The Non-Ordinary Laws of Physics Describing Life

1. Introduction

The existence of life is still a mystery for science. As E. Schrödinger formulated, "the construction [of living matter] is different from anything we have yet tested in the physical laboratory ...it is working in a manner that cannot be reduced to the ordinary laws of physics" [1]. Why he desperately inserted the highlighted word was his strong convincement that "And that not on the ground that there is any ’new force’ or what not, directing the behaviour of the single atoms within a living organism, but because the construction is different from anything we have yet tested in the physical laboratory." He attempted to find those laws in a disciplinary way. Maybe R.P.Feynman was right in saying that "We make no apologies for making these excursions into other fields, because the separation of fields, as we have emphasised, is merely a human convenience, and an unnatural thing. Nature is not interested in our separations, and many of the interesting phenomena bridge the gaps between fields." Maybe we must make ’excursions’ to find those non-ordinary (in other words, non-disciplinary) laws that describe life? Or, as Schrdinger implicitly suggested, we must revisit our approximations leading to classical physics (where we derive the well-known ’ordinary’ laws in one approximation for the construction the non-living matter represents)? Maybe the "different construction" the living matter represents needs different approximations, and then laws based on the same first principles in a different approximation (in a non-disciplinary approach) can describe life? We scrutinize the "construction" and its "working" in a non-disciplary way, using non-ordinary approximations and abstractions. We show that life is not against laws of science; it is only against discussing it in terms of a single grasped science discipline.

2. Laws of Motion

Science laws about separate interactions of masses and charges are based on abstractions, which enable and need approximations and omissions. While we understand that the speeds of electrical and gravitational interactions are finite, we can use the ’instant interaction’ approximation in classical physics. One effect of the first particle reaches the second particle simultaneously with the other effect, leading to the absence of a time-dependent term in the mathematical formulation. However, this is not the case in electrodiffusion, where the mass transfer is significantly slower than the transfer speed of the electromagnetic field.

From a physical point of view, ionic solutions are confined to a well-defined volume, with no interaction with the rest of the world. What complicates things is that their volume is finite, so we must adapt the corresponding laws to the case of finite resources. At a microscopic level, on the one hand, we use the abstraction that they consist of charge-less and size-less simple balls with mass, have thermal (kinetic) energy, and collide with each other, as thermodynamics excellently describes it. On the other hand, we use another abstraction, which is mass-less and size-less charged points with mutual repulsion. At a macroscopic level, we use the abstraction that the respective volume is filled with a continuous medium with uniformly distributed macroscopic parameters such as temperature, pressure, concentration, and potential.

One can parallelize describing how ions change their position with Newton’s laws of motion, relating an object’s motion to the forces acting on it. The first and third laws are static, and the second is dynamic. We can translate the first law to ions that without external invasion, their volume at rest will remain at rest. The third law, for ions’ volume, essentially states that in a resting state at every point, the electric and thermodynamic forces are equal; the Nernst-Planck electrodiffusion equation (without transport) expresses this. The second law, for mechanics, expresses the time course of the object: the position’s time derivative. Notice that in this case, we make one abstraction that the object (the carrier) has one attribute, its mass. (Recall how important it was for the special theory of relativity that the accelerated mass and the gravitational mass were identical; i.e., they could be described as one abstraction.)

For ions, we have two abstractions, and two attributes ’charge’ and ’mass’, and the two forces act on the two attributes that science classified to belong to different science disciplines. We cannot express easily how the electric and thermodynamic forces will change the object’s position because those forces act differently on different attributes. No time derivatives are known, only position derivatives. Due to this hiatus, physics (and consequently physiology) cannot describe the electrochemical processes: the second law of motion for electrodiffusion is missing. As a consequence of the instant interaction, classical science has no mechanism for handling the case when two different force fields (gradients) having different propagation speeds act on an object and two different abstractions (charge and mass), belonging to different science disciplines, translate the force into acceleration.

When describing processes (i.e., dynamical systems), we must have one or more equations of motion: how the time gradient of the fundamental entities change in the function of the fundamental entities. In classical science, we have only one fundamental entity: the position and also the driving forces depend only on the position. The (Newtonian) laws of motion do not depend on a time gradient. In the Einsteinian world, speed explicitly appears when describing the interrelation of fundamental entities mass, position, and time. Actually, a second entity (position and time) appears; and the speed connects them.

In our ’extraordinary’ science, we have two abstractions; correspondingly, we have two laws of motion, corresponding to the two disciplines or the two abstractions. In our laws of motion (see Eq.(5) and Eq(8)) we also have explicit speed dependence in describing the interrelation of concentration and potential. Actually, we are thinking in two entities (concentration and potential), but both of them are parametrized by position x. In line with the Einsteinian case, the time shines up and the speed connects those entities. However, the different interaction speeds act on the coordinates differently, that is, the effects of changed entities cannot be separated. This is why we need ’extraordinary’ laws. (Given that our thermodynamical speed is always by orders of magnitude lower than the electrical (limiting) speed, we neglect transforming the time.)

In all cases, the law has the form of a differential equation; i.e., we can derive the fundamental entities by integration. In ’ordinary’ science, we have a single-abstraction interaction; so we have one law of motion; an analytical solution is possible. In non-ordinary (non-disciplinary) case, we have dual-abstraction interaction, and only numerical solution is possible.

3. Steady State

The ions experience two effects in those two abstractions in volumes containing ions. When an invasion in the volume happens, electric potential, pressure, temperature, or concentration changes locally; dynamic changes begin to restore its balanced steady state. When the invasion persists, the system finds another steady state. The observer experiences that changing one macroscopic parameter of the system causes an unexpected (and unexplainable) local change in another macroscopic parameter. The microscopic world maps the changes from one abstraction to the other. Experimentally, the microscopic world maps the change from the world of electric abstraction to the world of thermodynamic abstraction and vice versa. Theoretically, we can do the exact mapping of macroscopic electrical and thermodynamical parameters using microscopic universal constants.

The phenomenon of invasion called ’electrodiffusion’ means that when a potential gradient is created in a volume with ions (while its thermodynamical parameters, such as its volume and temperature, are constant), it creates a concentration gradient. Conversely, a created concentration gradient creates a potential gradient. Two driving forces act on the ions: thermodynamical and electrical ones. In a steady state, at every spatial point of the segment, the two driving forces are equal, and the ions will not move. We can describe the equilibrium state (the mutual dependence of the spatial gradients of the electrical and thermodynamical fields on each other) using the Nernst-Planck electrodiffusion equation

$${\displaystyle \frac{d}{dx}}{V}_m\left(x\right)=-{\displaystyle \frac{RT}{q*F}}{\displaystyle \frac{1}{C_k\left(x\right)}}{\displaystyle \frac{d}{dx}}{C}_k\left(x\right)$$

(1)

In good textbooks (see, for example, [2], Eq (11.28)), its derivation is exhaustively detailed. In the equation, x is the spatial variable across the direction of the changed invasion parameter, R is the gas constant, F is the Faraday’s constant, T is the temperature, q the valence of the ion, $V_m\left(x\right)$ the potential, and $C_k\left(x\right)$ the concentration of the chemical ion. In simple words, it states that the change in the concentration of ions creates a change in the electric field (and vice versa), and in a stationary state, they remain unchanged. However, in classical science, there is no way to take into account the field’s propagation speed.

It is one of the rare cases when the starting point was wrong, but the conclusion was correct. The equation is a rearranged flux equation, where an identical speed for all interactions was assumed. The identical speed was calculated as a "mean-field" speed, where the "mean" stands for some average of some interaction speeds differing by several orders of magnitude. Not suitable for describing flux. However, in an equilibrium state, the actual value of both interaction speeds is zero, so they have the same value.

4. Time Derivatives

Eq.(1) describes a stationary state with no ionic movement. Deriving a time course (time derivatives) from the position derivatives is impossible in a strict mathematical sense: the interaction is instant. However, we can provide it using physical principles. We consider the electric ion current represented by viscous charged fluids [3]. As expected, selecting the speed (aka calculating the appropriate value of the macroscopic speed, see Eq.(9)) plays a key role, especially since we are at the boundaries of physics abstractions; furthermore, we are mixing microscopic and macroscopic notions.

In classical physics, because of the lack of time-dependent terms in the expressions, the changes are described by position-dependent terms (position derivatives), both in the case of electromagnetical and electrodiffusional interactions. In classical (’instant interaction’) science, the time derivatives are either not interpreted or can be derived through the externally derived joint interaction speed. As explained, we can extend the idea to enormously different speeds and derive time derivatives if we consider the faster interaction to be instant.

In timeless classical physics, there is no explicit dependence on the time: everything happens simultaneously. In a resting state, the Maxwell equations follow from the conservation of energy. One form of energy transforms into another form, and the system arrives in another balanced state. The carriers of force fields are continuous, so one can calculate and make infinitesimal changes in the driving forces; they do not change the system’s energy. If one gradient changes, the other automatically (per definitionem) changes in the opposite direction. In another words: the driving forces are permanently balanced, the magnetical and electrical forces act instantly ("simultaneously"), and are always of opposite sign. A time derivative cannot be interpreted: everything happens at the same time. In other words: at the same space-time (in the classical interpretation, the time is the same at any point).

In an electrodiffusional process, we start with the same point of view. We assume that the thermodynamic and electrical driving forces are equal in equilibrium. That assumption results in the Nernst-Planck equation. On the one side, we use a macroscopic parameter, the potential. On the other side, we use another macroscopic parameter, the concentration. The equation bridges those macroscopic parameters by using universal constants from the microscopic world. However, unlike electromagnetism, we cannot make infinitesimally small changes in the gradient since the carrier of the force fields is "atomic". Furthermore, moving it infinitesimally (changing only its position coordinates), the changes in the electric and thermodynamic gradients do not result in a new balanced state. The effect of ion’s charge has an immediate effect on the volume, but ion’s mass has a delayed effect. The infinitesimally small change in the position results in an infinitesimally small increase in the energy of the system, given that moving a carrier changes the potential and the concentration in the same direction, and we did not consider that time changes. In the Newtonian world, everything happens at the same time, so we cannot handle instant and finite interaction speeds simultaneously. The infinitesimally small change disappears only when the slower interaction reaches the other carriers in the volume. When interaction speeds differ, energy conservation is valid only if we use space-time.

Fortunately, we can derive the infinitely small change where the time and space (position) coordinates are connected, essentially, in the same way as in the special theory of relativity. Let us assume that the gradients act on the mass and the charge, but the ion’s effects on the gradients are negligible. According to the principle of relativity, the phenomena must remain the same in a reference frame moving with a constant speed relative to the first one, and we choose the one that moves together with the ion. In the second frame, no ionic movement occurs along the movement’s direction. In line with that the speed of the light is independent of the reference frame; we assume that the higher interaction speed remains the same in both systems: it is instant. The observers in both reference frames must see that the system is balanced. The difference is that in the first frame, the system is statically balanced (no change in the gradients, but the ion is moving), and in the second one, it is dynamically balanced (the gradients change to keep the ion in rest). The gradients the moving ion experiences are the ones that the standing ion experiences at another time (depending on its speed). In this way, we can provide the needed time course of the process.

Compared with the electromagnetic case, we see crucial differences. First, the mass’s propagation speed is millions of times slower than the charge’s. Second, the moving ion simultaneously represents mass transport and charge transport. Third, when deriving position derivatives, we conclude from the assumption that there is no movement (in other words, no explicit dependence on the time): the effect of the electric and magnetic driving forces are equal, whatever time is needed to reach that balanced state. In contrast, in electrodiffusion, the velocity changes the concentration gradient, and simultaneously, the potential gradient.

We assume that equation (1) is valid for a given time t. At time $t+dt$, in another steady state, the two interactions manifest at different times: we have

$${\displaystyle \frac{d}{dx}}{V}_m(x+v\left(x\right)*dt)=-{\displaystyle \frac{RT}{q*F}}{\displaystyle \frac{1}{C_k\left(x\right)}}{\displaystyle \frac{d}{dx}}{C}_k\left(x\right)$$

(2)

or, equivalently, it can be expressed as

$${\displaystyle \frac{d}{dx}}{C}_k(x-v\left(x\right)*dt)=-{\displaystyle \frac{q*F}{RT}}{C}_k\left(x\right){\displaystyle \frac{d}{dx}}{V}_m\left(x\right)$$

(3)

The concentration at position x determines the potential (apart from an integration constant) at position:

$$d{V}_m\left(x\right)=dx*{\displaystyle \frac{d}{dx}}{V}_m\left(x\right)=-dx{\displaystyle \frac{RT}{q*F}}{\displaystyle \frac{1}{C_k\left(x\right)}}{\displaystyle \frac{d}{dx}}{C}_k\left(x\right)$$

(4)

so (and here the constant disappears) the time derivative is

$${\displaystyle \frac{d}{dt}}{V}_m\left(x\right)=v\left(x\right)*{\displaystyle \frac{d}{dx}}{V}_m\left(x\right)=-v\left(x\right)*{\displaystyle \frac{RT}{q*F}}{\displaystyle \frac{1}{C_k\left(x\right)}}{\displaystyle \frac{d}{dx}}{C}_k\left(x\right)$$

(5)

or

$${\displaystyle \frac{d}{dt}}V\left(x\right)={\displaystyle \frac{D*R}{F}}*C\left(x\right)*{\displaystyle \frac{dC}{dx}}*{\displaystyle \frac{RT}{q*F}}{\displaystyle \frac{1}{C\left(x\right)}}{\displaystyle \frac{d}{dx}}{C}_k\left(x\right)$$

(6)

Similarly, at time $t-dt$, in another steady state, we have

$$d{C}_k(x-v\left(x\right)*dt)=dx*{\displaystyle \frac{d}{dx}}{C}_k\left(x\right)=-dx{\displaystyle \frac{q*F}{RT}}{V}_m\left(x\right){\displaystyle \frac{d}{dx}}{V}_m\left(x\right)$$

(7)

$${\displaystyle \frac{d}{dt}}{C}_k\left(x\right)=v\left(x\right)*{\displaystyle \frac{d}{dx}}{C}_k\left(x\right)=-v\left(x\right){\displaystyle \frac{q*F}{RT}}{V}_m\left(x\right){\displaystyle \frac{d}{dx}}{V}_m\left(x\right)$$

(8)

We expressed the dependence of gradients on each other using the ion’s speed v as an intermediate variable, that can be expressed by the Stokes-Einstein relation as

$$v=-{\displaystyle \frac{D*R}{F}}*C\left(x\right)*{\displaystyle \frac{dC}{dx}}$$

(9)

After simplifying the expression

$${\displaystyle \frac{dV}{dt}}=\left({\displaystyle \frac{T*R^2}{q*F^2}}\right)*D*{\displaystyle \frac{d^{2}C}{d{x}^2}}$$

(10)

For practical calculations, the voltage’s time derivative can be calculated directly from the input current, which directly considers the current production mechanism.

5. Fick’s Law

Given that

$${\displaystyle \frac{dV}{dt}}=D*{\displaystyle \frac{d^{2}C}{d{x}^2}}$$

(11)

expresses Fick’s Second Law of Diffusion, we can derive the ratio between the electric and thermodynamic temporal gradients. Using the values of universal constants

$${\displaystyle \frac{dV}{dt}}=2.23*{10}^{-6}*D*{\displaystyle \frac{d^{2}C}{d{x}^2}}=\mathbf{2}.\mathbf{23}*{\mathbf{10}}^{-\mathbf{6}}{\displaystyle \frac{dC}{dt}}$$

(12)

We note that non-dedicated experimental results (measuring concentration invasion and voltage invasion) result in an experimental value $\approx 2*{10}^{-6}$.

6. Summary

Onsager’s reciprocal thermodynamical relations resulted in a Nobel prize, but their mathematical handling is not yet solved. In physics, interactions of different fields are usually solved in a "mean-field approximation". However, in the case of electrodiffusion, one needs to average quantities deviating by six orders of magnitude, resulting in very inaccurate transport equations. Our approach to speed handling results in an exact derivation of the time courses of the concentration and voltage of electrodiffusion processes, which are needed for many practical transport processes based on electrodiffusion. By calculating the ratio of coefficients of electrodiffusion and diffusion, we provide a way to determine the electrodiffusion’s diffusion and/or viscosity parameters that are not directly accessible due to experimental difficulties.

Those equations are also the laws of motion for biological processes, that is, of life. Among others, they explain how in neuronal processes, an ion packet moves without external potential on the surface of the membrane and along the axon. Using them, one can quantitatively describe at which parameter combination a system composed of non-living components (such as a closed volume having two segments with largely differing concentrations of electrolyte, separated by a semipermeable membrane with ion channels, plus a current drain (such as AIS) and current sources (such as synapses)) shows symptoms that biology calls "action potential".

Appendix D.1. Speed of Light

In 1676, the Danish astronomer Ole Roemer was making meticulous observations of Jupiter’s moon Io and concluded not only that the speed of light is finite, but he measured its value with sufficient accuracy. Roemer never published a formal description of his method, possibly because of his bosses, Cassini and Picard, opposing his ideas. Cassini knew Rmer’s idea and the measurement data.

However, the theory of finite speed quickly gained support among other natural philosophers of the period, such as Christiaan Huygens and Isaac Newton. Although Newton surely knew that the observation speed was finite, in his "Philosophiae Naturalis Principiai Mathematica" [13], published in 1687, he decided to refer to observations that happened "at the same time" despite knowing that what we observe at the same times, happen at different times. Using instant interaction results in "nice’" mathematical laws and enables us to describe most of nature’s experiences with sufficient accuracy.

Einstein, in 1905, discovered [14] that the speed of observation (in moving reference frames) may play a decisive role in interpreting scientific phenomena. The results he derived using Minkowski-coordinates [8] were counter-intuitive, with many unexpected consequences. Instead of introducing improvement(s) or correction(s) to the existing classic principles and methods, he introduced a new principle: the finite (limiting) interaction speed. "The disciplinary analysis of the reception of Minkowski’s Cologne lecture reveals an overwhelmingly positive response on the part of mathematicians, and a decidedly mixed reaction on the part of physicists" [15] has turned to the exact opposite. Today, physics generally accepts the description, that is, the existence of finite interaction speed (resulting in the birth of a series of modern science disciplines). However, other science disciplines, including biology and computing science, refute (or at least do not use) it despite its evident effects.

Appendix D.2. Speed in Neuroscience

Helmholtz, in 1850, sent a short report off to the Academy [16] "I have found that a measurable time passes when the stimulus exerted by a momentary electric current on the hip plexus (Hftgeflecht) of a frog propagates itself to the nerves of the thigh and enters the calf muscle." His teacher "had thought that the speed of nervous conduction might be in excess of the speed of light and could probably never be measured. Helmholtz’s father, on hearing of the experiment and the surprisingly slow measured speed, wrote to his son that he would as soon believe this result as that one can see the light of a star that burned out a million years ago" [17].

With the development of measurement technology, it became evident that finite speed is a general feature of the "nervous connection". (Somehow, "the speed of nervous conduction" has been renamed to "conduction velocity", neglecting the clear distinction that physics makes between the two wording.) The experimental research also quickly (re-)discovered that those wires forward signals in a particular way; the speed of the potential wave is finite. Furthermore, the axons are not equipotential during transmission. Although its structure is practically identical with axons, biology assumes that, unlike an axon, the membrane remains equipotential during its operation, although the evidence shows the opposite: ’the action potential spreads as a traveling wave from the initial site of depolarization to involve the entire plasma membrane’ [18].

Appendix D.3. Finite-Speed Interactions

When speaking about speed, especially about the speed of charged objects inside biological objects, one needs to consider microscopic and macroscopic levels of understanding. On the boundaries of the two levels, we need to make distinctions between different kinds of speeds, among others (in units of $m/s$), the propagation speed ${10}^8$ of the electric field (aka potential gradient), the speed ${10}^5$ of thermal motion and potential-accelerated motion, the apparent speed of current (potential-assisted speed of a macroscopic stream, both in metals and electrolytes, mainly due to the repulsion of nearby ions in the stream) ${10}^1$, diffusion speed of electrons in a wire ${10}^{-4}$, drift speed of the individual carriers in aqueous solutions ${10}^{-7}$. Fortunately, in most but not all cases, different mechanisms (such as the Grotthuss mechanism or free electron model; for a review, see [19]) at the level of microscopic structure helps to create the illusion of a high macroscopic propagation speed (a million times higher than its microscopic carriers). The same carrier can have macroscopic speeds differing by orders of magnitude, depending on the context; see a biological example at ion channels. When more than one of those speeds play a role in the phenomenon we study, we must carefully consider its context and prepare for handling fast and slow effects, furthermore, their mixing.

When an object can interact with another in a way abstracted by science as more than one interaction type, we need to find the relation (the ’extraordinary’ law). Such a famous case is electricity and magnetism. Their interrelation is defined by the Maxwell equations: how an electrical field creates a magnetic one and vice versa (notice that the law is about their space derivatives, aka space gradients instead of the entities themselves). While we understand that the speeds of electromagnetic and gravitational interactions are finite, we can use the ’instant interaction’ approximation in classical physics because one effect of the first particle reaches the second particle simultaneously with the other effect, leading to the absence of a time-dependent term in the mathematical formulation.

An apparently similar case is found in electrodiffusion, where ions can be abstracted as mass and charge, one belonging to thermodynamics and the other to electricity. There is, however, an essential difference between those cases: the interaction speeds are the same in the first case (moreover, in the spirit of classical physics, the interactions are instant) and differ by several orders of magnitude in the second one. Of course, the Maxwell equations can be nicely solved and also modeled for biology if one introduces [20] that the axial currents have the same speed (BTW: its measured value $20m/s$) as the electric and magnetic waves, furthermore the longitudinal current is (?)defined(?) to have no attenuation. Furthermore, it is likely also defined that the current needs no driving force. This is why the positive and negative ions flow in the same direction. It is really a novel paradigm leading to "(mis)understanding cell interactions" but describes some alternative nature.

Appendix D.4. Speed in laws of science

The famous Coulomb’s Law is expressed as

$${\displaystyle \frac{F\left(t\right)}{Q_1}}=k{\displaystyle \frac{Q_2}{{\overrightarrow{r}}^2}}$$

(A1)

$\overrightarrow{r}$ is a space-time distance. That is, in the Newtonian approximation, time is identical at all places, so we used to omit it, and use the commonly known space coordinate instead. Considering the finite field propagation speed requires revisiting the fundamental physical laws. (in a Lorentz-transformed form, at zero speed) should be written as

$${\displaystyle \frac{F\left(t\right)}{Q_1}}=k{\displaystyle \frac{Q_2}{r^2}}\left(t-{\displaystyle \frac{r}{c}}\right)$$

(A2)

The electrostatic field that the charge $Q_1$ experiences due to the finite propagation speed c of the electric field (or interaction) corresponds to that $Q_2$ at a distance rgenerated ${\displaystyle \frac{r}{c}}$ time ago (k is the constant describing the electric interaction). The effect of delayed interaction has experimental proof in the case of gravitational interaction (gravitational waves). This term has no role if the two charges do not change their position; similarly to that in the special theory of relativity, only the relative movement leads to complications.

This speed term pops another law from classical physics into our minds: Kirchhoff’s junction rule. Given that the law expresses charge conservation, and the current is defined in differential form as $dQ/dt$, the law is perfect at any point of the circuit. However, for an extended object, it is only valid in the approximation ’instant interaction’ that classical physics uses, but not for biology. For extended biological objects, an input current arriving with finite speed leaves the object later (in the order of $msec$). The law is invalid for an extended object or only if we use time-space coordinates. In addition, when charges are "created" inside biological objects (ions diffuse into the junction; see the role of ion channels in the wall of membranes), it vitiates the law. Using a wrong definition of current means assuming ’instant interaction’, that is, that neural signals propagate with the speed of light. The currents (and the voltages), measured at two points in space-time, differ. Consequently, for extended objects (such as a line-like finite-size neuron), it is valid only with a time delay

$$I_{out}\left(t\right)=-I_{in}(t-\Delta t)$$

(A3)

The time delay in biology is in the $1msec$ range. We must not describe the axon or the membrane by the non-differential form Kirchoff equation: the input and the output currents flow at different times (the charge carriers need time to reach from input to output). We must not describe the axon or the membrane with the non-differential form of the Kirchoff equation: the input and the output currents flow at different times (the charge carriers need time to reach from input to output); only the differential equation form expresses charge conservation (furthermore, in the case of "producing" ions, even by the differential form is invalid). Studying electric phenomena on structured media, such as biological cells, needs much care. We must not apply laws derived from entirely different conditions (mainly metals).