4. Discussion: From Electrodynamics to Biophysics and Back
A fundamental question arises with the updated version of Maxwell's equations. How is the phenomenon of polarization included in Equation (11) and Equation (14)?
To answer this question, we first need a general paradigm to define polarization, even when dielectrics are far from ideal, when they might be time and frequency dependent, and voltage dependent as well. We need a paradigm that describes how the charge distribution varies with the electric field in as general a system as possible, including systems with charge movement driven by forces not in the Maxwell equations at all, such as convection and diffusion.
It seems obvious that a general paradigm cannot be found. After all the motions of matter in response to a change in electric field are more or less as complex as the motions of matter itself! Nonetheless, a paradigm of that may be helpful in many cases has been in use for many years, even if it is not perfectly general.
This problem has been addressed in membrane biophysics. A community of scholars has studied the nonlinear currents that control the opening of voltage sensitive protein channels for nearly fifty years, inspired by [6]. They have developed protocols that may be useful in other systems, as they have been in biophysics. Schneider and Chandler followed by Bezanilla and Armstrong are responsible for this paradigm, more than anyone else.
The basic setup used in these experiments is that of an electrochemical cell modified to deal with a cylindrical cell as shown in Figure 1. Membrane potential is measured across a biological membrane, with defined concentrations on both sides of the membrane. Current is applied through electrodes to control the potential, in the classical voltage clamp set up of Cole and Hodgkin, Huxley, and Katz. It is best to apply that current in electrodes different from those that record membrane potential using a so-called four electrode setup , like those described in textbooks of electrochemistry.
I propose using the operational definition of 'gating current' used to define nonlinear, time and voltage dependent polarization by biophysicists as a useful setup and definition of many types of polarization. Obviously, this definition is not general, but the hope is that it may be generally useful.
The basic idea is to apply a set of step functions of potential across the system—in biology across the membrane—and observe the currents that flow. The currents observed are transients that decline to a steady value, often to near zero after a reasonable (biologically relevant) time. The measured currents are perfectly reproducible. If a pulse is applied, the charge moved (the integral of the current) can be measured when the voltage step is applied. The integration goes on until t1 when the current ileak is nearly independent of time, often nearly zero. That integral is called the ON charge QON.
When the voltage is returned to its initial value (the value that was present before the ON pulse), another current is observed that often has quite different time course , much more so than in Figure 1. The integral of that current is the OFF charge QOFF.
If QON=QOFF, and the physical processes involved depend fundamentally on potential and not its time derivative, the biophysical paradigm is likely to be useful. In other cases, another paradigm is needed. If the current produced by the step in potential is in fact actually transient, the steady current will be what it was before the voltage step was applied. The transient will disappear with time as the word 'transient' implies. In that case it seems that the biophysical paradigm is not only useful but may even provide a unique definition of gating current and the corresponding polarization.
Gating current as measured in biophysical experiments depends on the membrane voltage before the step, as well as the voltage just after and during the step. It also depends separately on the voltage after the step, although Figure 1 does not illustrate the dependence documented in the literature. The voltage and time dependence arises from the molecular motions underlying the gating current. The voltage and time dependence defines the mean molecular motions and is called 'the gating current' in the biophysics literature.
If the ON charge is found experimentally to equal the OFF charge, for a variety of pulse sizes and range of experimental conditions, the current is said to arise in a nonlinear (i.e., voltage dependent) polarization capacitance and is interpreted as the movement of charged groups in the electric field. The charged groups move to one location after the ON pulse, and return to their original location following the OFF pulse. The charge is called 'gating charge', and the current that carries the charge is called 'gating current'.
The macroscale current observed in the set-up is equal to the sum of the micro (actually atomic scale) currents carried by the charged groups inside a channel protein, even though the recording electrodes are remote from the protein. Indeed, there might be 1018 charged atoms (ions) between the electrodes and the protein.
The currents in the electrodes and the channel protein are equal because the setup is designed to be an unbranched one-dimensional circuit with everything in series. In a one-dimensional series setup the total current is equal everywhere in the series system at any one time, even though the total current varies significantly with time. The Maxwell equations guarantee spatial uniformity of total current (including the ethereal current ε0∂E/∂t) independent of the microphysics of movement of charge (with mass): Figure 2 of [73], and [43,75,87]. The equality of current can be checked by measuring current in different locations in the experiment. The spatial equality of current needs also to be checked in simulations as in [18,21,88] because tiny inadvertent errors in numerical procedures or coding can produce substantial deviations from spatial equality and thus misleading artifacts. Imposing periodic boundary conditions on nonperiodic systems is another possible source of such artifacts.
If the currents reach a steady value independent of time, but not equal to zero, as in Figure 1, the signal is not transient, in the strict meaning of the word. In biophysics, the steady current ileak is then usually considered to flow in a resistive path that is time independent, but perhaps voltage dependent, in parallel with the path or device in which the gating charges QON and QOFF flow. If the current does not reach a steady value, or if the areas in Figure 2 are not equal, the currents are not considered 'capacitive' and are interpreted as those through a time and voltage dependent 'resistor'. This is a biological and biophysical assumption. It is not a physical or mathematical necessity. Thus, it is important to investigate the properties of the currents through the resistive path—e.g., those that are not transient and do not return to zero and those that make QON≠QOFF by independent methods to see if they are time independent. In biophysics, currents can be done by blocking the resistive path with drugs, or with mutations of the channel protein. If the resistive currents are not time independent, the definition of QON and QOFF in Figure 1 needs to be changed. Indeed, experiments of another type must be designed that allow separation of polarization from conduction currents. The simplest version of the biophysics paradigm then needs to be extended.
Clearly, this approach will only work if step functions can reveal all the properties of the underlying mechanism. If the underlying mechanisms depend on the time rate of change of voltage, step functions are clearly insufficient because ∂V/∂t, is zero or infinity but nothing else in a step function. In the classical language of membrane biophysics, the ionic conductances gNa and gK must not depend on the rate of change of voltage.
Much work has been conducted showing that step functions are enough to understand the voltage dependent mechanisms in the classical action potential of the squid axon , starting with [78], Figure 10 and Equation (11). Hodgkin kindly explained the significance of this issue to colleagues, including the author (around 1970). He explained the possible incompleteness of step function measurements: if sodium conductance had a significant dependence on ∂V/∂t, the action potential computed from voltage clamp data would differ from experimental measurements. He mentioned that this possibility was an important motivation for Huxley's heroic hand integration of the Hodgkin Huxley differential equations. Huxley confirmed this in a separate personal communication, Huxley to Eisenberg. Those computations and many papers since [89,90,91] have shown that voltage clamp data (in response to steps) is enough to predict the shape and propagation of the action potential in nerve and skeletal muscle. It should be clearly understood that such a result is not available for biological systems in which the influx of Ca++ drives the action potential and its propagation.
The conductance of the voltage activated calcium channel has complex dependence on the current through the channel because the concentration of Ca++ in the cytoplasm is so low (~10−8 M at rest) that the current almost always changes the local concentration in and near the channel on the cytoplasmic side. Those concentration changes, in turn, alter the gating and selectivity characteristics of the channel protein, as calcium ions are prone to do int many physical and biological systems, particularly at interfaces.
It seems unlikely that the resulting properties of voltage dependent calcium channels can be comfortably described by the same formalism used for voltage-controlled sodium and potassium channels of nerve and skeletal muscle. That formalism uses variables that depend on membrane potential and not membrane current because Cole and Hodgkin guessed that neuronal action potentials were essentially voltage dependent, not current dependent. They found action potentials in 'space clamped' axons with wires down their middle that ensured spatial uniformity of potential. These axons had very different patterns of current flow from normal axons, and so Cole and Hodgkin were confirmed in their view that the membrane processes generating the action potential were voltage dependent, much more than current dependent (personal communication Cole to Eisenberg 1960; Hodgkin to Eisenberg 1961, et al.).
Hodgkin, Huxley, Katz, and Cole did not know of action potentials driven by calcium channels , nor of the extraordinarily small concentration of calcium ions inside cells. There may of course be other reasons the formalism is inadequate. In summary, experiments, theory, computations and perhaps simulations are needed to show that responses to steps of voltage allow computation of a calcium driven action potential.
The polarization protocol described here can be applied to simulations of polarization as well as experimental measurements of polarization. Indeed, the operational definition of polarization has been applied even when theories or simulations are enormously complicated by atomic detail that includes the individual motions of thousands of atoms.
Another question of general interest is how does the polarization defined this way correspond to the polarization −P=(εr−1)ε0E in the classical formulation of the Maxwell Equations (7) and implied in Equations (1) and (6)? Does the estimated polarization equal P?
The answer is not pleasing. Polarization cannot be defined in general. The variety of possible responses of matter to a step of potential prevents a general answer. Indeed, a main point of this paper is that polarization must be defined by a protocol in a specific setting that specifies how the local electric field changes the distribution of charge.
Polarization cannot be defined in general because there are too many possible motions of mass with charge in response to a change in the electric field. Every possible motion of mass (with charge), including rotations and translations and changes of shape and density of charge, would produce a polarization. Polarization currents can be as complicated as the motions of matter.
In mechanical systems in general these issues do not attract much attention. It seems obvious that one must have a model and theory of how a system changes shape (and distribution of mass) when forces are applied. Seeking a general treatment is silly. In electrodynamics, for illogical reasons of history, tradition, and respect for our elders, scientists have sought the general treatment that would be considered silly for mechanical systems.