Question

A cell membrane has a resistance and a capacitance and thus a characteristic time constant.

What is the time constant of a 8.3 nm -thick membrane surrounding a 3.6×10−2 mm -diameter spherical cell? Assume the resistivity of the cell membrane as 3.6×106 Ω⋅m and the dielectric constant is approximately 9.0.

Answer 1

Given that :

thickness of cell membrane, L = 8.3 x 10^-9 m

diameter of the spherical cell, D = 3.6 x 10^-2 mm

radius of spherical cell, r = 1.8 x 10^-5 m

resistivity of cell membrane, = 3.6 x 10⁶.m

dielectric constant, k = 9

Using an equation,                C = k ₀ A / d                                                                { eq.1 }

where, A = area of spherical cell = 4r²

C = (9) (8.85 x 10^-12 C²/Nm²) [4 (3.14) (1.8 x 10^-5 m)² / (8.3 x 10^-9 m)

C = (79.65 x 10^-12 C²/Nm²) (42.9 x 10^-10 m²) / (8.3 x 10^-9 m)

C = (3416.9 x 10^-22 C²/N) / (8.3 x 10^-9 m)

C = 411.6 x 10^-13 F

C = 41.1 x 10^-12 F

And resistance may be defined by,     R = (L / A)                                                       { eq.2 }

inserting the values in eq.2

R = (3.6 x 10⁶.m) (8.3 x 10^-9 m) / (42.9 x 10^-10 m²)

R = (29.8 x 10^-3.m²) / (42.9 x 10^-10 m²)

R = 0.694 x 10⁷

R = 6.94 x 10⁶

The time constant for a cell membrane which will be given as :

= R C                                                                                { eq.3 }

inserting the values in eq.3,

= (6.94 x 10⁶) (41.1 x 10^-12 F)

= 285.2 x 10^-6 sec

= 0.285 ms