Resolvability of fluorescence lifetime distributions using phase fluorometry.

The analysis of the fluorescence decay using discrete exponential components assumes that a small number of species is present. In the absence of a definite kinetic model or when a large number of species is present, the exponential analysis underestimates the uncertainty of the recovered lifetime values. A different approach to determine the lifetime of a population of molecules is the use of probability density functions and lifetime distributions. Fluorescence decay data from continuous distributions of exponentially decaying components were generated. Different magnitudes of error were added to the data to simulate experimental conditions. The resolvability of the distributional model was studied by fitting the simulated data to one and two exponentials. The maximum width of symmetric distributions (uniform, gaussian, and lorentzian), which cannot be distinguished from single and double exponential fits for statistical errors of 1 and 0.1%, were determined. The width limits are determined by the statistical error of the data. It is also shown that, in the frequency domain, the discrete exponential analysis does not uniformly weights all the components of a distribution. This systematic error is less important when probability and distribution functions are used to recover the decay. Finally, it is shown that real lifetime distributions can be proved using multimodal probability density functions. In the companion paper that follows we propose a physical approach, which provides lifetime distribution functions for the tryptophan decay in proteins. In the third companion paper (Alcala, J.R., E. Gratton, and F.J. Prendergast, 1987, Biophys. J., in press) we use the distribution functions obtained to fit data from the fluorescence decay of single tryptophan proteins.     

Inversion of Markov processes to determine rate constants from single-channel data.

The determination of rate constants from single-channel data can be very difficult, in part because the single-channel lifetime distributions commonly analyzed by experimenters often have a complicated mathematical relation to the channel gating mechanism. The standard treatment of channel gating as a Markov process leads to the prediction that lifetime distributions are exponential functions. As the number of states of a channel gating scheme increases, the number of exponential terms in the lifetime distribution increases, and the weights and decay constants of the lifetime distributions become progressively more complicated functions of the underlying rate constants. In the present study a mathematical strategy for inverting these functions is introduced in order to determine rate constants from single-channel lifetime distributions. This inversion is easy for channel gating schemes with two or fewer states of a given conductance, so the present study focuses on schemes with more states. The procedure is to derive explicit equations relating the parameters of the lifetime distribution to the rate constants of the scheme. Such equations can be derived using the equality between symmetric functions of eigenvalues of a matrix and sums over principle minors, as well as expressions for the moments, derivatives, and weights of a lifetime distribution. The rate constants are then obtained as roots to this system of equations. For a gating scheme with three sequential closed states and a single gateway state, exact analytical expressions were found for each rate constant in terms of the parameters of the three-exponential closed-time distribution. For several other gating schemes, systems of equations were found that could be solved numerically to obtain the rate constants. Lifetime distributions were shown to specify a unique set of real rate constants in sequential gating schemes with up to five closed or five open states. For kinetic schemes with multiple gating pathways, the analysis of simulated data revealed multiple solutions. These multiple solutions could be distinguished by examining two-dimensional probability density functions. The utility of the methods introduced here are demonstrated by analyzing published data on nicotinic acetylcholine receptors, GABA(A) receptors, and NMDA receptors.     

Statistical analysis of channel current from a membrane patch. II. A stochastic theory of a multi-channel system in the steady-state

A general stochastic theory is presented for analysis of current records of a patch containing an arbitrary number (N) of independent homologous channels in the steady-state. We give the "basic theorem" that at the instant of any open (or shut) transition of a channel, the other N-1 channels are located in each state with a probability equal to those in the steady-state, if enough transitions are observed. Using the "basic theorem", we derived: (a) the time-dependent open and shut frequencies after a definite type of transition, and (b) the probability density functions (pdf) of the duration of any period between two successive transitions. Briefly, the main results obtained were: (1) The time-dependent open (or shut) transition frequency after every shut (or open) transition at t = 0 in an N-channel patch, fJSh,Op(t)(N) (or fJOp,Sh(t)(N)), is the same as that of a one-channel patch except for the value of the constant. (2) In the all-shut (or all-open) period of a patch, the average duration of the period is 1/N, and the slowest exponential decay constant contained in the pdf is N times those of a single channel patch, respectively. (3) An example calculation for small N showed that the stochastic properties of a single channel can be obtained even when N is uncertain, if the channel open probability is small and exponential decay constants are separated. (4) When the channels are in equilibrium, the pdf of duration of every type of period in the patch is described by a sum of exponential terms with positive coefficients. This also holds for fJSh,Op(t)(N) and fJOp,Sh(t)(N).     

Statistical properties of single sodium channels

Single channel currents were obtained from voltage-activated sodium channels in outside-out patches of tissue-cultured GH3 cells, a clonal line from rat pituitary gland. In membrane patches where the probability of overlapping openings was low, the open time histograms were well fit by a single exponential. Most analysis was done on a patch with exactly one channel. We found no evidence for multiple open states at -25 and -40 mV, since open times, burst durations, and autocorrelation functions were time independent. Amplitude histograms showed no evidence of multiple conductance levels. We fit the gating with 25 different time-homogeneous Markov chain models having up to five states, using a maximum likelihood procedure to estimate the rate constants. For selected models, this procedure yielded excellent predictions for open time, closed time, and first latency density functions, as well as the probability of the channel being open after a step depolarization, the burst duration distribution, autocorrelation, and the distribution of number of openings per record. The models were compared statistically using likelihood ratio tests and Akaike's information criterion. Acceptable models allowed inactivation from closed states, as well as from the open state. Among the models eliminated as unacceptable by this survey were the Hodgkin-Huxley model and any model requiring a channel to open before inactivating.     

Statistical analysis of channel current from a membrane patch. I. Some stochastic properties of ion channels or molecular systems in equilibrium

The stochastic behavior of single-channel current in a steady-state has been interpreted as the channel's state transitions between several open and shut states, and these transitions have been regarded as a homogeneous Markov process. When a channel is in equilibrium, the principle of detailed balance holds for every step in the state transition scheme. Here we show two stochastic properties of a channel, or any molecule obeying a reversible state transition scheme, under the constraint of detailed balance. First, the distribution functions and the probability density functions of shut or open dwell-time are expressed by the sum of exponential terms with positive coefficients. The same holds for the time-dependent open (or shut) frequency after the shut (or open) transition. Second, the time course of state transition from the state SI to SJ (PI,J(t] is proportional to its reverse transition time course (PJ,I(t], even if SI and SJ are widely separated. The same relation holds also for a transition scheme having transition pathways to the absorbing states. If analysis of a channel current record shows it to be incompatible with either of these two properties, the channel is not in equilibrium but in a steady-state with an energy-consuming cyclic flow. These two properties are also useful for the analysis of any molecular process obeying a homogeneous Markov process or a network of first-order chemical reactions.     

Mechanism of Maxi-K Channel Activation by Dehydrosoyasaponin-I

Dehydrosoyasaponin-I (DHS-I) is a potent activator of high-conductance, calcium-activated potassium (maxi-K) channels. Interaction of DHS-I with maxi-K channels from bovine aortic smooth muscle was studied after incorporating single channels into planar lipid bilayers. Nanomolar amounts of intracellular DHS-I caused the appearance of discrete episodes of high channel open probability interrupted by periods of apparently normal activity. Statistical analysis of these periods revealed two clearly separable gating modes that likely reflect binding and unbinding of DHS-I. Kinetic analysis of durations of DHS-I-modified modes suggested DHS-I activates maxi-K channels through a high-order reaction. Average durations of DHS-I-modified modes increased with DHS-I concentration, and distributions of these mode durations contained two or more exponential components. In addition, dose-dependent increases in channel open probability from low initial values were high order with average Hill slopes of 2.4–2.9 under different conditions, suggesting at least three to four DHS-I molecules bind to maximally activate the channel. Changes in membrane potential over a 60-mV range appeared to have little effect on DHS-I binding. DHS-I modified calcium- and voltage-dependent channel gating. 100 nM DHS-I caused a threefold decrease in concentration of calcium required to half maximally open channels. DHS-I shifted the midpoint voltage for channel opening to more hyperpolarized potentials with a maximum shift of −105 mV. 100 nM DHS-I had a larger effect on voltage-dependent compared with calcium-dependent channel gating, suggesting DHS-I may differentiate these gating mechanisms. A model specifying four identical, noninteracting binding sites, where DHS-I binds to open conformations with 10–20-fold higher affinity than to closed conformations, explained changes in voltage-dependent gating and DHS-I-induced modes. This model of channel activation by DHS-I may provide a framework for understanding protein structures underlying maxi-K channel gating, and may provide a basis for understanding ligand activation of other ion channels.     

Gating kinetics of four classes of voltage-dependent K+ channels in pheochromocytoma cells

Clonal pheochromocytoma (PC-12) cells have four different types of voltage-dependent K+ channels whose activation does not require high concentrations of Ca++ on the cytoplasmic side of the membrane (Hoshi, T., and R. W. Aldrich, 1988, Journal of General Physiology, 91:73-106). The durations of open and closed events of these four different types of voltage-dependent K+ channels were measured using the excised configuration of the patch-clamp method. The open durations of a class of K+ channels termed the Kz channel, which activates rapidly and inactivates slowly in response to depolarizing pulses, had two exponential components. The closed durations of the Kz channel had at least four exponential components. The time constants of the fastest of the two exponential components in the closed durations were very similar to those of the two exponential components present in the first-latency distribution. The first latencies of the Kz channel decreased steeply with depolarization, contributing to the increased probability of the channel being open with depolarization. The Kz channel also had a very slow gating process that resulted in a clustering of blank sweeps. A gating scheme containing two open states and five closed states is consistent with the observations. The Ky channel had one exponential component in the open durations and three exponential components in the closed durations. The first latencies varied greatly depending on the prepulse voltage and duration. The results were consistent with a sequential model with a large number of closed states and one open state. The Kx channel, which required large hyperpolarizing prepulses to remove steady state inactivation and did not show inactivation with maintained depolarization, had two exponential components in the open durations and three exponential components in the closed durations. The burst behavior of the Kx channel involved many more than two states. The transient Kw channel had one exponential component in the open durations and the mean open time increased with depolarization. The first latencies of the Kw channel were steeply dependent on the voltage, decreasing with depolarization.     

Process‐based functions for seed retention on animals: a test of improved descriptions of dispersal using multiple data sets

Studies of external seed transport on animals usually assume that the probability of detachment is constant, so that seed retention should show a simple exponential relationship with time. This assumption has not been tested explicitly, and may lead to inaccurate representation of long distance seed dispersal by animals. We test the assumption by comparing the fit to empirical data of simple, two‐parameter functions. Fifty‐two data sets were obtained from five published studies, describing seed retention of 32 plant species on sheep, cattle, deer, goats and mice. Model selection suggested a simple exponential function was adequate for data sets in which seed retention was followed for short periods ( <48 h). The data gathered over longer periods (49–219 days) were best described by the power exponential function, a form of the stretched exponential which allows a changing dropping rate. In these cases the power exponential showed that seed dropping rate decreased with time, suggesting that seeds vary in attachment, with some seeds becoming deeply buried or wound up in the animal's coat. Comparison of fitted parameters across all the data sets also confirmed that seeds with adhesive structures have lower dropping rates than those without. We conclude that the seed dropping rate often changes with time during external transport on animals and that the power exponential is an effective function to describe this change. We advise that, to analyse seed dropping rates adequately, retention should be measured over reasonable time periods – until most seeds are dropped – and both the simple and power exponential functions should be fitted to the resulting data. To increase its utility, we provide functions describing the seed dropping rate and the dispersal kernel resulting from the power exponential relationship.     

Growth of cell populations with arbitrarily distributed cycle durations. I. basic model

A discrete model is proposed describing the growth of cell populations with arbitrary frequency distributions of cycle durations. The model assumes that each cell divides into two cells at the end of its cycle, and that each new cell is assigned an individual cycle duration according to a probability distribution that can be arbitrarily defined. The increase in the cell number is calculated, either from the numbers of cells at earlier time points or from the initial conditions of the population, by a recurrence formula; it is also approximated by the optimal exponential function, whose parameters are determined by the initial conditions. The appropriate average cycle duration is shown not to be the arithmetic or geometric mean, but rather the solution to a more complex equation. Age distributions are calculated and compared with those found in the literature. The results of the model calculations are compared with computer simulations and with observed data on populations of the ciliate Tetrahymena geleii.     

Deviations from Michaelis-Menten kinetics. Computation of the probabilities of obtaining complex curves from simple kinetic schemes.

1. It is possible to calculate the intrinsic probability associated with any curve shape that is allowed for rational functions of given degree when the coefficients are independent or dependent random variables with known probability distributions. 2. Computations of such probabilities are described when the coefficients of the rational function are generated according to several probability distribution functions and in particular when rate constants are varied randomly for several simple model mechanisms. 3. It is concluded that each molecular mechanism is associated with a specific set of curve-shape probabilities, and this could be of value in discriminating between model mechanisms. 4. It is shown how a computer program can be used to estimate the probability of new complexities such as extra inflexions and turning points as the degree of rate equations increases. 5. The probability of 3 : 3 rate equations giving 2 : 2 curve shapes is discussed for unrestricted coefficients and also for the substrate-modifier mechanisms. 6. The probability associated with the numerical values of coefficients in rate equations is also calculated for this mechanism, and a possible method for determining the approximate magnitude of product-release steps is given. 7. The computer programs used in the computations have been deposited as Supplement SUP 50113 (21 pages) with the British Library Lending Division, Boston Spa, Wetherby, West Yorkshire LS23 7BQ, U.K., from whom copies can be obtained on the terms indicated in Biochem, J. (1978) 169, 5.     

The 20-pS chloride channel of the human airway epithelium.

The single-channel inside-out patch clamp technique was used to characterize chloride channels in the apical membranes of human airway epithelial cells maintained in primary culture. Patches were obtained from single isolated cells or from cells at the edges of confluent groups. The channel seen most often, in 24% of all patches, had a conductance of approximately 20 pS and had a linear current-voltage relationship in symmetric chloride solutions. The anion selectivity sequence for the channel was NO3- greater than Cl- greater than HCO3-, and it was impermeable to gluconate ions, indicating that the channel diameter lies between 4.7 and 5.5 A. Current through the channel saturated at high chloride concentrations, and the relationship between channel current and chloride concentration could be approximated by the Michaelis-Menten equation. Analysis of the channel's anion permeability and its current vs. concentration relationship indicates that it can be described by the one-ion channel theory, with a relatively weak binding site inside the channel. Histograms of channel open and closed durations were constructed using the log binning technique and could be well fitted by triple exponential distributions, suggesting that the channel has at least three open and three closed states.     

Fractal and Markov behavior in ion channel kinetics

Kinetic analysis of ion channel recordings attempts to distinguish the number and lifetimes of channel molecular states. Most kinetic analysis assumes that the lifetime of each state is independent of previous channel history, so that open and closed durations are Markov processes whose probability densities are sums of exponential decays. An alternative approach assumes that channel molecules have many configurtions with widely varying lifetimes. Rates of opening and closing then vary with the time scale of observation, leading to fractal kinetics. We have examined kinetic behavior in two types of channels from human and avian fibroblasts, using a maximum likehood method to test the dependence of rates on observational time scale. For both channels, openings showed mixed fractal and Markov behavior, while closings gave mainly fractal kinetics.     

Cardiac sodium channel Markov model with temperature dependence and recovery from inactivation

A Markov model of the cardiac sodium channel is presented. The model is similar to the CA1 hippocampal neuron sodium channel model developed by Kuo and Bean (1994. Neuron. 12:819-829) with the following modifications: 1) an additional open state is added; 2) open-inactivated transitions are made voltage-dependent; and 3) channel rate constants are exponential functions of enthalpy, entropy, and voltage and have explicit temperature dependence. Model parameters are determined using a simulated annealing algorithm to minimize the error between model responses and various experimental data sets. The model reproduces a wide range of experimental data including ionic currents, gating currents, tail currents, steady-state inactivation, recovery from inactivation, and open time distributions over a temperature range of 10 degrees C to 25 degrees C. The model also predicts measures of single channel activity such as first latency, probability of a null sweep, and probability of reopening.     

A method for the estimation of parameters for natural stage-structured populations

A relatively simple method is proposed for the estimation of parameters of stage-structured populations from sample data for situation where (a) unit time survival rates may vary with time, and (b) the distribution of entry times to stage 1 is too complicated to be fitted with a simple parametric model such as a normal or gamma distribution. The key aspects of this model are that the entry time distribution is approximated by an exponential function withp parameters, the unit time survival rates in stages are approximated by anr parameter exponential polynomial in the stage number, and the durations of stages are assumed to be the same for all individuals. The new method is applied to four Zooplankton data sets, with parametric bootstrapping used to assess the bias and variation in estimates. It is concluded that good estimates of demographic parameters from stagefrequency data from natural populations will usually only be possible if extra information such as the durations of stages is known.     

Fossil preservation and the stratigraphic ranges of taxa

The incompleteness of the fossil record hinders the inference of evolutionary rates and patterns. Here, we derive relationships among true taxonomic durations, preservation probability, and observed taxonomic ranges. We use these relationships to estimate original distributions of taxonomic durations, preservation probability, and completeness (proportion of taxa preserved), given only the observed ranges. No data on occurrences within the ranges of taxa are required. When preservation is random and the original distribution of durations is exponential, the inference of durations, preservability, and completeness is exact. However, reasonable approximations are possible given non-exponential duration distributions and temporal and taxonomic variation in preservability. Thus, the approaches we describe have great potential in studies of taphonomy, evolutionary rates and patterns, and genealogy. Analyses of Upper Cambrian-Lower Ordovician trilobite species, Paleozoic crinoid genera, Jurassic bivalve species, and Cenozoic mammal species yield the following results: (1) The preservation probability inferred from stratigraphic ranges alone agrees with that inferred from the analysis of stratigraphic gaps when data on the latter are available. (2) Whereas median durations based on simple tabulations of observed ranges are biased by stratigraphic resolution, our estimates of median duration, extinction rate, and completeness are not biased.(3) The shorter geologic ranges of mammalian species relative to those of bivalves cannot be attributed to a difference in preservation potential. However, we cannot rule out the contribution of taxonomic practice to this difference. (4) In the groups studied, completeness (proportion of species [trilobites, bivalves, mammals] or genera [crinoids] preserved) ranges from 60% to 90%. The higher estimates of completeness at smaller geographic scales support previous suggestions that the incompleteness of the fossil record reflects loss of fossiliferous rock more than failure of species to enter the fossil record in the first place.     

Relationship between statistical properties of single ionic channel recordings and the thermodynamic state of the channels

Recordings of the electric conductivity of a single ionic channel usually exhibit two levels of conductance: a zero and a finite level. The channel may, however, be in a few states which have the same conductivity level, and the distribution of dwell time durations at this conductivity level is thus not monoexponential. It is shown that the joint probability p(tc,to) of the occurrence of a time interval tc during which the channel is not conducting, immediately followed by a time interval to during which the channel is conducting may or may not be equal to the joint probability pr(tc,to) of the occurrence of a non-conducting interval tc preceded by a conducting interval to. If the interconversions between the various states in which the channel can exist obey detailed balance, i.e., if the channel behaves like a system at thermodynamic equilibrium, then p(tc,to) = pr(tc,to). This should help to reveal whether irreversible processes, like metabolic reactions or flows of substances across the membrane, are coupled to the gating process of the ionic channels.     

Cycle-time and residence-time density approximations in a stochastic model for circulatory transport

The concentration of a drug in the circulatory system is studied under two different elimination strategies. The first strategy—geometric elimination—is the classical one which assumes a constant elimination rate per cycle. The second strategy—Poisson elimination—assumes that the elimination rate changes during the process of elimination. The problem studied here is to find a relationship between the residence-time distribution and the cycle-time distribution for a given rule of elimination. While the presented model gives this relationship in terms of Laplace-Stieltjes transform, the aim here is to determine the shapes of the corresponding probability density functions. From experimental data, we expect positively skewed, gamma-like distributions for the residence time of the drug in the body. Also, as some elimination parameter in the model approaches a limit, the exponential distribution often arises. Therefore, we use laguerre series expansions, which yield a parsimonious approximation of positively skewed probability densities that are close to a gamma distribution. The coefficients in the expansion are determined by the central moments, which can be obtained from experimental data or as a consequence of theoretical assumptions. The examples presented show that gamma-like densities arise for a diverse set of cycle-time distributions and under both elimination rules.     

The Role of Auxiliary Subunits for the Functional Diversity of Voltage‐Gated Calcium Channels

Voltage‐gated calcium channels (VGCCs) represent the sole mechanism to convert membrane depolarization into cellular functions like secretion, contraction, or gene regulation. VGCCs consist of a pore‐forming α₁ subunit and several auxiliary channel subunits. These subunits come in multiple isoforms and splice‐variants giving rise to a stunning molecular diversity of possible subunit combinations. It is generally believed that specific auxiliary subunits differentially regulate the channels and thereby contribute to the great functional diversity of VGCCs. If auxiliary subunits can associate and dissociate from pre‐existing channel complexes, this would allow dynamic regulation of channel properties. However, most auxiliary subunits modulate current properties very similarly, and proof that any cellular calcium channel function is indeed modulated by the physiological exchange of auxiliary subunits is still lacking. In this review we summarize available information supporting a differential modulation of calcium channel functions by exchange of auxiliary subunits, as well as experimental evidence in support of alternative functions of the auxiliary subunits. At the heart of the discussion is the concept that, in their native environment, VGCCs function in the context of macromolecular signaling complexes and that the auxiliary subunits help to orchestrate the diverse protein–protein interactions found in these calcium channel signalosomes. Thus, in addition to a putative differential modulation of current properties, differential subcellular targeting properties and differential protein–protein interactions of the auxiliary subunits may explain the need for their vast molecular diversity. J. Cell. Physiol. 999: 00–00, 2015. © 2015 The Authors. Journal of Cellular Physiology Published by Wiley Periodicals, Inc. J. Cell. Physiol. 230: 2019–2031, 2015. © 2015 Wiley Periodicals, Inc.     

Statistical properties of ion channel records. Part II: estimation from the macroscopic current

Macroscopic ion channel current is the summation of the stochastic records of individual channel currents and therefore relates to their statistical properties. As a consequence of this relationship, it may be possible to derive certain statistical properties of single channel records or even generate some estimates of the records themselves from the macroscopic current when the direct measurement of single channel currents is not applicable. We present a procedure for generating the single channel records of an ion channel from its macroscopic current when the stochastic process of channel gating has the following two properties: (I) the open duration is independent of the time of opening event and has a single exponential probability density function (pdf), (II) all the channels have the same probability to open at time t. The application of this procedure is considered for cases where direct measurement of single channel records is difficult or impossible. First, the probability density function (pdf) of opening events, a statistical property of single channel records, is derived from the normalized macroscopic current and mean channel open duration. Second, it is shown that under the conditions (I) and (II), a non-stationary Markov model can represent the stochastic process of channel gating. Third, the non-stationary Markov model is calibrated using the results of the first step. The non-stationary formulation increases the model ability to generate a variety of different single channel records compared to common stationary Markov models. The model is then used to generate single channel records and to obtain other statistical properties of the records. Experimental single channel records of inactivating BK potassium channels are used to evaluate how accurately this procedure reconstructs measured single channel sweeps.