BC-535 Preliminary, Rev. 060126
28
Membrane capacitance calculations
It is possible to theoretically derive an equation to determine the size of the bilayer formed across
the aperture. While this equation will probably not yield an exact result (most likely due to variation in
the dielectric constant of your lipid mixture), it will give a reasonably approximate result.
Recall that we examine the formation of the bilayer by applying a triangular wave to the
membrane and observing a square wave at the I
m
output. The reason we see a square wave is that a
capacitor returns the derivative of the applied voltage, as shown in the equations on the next page.
However, under normal circumstances you will most likely dispense with a calculation and visually
determine if the membrane size is appropriate by examining the amplitude of the square wave on the
oscilloscope.
From physics, we know that the equation describing the capacitance of a parallel plate capacitor
in the MKS system of units is
d
A
C
ε
=
(1)
where C is the capacitance (in Farads),
ε
is the dielectric constant of the material between the plates,
A is the area of the plates and d is the plate separation (both in meters). Likewise, we know that that
the steady-state charge on a capacitor can be expressed as
CV
q
=
(2)
where q is the charge on one capacitor plate (in Coulombs) and V is the potential between the plates
(in Volts). Equation (2) can be dynamically expressed by taking the time derivative of the charge, thus
dt
dV
C
i
=
(3)
where the current
dt
dq
i
=
is defined as the time rate of change of the charge. Substituting equation
(1) into equation (3) yields the general equation,
dt
dV
d
A
i
ε
=
.
(4)
Recall that a bilayer membrane is electronically represented as a capacitor, and that we monitor
the forming bilayer through the application a triangular wave. Since a triangular wave, by definition,
has a constant rate of change of applied voltage,
dt
dV
is constant. Likewise, since
ε
is an intrinsic
property of the lipid mixture, it is also constant.
Now consider the forming bilayer membrane. Once a sufficient quantity of lipids have drained
away from the aperture, the remaining lipids begin to form a bilayer. Since the distance, d, separating
both sides of the membrane (the plates of our hypothetical capacitor) is fixed by the length of the lipid
tails, this term will also become constant. Therefore, the only remaining variable on the right side of
equation (4) is the area, A, of the forming bilayer. Thus we can express our equation as
kA
i
=
(5)
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