell membrane that determine echinocyte deformability

Abstract

The natural biconcave shape of red blood cells (RBC) may be altered by injury or environmental conditions into a spiculated form (echinocyte). An analysis is presented of the effect of such a transformation on the resistance of RBC to entry into capillary sized cylindrical tubes. The analysis accounts for the elasticity of the membrane skeleton in dilation and shear, and the local and nonlocal resistance of the bilayer to bending, the latter corresponding to different area strains in the two leaflets of the bilayer. The shape transformation is assumed to be driven by the equilibrium area difference (ΔA ₀, the difference between the equilibrium areas of the bilayer leaflets), which also affects the energy of deformation. The cell shape is approximated by a parametric model. Shape parameters, skeleton shear deformation, and the skeleton density of deformed membrane relative to the skeleton density of undeformed membrane are obtained by minimization of the corresponding thermodynamic potential. Experimentally, ΔA ₀ is modified and the corresponding discocyte–echinocyte shape transition obtained by high-pressure aspiration into a narrow pipette, and the deformability of the resulting echinocyte is examined by whole cell aspiration into a larger pipette. We conclude that the deformability of the echinocyte can be accounted for by the mechanical behavior of the normal RBC membrane, where the equilibrium area difference ΔA ₀ is modified.

Appendices

Appendix A. Elimination of cell shape parameters r _b, L _o, and R _c

The shape of the echinocyte is defined parametrically by six parameters (L _o, r _b, R _c, N _b, r _c, and L _p). Three of them can be derived from the geometrical conditions: the cell surface area is constant, the cell volume is constant, and spicules connect smoothly to the spherical cell body with a radius R _c. The remaining three parameters are determined with the minimization of thermodynamic potential (Eq. 16).

It can be shown that it is more convenient to derive L _o, r _b, and R _c from geometrical conditions that are expressed by the following equations:

$$ {A_{0} = A_{{\rm{p}}} + A_{{\rm{s}}} + N_{{\rm{b}}} A_{{\rm{b}}} ,} $$

(23)

$$ {V_{0} = V_{{\rm{p}}} + V_{{\rm{s}}} + N_{{\rm{b}}} V_{{\rm{b}}} ,} $$

(24)

$$ {{{{\rm{d}}z_{{\rm{b}}} } \over {{\rm{d}}r}}\left| {_{{r_{{\rm{b}}} }} } \right. = - {{r_{{\rm{b}}} } \over {{\sqrt {R^{2}_{{\rm{c}}} - r^{2}_{{\rm{b}}} } }}}.} $$

(25)

In Eq. (23), A _p is the surface area of the aspirated membrane projection, A _s is the surface area of the spherical portion of the cell body (excluding the bases of all the spicules and the pipette entrance), and A _b is the surface area of each spicule:

$$ {A_{{\rm{p}}} = 2\pi R_{{\rm{p}}} L_{{\rm{p}}} ,} $$

(26)

$$ {A_{{\rm{s}}} = 4\pi R^{2}_{{\rm{c}}} - N_{{\rm{b}}} \cdot 2\pi R_{{\rm{c}}} {\left( {R_{{\rm{c}}} - {\sqrt {R^{2}_{{\rm{c}}} - r^{2}_{{\rm{b}}} } }} \right)} - 2\pi R_{{\rm{c}}} {\left( {R_{{\rm{c}}} - {\sqrt {R^{2}_{{\rm{c}}} - R^{2}_{{\rm{p}}} } }} \right)},} $$

(27)

$$ {A_{{\rm{b}}} = 2\pi {\int\limits_0^{r_{{\rm{b}}} } r } \cdot {\sqrt {1 + {\left( {{{{\rm{d}}z_{{\rm{b}}} } \over {{\rm{d}}r}}} \right)}^{2} } } \cdot {\rm{d}}r.} $$

(28)

Similarly, in Eq. (24), V _p is the volume of the cell portion aspirated into the pipette, V _s is the volume of the spherical cell body (adjusted for the volume of the portion that is in pipette entrance), V _b is the volume of the spicule (adjusted for a overlapping volume between the spicule and the spherical cell body):

$$ {V_{{\rm{p}}} = \pi R^{2}_{{\rm{p}}} {\left( {L_{{\rm{p}}} - {{R_{{\rm{p}}} } \over 3}} \right)},} $$

(29)

$$ {V_{{\rm{s}}} = {4 \over 3}\pi R^{3}_{{\rm{c}}} - \pi h^{2}_{1} {\left( {R_{{\rm{c}}} - h_{1} /3} \right)},} $$

(30)

$$ {\eqalign{ & V_{{\rm{b}}} = {1 \over 2}r^{{\rm{2}}}_{{\rm{C}}} \pi L{\left[ {\pi - 2\arctan {\left( {1 - {\sqrt 2 }{{r_{{\rm{b}}} } \over {r_{{\rm{C}}} }}} \right)} - 2\arctan {\left( {1 + {\sqrt 2 }{{r_{{\rm{b}}} } \over {r_{{\rm{C}}} }}} \right)}} \right]} \cr & - \pi r^{2}_{{\rm{b}}} z_{{\rm{b}}} (r_{{\rm{b}}} ) - \pi h^{2}_{2} {\left( {R_{{\rm{c}}} - h_{2} /3} \right)}, \cr} } $$

(31)

where h ₁ and h ₂ are:

$$ {h_{1} = R_{{\rm{c}}} - {\sqrt {R^{{\rm{2}}}_{{\rm{c}}} - R^{2}_{{\rm{p}}} } },} $$

(32)

$$ {h_{2} = R_{{\rm{c}}} - {\sqrt {R^{{\rm{2}}}_{{\rm{c}}} - r^{2}_{{\rm{b}}} } }.} $$

(33)

The third constraint, related to the smooth joining of the spicule to the spherical cell body (Eq. 25), leads to an expression for the spherical cell radius R _c:

$$ {R_{{\rm{c}}} = r_{{\rm{b}}} {\sqrt {1 + {{r^{8}_{{\rm{C}}} L^{4} } \over {16{\left[ {r_{{\rm{b}}} + z_{{\rm{b}}} (r_{{\rm{b}}} )} \right]}^{6} }}} }.} $$

(34)

The variable L is a difference between the maximal and minimal value of the contour function (Eq. 10) and it is related to the spicule height L ₀:

$$ {L = L_{0} + z_{{\rm{b}}} {\left( {r_{{\rm{b}}} } \right)} - {\sqrt {R^{2}_{{\rm{c}}} - r^{2}_{{\rm{b}}} } } + R_{{\rm{c}}} .} $$

(35)

In contrast to the equation for R _c (Eq. 34), Eqs. (23) and (24) cannot be solved explicitly for the parameters L _o and r _b. Therefore, the equations were solved numerically by a simple bisection method. For specified values of N _b, r _c, and L _p, the parameter r _b was modified to find a root of the "area equation" (Eq. 23). For each value of r _b, a value for L _o was calculated from the "volume equation" (Eq. 24). Then R _c could be expressed explicitly with Eq. (34).

Appendix B. Bending energy of the bilayer

Contributions to the local bending energy W _b in Eq. (12) are calculated according to Eq. (7). For the portion of the cell in the pipette, and for the spherical cell body, these resolve to:

$$ {W^{{\rm{p}}}_{{\rm{b}}} = 4\pi k_{{\rm{c}}} {\left( {1 - {{L_{{\rm{p}}} - R_{{\rm{p}}} } \over {4R_{{\rm{p}}} }}} \right)},} $$

(36)

$$ {W^{{\rm{s}}}_{{\rm{b}}} = 2k_{{\rm{c}}} {{A_{s} } \over {R^{{\rm{2}}}_{{\rm{c}}} }}.} $$

(37)

For the spicules, ${W^{{\rm{b}}}_{{\rm{b}}} }$ is obtained by numerical integration of Eq. (7) along the spicule surface. The principal curvatures are defined as follows:

$$ c_{{\text{p}}} = \frac{{\sin \theta }} {r}\;\;\;{\text{and}}\;\;\;c_{{\text{m}}} = \frac{{{\text{d}}\theta }} {{{\text{d}}s}}, $$

(38)

where θ is the angle between the local normal and the symmetry axis. The shape of the cell contour z(r) is known, and therefore it is convenient to rewrite Eqs. (38) as follows:

$$ c_{{\text{p}}} = \frac{1} {{r \cdot P}} \cdot \frac{{{\text{d}}z}} {{{\text{d}}r}}\;\;\;{\text{and}}\;\;\;c_{{\text{m}}} = \frac{1} {{P^{2} {\text{d}}z/{\text{d}}r}} \cdot \frac{{{\text{d}}P}} {{{\text{d}}r}}, $$

(39)

$$ {P = {{{\rm{d}}s} \over {{\rm{d}}r}} = {\sqrt {1 + {\left( {{{{\rm{d}}z} \over {{\rm{d}}r}}} \right)}^{2} } }.} $$

(40)

Evaluating the derivatives leads to the following expressions for the principal curvatures c _m and c _p:

$$ {c_{{\rm{m}}} = {{12r^{2} } \over {r^{{\rm{2}}}_{{\rm{C}}} L^{2} }}{\left[ {1 - {5 \over 3}{\left( {{r \over {r_{{\rm{C}}} }}} \right)}^{4} } \right]} \cdot {\left( {{{z_{{\rm{b}}} (r)} \over {P_{b} }}} \right)}^{3} } $$

and

$$ {c_{{\rm{p}}} = {{4r^{2} z_{{\rm{b}}} (r)^{2} } \over {r^{{\rm{4}}}_{{\rm{C}}} LP_{{\rm{b}}} }}} $$

(41)

(P _b stands for P (Eq. 40) calculated for the spicule contour).

Calculation of the nonlocal bending energy requires evaluation of Δa (Eq. 13). For the pipette projection and the spherical portion of the cell body, the relevant expressions are:

$$ {\Delta a^{{\rm{p}}} = {{L_{{\rm{p}}} + R_{{\rm{p}}} } \over {4R_{{\rm{0}}} }},} $$

(42)

$$ {\Delta a^{{\rm{s}}} = {{R_{{\rm{c}}} } \over {4\pi R_{0} }}A_{{\rm{s}}} \cdot } $$

(43)

For the spicules, Δa ^b was calculated by numerical integration of Eq. (4) using the expressions for the principal curvatures given in Eq. (41). An additional term, Δa ^T, arises from a leaflet area difference of the junction between the spherical cell body and the cell projection in the pipette. The leaflet area difference of that small region of high curvature was estimated by treating the junction as a torus that matches slopes with the cylindrical projection in the pipette and the spherical outer section. Thus, the larger radius of the torus is equal to R _p. The expression for Δa ^T was obtained by taking the limit as the smaller radius approaches zero:

$$ {\Delta a^{{\rm{T}}} = - {{R_{{\rm{p}}} } \over {4R_{{\rm{0}}} }}{\left( {{\pi \over 2} - \arcsin {{R_{{\rm{p}}} } \over {R_{{\rm{c}}} }}} \right)},} $$

(44)

Appendix C. Elastic energy of the skeleton

Deformation of the membrane skeleton is characterized by the mapping of a small skeleton portion from its resting location to the deformed one, where the mapping function is obtained by minimizing the deformation energy of the skeleton (W _sk). As described in the Theory section, the resting shape of the membrane is assumed to be discoid and the shape of the deformed membrane is constructed from different geometric units: a spherical cell body, spicules, and a cell projection (Fig. 9). Both the resting and deformed shapes of the membrane skeleton are specified by contour functions r ₀(s ₀) and r(s). These are shown in Table 1 (cell projection formation) and in Table 2 (spicule formation). Axisymmetric mapping of a small portion of the skeleton from its resting location at (s ₀, r ₀) to the deformed one at (s, r), can be defined by a single parametric mapping function:

Fig. 9.

Membrane regions of the cell contour for membrane deformation due to pipette aspiration (a) and spicule formation (b). The different regions are distinguished by dashed and solid lines and designated by I₀, I, etc. The position of the material point is defined by radial and meridional coordinates, r ₀ and s ₀ for the resting contour (left side of the figure), r and s for the deformed contour (right side of the figure). Axes of symmetry are indicated by dotted lines

Table 1. Parametric description of the cell projection contour and the corresponding resting membrane contour; see Fig. 9a

Table 2. Parametric description of the spicule contour and the corresponding resting membrane contour; see Fig. 9b

$$ {s_{0} = s_{0} {\left( s \right)}.} $$

(45)

Deformation of that small portion of the skeleton is described by material extension ratios defined by Eq. (14):

$$ \lambda _{1} = \frac{{r(s)}} {{r_{{\text{0}}} (s_{0} )}}\;\;\;{\text{and}}\;\;\;\lambda _{2} = \frac{1} {{{s}'_{0} }}, $$

(46)

where s′₀=ds ₀/ds. The mapping function s ₀(s) is obtained by the minimization of the elastic energy of the skeleton W _sk (Eqs. 8 and 9), which can be expressed as a function of s ₀(s) and its derivative only. W _sk is obtained by integration over deformed membrane surface:

$$ {W_{{{\rm{sk}}}} (s_{{\rm{0}}} ,{s}'_{{\rm{0}}} ) = 2\pi {\int {{\ell }(s,s_{{\rm{0}}} ,{s}'_{{\rm{0}}} ) \cdot {\rm{d}}s} },} $$

(47)

where ${{\ell }(s,s_{{\rm{0}}} ,{s}'_{{\rm{0}}} )}$ takes the form:

$$ {{\ell }(s,s_{{\rm{0}}} ,{s}'_{{\rm{0}}} ) = {\left\{ {{\mu \over 2} \cdot {\left[ {{{r(s)} \over {r_{{\rm{0}}} (s_{0} )}} - {1 \over {{s}'_{0} }}} \right]}^{2} + {K \over 2} \cdot {\left[ {{{r(s)} \over {r_{{\rm{0}}} (s_{0} )}} + {1 \over {{s}'_{0} }} - 2} \right]}^{2} } \right\}}r_{{\rm{0}}} (s_{0} ){s}'_{0} .} $$

(48)

In order to minimize W _sk (Eq. 47), s ₀(s) is derived by solving the Euler–Lagrange equation

$$ {{{\partial {\ell }(s,s_{{\rm{0}}} ,{s}'_{{\rm{0}}} )} \over {\partial s_{{\rm{0}}} }} - {{\rm{d}} \over {{\rm{d}}s}}{{\partial {\ell }(s,s_{{\rm{0}}} ,{s}'_{{\rm{0}}} )} \over {\partial {s}'_{{\rm{0}}} }} = 0.} $$

(49)

Separate numerical solutions are obtained for the aspirated projection and for the outer cell portion. To solve the Euler–Lagrange equation, the values of the mapping function and its derivative at the initial point are required. They must satisfy boundary conditions. The central point of the resting skeleton must map onto the apex of the deformed shape (either the tip of the spicule located on the axis of symmetry or the tip of the cell projection). The total surface area of the skeleton must not change during its deformation, and the lateral tension of the aspirated part must match the lateral tension of the outer part of the cell. Note that the relative proportion of the resting skeleton that is either aspirated into the pipette or deformed into the outer sphero-echinocytic portion is not fixed. In our approach the relative proportion of resting skeleton in the pipette or on the cell body was varied to obtain the minimal skeleton deformation energy, and thus to assure matching of the lateral tensions.