Internal motions in proteins and gating kinetics of ionic channels.

Single-channel current recordings have revealed a complex kinetic behavior of ionic channels. Many channels exhibit closed-time distributions in which long waiting times occur with a much higher frequency than predicted by a simple exponential decay function. In this paper a model for opening-closing transitions that accounts for internal motions in the protein matrix is discussed. The model is based on the notion that the transition between a conductive and a nonconductive state of the channel represents a local process in the protein, such as the movement of a small segment of a peptide chain or the rotation of a single amino-acid residue. When the blocking group moves into the ion pathway, a structural defect is created consisting in a region of loose packing and/or poor hydrogen bonding. By rearrangements of neighboring groups, the defect may migrate within the protein matrix, carrying out a kind of random walk. Once the defect has moved away from the site where it was formed, a transition back to the open state of the channel is possible only when the defect has returned by chance to the original position. The kinetic properties of this model are analyzed by stochastic simulation of defect diffusion in a small domain of the protein. With a suitable choice of domain size and diffusion rate, the model is found to predict closed-time distributions that agree with experimental observations.     

Characterization of single channel currents using digital signal processing techniques based on Hidden Markov Models.

Techniques for extracting small, single channel ion currents from background noise are described and tested. It is assumed that single channel currents are generated by a first-order, finite-state, discrete-time, Markov process to which is added 'white' background noise from the recording apparatus (electrode, amplifiers, etc). Given the observations and the statistics of the background noise, the techniques described here yield a posteriori estimates of the most likely signal statistics, including the Markov model state transition probabilities, duration (open- and closed-time) probabilities, histograms, signal levels, and the most likely state sequence. Using variations of several algorithms previously developed for solving digital estimation problems, we have demonstrated that: (1) artificial, small, first-order, finite-state, Markov model signals embedded in simulated noise can be extracted with a high degree of accuracy, (2) processing can detect signals that do not conform to a first-order Markov model but the method is less accurate when the background noise is not white, and (3) the techniques can be used to extract from the baseline noise single channel currents in neuronal membranes. Some studies have been included to test the validity of assuming a first-order Markov model for biological signals. This method can be used to obtain directly from digitized data, channel characteristics such as amplitude distributions, transition matrices and open- and closed-time durations.     

Structural determinants of gating in inward-rectifier K+ channels

The gating characteristics of two ion channels in the inward-rectifier K+ channel superfamily were compared at the single-channel level. The strong inward rectifier IRK1 (Kir 2.1) opened and closed with kinetics that were slow relative to those of the weakly rectifying ROMK2 (Kir 1.1b). At a membrane potential of -60 mV, both IRK and ROMK had single-exponential open-time distributions, with mean open times of 279 +/- 58 ms (n = 4) for IRK1 and 23 +/- 1 ms (n = 7) for ROMK. At -60 mV (and no EDTA) ROMK2 had two closed times: 1.3 +/- 0.1 and 36 +/- 3 ms (n = 7). Under the same conditions, IRK1 exhibited four discrete closed states with mean closed times of 0.8 +/- 0.1 ms, 14 +/- 0.6 ms, 99 +/- 19 ms, and 2744 +/- 640 ms (n = 4). Both the open and the three shortest closed-time constants of IRK1 decreased monotonically with membrane hyperpolarization. IRK1 open probability (Po) decreased sharply with hyperpolarization due to an increase in the frequency of long closed events that were attributable to divalent-cation blockade. Chelation of divalent cations with EDTA eliminated the slowest closed-time distribution of IRK1 and blunted the hyperpolarization-dependent fall in open probability. In contrast, ROMK2 had shorter open and closed times and only two closed states, and its Po was less affected by hyperpolarization. Chimeric channels were constructed to address the question of which parts of the molecules were responsible for the differences in kinetics. The property of multiple closed states was conferred by the second membrane-spanning domain (M2) of IRK. The long-lived open and closed states, including the higher sensitivity to extracellular divalent cations, correlated with the extracellular loop of IRK, including the "P-region." Channel kinetics were essentially unaffected by the N- and C-termini. The data of the present study are consistent with the idea that the locus of gating is near the outer mouth of the pore.     

The Effect of an Upper Limit to Population Size on Persistence Time

For a population with density-independent vital rates in a randomly varying environment, previous authors have calculated the probability that population size will first drop to some specified (arbitrary) low level at a given time (the first passage time distribution (FPTD), which may be interpreted as a distribution of extinction times). In this paper, we study the FPTD For a stochastic model of density-independent population growth which includes a hard upper limit to population size. We discuss the conditions under which this distribution may be approximated by the FPTD of a Wiener process with a reflecting boundary condition, for which an exact calculation is presented in an appendix. We compare the FPTD of the new model with its counterpart in the model without an upper limit. The most important effects of introducing the upper limit are: (a) ultimate extinction becomes certain; (b) if the long run growth rate in the absence of the upper boundary was small but positive, extinction within ecologically significant times is likely; (c) for larger values of the long run growth rate, persistence over ecologically significant times is almost certain. We discuss the implications of result (b) for conservation. Result (c) establishes that "density-vague" regulation can produce persistent, but bounded, populations.     

Activation of a nicotinic acetylcholine receptor.

We studied activation of the nicotinic acetylcholine (ACh) receptor on cells of a mouse clonal muscle cell line (BC3H1). We analyzed single-channel currents through outside-out patches elicited with various concentrations of acetylcholine (ACh), carbamylcholine (Carb) and suberyldicholine (Sub). Our goal is to determine a likely reaction scheme for receptor activation by agonist and to determine values of rate constants for transitions in that scheme. Over a wide range of agonist concentrations the open-time duration histograms are not described by single exponential functions, but are well-described by the sum of two exponentials, a brief-duration and a long-duration component. At high concentration, channel openings occur in groups and these groups contain an excess number of brief openings. We conclude that there are two open states of the ACh receptor with different mean open times and that a single receptor may open to either open state. The concentration dependence of the numbers of brief and long openings indicates that brief openings do not result from the opening of channels of receptors which have only one agonist molecule bound to them. Closed-time duration histograms exhibit a major brief component at low concentrations. We have used the method proposed by Colquhoun and Sakmann (1981) to analyze these brief closings and to extract estimates for the rates of channel opening (beta) and agonist dissociation (k-2). We find that this estimate of beta does not predict our closed-time histograms at high agonist concentration (ACh: 30-300 microM; Carb: 300-1,000 microM). We conclude that brief closings at low agonist concentrations do not result solely from transitions between the doubly-liganded open and the doubly-liganded closed states. Instead, we postulate the existence of a second closed-channel state coupled to the open state.     

Blocking kinetics of Cl- channels in colonic carcinoma cells (HT29) as revealed by 5-nitro-2-(3-phenylpropylamino)benzoic acid (NPPB)

The blocking effect of 5-nitro-2-(3-phenylpropylamino)benzoic acid (NPPB) was investigated on single Cl- channels of the cultured human colon carcinoma cells, HT29. In the absence of NPPB, the open-time histogram yielded two time constants, with 0.9 ms and 33 ms, whereas the closed-time distribution could be fitted by a single exponential with a time constant of 0.7 ms. Addition of NPPB in the range 1-50 microM induced brief closing events of the single-channel current. This resulted in a decrease of the long open-time constant to 2.1 ms and in an increase of the closed-time constant to 1.8 ms at 50 microM NPPB concentration. The short open-time constant did not change at low blocker concentration (1 microM), but could no longer be resolved at higher concentrations. The open-state probability decreased from 0.9 (control conditions) to 0.5 at 50 microM NPPB. The Hill plot yielded a Hill coefficient of about 0.7, compatible with one NPPB molecule inhibiting one channel molecule. The kinetics of channel gating are described by a sequential model with one closed and two open states. Since in the presence of NPPB no additional time constant appeared in the time histograms, we assumed the same kinetic scheme as under control conditions, and hypothesize that NPPB has an influence on rate constants.     

Statistical inference from single channel records: two-state Markov model with limited time resolution

Though stochastic models are widely used to describe single ion channel behaviour, statistical inference based on them has received little consideration. This paper describes techniques of statistical inference, in particular likelihood methods, suitable for Markov models incorporating limited time resolution by means of a discrete detection limit. To simplify the analysis, attention is restricted to two-state models, although the methods have more general applicability. Non-uniqueness of the mean open-time and mean closed-time estimators obtained by moment methods based on single exponential approximations to the apparent open-time and apparent closed-time distributions has been reported. The present study clarifies and extends this previous work by proving that, for such approximations, the likelihood equations as well as the moment equations (usually) have multiple solutions. Such non-uniqueness corresponds to non-identifiability of the statistical model for the apparent quantities. By contrast, higher-order approximations yield theoretically identifiable models. Likelihood-based estimation procedures are developed for both single exponential and bi-exponential approximations. The methods and results are illustrated by numerical examples based on literature and simulated data, with consideration given to empirical distributions and model control, likelihood plots, and point estimation and confidence regions.     

Directional Migration of Recirculating Lymphocytes through Lymph Nodes via Random Walks

Naive T lymphocytes exhibit extensive antigen-independent recirculation between blood and lymph nodes, where they may encounter dendritic cells carrying cognate antigen. We examine how long different T cells may spend in an individual lymph node by examining data from long term cannulation of blood and efferent lymphatics of a single lymph node in the sheep. We determine empirically the distribution of transit times of migrating T cells by applying the Least Absolute Shrinkage & Selection Operator () or regularised to fit experimental data describing the proportion of labelled infused cells in blood and efferent lymphatics over time. The optimal inferred solution reveals a distribution with high variance and strong skew. The mode transit time is typically between 10 and 20 hours, but a significant number of cells spend more than 70 hours before exiting. We complement the empirical machine learning based approach by modelling lymphocyte passage through the lymph node . On the basis of previous two photon analysis of lymphocyte movement, we optimised distributions which describe the transit times (first passage times) of discrete one dimensional and continuous (Brownian) three dimensional random walks with drift. The optimal fit is obtained when drift is small, i.e. the ratio of probabilities of migrating forward and backward within the node is close to one. These distributions are qualitatively similar to the inferred empirical distribution, with high variance and strong skew. In contrast, an optimised normal distribution of transit times (symmetrical around mean) fitted the data poorly. The results demonstrate that the rapid recirculation of lymphocytes observed at a macro level is compatible with predominantly randomised movement within lymph nodes, and significant probabilities of long transit times. We discuss how this pattern of migration may contribute to facilitating interactions between low frequency T cells and antigen presenting cells carrying cognate antigen.     

Population dynamics in variable environments I. Long-run growth rates and extinction

A model of population growth is studied in which the Leslie matrix for each time interval is chosen according to a Markov process. It is shown analytically that the distribution of total population number is lognormal at long times. Measures of population growth are compared and it is shown that a mean logarithmic growth rate and a logarithmic variance effectively describe growth and extinction at long times. Numerical simulations are used to explore the convergence to lognormality and the effects of environmental variance and autocorrelation. The results given apply to other geometric growth models which involve nonnegative growth matrices.     

On the use of time constants as estimates of mean lifetimes in single channel data analysis

For Markov models of single channel kinetics, a sojourn time in a class of states has a density function which is usually a linear combination of exponential densities. There are many instances in the single channel literature where the time constants of exponentials fitted to sojourn time data have been used as estimated mean sojourn times in individual states, though the two may be very different. In the present study the nature and magnitude of this difference in the case of a two state class is illustrated analytically and numerically. The time constants should be viewed at best as approximations, possibly poor, to the estimated mean sojourn times. Estimates of kinetic parameters cannot in general be obtained explicitly from the fitted parameters of the density alone. However, this is shown to be possible in some special cases and enables direct estimation of, for example, the channel opening rate constant (or an upper limit to the estimate of in the case of multiple channels) in standard sequential three or four state models of nicotinic receptor kinetics, using only the fitted parameters of the closed-time density. Offprint requests to: R. O. Edeson     

Stochastic modelling of a single ion channel: interdependence of burst length and number of openings per burst

Previous modelling of single channel behaviour based on Markov processes has been concerned mainly with means and marginal distributions of particular quantities. The present study derives the joint distribution, conditional distributions, and associated mean values for the burst length (T) and the number (N) of openings per burst in two simple three-state models in which bursting is possible, one for an agonist-only and one for a channel blocking mechanism. In both models the conditional mean burst length (E(T/N = r)) increases linearly as a function of the number of openings per burst, while the conditional mean number of openings per burst (E(N/T = x)) is a nonlinear strictly increasing function of burst length, which is asymptotically linear for large burst length. The asymptotic intercept for each model is shown to be less than, equal to, or greater than unity according as mean channel closed-time is less than, equal to, or greater than mean open-time. For parameter values typical of the nicotinic receptor, this intercept is less than unity for the agonist-only model and greater than unity for the blocking model. As a result of the dependence between the number of openings per burst and burst length, it is shown that experimental estimates of the unconditional mean number of openings per burst may be biased if bursts of only short duration are collected.     

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.     

Divergence time uncertainty and historical biogeography reconstruction – an example from Urophylleae (Rubiaceae)

Aim When hypotheses of historical biogeography are evaluated, age estimates of individual nodes in a phylogeny often have a direct impact on what explanation is concluded to be most likely. Confidence intervals of estimated divergence times obtained in molecular dating analyses are usually very large, but the uncertainty is rarely incorporated in biogeographical analyses. The aim of this study is to use the group Urophylleae, which has a disjunct pantropical distribution, to explore how the uncertainty in estimated divergence times affects conclusions in biogeographical analysis. Two hypotheses are evaluated: (1) long‐distance dispersal from Africa to Asia and the Neotropics, and (2) a continuous distribution in the boreotropics, probably involving migration across the North Atlantic Land Bridge, followed by isolation in equatorial refugia. Location Tropical and subtropical Asia, tropical Africa, and central and southern tropical America. Methods This study uses parsimony and Bayesian phylogenetic analyses of chloroplast DNA and nuclear ribosomal DNA data from 56 ingroup species, beast molecular dating and a Bayesian approach to dispersal–vicariance analysis (Bayes‐DIVA) to reconstruct the ancestral area of the group, and the dispersal–extinction–cladogenesis method to test biogeographical hypotheses. Results When the two models of geographic range evolution were compared using the maximum likelihood (ML) tree with mean estimates of divergence times, boreotropical migration was indicated to be much more likely than long‐distance dispersal. Analyses of a large sample of dated phylogenies did, however, show that this result was not consistent. The age estimate of one specific node had a major impact on likelihood values and on which model performed best. The results show that boreotropical migration provides a slightly better explanation of the geographical distribution patterns of extant Urophylleae than long‐distance dispersal. Main conclusions This study shows that results from biogeographical analyses based on single phylogenetic trees, such as a ML or consensus tree, can be misleading, and that it may be very important to take the uncertainty in age estimates into account. Methods that account for the uncertainty in topology, branch lengths and estimated divergence times are not commonly used in biogeographical inference today but should definitely be preferred in order to avoid unwarranted conclusions.     

大鼠大脑皮层神经元单离子钾通道门控动力学

采用膜片钳制技术,对新生大鼠大脑皮层神经元作了细胞贴附和内膜向外两种模式的单通道电流记录。通道开放和关闭事件的转换过程为一随机过程,开关时间服从指数分布。细胞膜单离子通道的时间常数和对离子通透性接近的构象组成构象集合态。由残差法获得模型参数初值,由非线性最小二乘法获得修正值。新生大鼠大脑皮层神经元在-40mV钳制电位和电极内外对称高钾溶液下,细胞贴附和内膜向外的两种膜片钳制模式的单通道电流的动力学特征有明显差异。通道开放时间分布接近一状态分布。细胞贴附时的通道平均开放时间为2.53ms。内膜向外时的通道平均开放时间为2.04ms。关闭时间分布接近三状态分布,细胞贴附时通道平均关闭时间为3.36ms,内膜向外时通道的平均关闭时间为7.58ms。细胞贴附下,通道关闭时主要处于第一和第二关闭态;内膜向外下,通道关闭时主要处于第一关闭态。经初值估计和参数修正,得到各状态间的转移概率密度常数。     

Variance of ventilation during exercise.

Expired gas concentrations were measured during a multibreath washin of He in one female and seven male subjects at rest (seated) and during cycle exercise at work rates of 70-210 W. In a computational model, the ventilation distribution was represented as a log-normal distribution with standard deviation (sigmaV); values of sigmaV were obtained by fitting the output of the model to the data. At rest, sigmaV was 0.89 +/- 0.18; during exercise, sigmaV was 0.60 +/- 0.13, independent of the level of exercise. These values for the width of the functional ventilation distribution at the scale of the acinus are approximately two times larger than those obtained from anatomic measurements in animals at a scale of 1 cm3. The values for sigmaV, together with data from the literature on the width of the functional ventilation-perfusion distribution, show that ventilation and perfusion are highly correlated at rest, in agreement with anatomic data. The structural sources of nonuniform ventilation and perfusion and of the correlation between them are unknown.     

Recent Retrotransposon Insertions Are Methylated and Phylogenetically Clustered in Japonica Rice (Oryza sativa spp. japonica)

In plants, the genome of the host responds to the amplification of transposable elements (TEs) with DNA methylation. However, neither the factors involved in TE methylation nor the dynamics of the host-TE interaction are well resolved. Here, we identify 5,522 long terminal repeat retrotransposons (LTR-RT) in the genome of Oryza sativa ssp. japonica and then assess methylation for individual elements. Our analyses uncover three strong trends: long LTR-RTs are more highly methylated, the insertion times of LTR-RTs are negatively correlated with methylation, and young LTR-RTs tend to be closer to genes than older insertions. Additionally, a phylogenetic examination of the gypsy-like LTR-RT superfamily revealed that methylation is phylogenetically correlated. Given these observations, we present a model suggesting that the phylogenetic correlation among related LTR-RTs is a primary mechanism driving methylation. In this model, bursts of transposition produce new elements with high sequence similarity. The host machinery identifies proliferating elements as well as closely related LTR-RTs through cross-homology. In addition, our data are consistent with previous hypotheses that methylated LTR-RT elements are removed preferentially from regions near genes, explaining some of the observed age distribution.     

Kinetics of sickle hemoglobin polymerization. III. Nucleation rates determined from stochastic fluctuations in polymerization progress curves

The polymerization kinetics of sickle cell hemoglobin are found to exhibit stochastic variations when observed in very small volumes (approximately 10(-10) cm3). The distribution of progress curves has been measured at several temperatures for a 4.50 mM-hemoglobin S sample using a laser-photolysis, light-scattering technique. The progress curves at a given temperature are superimposable when translated along the time axis, showing that the variability of the kinetic progress curves results primarily from fluctuations in the time at which polymerization is initiated. The shapes of the initial part of the progress curves are well-fitted using the functional form I(t) = Io + As exp (Bt), derived from a dual nucleation model. When the distribution of the measured tenth times is broad, the rate of homogeneous nucleation can be obtained by fitting the exponential tail of the distribution. As the distribution sharpen, the rate of homogeneous nucleation can be estimated by modelling the width of the distribution function using a simple Monte-Carlo simulation of the polymerization kinetics. Using the rates of homogeneous nucleation obtained from the distributions, the rates of heterogeneous nucleation and polymer growth can be obtained from the experimental parameters As and B. The resulting nucleation rates are roughly 1000 times greater than those obtained from an analysis of bulk kinetic data. The results provide strong support for the dual-nucleation mechanism and show that the distribution of progress curves provides a powerful independent method for measuring the rate of homogeneous nucleation and thereby obtaining values for the other principal rates of the mechanism.     

A comparison of the distribution of 239-Pu and calcein in the illium of the female CBA mouse

Conclusions As a model for the behaviour of plutonium in bone, calcein data must be treated with some care. It does not label the same surfaces as plutonium which results in different distribution patterns at later times. However, it might be that if in man or a long lived animal, labelling occured over a period of weeks, so that most surfaces become labelled, the resultant distribution pattern at late times would more nearly model plutonium behaviour. The single biggest difference between the behaviour of the two substances is the build up of plutonium in the bone marrow, an effect seen only slightly with calcein. The other differences noted was the redeposition of plutonium, as a consequence of recycling of the radionuclide, maintaining a concentration of plutonium on endosteal surface.     

Epidermal cell proliferation. I. Changes with time in the proportion of isolated, paired and clustered labelled cells in sheets of murine epidermis

A new technical approach to analysing labelled cells in sheets of epidermis is presented. The changes in the proportion of isolated single labelled cells, paired or clusters of 3, 4, or more than 4, labelled cells in sheets of epidermis from the back of the mouse have been analysed at various times up to 500 h after 3HTdR administration at either 03.00 h or 15.00 h. The technique is not dependent on the relative number of labelled cells (i.e. the labelling index) but on the spatial distribution of labelled cells. The data cannot be adequately explained on the basis of a simple homogeneous stem cell population in the basal layer but can be better understood on the basis of an hierarchical stem cell-dividing transit proliferative model. The data are consistent with an average cell cycle time of about 100 h but there are suggestions of considerable cell kinetic heterogeneity. The data also suggest that the amount of lateral cell movement within the basal layer is small. The results may suggest that some stem cells either loose label in a manner similar to that suggested by Cairns (1975) i.e. through a process of selective segregation of their DNA strands, or that they have an extremely short S phase duration as postulated earlier (Potten et al. 1982). The present data have been extensively mathematically modelled in an accompanying paper. The model which best fits all the data is an hierarchical scheme with three cell divisions in the transit population but some branches of the lineage may be prematurely terminated by the early production of post-mitotic cells.(ABSTRACT TRUNCATED AT 250 WORDS)     

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.