*2.5. The Rate of the Planar Lipid Bilayer Capacitance Change at tbr*

Current *i*(*t*) which charges planar lipid bilayer can be represented as

$$i(t) = \frac{u(t)}{\mathcal{R}(t)} + \frac{d\mathcal{Q}(t)}{dt} = \frac{u(t)}{\mathcal{R}(t)} + \frac{d(\mathcal{C}(t)u(t))}{dt} = \frac{u(t)}{\mathcal{R}(t)} + \mathcal{C}(t)\frac{du(t)}{dt} + u(t)\frac{\mathcal{C}(t)}{dt}.\tag{3}$$

During voltage-controlled experiments, voltage *u*(*t*) rises with time linearly in accordance with the slope *ku*. Therefore, Equation (3) changes to

$$i(t) = \frac{k\_{\rm ll}}{R(t)}t + \mathcal{C}(t)k\_{\rm ll} + k\_{\rm u}t\frac{d\mathcal{C}(t)}{dt}.\tag{4}$$

*i*(*t*) at the moment *tbr* is

$$i(t\_{br}) = \frac{k\_{\rm u}}{R(t\_{br})} t\_{br} + \mathcal{C}(t\_{br}) k\_{\rm u} + k\_{\rm u} t\_{br} \frac{d(\mathcal{C}(t))}{dt} \bigg|\_{t\_{br}}.\tag{5}$$

By using experimentally obtained values for *tbr* and *i*(*tbr*), *d*(*C*(*t*)) *dt* |*tbr* can be calculated. We assumed that the changes in *C* due to electrostrictive thinning are small enough to be neglected. Therefore, for each lipid composition, *C* measured before exposure of the planar lipid bilayer to linearly rising signal was used in calculations. As a result that we cannot measure *R* before the membrane breakdown occurs, we calculated it for each lipid composition from current-controlled experiments, performed with the slowest linearly rising current (*k<sup>i</sup>* = 0.5 µA) at the moment *tbr*.

We assumed that the last term in Equation (3) is zero, because *C* has not been changed yet, and the current rises linearly in time according to *i*(*t*) = *k<sup>i</sup>* · *t*. Differential equation for transmembrane voltage *u*(*t*) was solved using Laplace transform. The solution is

$$u(t) = k\_l \cdot R \cdot \left(t - R\mathbb{C} + (R\mathbb{C})^2 \cdot e^{-\frac{t}{R\mathbb{C}}}\right), \quad t > 0. \tag{6}$$

The equation is valid also at the moment *tbr*. In the case of slowly linearly rising current, the product *RC* is at least 10 times smaller than *tbr*. Thus, the last summand in the parenthesis in Equation (6) is much smaller than the other two and can be neglected. The Equation (6) can be simplified to

$$\mathbf{C} \cdot \mathbf{R}^2 - t\_{br} \cdot \mathbf{R} + \frac{u(t\_{br})}{k\_i} = \mathbf{0},\tag{7}$$

and solved for *R*.
