Cellular information dynamics through transmembrane flow of ions

We propose cells generate large transmembrane ion gradients to form information circuits that detect, process, and respond to environmental perturbations or signals. In this model, the specialized gates of transmembrane ion channels function as information detectors that communicate to the cell through rapid and (usually) local pulses of ions. Information in the ion “puffs” is received and processed by the cell through resulting changes in charge density and/or mobile cation (and/or anion) concentrations alter the localization and function of peripheral membrane proteins. The subsequent changes in protein binding to the membrane or activation of K+, Ca2+ or Mg2+-dependent enzymes then constitute a cellular response to the perturbation. To test this hypothesis we analyzed ion-based signal transmission as a communication channel operating with coded inputs and decoded outputs.

By minimizing the Kullback-Leibler cross entropy \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{H}}}_{{\boldsymbol{K}}{\boldsymbol{L}}}({\boldsymbol{p}}||{\boldsymbol{q}})$$\end{document}HKL(p||q) between concentrations of the ion species inside \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{p}}}_{{\boldsymbol{i}}}({\boldsymbol{t}}){\boldsymbol{,}}{\boldsymbol{i}}={\boldsymbol{1}},\ldots ,{\boldsymbol{N}}$$\end{document}pi(t),i=1,…,N and outside \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,{{\boldsymbol{q}}}_{{\boldsymbol{i}}}({\boldsymbol{t}})$$\end{document}qi(t) the cell membrane, we find signal transmission through transmembrane ion flow forms an optimal Shannon information channel that minimizes information loss and maximizes transmission speed. We demonstrate the ion dynamics in neuronal action potentials described by Hodgkin and Huxley (including the equations themselves) represent a special case of these general information principles.

Investigation of information dynamics in living systems typically focus on the role of the genome in transgenerational flow of information and, via epigenetics, variations in gene transcription in response to internal or external signals. Here we propose an additional mechanism of information dynamics that is based on ion fluxes at the cell membrane. As outlined below, we posit these dynamics are localized around the cell membrane and are independent of additional input from the genome, thus allowing both rapid and local responses to environmental perturbations or signals.

In brief, our hypothesis begins with the observation that virtually all cells generate transmembrane ion gradients produced by ATP-dependent membrane pumps. These gradients form high information1,2 states (e.g., large Shannon entropy) and possess high potential energy (similar to a coiled spring) initial conditions (Fig. 1A). External perturbations are detected by specific gates in the transmembrane ion channels. Information triggered by the external cue is then communicated to the cytoplasm through a local flow of ions that is controlled by opening the gates in these channels (Fig. 1B). The information from the “pulse” of ions (similar to Ca2+ puffs3) around one or more membrane channels is then translated into signaling circuits by changing local charge density and/or ion concentration. We note that the ion could flow into the cells creating the equivalent of the Ca2+ puffs or out of the cell generating a “negative puff” caused by absent ions (Fig. 1B).Figure 1Summary of ion-based information dynamics around the cell membrane (A). Baseline state. The initial conditions of the cell membrane produced through energy-dependent ion pumps. Concentrations of Na+ and Cl− are much higher in the extracellular space (left) than in the intracellular space (right) and vice versa for K+. Eukaryotic cells can use up to 40% of their energy budget to produce ion gradients. Specific transmembrane ion channels are present with their gates closed, thus allowing limited ion flow along concentration gradients. Peripheral membrane proteins included structural elements that support the membrane (proteins A) and enzymes (proteins B). Negative charges on enzymes (proteins B) are shielded by K+ ions causing the enzyme to be inactive. At the inner leaflet of the cell membrane primarily K+ ions shield proteins from the negative chargers of the membrane.

As a result the positively charged protein A cannot bind. Note that K+ ions are about twice the volume of Na+ ions. In our model, this subtle difference is posited to be sufficient to alter the shape of the active site and, therefore the activity of some enzymes (as shown in enzyme B). (B) Ion flux due to perturbation. An external cue causes the gate in the K+ channel to open such that K+ ions flow along concentration gradients out of the cell. This results in decreased shielding of the negative charges in the inner membrane leaflet so that positive charges on protein A can form electrostatic bonds. The binding of the structural protein to the lipid membrane stabilizes it as a response to the perturbation. Further, loss of K+ ions causes the enzyme (protein B) to change in both position and shape so that it binds to its substrate to catalyze the reaction S → P1 + P2. Note that biological enzymes can increase the reaction rate by as much as 15 orders of magnitude allowing the response to be both local and rapid. (C) Ion-Flux Induces response to perturbation. Activation of the enzyme B results in rapid increase in the local concentrations of the reaction products P1 and P2. In this model, P2 blocks the actions of the perturbation closing the gate on the K channel. The other product, P1, opens the Na channel allowing rapid flow of cations, which restore the shielding on the inner leaf of the cell membrane, displacing the A proteins from the inner leaflet of the cell membrane back into the cytoplasm.

The change in ions shielding negative charges on the inner leaf of the cell membrane can alter the binding of local peripheral membrane proteins4 thus altering the membrane structure and function (Fig. 1C). Similarly, opening of the potassium channel receptor by ligand binding to the Epidermal Growth Factor Receptor5,6 results in loss of K+ shielding of negative charges on the inner leaf of the cell membrane allowing “recruitment”7,8 of positively charged RAF proteins to the GTP-bound RAS proteins (which are also negatively charged).

An additional cellular response is elicited because the functions of many enzymes9 is dependent on local cation concentrations (e.g. Na+, Mg2+, Ca2+,or K+). The subsequent change in localization, activation or repression of local enzymes can produce an additional response to the local perturbation (Fig. 1C). When a successful response to the perturbation is complete, the ion gates close and the initial conditions (i.e. large transmembrane ion gradients) are restored to respond to any subsequent environmental changes.

These ion dynamics are similar to those observed in propagating action potentials of neurons10. However, we note that the steep transmembrane ion gradients that permit rapid flow of ions through the transmembrane channels of neurons are widely present in eukaryotes. Hence, we hypothesize that the transmembrane ion flows observed and modeled by Hodgkin and Huxley represent a special case of a broader phenomenon in which cells use transmembrane ion gradients to access, process and respond to information cues in the environment.

Here we assessed the potential of pulses of ions through cell membrane channels as a mode of information transmission, and define a general mathematical expression of these dynamics. We demonstrate that communication through ion fluxes represents an optimal mechanism for information transmission that both minimizes information loss and maximizes the speed of information transmission. Furthermore, our analyses support the notion that the ion dynamics observed in the Hodgkin-Huxley experiments and, indeed, the Hodgkin-Huxley equations themselves, represent a special case of these broader principles that are valid for a wide range of eukaryotic cells.

Consider a problem of determining, over a time interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${t}=({0},{\rm{\infty }})$$\end{document}t=(0,∞), the generally varying change in the intracellular ion concentrations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t}),{i}={1},\ldots ,{N}$$\end{document}pi(t),i=1,…,N of N types of ions that result from flow of ions through channels in the cell membrane. Examples might be the ions Na+, K+, Cl−, Ca+2, etc. Thus, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})$$\end{document}pi(t) are the probability density functions over the time interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${t}=({0},{\rm{\infty }})$$\end{document}t=(0,∞) and so can also represent a rate or flow. We also use this terminology.

Thus each type i of ion obeys normalization1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\int }_{0}^{\infty }{p}_{i}(t)dt=1,$$\end{document}∫0∞pi(t)dt=1,Also, definite boundary values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,{{p}}_{{i}}({0}),{{p}}_{{i}}({\rm{\infty }})$$\end{document}pi(0),pi(∞) are taken to hold. What are the unknown fluctuations of ion probability density functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})$$\end{document}pi(t)?

Here we assume that, over the time domains in which a model functions, the probability densities of the ions outside of the membrane are fixed. That is, ATP-dependent membrane ions pumps will use energy to move ions into or out of the cell. This will have the effect of changing the intracellular ion concentrations but not those of extracellular ions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{q}}_{{i}}({t})$$\end{document}qi(t), which are approximately constant2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${q}_{i}(t)\equiv {q}_{i}=constant,i=1,\ldots ,N,$$\end{document}qi(t)≡qi=constant,i=1,…,N,in comparison to the generally varying probability densities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})\,\,$$\end{document}pi(t)of ions within the cell. Note that this constancy, or uniformity, is mathematically equivalent to the assumption of an ideal enzyme of uniform density, as is often used to characterize the Arrhenius law11.

Constancy (2) arises as follows:In the case of a single-cell organism the extracellular space is effectively the ocean, wherein ion concentrations do not change in biologically relevant time. OrFor a multicellular organism, the extracellular space is in equilibrium with the interstitial fluid of the entire organisms (including, for example, the blood). Therefore the ion concentrations in the stable physiological state of a healthy individual should likewise tend to be constant. While there may be a small local changes in concentration exterior to the cell due to ion flow across the membrane, this will be quickly dissipated such that the extracellular ion space remains constant, serving as a source or sink for intracellular ions.

With the assumption that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{q}}_{{i}}({t})\,\equiv\,{{q}}_{{i}}\,=\,{c}{o}{n}{s}{t}{a}{n}{t},\,{i}\,=\,1,\ldots ,{N},$$\end{document}qi(t)≡qi=constant,i=1,…,N, we can therefore assume that changes in intracellular ion concentrations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t}),{i}\,=\,{1},\ldots ,{N}$$\end{document}pi(t),i=1,…,N are solely dependent on functions within the cell membrane. That is, membrane ion pumps generate transmembrane gradients by varying the intracellular ion concentrations relative to fixed extracellular concentrations. Further, when the channel gates open, the rapid flow of ions (measured to be about 105 ions/second) of ions allows \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})\to {{q}}_{{i}}$$\end{document}pi(t)→qi. We propose this ion-based communication channel represents a method for information transfer that both minimizes loss and maximizes speed..

We first seek a general solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t}),{i}={1},\ldots ,{N}$$\end{document}pi(t),i=1,…,N for intracellular ion concentration changes that is consistent with the constant concentrations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{q}}_{{i}}$$\end{document}qi outside the cell. We first useful to consider how informative flow rates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})$$\end{document}pi(t) occur.

We start with the assumption that ATP-dependent membrane pumps transfer ions across the cell membrane. Because of these ion-specific pumps, the intracellular concentrations of Na+ and Cl− are much lower than those in the extracellular space, and vice versa for the concentrations of K+ (Fig. 1). The non-equilibrium condition is manifested by an often large transmembrane potential V. Notably, the asymmetric distribution of ions produced by these pumps (compared to random distribution in the absence of those pumps) encodes on the order of 1014 bits of Shannon12 information1,2,13,14.

Suppose the flow of some number of ions from outside to inside the cell membrane can signal the occurrence of an environmental event. For this to occur, the event must trigger the gates in one or more of the ion-specific membrane channels. These gates permit ion flow along the pre-existing concentration gradients between the inside and outside of the cell. Several hundred types of channel gates are encoded in the human genome, permitting responses to a potentially large range of external perturbations. The role of ion flow in cell functions is most well-studied in neurons. To propagate nerve impulses, the channel gates along the axis of the neuron respond to voltage fluctuations cause by the local arrival of the action potential. When the gates open, the transmembrane gradient permits rapid ion flow along steep concentration gradients producing the action potential described by the Hodgkin Huxley equations.

We assume that evolution selects for optimal means for addressing risk and capitalizing on opportunities in the environment. We propose that this optimization principle selects for information dynamics at the membrane, which conveys maximum Shannon information regarding the occurrence times of these events. Of course the system is passive so that, in general, there can only be a loss of such information relative to the constant ion density probability functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{q}}_{{i}},{i}={1},\ldots ,{N}$$\end{document}qi,i=1,…,N outside. Accordingly, the maximal Shannon information is conveyed when the resulting intracellular ion probability density \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\,{{p}}_{{i}}({t})$$\end{document}pi(t) is minimally different the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${c}{o}{n}{s}{t}{a}{n}{t}\equiv {{q}}_{{i}}({t})\equiv {{q}}_{{i}}$$\end{document}constant≡qi(t)≡qi of Eq. (2). As noted below, this also defines how fast the information from an environmental source can be decoded by the cell surface.

A reasonable assumption is that an environmental change is sensed at the cell membrane. This could represent a change in a physical interaction with an adjacent cell or other surface or a change in the local concentration of some substance (e.g., substrate molecules in the case of a free swimming eukaryote or, in the case of neurons, a neurotransmitter). How is this information communicated into the cell? We propose that eukaryotic cells have efficient mechanisms for sensing and reacting to environmental change through gated ion channels on the cell membrane. ATP-dependent membrane ion pumps typically generate large transmembrane gradients but the flow of ions along such gradients is ordinarily very slow (“leakage current”) due to high resistance in nonpolar regions of the cell membrane. Opening of channel gates by an external perturbation has the effect of greatly expanding the transmembrane flow of the specific ion that passes through that channel. Indeed, it has been noted that the actions of the channel in the membrane are analogous to a catalyst15 for diffusion, in the sense that the acceleration of the ion flow are qualitatively and quantitatively similar to enzyme-induced acceleration of a chemical reaction rate.

What can we learn by considering such an analogy? As will be shown, it leads to derivation of the Hodgkin-Huxley equations that govern the propagation of ions across the neural cell membrane. Moreover, the derivation is sufficiently general to apply to ion flow through gates in a wide class of cell membranes.

In general the passage of a given type of ion, say K+, is restricted to a specific channel just as each of the key ions (e.g., Na+, K+, Cl−, Ca2+, Mg2+) can pass through one and only one type of channel16. Each such ion then represents a distinct component of the transmembrane information flow. The channels contain gates that are ordinarily closed, thus preventing ion flow. However, this still allows a “leakage” current, which is the usually small flow of ions through the membrane. Once a transmembrane ion gradient is established through ATP-dependent membrane ion pumps, maintenance of that gradient requires only that the pumps replace the small loss of ions due to leakage current. A detailed description of the gate channels can be found in16.

Finally, the passage of a given type of ion, say K+, requires that more than one gate in the channel opens. For example, the K+ channel allows ion flow only when all 3 channel gates are open. These appear to form a ring shaped gap through which the required K+ ion passes17,18.

The cell membrane provides a ‘window’ to the world outside for each cell. Some signals that arrive at the membrane, such as growth factors, require transmission through well recognized cytoplasmic signaling pathways to the nucleus for further processing and action. However, others, particularly life-threatening perturbations, may require an immediate response that is mediated entirely by the molecular machinery in and around the cell membrane.

The rapid transmembrane flow of ions within one or more channels allows cells to receive, locally process and respond to information in the environment. Thus, ions of type i = 1,.., N may pass through the cell membrane, each potentially signaling vital information to the cell. Hence the changing concentration, and correct identification, of the various ion types is vital to cell survival and proliferation. We shall assume that natural selection, acting through geological time, has optimized the speed and accuracy of identifying i on average. This implies that the intracellular probability functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})$$\end{document}pi(t) are formed, coded, and decoded at the membrane according to a well-defined optimization principle. We use minimum average code length as the criterion19, as follows (in brief, see Eqs (3), (4)).

Claude Shannon12,20 posed the following question of optimization: If we are given a probability \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{q}}_{{i}}({t})$$\end{document}qi(t) on event times t for events i, what code lengths \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{l}}_{{i}}({t})$$\end{document}li(t), i = 1, .., N achieve their minimum expected values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ < {{l}}_{{i}}({t}) > $$\end{document}<li(t)> over all times t? This defines a maximally efficient code.

Shannon and Fano showed that if symbols (say letters of an alphabet) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{a}}_{{i}}$$\end{document}ai, i = 1, .., N occur with respective frequencies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{q}}_{{i}}$$\end{document}qi these are coded with maximum efficiency (minimum average code length) if each symbol i is coded with a length \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{l}}_{{i}}=\,{\rm{l}}{\rm{o}}{\rm{g}}(\frac{{1}}{{{q}}_{{i}}})$$\end{document}li=log(1qi). Thus, the most frequently occurring symbols \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{a}}_{{i}}$$\end{document}ai would be coded with minimum lengths l. Or, minimum code lengths are required, and this is independent of time. This is intuitive for maximally fast response.

Thus, the information dynamics at the cell membrane in which the external ion concentrations (expressed as probabilities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{q}}_{{i}}({t})$$\end{document}qi(t)) are actually constants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{q}}_{{i}}$$\end{document}qi, manifest the conditions for optimum coding.

However, the same constancy does not hold in practice for data. Their occurrence rates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{p}}_{{i}}({t})$$\end{document}pi(t) randomly differ rates from the ideal rates \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{q}}_{{i}}$$\end{document}qi. (Think of the letter rates on a single page of print). Hence, if these are coded according to Shannon and Fano’s scheme, the result is a wasteful excess of code lengths by amount3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${l}_{q}(t)\,-\,{l}_{i}(t)=\,\mathrm{log}(\frac{1}{{q}_{i}})\,-\,\mathrm{log}(\frac{1}{{p}_{i}(t)})=\,\mathrm{log}(\frac{{p}_{i}(t)}{{q}_{i}})$$\end{document}lq(t)−li(t)=log(1qi)−log(1pi(t))=log(pi(t)qi)at each time. The expected value of the excess code length over all time for each ion type i is then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ < {\rm{l}}{\rm{o}}{\rm{g}}(\frac{{{p}}_{{i}}({t})}{{{q}}_{{i}}}) > $$\end{document}<log(pi(t)qi)>. This average excess code length translates into excessive total average communication time required for responding to the environmental event being reported on by the ion.

In turn, this implies excess reaction time for coping with that event. Excess reaction time to an external threat, for example, can be deadly for the cell. Hence we assume that natural selection minimizes reaction time.

The above models focus on a single specific channel and channel state. This is clearly an oversimplification as it is likely that ion flows into mammalian cells are the result of an ensemble of single channels that superimpose at various times with various weights to form the net channel state21. This may vary with time both stochastically and with subtle changes in the environmental condition. These ensemble dynamics leading to a net channel can be accommodated using KL principle because it is a linear differential equation. Thus, the net channel state P n(t) is, by hypothesis, the statistically average state. Therefore any linearly weighted superposition of single channel outputs will also satisfy our KL principle so that the net state P n(t) obeys KL principle. However, the detailed shape of the curve P n(t) will depend upon what the times \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau }_{n}$$\end{document}τn, n = 1, …, 4 are in the various channel states. Nevertheless, even allowing for a quite complex ensemble of independent channels, the total ion flux gives an output activation wave obeying the KL principle.