Similarities between action potentials and acoustic pulses in a van der Waals fluid

An action potential is typically described as a purely electrical change that propagates along the membrane of excitable cells. However, recent experiments have demonstrated that non-linear acoustic pulses that propagate along lipid interfaces and traverse the melting transition, share many similar properties with action potentials. Despite the striking experimental similarities, a comprehensive theoretical study of acoustic pulses in lipid systems is still lacking. Here we demonstrate that an idealized description of an interface near phase transition captures many properties of acoustic pulses in lipid monolayers, as well as action potentials in living cells. The possibility that action potentials may better be described as acoustic pulses in soft interfaces near phase transition is illustrated by the following similar properties: correspondence of time and velocity scales, qualitative pulse shape, sigmoidal response to stimulation amplitude (an ‘all-or-none’ behavior), appearance in multiple observables (particularly, an adiabatic change of temperature), excitation by many types of stimulations, as well as annihilation upon collision. An implication of this work is that crucial functional information of the cell may be overlooked by focusing only on electrical measurements.


Supplemental materials
The supplemental materials are intended to provide interested readers with further mathematical details (section A), additional solutions (section B) and caption to the supplemental movie (section C).

A Model Equations
Model variables and parameters. The fluid considered in this letter is described by six dynamic variables of space and time, that are coupled to one another by three conservation laws (Eq. (1)), two constitutive equations (Eq. (4)), and one inverse relation, ρ = w −1 . The six variables are listed in Table S1. In addition, the model depends on seven constant parameters, all are measurable physical quantities (Table S2). The van der Waals parameters (m, a, b) are related to the critical point according to The scaling of the spatial coordinate h (in the Lagrange frame) was defined according to the critical density (w = w c ). Finally, k B ∼ = 1.38 · 10 −23 J/K is the Boltzmann constant.
Dimensionless model equations. The critical point and the fluid viscosity parameters were used (Eq. (5)) to introduce the following reduced variables and parameters: The spatial dimension in the Lagrange frame was normalized by the choicew = w c . The dimensionless model equations are The model equations (S3)-(S5) depend on three dimensionless parameters: heat capacity, thermal conductivity and capillarity coefficient Note that a physically meaningful solution requires 1/3 <w and 0 <θ.
Model equations in the Euler frame. The model equations in the Lagrange frame are given in Eq.
(1). These equations were transformed into the Euler frame using Eq. (2); i.e., ∂x/∂h = w/w, and by replacing the and In comparison, the classical Navier-Stokes equations in the Euler frame are with    Figure S1: Comparison of pulse shape at two different values of the capillarity coefficient, C/ζ 2 = 10 −20 (solid line) and 1 (dotted-dashed line). Excitation amplitude waspexc = 120 and 300, respectively. Duration of excitation wast 0 = 0.3 for both curves. Pulse was measured at x/L=0.5 and 0.8, respectively. Other parameters appear in caption of Fig. 2.

B Pulse solution in a vdW fluid model
Capillarity coefficient and the line tension.
Gibbs approached the phase boundary as an infinitesimal line that separates the two phases. The energy stored in a phase boundary of length L and line tension γ is E = γ(l)dl = γL. In contrast, van der Waals approached the phase transition as a continuous change, and the phase boundary, therefore, has a finite width. Van der Waals suggested an energy term that depends on the density gradient For a phase boundary of width and a difference in density ∆ρ between the two phases, the energy stored in the phase boundary is E ∼ K ∆ρ 2 2 L. Therefore, the capillarity coefficient is related to the line tension according to Using Slemrod's ansatz, the capillarity coefficient is C ∼ γ ρ 3 c ∆ρ 2 from scaling arguments. Assuming ∼ 10 −9 − 10 −6 m, ∆ρ ∼ ρ c ∼ 10 −6 kg/m 2 and γ ∼ 10 −12 N (Benvegnu and McConnell, J Phys Chem, 1992) we obtain C ∼ 10 −27 − 10 −24 kg 2 /s 2 . However, in the numerical calculation we have used a larger value (C/ζ 2 ∼ 1), to avoid sharp gradients in density. This did not have qualitative effect on the properties of sound pulses near phase transition as evident in Fig. S1.
Effect of stimulation parameters on the pulse characteristics. Figure S2a shows the effect of stimulation duration on the shape of the density pulse, as measured at distance x/L=1. It was evident that the compression part of the density pulse was not affected much and maintained a pulse duration of ≈0.1T. On the other hand, the subsequent rarefaction monotonically decreased its density with stimulation duration. The density, pressure and temperature field solutions as a function of space (x-axis) and time (y-axis) are shown in Figs S2b,c,d, respectively, for stimulation duration of t 0 /T=0.4. It was evident that the pulse maintained a recognizable shape for distances larger than 10L. An increase of the stimulation width (λ) resulted in similar observations (Fig. S2e).

C Caption of supplemental movie
Movie S1: Collision between two propagating pulses is demonstrated in the density, pressure and temperature fields. At the peak of the collision a short burst of pressure is obtained (t/T=0.2). The pulses annihilate afterwards. Computational details are given in caption of Fig. 5. Figure S2: (a) Variation of stimulation duration (t 0 ) did not affect the duration of the compression part of the density pulse, as measured at distance x/L=1 from the excitation point. The subsequent rarefaction, however, was altered. (b) Density, (c) pressure and (d) temperature solutions as a function of space (x-axis) and time (y-axis) upon stimulation with t 0 /T=0.4. Pulse maintains a recognizable shape for distances larger than 10L. (e) The shape of the pulse did not change much with an increase in stimulation width. Initial density was ρ 0 = 0.67, and numerical calculation was conducted with 2048 grid points, x-domain [−5π, 5π] and dt=10 −3 . Other parameters appear in the caption of Fig. 2.