It is clear that this normalisation induces non-standardI−V andI−Cbehaviour.
It needs to be tested against experimental recordings with a description of proper- ties such as rectication. In gure 7.7 we compare two state conduction with our full state space conduction in a physiological voltage rangeφ =−0.2V :→+0.2V.
It is clear that the two curves coexist within this small voltage range, and so the two state reduction exactly describes the conduction. The parameter ∆¯µhas
Concentration [M] 0 0.5 1 1.5 2 Current [pA] 0 5 10 15 20 25 ∆µ0 = 5 kT ∆µ0 = 0 kT ∆µ0 = -5 kT
Figure 7.6: The standard tting parameters are used withφ = 0.2V. The curves
are colour coordinated with the value of ∆¯µ0, meanwhile the black dashed curves
representIM M.
quite a profound eect in shaping the current because if this parameter is small (or negative) or very large then current has a much smaller magnitude and is quasi- exponential in its growth which will be important when discussing rectication (see later).
From gure (7.7) it is clear that up to∼200mV the only conducting energy levels
are the optimal transport regime {2K+},{3K+} and so we can reduce the state
space to these states. Current in this two state system reduces as,
I = q 2 Γ L 23−Γ R 23 , (7.33)
with the expanded form,
I = q 2D c /L2xRe−(∆E−∆¯µR−zq(0−χ)φ)/kT −xLe−(∆E−∆¯µL−zq(1−χ)φ)/kT× h xLxRe−(2∆E−∆¯µR−∆¯µR−zq(1−2χ)φ)/kT +xRe−(∆E−∆¯µR−zq(0−χ)φ)/kT+ xLe−(∆E−∆¯µL−zq(1−χ)φ)/kT + 1i −1 (7.34)
φ [V] 0 0.05 0.1 0.15 0.2 I [pA] 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 ∆µ=-5kT (A) φ [V] 0 0.05 0.1 0.15 0.2 I [pA] -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 ∆µ=0kT (B) φ [V] 0 0.05 0.1 0.15 0.2 I [pA] -5 0 5 10 15 20 25 ∆µ=5kT (C) φ [V] 0 0.05 0.1 0.15 0.2 I [pA] -2 0 2 4 6 8 10 12 ∆µ=10kT (D)
Figure 7.7: Comparison of theoretical I −V curves between full and reduced
state-space currents. The curves coexist exactly for the full range of ∆¯µ values
suggesting the reduced states space model exactly describes conduction. Varying
∆¯µ had a profound eect on the shape and amplitude of the current.
If we consider a symmetrical lter such thatχ= 1/2 and concentrations in either
bulk are equal: ηL=ηR then we can collect terms and write current as,
I =qDc/L2 × x e
(∆E−∆¯µ+qzφ/2)/kT −e(∆E−∆¯µ−qzφ/2)/kT
2x2+ 2xe(∆E−∆¯µ+qzφ/2)/kT + 2xe(∆E−∆¯µ−qzφ/2)/kT + 2e(2∆E−2∆¯µ)/kT (7.35) If we consider the large voltage regime then it saturates to,
lim
φ→±∞I =±
qDc
to a MM form given suitable tting parameters. It is further complicated by the non-linear concentration dependence in∆¯µvia the Debye-Hückel term. If we
neglect the importance of this dependence for now, then we can write a conditional current. When the tting parameters are such that,
2x2 <2xe(∆E−∆¯µ+qzφ/2)/kT + 2xe(∆E−∆¯µ−qzφ/2)/kT + 2e(2∆E−2∆¯µ)/kT (7.37)
we can write the current as,
I = xkm x+K (7.38) km = q 2D c/L2× e (∆E−∆¯µ+zqφ/2)/kT −e(∆E−∆¯µ−zqφ/2)/kT (e(∆E−∆¯µ+zq)φ/2)/kT +e(∆E−∆¯µ−zqφ/2)/kT ) (7.39)
K =e(2∆E−2∆¯µ)/kT × e(∆E−∆¯µ+zqφ/2)/kT +e(∆E−∆¯µ−zqφ/2)/kT−1. (7.40)
where km and K are the voltage dependent maximum permeation rate and the Michaelis mole fraction respectively. Of course this can only describe quasi-MM behaviour in any case because of the non-linear concentration dependence in ∆¯µ.
In gure 7.8 a comparison is given between pure two-state kinetic equation current (solid) from equation (7.35), and our reduced current (dash-dot) in MM form from equation (7.38). Only minor dierences can be observed for the largest ∆¯µ with
a peak dierence in current of ∼ 2pA and so it is unlikely to detract from the
quality of tting.
Rectication
Rectication of current is described by small non-ohmic current at relatively large voltages. It is often asymmetrical and therefore requires an electrical asymmetry introduced via χ, when it takes values0 ≤χ < 1/2 and 1/2< χ ≤1. To discuss
Concentration [M] 0 0.5 1 1.5 2 Current [pA] 0 5 10 15 20 25 ∆µ0 = 5 kT ∆µ0 = 0 kT ∆µ0 = -5 kT
Figure 7.8: A comparison between equations (7.35) and (7.38) (black dashed curve) describing single-species current vs. concentration. The kinetic equation
solution only diered with its reduced MM form by ∼ 2pA suggesting that it
should result in good tting to data.
source of asymmetry is fromχ. If we dene the constant A,
Ab = exp(∆E −kTln xb−∆¯µb)/kT, (7.41)
then we can rewrite current as,
I =q/2Dc/L2 1 1 +Aexp[(−qz(1−χ)φ)/kT] − 1 1 +Aexp[(−qz(0−χ)φ)/kT] (7.42) There are two distinct regimes now to obtain rectication eitherA1orA1.
Thus if we rstly consider the largeAlimit we approximate the sigmoidal function
as exponential growth,
I =q/2Dc/L2 A−1exp[(qz(1−χ)φ)/kT]−A−1exp[(qz(0−χ)φ)/kT]
main depending on χ. If we are in the non-rectied domain then the approxima-
tion will break down whenφ becomes large because it becomes quasi-exponential
growth. If we consider the positive φ domain then current will be dominated by
this rst term, and if χ > 0.5 it will be rectied in the positive voltage domain.
With rectication in the negative voltage domain found whenχ <0.5.
If we now consider the reverse limit such thatA1the current can be expanded
as,
I =q/2Dc/L2(Aexp[(−qz(0−χ)φ)/kT]−Aexp[(−qz(1−χ)φ)/kT]). (7.44)
The value of χ needed for rectication is now reversed, such that whenχ <0.5 it
recties in the positive voltage domain.
These two expressions for rectied current are similar but not exact and thus oer a distinct form of rectication. Physically this rectication requires an asymmetry in the position of the binding site and an energy barrier/well to the binding energy given from A.
Figure 7.9 displays the normal and rectied current under standard conditions, for large A (left) and small A (right). In both gures the dashed lines indicate the
theoretical current calculated from the approximate expressions. In both gures the approximations hold well for the rectied domain but break-down in the op- posite domain at ∼ 0.1V corresponding to ∼ 4kT because this contribution is of
the order of A. This conrms that varying∆¯µand χcan produce rectication.