Collision and annihilation of nonlinear sound waves and action potentials in interfaces

Abstract

Nerve impulses, previously proposed as manifestations of nonlinear acoustic pulses localized at the plasma membrane, can annihilate upon collision. However, whether annihilation of acoustic waves at interfaces takes place is unclear. We previously showed the propagation of nonlinear sound waves that propagate as solitary waves above a threshold (super-threshold) excitation in a lipid monolayer near a phase transition. Here we investigate the interaction of these waves. Sound waves were excited mechanically via a piezo cantilever in a lipid monolayer at the air–water interface and their amplitude is reported before and after a collision. The compression amplitude was observed via Förster resonance energy transfer between donor and acceptor dyes, measured at fixed points along the propagation path in the lipid monolayer. We provide direct experimental evidence for the annihilation of two super-threshold interfacial pulses upon head-on collision in a lipid monolayer and conclude that sound waves propagating in a lipid interface can interact linearly, nonlinearly, or annihilate upon collision depending on the state of the system. Thus we show that the main characteristics of nerve impulses, i.e. solitary character, velocity, couplings, all-or-none behaviour, threshold and even annihilation are also demonstrated by nonlinear sound waves in a lipid monolayer, where they follow directly from the thermodynamic principles applied to an interface. As these principles are equally unavoidable in a nerve membrane, our observations strongly suggest that the underlying physical basis of action potentials and the observed nonlinear-pules is identical.

1. Introduction

In the 80s and 90s, K. Kaufmann first pointed out how to derive processes that we would commonly assign to living systems from the 2nd law of thermodynamics [1–5]. In his work, he followed Einstein's approach to thermodynamics [6–8] and consequently applied it to the interface since the living cell is packed with hydrated interfaces that need to obey the 2nd law. His approach to neuroscience included a brain theory in which collisions play a crucial role. This study is based on and inspired by Kaufmann's work.

Any perturbation of the equilibrated hydration layer experiences a restorative force quantitatively extracted from the slope of the entropy potential (appendix A). This potential combined with the reversion of Boltzmann's principle, i.e. start from the ‘empirical ascertained thermal behaviour’, forms the basis of Einstein's approach to thermodynamics [6,9,10]. It puts one in an advantageous position not only because no model assumptions are necessary, but also because the 2nd law of thermodynamics cannot be violated, not even by mistake.

The ‘empirical ascertained thermal behaviour’ of interfaces, and hence the slope and/or curvature of the entropy potential, is characterized by their state diagrams (e.g. surface pressure versus area (π, a), surface potential versus charge, pH/chemical potential versus protonation etc.) obtained experimentally using, for instance, the Langmuir technique [11]. Kaufmann showed [1,2,5], that while the second derivatives of the entropy potential ∂²S predict current fluctuations [1] and catalytic activity [12], the first derivatives ∂S, which are the thermodynamic forces, resemble driving forces for oscillations and propagation [2] in a continuum interface (appendix A). Owing to the conservation of entropy along with mass, momentum and energy, a perturbation on the interface propagates, and the propagation velocity is given by the adiabatic state diagram of the interface. Membranes thus provide quasi-two dimensional continua that can orchestrate coupling between intracellular regions via acoustic pulses [13–15].

We have therefore recently focused on the existence of such pulses at interfaces. We provided direct evidence of their existence [14] and couplings by demonstrating instantaneous changes in pressure [14,16], voltage [17], optical [16] and chemical [13] properties, i.e. these pulses which follow from the same law as acoustic pulses are not purely mechanical pulses but also mechanical. In particular, we have found nonlinear pulses near phase transitions, which show threshold (all-or-none type) behaviour, i.e. only above a certain level of excitation strength can these (solitary) pulses exist and propagate with velocities of approximately 0.5 m s⁻¹, which is similar to those of action potentials in non-myelinated cells [18,19]. Furthermore, we have shown that there are significant state changes across the wavefront that can change the local phase state of the lipids within a pulse. These findings have supported our hypothesis regarding linear and nonlinear sound waves along interfaces being a possible basis for biological signalling.

While linear waves are by definition incapable of interaction, this is not the case for nonlinear waves. Collisions of nonlinear waves have been described theoretically by nonlinear equations (such as Korteweg–de Vries (KdV)) or by classical shock physics. For example, nonlinear waves known as solitons, strictly speaking, are the solutions for first order KdV equations, and they do NOT interact upon collision. However, this is not true in general and interaction is indeed observed in solutions of higher order KdV equation [20]. General predictions for collisions in shock physics also exist, but they do not apply near phase transitions due to the violation of the convexity of the state diagrams [21–23]. While experimental studies are rare, they have investigated the interaction of nonlinear waves in dissimilar systems. Interactions of solitary waves in water were found to be weak even at high amplitudes [24], but dispersive shock waves in Bose–Einstein condensate showed significant annihilation upon collision [25,26]. Altogether from both theoretical and experimental approaches, the outcomes of colliding nonlinear waves can vary widely, and general predictions are absent.

In order to introduce linear and/or nonlinear pulses as a basis for biological communication (signalling), it is important to understand the interaction of such pulses (if any interaction exists). For example, interactions of impulses in neurons are particularly studied to understand (i) linear summation, (ii) supralinear summation, i.e. the response is greater than the sum of the amplitude of the interacting pulses, or (iii) sublinear summation, i.e. the response is smaller than the sum of the amplitude of interacting pulses [27–29]. Given the constant perturbations that any biological system is subjected to, many pulses would be excited on a biological interface. But how would they interact? To investigate this question, we present our experimental findings on the collision of nonlinear pulses in lipid monolayers.

2. Experiments and results

Figure 1 describes the principle set-up and execution of the experiment, in which pulses are excited and detected as outlined in our early studies of nonlinear pulse propagation at interfaces [18]. Briefly, the DPPC (1,2-dipalmitoyl-sn-glycero-3-phosphocholine) monolayer conjugated with FRET-pair dyes is spread on a Langmuir trough, and its state is isothermally clamped slightly outside the liquid-expanded (LE) and liquid-condensed (LC) transition region (figure 1c). A pulse is generated using a razor blade attached to a piezo cantilever that imposes small (100–1000 µm, 10 ms) compressional displacements or ‘kicks’ onto the monolayer. Two pulse sources placed 1 cm across from each other generate pulses towards each other at varying delay time and strength to test head-on collision of pulses. The response of the interface is measured optically at a single fixed position (figure 1a,b, detector) through the changes in the FRET signal, acquired simultaneously via two photo-multiplier tubes (amplifier bandwidth 8 MHz) at 100 kS s⁻¹. The waveforms in the figures below have been plotted after applying 20 point moving average.

Figure 1.

Experiment design. The two pulses travel a given distance towards each other before colliding, and the outcome is recorded at a fixed location (detector) optically between the two blades that excite the respective pulses. Possible scenarios and their results are shown in (a) and (b). (a) If the delay between two excitations is sufficiently long t_delay > 2x/c_avg, the pulse travelling in +x (blue) direction would pass the second blade before it is excited. Hence there would be no collision in space. The one travelling in +x direction appearing first t = t₁ followed by the one travelling in −x (red) direction at t = t₂ = t₁ + t_delay. (b) However for a suffiently short time delay t_delay < 2x/c_avg the two pulses will collide between the detector and the second blade at x > 0. Temporally the possible outcomes are shown again. The first pulse travelling in +x direction is observed at t = t₁ before collision, while the pulse travelling in −x direction is observed after collision at t = t₂ = t₁ + t_delay and will show a loss of amplitude if there was any annihilation. (c) The initial state of the lipid monolayer is marked on the pressure-area isotherm. (Online version in colour.)

To test whether any pulse interaction can occur, pulses are first excited at equal strength but at different time points. As control experiment, we consider the case when two pulses are excited at a long delay from each other (figure 1a). Here, pulses do not meet each other since the first pulse, detected at t₁, leaves the propagation path and disappears before the second pulse is triggered. Both pulses would then be detected (figure 1a, solid line), which are compared to the hypothetical signal that shows the case when the pulses excited individually are superimposed (figure 1a, dotted line). Our measurements indicate that when the second pulse is excited 180 ms after the first one, both pulses appear as expected to the hypothetical case (figure 2a). Thus, we could establish that pulses at long delay neither transmit through each other nor collide and annihilate with each other. In the subsequent collision experiment, we reduce the delay time significantly. In this scenario, the second pulse meets the first pulse after the first is detected, and if both pulses pass each other (no interaction), then the second pulse could emerge unchanged compared to the hypothetical signal. However, if the two pulses underwent some interaction, then the second pulse would appear differently (figure 1b). Figure 2b demonstrates, that when the delay time is reduced to 10 ms, the amplitude of the second pulse that underwent collision significantly declines, which demonstrates that some annihilation indeed occurs.

Figure 2.

Annihilation measurements. The two colliding pulses have been referred to as incoming and outgoing, where incoming refers to the pulse travelling in +x direction that is excited first and is observed before the collision while outgoing pulse is the delayed pulse travelling in −x direction that is observed after the collision. (a) Experiment corresponding to figure 1a (no collision). Upper (excited at t = 0) and middle figure (excited at t = 180 ms) present the signal at the detector when only one blade is excited at a time. The lower figure shows the signal at the detector (black solid) when both blades are excited (second excited with a 180 ms delay). Superimposing the control signals (red dashed) completely overlaps that observed signal (black). (b) Experiment corresponding to figure 1b. Again upper (excited at t = 0) and middle figure (excited at t = 10 ms) represent the individual pulses. The lower figure shows the result of collision (solid black) compared to the signal expected from linear superimposition of the control signals (red dashed). The mismatch represents a measure for the degree of annihilation. (Please see electronic supplementary material for repeatability.) (Online version in colour.)

We then keep the delay time short but adjust the pulse amplitude to investigate whether different degrees of annihilation can occur. The delay time between the two pulses are clamped at 8 ms to ensure that pulses undergo interaction as shown in figure 2b, but the excitation of the first pulse is increased in steps (figure 3, left) while the strength of the second pulse is kept constant (figure 3, middle). This enables collision of subthreshold and super-threshold pulses, where super-threshold pulses are indicated by a sudden and drastic increase in pulse amplitude by 1–2 orders of increase in magnitude from those in subthreshold pulses. The right column of figure 3 illustrates how the amplitude and the sub/super-threshold nature of the colliding pulses affect the level of annihilation. When sub- and super-threshold pulses collide, the second pulse emerges similar to the hypothetical case, which indicates that no annihilation occurs (figure 3, row 1 and 2). When both the pulses are super-threshold, (row 3 to 4), partial annihilation appears (approx. 50%), and increasing the amplitude of the first pulse further increases the degree of annihilation to approximately 80% (figure 3, row 5). These results altogether demonstrate that the pulse amplitude between the collided pulses can affect the degree of annihilation and that annihilation occurs generally between collision of nonlinear pulses.

Figure 3.

Annihilation as a function of pulse amplitude. The collision is further resolved with a time delay of only 5 ms. Also while the excitation strength of the pulse travelling in −x direction is kept constant (middle panel, control), the amplitude of the pulse travelling in +x (left panel, control) is increased from subthreshold (Gain = 1) to super-threshold (Gain = 5). The result of corresponding collision experiments is reported on the right (black solid). Again the results expected from superposition of control signals in the left and middle panels are plotted (red dash) for comparison. (Online version in colour.)

3. Discussion

We have shown that super-threshold nonlinear sound waves propagating in a lipid interface are stable against small (subthreshold) perturbations, but they exhibit significant annihilation upon collision with another super-threshold wave. Based on thermodynamic and hydrodynamic arguments, Heimburg et al. have previously modelled nonlinear acoustic waves in lipid membranes as solitons [30]. Fundamentally we think along similar lines, in particular the central role played by the nonlinearity and dispersion that results from a phase transition. However, there are a few crucial discrepancies between what is—generally—believed to be soliton-like behaviour and our observations. For example, we observe an increase in velocity with an increase in amplitude, while Heimburg's soliton model predicts that velocity decreases with an increase in amplitude [30]. We believe that the qualitative disagreement in the observed nonlinearity indicates a fundamental difference in the dispersion relation assumed in the soliton model and its applicability to a lipid monolayer [31]. With respect to collisions in particular, the predictions of the theoretical analysis of colliding pulses reveals no annihilation (less than 4%) in soliton models [32,33], which is in clear contrast to our results that demonstrate significant (up to 80%) decrease in amplitude upon collision. In fact we observe annihilation of colliding pulses both in lipid monolayers as well as Algae [34]. In addition to correctly accounting for nonlinearity and dispersion, internal heat transfer in the colliding pulses is also central to an understanding of the annihilation phenomenon, which requires incorporating the nonlinear viscosity resulting from relaxation of phase change in the hydrodynamic equations as well [22,31]. Of course, it is unlikely that biology narrowed itself down to one particular type of pulse. Presumably, all kinds of pulses exist and will be observed [35]. For example, slower excitation methods such as acidic puffs [13] or solvent droplets [14,16,17] do not demonstrate all-or-none excitation of acoustic pulses in the lipid monolayer [36] and these pulses also interact linearly [37].

In order to propose a possible outline for the mechanism, we briefly summarize the properties of the amplitude of the observed nonlinear pulses from our earlier work [19]. (1) Solitary waves of significant amplitude travel only beyond a certain threshold compression rate. (2) Beyond the threshold, the amplitude increases abruptly with increasing compression rate, but it quickly approaches a maximum asymptotic amplitude (saturation) while further increase in compression only broadens (increased dispersion) the pulse. (3) Nonlinear pulses can split based on the distance travelled. These observations underline the nature of state changes within a pulse that are allowed in the presence of discontinuities in the state diagram. The existence of both nonlinear response and saturation allows for at least four possible scenarios for the interaction of two colliding pulses (figure 4). (i) In the limit of small amplitude the pulses interact linearly where both the compressibility and dispersion are not altered significantly by the amplitude. (ii) On increasing the amplitude the pulses may at first add sub-linearly (|A + B| < |A| + |B|) with the interaction being dominated by dispersion due to phase change. (iii) If the colliding pulses can provide an excitation strength close to the threshold excitation, then the collision will be supralinear (|A + B| > |A| + |B|). (iv) Finally, if the interacting pulses compress the lipid beyond the saturation limit, the pulse cannot absorb any more energy, dissipating and/or dispersing the excess energy. The thermodynamic basis for the observed upper limit of the amplitude needs further investigation. As found by others, we expect that the relaxation processes associated with non-equilibrium phase change play a significant role in the dissipation and dispersion process during collision [22,23]. Such a mechanism would be in agreement with the observations of Tasaki, which showed that the mechanism is associated with the front of the wave and not with the refractive period that follows the pulse [29].

Figure 4.

Threshold and saturation of pulse amplitude. Maximum amplitude as a function of excitation strength is shown for the observed pulses near phase transition, adapted from fig. 1 in [19] using a sigmoidal fit. A threshold value of magnitude 1 has been assigned to the excitation strength that corresponds to 50% of the saturation amplitude of magnitude 1. Thus excitation of magnitude less than 1 can be considered subthreshold pulses.

This study has laid down the key features of the interaction of nonlinear pulses in a lipid monolayer, which can have significant implications for their proposed role in biological signalling [13]. We have shown that super-threshold nonlinear pulses are stable against small (subthreshold) perturbations. Such a feature is important for the conservation of the information, in terms of the amplitude and the shape (frequency components) of the pulse against constant perturbations experienced in the form of various stimuli by biological systems. In contrast, collision with another super-threshold pulse leads to annihilation. The irreversibility arising from such collisions means that a reversible (and hence less dissipative) path for communication can only be established if nonlinear collisions are avoided. This criteria may be important for the role that nonlinear collisions play in the adaptation and the evolution of the system. In order to adapt, the system should be capable of changing [38], which in turn can be facilitated by apparent irreversible collisions.

In summary, we outline and suggest an approach to understand biological communication from physical principles of the ubiquitous hydrated interface. Transitions, pulses and the 2nd law are the keys. Together with annihilation upon collision, we have shown so far that these pulses demonstrate a series of properties similar to action potentials:¹ electro-mechanic-opto-chemical coupling [13,14,16,17], all-or-none type excitation [18,19], saturation, collision of subthreshold pulses to give a super-threshold pulse [39] and propagation velocities typical for non-myelinated nerves [18,19]. Furthermore, we have shown that qualitative characteristics of the entire phenomena can be derived solely from the state diagram of the system without invoking any molecular models or fit parameters. Taken together all the similarities of such rather unconventional (nonlinear) observations leaves little room but to conclude that the underlying cause between action potentials and the observed nonlinear-pules is identical, i.e. that the origin of the action potentials is the same as the one for sound just as proposed by Kaufmann in 1989 [2].

Appendix A. The 2nd law and sound

To clarify how we see the relation between sound and the 2nd law, we start with the Euler equation. Following from the continuum condition (momentum conservation), the equation in one dimension reads

It is then easy to see that a sound wave will emerge only if a (lateral) stress/pressure gradient (r.h.s) exists. The pressure, however, is a thermodynamic quantity and related to the entropy potential via

i.e. the change in pressure at one point of the continuum arises as a result of a local perturbation, and hence the resulting pressure gradient ∂π/∂x is determined by the slope of the entropy potential S(A) of the interface. This potential (and its derivatives) is, in Einstein's formulation, a consequence of the 2nd law, and its slope is represented by the thermodynamic state, i.e. the state diagrams of the interface π(A).

Data accessibility

Data have been made available on Dryad data repository under the title of the manuscript with doi:10.5061/dryad.f06d9k2 [40].

Authors' contributions

M.F.S. supervised the overall project. S.S. designed the experiments. K.H.K. performed the experiments. All authors contributed to the preparation of the manuscript.

Competing interests

We declare we have no competing interests.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.