Question

Consider the cortex of a red blood cell. Develop a model to predict the membrane mechanics of a two-dimensional array of cross-linked spectrin polymers arranged in a simple rectangular lattice. The lattice spacing is L_e. Each one of the polymers has a persistence length l_p, contour length L, and diameter d.

Using optical tweezers, researchers have found that the persistence length of spectrin is 10nm. Furthermore, the contour length of each spectrin molecule is 200 nm, and L_e is approximately 70 nm.

a) Assuming that the deformations are small, what regime (ideal chain, semiflexible polymer, continuum mechanics) are these polymers in and why? What model would be appropriate to describe their behavior?

b) Now derive an expression for the membrane area expansion modulus K_A for the model.

This question is from the book : Introduction to Cell Mechanics and Mechanobiology
By; Jacobs, Huang, and Kwon ISBN 978-0-8153-4425-4. (Chapter 9, question 6)

Answer 1

a)

Let us consider a 2D model consisting of the spectrin polymers connected to the nodes which are cross linked in an array.The below model would be appropriate.

The Lattice spacing of the structure is given by Le which is 70nm. It is a space between the adjacent nodes in the lattice structure.

Persistence length given by lp which is 10 nm. It is a point at which a polymer can be ceased to act elastically and should be treated static.

and have contour length of 200 nm. It is a maximum length up to which a polymer can be extended.

b)

  

Let us assume that spectrin is isotropic elastic and homogeneous. We will consider the strain energy which will be stored in spectrin due to elastic deformation. This will have similar effect as energy stored in a spring.

Let us consider spring is linear and have stiffness k, the force F required to pull it from x₀ to x is k(x-x₀), then the energy is

^{U ==   =k (x-xo)2}

Now consider a section of an elastic spectrin of thickness d and lengths dx1 and dx2 subjected to a uniaxial tension stress σ1.

As it is stretched gradually increased, the gain in the elastic energy is equal to the work done. So work done is-

^{dU ==}

By law of conservation of energy, work done in stretching the spectrin must be stored . The right-hand-side represents the strain energy in the volume dx₁dx₂dx₃, then the energy per unit area of spectrin polymer is

U_od = Ehε₁²  / 2 (per unit area)

If strain is isotropic then, ε1 = ε2 = ε, can also be expressed as areal strain -

U_od = Ed /4 (1-v) * (ΔA/ A_o)²

Which can also be expressed in the area expansion modulus and has the form:

KA = Ed / 2 (1-v)