Correction: Outer hair cell electromotility is low-pass filtered relative to the molecular conformational changes that produce nonlinear capacitance

Vol. 151, No. 12 | https://doi.org/10.1085/jgp.201812280 | November 1, 2019

In the Appendix of this paper, two transition rates, k₊ and k₋, did not include the effect of the energy barrier between the two conformational states. The inclusion of this barrier effect required redefinition of those two transition rates. In addition, there were typographical errors in Eqs. 38 and 41, which have been corrected. The corrections appear in bold below. In addition, the values on the x axis in Fig. A3 were changed to kilohertz. These errors were fixed in the online article but appear in print and in the PDF.

Motile element with two states

Consider a membrane molecule with two discrete conformational states, C₀ and C₁, and let the transition rates k₊ and k₋ between them be schematically expressed as

C 0 \begin{array}{c} k_{+} \\ \leftrightarrow \\ k_{-} \end{array} C 1 .

Let P₁ be the probability that the molecule in state C₁. Then, the probability P₁ can be expressed by the transition rates

\frac{P_{1}}{1 - P_{1}} = \frac{k_{+}}{k_{-}} = \exp {- β [q (V - V_{1 / 2})]},

(A1)

where q is the charge transferred across the membrane during conformational changes, V the membrane potential, V_1/2 the half-point voltage of the transition, and β = 1/k_BT with the Boltzmann constant and T the temperature.

If q is positive, the energy level of the state C₁ is higher, reducing P₁ as the membrane potential V rises. For prestin in outer hair cells, which shorten on depolarization, if we choose C₁ as the shortened state, the unit length change a on conformational change is negative, and then we have q < 0. Notice that the quantity a does not appear in Eq. A1.

The transition rates that satisfy can be given by

k_{+} = {\bar{k}}_{+} \exp [- α β q (V - V_{0})] \approx {\bar{k}}_{+} [1 - α β q (V - V_{0})],

(A2)

k_{-} = {\bar{k}}_{-} \exp [(1 - α) β q (V - V_{0})] \approx {\bar{k}}_{-} [1 + (1 - α) β q (V - V_{0})] .

(A3)

Here, α is a constant between 0 and 1 and

{\bar{k}}_{+}

and

{\bar{k}}_{-}

are transition rates at the operating voltage V₀, around which V changes with time. The exponential function can be linearized because we assume V − V₀ is small. These rates at the operation point are expressed by

{\bar{k}}_{+} = \bar{k} \exp [1 - α β q (V_{0} - V_{1 / 2})],

{\bar{k}}_{-} = \bar{k} \exp [(1 - α) β q (V_{0} - V_{1 / 2})],

where the transition rate $\bar{k}$ is due to an energy barrier between the two states, excluding the difference in the energy levels, which are voltage dependent.

The time dependence of P₁ can be expressed by the rate equation

\frac{d}{d t} P_{1} = k_{+} - (k_{+} + k_{-}) P_{1} .

(A4)

Now we introduce sinusoidal voltage changes of small amplitude

v

on top of constant voltage

\bar{V}

⁠, i.e.,

V = \bar{V} + v \exp [i ω t]

⁠, where ω is the angular frequency and i = √−1. Then the transition rates are time-dependent due to the voltage dependence . They satisfy

\frac{k_{+}}{k_{-}} = \frac{{\bar{k}}_{+}}{k_{-}} (1 - β q ν \exp [i ω t]) .

(A5)

Notice

{\bar{k}}_{-}

and

{\bar{k}}_{+}

are time independent, and we assume that v is small so that βqv << 1. A set of k₊ and k₋ that satisfies can be expressed

k_{+} = {\bar{k}}_{+} (1 - α β q ν \exp [i ω t]),

(A6)

k_{-} = {\bar{k}}_{-} {1 - (1 - α) β q ν \exp [i ω t]} .

(A7)

If we express

P_{1} = {\bar{P}}_{1} + p_{1} \exp [i ω t]

⁠, we have respectively for the 0th and first order terms (Iwasa, 1997)

{\bar{P}}_{1} = \frac{{\bar{k}}_{+}}{k_{+} + k_{-}},

(A8)

p_{1} = - \frac{{\bar{k}}_{+} {\bar{\bar{k}}}_{-}}{{\bar{k}}_{+} + {\bar{\bar{k}}}_{-}} \cdot \frac{β q ν}{i ω + {\bar{k}}_{+} + {\bar{k}}_{-}} .

(A9)

Notice that p₁ does not depend on the factor α.

leads to voltage-driven mechanical displacement ap₁ exp[iωt] with

{a p}_{1} = {- \bar{P}}_{\pm} \pm \cdot \frac{β a q v}{1 + i ω / ω_{g}},

(A10)

where

{\bar{P}}_{\pm} = {\bar{P}}_{1} (1 - {\bar{P}}_{1})

⁠. The amplitude |χ| of the motile response is given by

{| x |}^{2} = \frac{{(β a q {\bar{P}}_{\pm})}^{2}}{1 + {(ω / ω_{g})}^{2}} \cdot v^{2} .

(A11)

Charge displacement is expressed by qp₁ and the contribution to complex admittance Y(ω) is given by (q/v)(d/dt)p₁ exp[iωt] (Iwasa, 1997). The contribution to the membrane capacitance is C_nl(ω) = Im[Y(ω)]/ω and therefore

C_{n l} (ω) = \frac{β q^{2} {\bar{P}}_{1} ({\bar{P}}_{1})}{1 + {(ω / ω_{g})}^{2}} .

(A12)

This contribution to the membrane capacitance is commonly referred to as NLC because it shows marked voltage dependence. Notice also that the above derivation evaluates the contribution of a single unit of motile element. For a cell that contains N motile units, both |x| and C_nl need to be multiplied by N.

The roll-off frequency ω_g due to gating is expressed by

ω_{g} = {\bar{k}}_{+} + {\bar{k}}_{-} .

(A13)

With Eqs. A6 and A7, this means that 1/ω_g rises at both ends of the membrane potential because α can take any value between 0 and 1. That means ω_r can be asymmetric unless α = 1/2.

In the special case of α = 1/2,

{\bar{k}}_{+}

= 1/

{\bar{k}}_{-}

⁠. If we define

b (V) = \exp [- β q (\bar{V} - V_{0}) / 2]

⁠, then

ω_{g} = {\bar{k}}_{+} b (V) + {\bar{k}}_{-} / b (V),

(A14)

which resembles the bell-shaped voltage dependence of nonlinear capacitance at low frequencies (ω → 0).

Mechanoelastic coupling

For motile membrane proteins based on mechanoelectric coupling, charge transfer is affected by mechanical factors. Here, we assume the cell is cylindrical as in the case of cochlear outer hair cells and approximate it as a one-dimensional object (Fig. A1).

Supposing charge transfer q is associated with a change a in the length of the cell, should be replaced by

\frac{P_{1}}{1 - P_{1}} = \frac{k_{+}}{k_{-}} = \exp {- β [q (V - V_{1 / 2}) + a F]},

(A15)

where V_1/2 is the midpoint voltage of the Boltzmann function and F is the axial force. Eq. A16 and A17 for the transition rates should be

k_{+} = {\bar{k}}_{+} \exp [- α β [q (V - V_{0}) + a f]],

(A16)

k_{-} = {\bar{k}}_{-} \exp [(1 - α) β [q (V - V_{0}) + a f]],

(A17)

where f is a small change in the axial force F that corresponds to a small voltage change V − V₀. The transition rates ${\bar{k}}_{+}$ and ${\bar{k}}_{-}$ are redefined by including the effect of the axial force F₀ of the resting condition. For the rest of the present paper, the dependence on the value of the parameter α does not appear except for ω_g.

With a shorthand notation

{\bar{P}}_{\pm} (= {\bar{P}}_{1} (1 - {\bar{P}}_{1}))

⁠, the change of the conformational probability p₁ can be driven either by changes in the voltage as well as force:

p_{1} = - β {\bar{P}}_{\pm} \cdot \frac{q v + a f}{1 + i ω / ω_{g}} .

(A18)

If the motile element is driven by voltage changes, p₁ is proportional to v and mechanical displacement is given by ap₁.

Effect of viscous drag

Movement is driven by a deviation from Boltzmann distribution. When voltage changes with amplitude v is imposed, p₁ as expressed by is the goal of the drive. Since this force is countered by viscous drag (with drag coefficient η), the equation of motion in the frequency domain can be expressed by

i η ω a p = k a (p_{1} - p) .

(A19)

Notice here that the equilibrium transition rates here depend not only on $\bar{V}$ but also on $\bar{F}$ because the motile element based on piezoelectricity is sensitive to mechanical force as well as the membrane potential.

leads to

(1 + i ω / ω_{η}) p = - \frac{β {\bar{P}}_{\pm}}{1 + i ω / ω_{g}} \cdot q v,

(A20)

similar to the previous treatment for the special case of without inertial loading (Iwasa, 2016). Here, the viscoelastic relaxation frequency is defined by ω_η = k/η. It is essentially an equation for viscoelastic relaxation, adding a low pass filter to the motile mechanism. It is consistent with previous expressions in both extremes, i.e., ω_g → ∞ and ω_η → ∞.

The voltage dependence of NLC and that of motile response are identical. In the following, we show that mechanical load with complex relaxation can lead to discrepancy in their frequency dependences.

Complex mechanical relaxation

Let X represent the point that links a spring k₁ with a dashpot η₁. Let Y represent the point that joins the spring k₁ with the rest, which includes a spring k₂, a dashpot η₂, and a driver (Fig. A2). The equations of motion of this system driven by force F generated at the location P can be expressed

η_{1} \frac{d X}{d t} = - k_{1} (X - Y),

(A21)

F + k_{2} Y + η_{2} \frac{d Y}{d t} = k_{1} (X - Y) .

(A22)

If the force generator operates at a frequency ω with small amplitude on top of its steady value

\bar{F}

⁠,F =

\bar{F}

+ f exp[iωt]. By letting the small amplitude components of X and Y with frequency ω represented respectively by x and y, and turn into

i ω η_{1} x = - k_{1} (x - y),

(A23)

f + (k_{2} + i ω η_{2}) y = k_{1} (x - y) .

(A24)

can be rewritten as

x = \frac{y}{1 + i ω / ω_{1}},

(A25)

which indicates that the quantity x is obtained by low-pass filtering y with roll-off frequency of ω₁(= k₁/η₁).

By introducing a characteristic frequency ω₂(= (k₁ + k₂)/η₂), can be transformed into

f + (k_{1} + k_{2}) [1 + i ω / ω_{2}] y = k_{1} x .

(A26)

Elimination of x from with the aid of leads to

y = f / G_{1} (ω),

(A27)

G_{1} (ω) = \frac{k_{1}}{1 + i ω / ω_{1}} - (k_{1} + k_{2}) (1 + i ω / ω_{2}) .

(A28)

An approach analogous to those in the previous sections lead to an equation

G_{1} (ω) a p = a k_{2} (p_{1} - p) .

(A29)

Since we have y = ap, this equation leads to

[G_{1} (ω) + k_{2}] y = - \frac{β {\bar{P}}_{\pm} k_{2} a q v}{1 + i ω / ω_{g}} .

(A30)

Eqs. A25 and A30 show that the relationship between y and v has three adjustable parameters, ω₁, ω₂, and k_r (= k₁/k₂). For an example of the frequency dependence of y, see Fig. A3.

The frequency dependence of NLC is the same as that of y. Motile response x is obtained by low-pass filtering y. The roll-off frequency of y is voltage-dependent due to the voltage dependence of ω_g.

With the connectivity of Fig. A2, it is difficult to make high frequency roll off of both quantities as similar as the experimental data. For y to roll off at relatively high frequency, ω₁ has to be small and ω₂ has to be large because G₁ must be small as required by Eq. A30. This requirement makes x roll off at a frequency much lower than y does (Fig. A3).

Modified complex mechanical relaxation

The model described above predicts a difference between x and y much larger than the experimentally observed frequency dependence. Let us add a spring across the upper dashpot (Fig. A4).

The set of equations that describe this configuration are

(η_{0} \frac{d}{d t} + k_{0}) X = k_{1} (Y - X),

(A31)

F + (η_{2} \frac{d}{d t} + k_{2}) Y = - k_{1} (Y - X) .

(A32)

If force F is driven at angular frequency ω with amplitude f, the equation is transformed into

(i ω η_{0} + k_{0}) x = k_{1} (y - x),

(A33)

f + (i ω η_{2} + k_{2}) y = - k_{1} (y - x),

(A34)

where variables in the lower case x and y are the complex amplitude of frequency ω.

and can be rewritten as

x = \frac{k_{1}}{k_{0} + k_{1} + i ω η_{0}} \cdot y,

(A35)

f = k_{1} x - (k_{1} + k_{2} + i ω η_{2}) y .

(A36)

In the manner similar to and in the previous case, these equations can be expressed as

y = f / G_{2} (ω),

(A37)

G_{2} (ω) = \frac{{k_{1}}^{2}}{k_{0} + k_{1} + i ω η_{0}} - (k_{1} + k_{2} + i ω η_{2}),

(A38)

which corresponds to G₁(ω) in the previous case.

Since force generation is associated with spring k₂ in the manner similar to the previous case, we obtain

[G_{2} (ω) + k_{2}] y = - \frac{β a q k_{2} {\bar{P}}_{\pm}}{1 + \frac{i ω}{ω_{g}}} \cdot v,

(A39)

and x is obtained with Eq. A35.

If the cell contains N motile units, a should be replaced by aN. For numerical analysis, the number of parameters can be reduced by introducing the ratios k₀₂(= k₀/k₂), k₁₂(= k₁/k₂), ω₀(= k₂/η₀), and ω₂(= k₂/η₂), and x and y are expressed by

x = \frac{k_{12}}{k_{02} + k_{12} + i ω / ω_{0}} \cdot y,

(A40)

y = \frac{β a q N {\bar{P}}_{\pm}}{1 + i ω / ω_{g}} \cdot \frac{1}{k_{12}^{2} / (k_{02} + k_{12} + i ω / ω_{0}) - (k_{12} + i ω / ω_{2})} \cdot v .

(A41)

The corresponding equations (Eq. 10) in the main text are expressed with linear frequency f instead of angular frequency ω. Because these equations depend only on frequency ratios, no extra factor appears.

View Original Article

2023

This article is available under a Creative Commons License (Attribution 4.0 International, as described at https://creativecommons.org/licenses/by/4.0/).

Motile element with two states

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Effect of viscous drag

Complex mechanical relaxation

Modified complex mechanical relaxation

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Correction: Outer hair cell electromotility is low-pass filtered relative to the molecular conformational changes that produce nonlinear capacitance

Motile element with two states

Mechanoelastic coupling

Effect of viscous drag

Complex mechanical relaxation

Modified complex mechanical relaxation

Data & Figures

Contents

Supplements

References

Related & Metrics

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