Monotone dynamics of two cells dynamically coupled by a voltage-dependent gap junction

We introduce and analyse a simple model for two non-excitable cells that are dynamically coupled by a gap junction, a plaque of aqueous channels that electrically couple the cells. The gap junction channels have a low and high conductance state, and the transition rates between these states are voltage-dependent. We show that the number and stability of steady states of the system has a simple relationship with the determinant of the Jacobian matrix. For the case that channel opening rates decrease with increasing trans-junctional voltage, and closing rates increase with increasing trans-junctional voltage, we show that the system is monotone, with tridiagonal Jacobian matrix, and hence every initial condition evolves to a steady state, but that there may be multiple steady states.     

Functional expression and biophysical properties of polymorphic variants of the human gap junction protein connexin37

Connexin37 (Cx37) forms gap junction channels between endothelial cells, and two polymorphic Cx37 variants (Cx37-S319 and Cx37-P319) have been identified with a possible link to atherosclerosis. We studied the gap junction channel properties of these hCx37 polymorphs by expression in stably transfected communication-deficient cells (N2A and RIN). We also expressed a third, truncated variant (Cx37-fs254Delta293) and Cx37 constructs containing epitope tags added to their amino or carboxyl termini. All Cx37 constructs were produced by the transfected cells as demonstrated by RT-PCR and immunoblotting and trafficked to appositional surfaces between cells as demonstrated by immunofluorescence microscopy. Dual whole cell patch-clamping studies demonstrated that Cx37-P319, Cx37-S319, and Cx37-fs254Delta293 had large unitary conductances ( approximately 300 pS). However, addition of an amino terminal T7 tag (T7-Cx37-fs254Delta293) produced a single channel conductance of 120-145 pS with a 24-30 pS residual state. Moreover, the kinetics of the voltage-dependent decline in junctional current for T7-Cx37-fs254Delta293 were significantly slower than for the wild type, implying a destabilization of the transition state. These data suggest that the amino terminus of Cx37 plays a significant role in gating as well as conductance. The carboxyl terminal tail has lesser influence on unitary conductance and inactivation kinetics.     

Analysing grouping of nucleotides in DNA sequences using lumped processes constructed from Markov chains

The most commonly used models for analysing local dependencies in DNA sequences are (high-order) Markov chains. Incorporating knowledge relative to the possible grouping of the nucleotides enables to define dedicated sub-classes of Markov chains. The problem of formulating lumpability hypotheses for a Markov chain is therefore addressed. In the classical approach to lumpability, this problem can be formulated as the determination of an appropriate state space (smaller than the original state space) such that the lumped chain defined on this state space retains the Markov property. We propose a different perspective on lumpability where the state space is fixed and the partitioning of this state space is represented by a one-to-many probabilistic function within a two-level stochastic process. Three nested classes of lumped processes can be defined in this way as sub-classes of first-order Markov chains. These lumped processes enable parsimonious reparameterizations of Markov chains that help to reveal relevant partitions of the state space. Characterizations of the lumped processes on the original transition probability matrix are derived. Different model selection methods relying either on hypothesis testing or on penalized log-likelihood criteria are presented as well as extensions to lumped processes constructed from high-order Markov chains. The relevance of the proposed approach to lumpability is illustrated by the analysis of DNA sequences. In particular, the use of lumped processes enables to highlight differences between intronic sequences and gene untranslated region sequences.     

Markov chains: computing limit existence and approximations with DNA

We present two algorithms to perform computations over Markov chains. The first one determines whether the sequence of powers of the transition matrix of a Markov chain converges or not to a limit matrix. If it does converge, the second algorithm enables us to estimate this limit. The combination of these algorithms allows the computation of a limit using DNA computing. In this sense, we have encoded the states and the transition probabilities using strands of DNA for generating paths of the Markov chain.     

Entropy and convergence in dynamics and demography

Demographic dynamics is formally equivalent to the dynamics of a Markov chain, as is true of some nonlinear dynamical systems. Convergence to demographic equilibrium can be studied in terms of convergence in the Markov chain. Tuljapurkar (1982) showed that population entropy (Kolmogorov-Sinai entropy) provides information on the rate of this convergence. This paper begins by considering finite state Markov chains, providing elementary proofs of the relationship between convergence rate and entropy, and discusses in detail the uses and limitations of entropy as a convergence measure; these results also apply to Markovian dynamical systems. Next, new qualitative and quantitative arguments are used to discuss the demographic meaning of entropy. An exact relationship is established giving population entropy in terms of the eigenvalues of the Leslie matrix characteristic equation. Finally, the significance of imprimitive and periodic limits is discussed in relation to population entropy.     

Markov chains with infinite transition rates

We investigate finite state, continuous time Markov chains, whose transition rates have different orders of magnitude. Such a chain may be approximated by a simple one, which is obtained by grouping together states whose mutual communication is of the highest order. Using this reduced chain, the computation of both transition probability matrix and stationary distribution of the original chain is simplified significantly. The method is applicable in enzyme kenetics, and presumably in mathematical ecology or population genetics.     

Evaluation and Estimation of Various Markov Models with Applications to Membrane Channel Kinetics

Hidden Markov modelling is a powerful and efficient digital signal processing strategy for extracting the maximum likelihood model from a finite length sample of noisy data. Assuming the number of states in the model is known, then the state levels, transition probabilities, initial state distribution and the noise variance can be estimated. We investigate the applicability of this technique in membrane channel kinetics not only as a parameter estimator, but also as an aid to discriminating between various model types according to their statistical likelihood. We survey three representative classes of channel dynamics, namely: aggregated Markov models, semi-Markov models (with asymptotically convergent transition probabilities), and coupled Markov models; reformulating each within a discrete-time hidden Markov model framework. We then provide numerical evidence of the effectiveness of the procedure using simulated channel data and hence show that the correct model, as well as the model parameters, can be discerned. We also demonstrate that the model likelihood can be used to indicate the approximate number of states in the model.     

Functional assembly of gap junction conductance in lipid bilayers: demonstration that the major 27 kd protein forms the junctional channel

Gap junctions isolated from rat liver were incorporated into planar lipid bilayers. A channel activity that was directly dependent on voltage was recorded. Changes of pH and (Ca2+) had no direct effect on channel activity; however, they modulated the voltage-dependent gating of the gap junction channels differently. Single-channel fluctuations showed large scatter with peak amplitudes of 140 and 280 picoSiemmens in 0.1 M NaCl. The major protein of gap junctions (Mr of 27 kd) was also reconstituted into bilayers, giving channel properties similar to those of intact gap junctions. Polyclonal antibodies specific for this protein caused inhibition of the junctional conductance in bilayers. These data provide direct evidence that the 27 kd protein is the molecular species responsible for gap junction communication between cells.     

Past states of continuous-time Markov models for ecological communities

Discrete-time Markov chains are often used to model communities of sessile organisms. The community is described by a set of discrete states, which may represent species or groups of species. Transitions between states are modelled using a stochastic matrix. A recent study showed how the time-reversal of such a Markov chain can be used to estimate the distribution of time since the last occurrence of some state of interest (such as empty space) at a point, given the current state of the point. However, if the underlying process operates in continuous time but is observed at regular intervals, this distribution describes the time since the last possible observation of the state of interest, rather than the time since its last occurrence. We show how to obtain the distribution of time since the last occurrence of a state of interest for a continuous-time homogeneous Markov chain. The expected time since the last occurrence of an initial state can be interpreted as a measure of the successional rank of a state. We show how to distinguish between different ways in which a state can have high successional rank. We apply our results to a marine subtidal community.     

Connexin45 gap junction channels in rat cerebral vascular smooth muscle cells

Cells in blood vessel walls express connexin (Cx)43, Cx40, and Cx37. We recently characterized gap junction channels in rat basilar artery smooth muscle cells and found features attributable not only to these three connexins but also to an unidentified connexin, including strong voltage dependence and single channel conductance of 30-40 pS. Here, we report data consistent with identification of Cx45. Immunofluorescence using anti-human Cx45 and anti-mouse Cx45 antibodies revealed labeling between alpha-actin-positive cells, and RT-PCR of mRNA from arteries after endothelial destruction yielded amplicons exhibiting 90-98% identity with mouse Cx45 and human Cx45. Dual-perforated patch clamping was performed after exposure to oligopeptides that interfere with docking of Cx43, Cx40, or Cx45. Cell pairs pretreated with blocking peptides for Cx43 and Cx40 exhibited strongly voltage-dependent transjunctional conductances [voltage at which voltage-dependent conductance declines by one-half (V1/2) = +/-18.9 mV] and small single channel conductances (31 pS), consistent with the presence of Cx45, whereas cell pairs pretreated with blocking peptide for Cx45 exhibit weaker voltage-dependent conductances (V1/2 = +/-37.9 mV), consistent with block of Cx45. Our data suggest that Cx45 is transcribed, expressed, and forms functional gap junction channels in rat cerebral arterial smooth muscle.     

Biophysical properties of gap junctions between freshly dispersed pairs of mouse pancreatic beta cells.

Coupling between beta cells through gap junctions has been postulated as a principal mechanism of electrical synchronization of glucose-induced activity throughout the islet of Langerhans. We characterized junctional conductance between isolated pairs of mouse pancreatic beta cells by whole-cell recording with two independent patch-clamp circuits. Most pairs were coupled (67%, n = 155), although the mean junctional conductance (gj) (215 +/- 110 pS) was lower than reported in other tissues. Coupling could be recorded for long periods, up to 40 min. Voltage imposed across the junctional or nonjunctional membranes had no effect on gj. Up to several hours of treatment to increase intracellular cAMP levels did not affect gj. Electrically coupled pairs did not show transfer of the dye Lucifer yellow. Octanol (2 mM) reversibly decreased gj. Lower concentrations of octanol (0.5 mM) and heptanol (0.5 mM) than required to uncouple beta cells decreased voltage-dependent K+ and Ca2+ currents in nonjunctional membranes. Although gj recorded in these experiments would be expected to be provided by current flowing through only a few channels of the unitary conductance previously reported for other gap junctions, no unitary junctional currents were observed even during reversible suppression of gj by octanol. This result suggests either that the single channel conductance of gap junction channels between beta cells is smaller than in other tissues (less than 20 pS) or that the small mean conductance is due to transitions between open and closed states that are too rapid or too slow to be resolved.     

Superposition properties of interacting ion channels.

Quantitative analysis of patch clamp data is widely based on stochastic models of single-channel kinetics. Membrane patches often contain more than one active channel of a given type, and it is usually assumed that these behave independently in order to interpret the record and infer individual channel properties. However, recent studies suggest there are significant channel interactions in some systems. We examine a model of dependence in a system of two identical channels, each modeled by a continuous-time Markov chain in which specified transition rates are dependent on the conductance state of the other channel, changing instantaneously when the other channel opens or closes. Each channel then has, e.g., a closed time density that is conditional on the other channel being open or closed, these being identical under independence. We relate the two densities by a convolution function that embodies information about, and serves to quantify, dependence in the closed class. Distributions of observable (superposition) sojourn times are given in terms of these conditional densities. The behavior of two channel systems based on two- and three-state Markov models is examined by simulation. Optimized fitting of simulated data using reasonable parameters values and sample size indicates that both positive and negative cooperativity can be distinguished from independence.     

Gap junction coupling and calcium waves in the pancreatic islet

The pancreatic islet is a highly coupled, multicellular system that exhibits complex spatiotemporal electrical activity in response to elevated glucose levels. The emergent properties of islets, which differ from those arising in isolated islet cells, are believed to arise in part by gap junctional coupling, but the mechanisms through which this coupling occurs are poorly understood. To uncover these mechanisms, we have used both high-speed imaging and theoretical modeling of the electrical activity in pancreatic islets under a reduction in the gap junction mediated electrical coupling. Utilizing islets from a gap junction protein connexin 36 knockout mouse model together with chemical inhibitors, we can modulate the electrical coupling in the islet in a precise manner and quantify this modulation by electrophysiology measurements. We find that after a reduction in electrical coupling, calcium waves are slowed as well as disrupted, and the number of cells showing synchronous calcium oscillations is reduced. This behavior can be reproduced by computational modeling of a heterogeneous population of β-cells with heterogeneous levels of electrical coupling. The resulting quantitative agreement between the data and analytical models of islet connectivity, using only a single free parameter, reveals the mechanistic underpinnings of the multicellular behavior of the islet.     

Gap junction channels formed by coexpressed connexin40 and connexin43

Many cardiovascular cells coexpress multiple connexins (Cx), leading to the potential formation of mixed (heteromeric) gap junction hemichannels whose biophysical properties may differ from homomeric channels containing only one connexin type. We examined the potential interaction of connexin Cx43 and Cx40 in HeLa cells sequentially stably transfected with these two connexins. Immunoblots verified the production of comparable amounts of both connexins, cross-linking showed that both connexins formed oligomers, and immunofluorescence showed extensive colocalization. Moreover, Cx40 copurified with (His)(6)-tagged Cx43 by affinity chromatography of detergent-solubilized connexons, demonstrating the presence of both connexins in some hemichannels. The dual whole cell patch-clamp method was used to compare the gating properties of gap junctions in HeLa Cx43/Cx40 cells with homotypic (Cx40-Cx40 and Cx43-Cx43) and heterotypic (Cx40-Cx43) gap junctions. Many of the observed single channel conductances resembled those of homotypic or heterotypic channels. The steady-state junctional conductance (g(j,ss)) in coexpressing cell pairs showed a reduced sensitivity to the voltage between cells (V(j)) compared with homotypic gap junctions and/or an asymmetrical V(j) dependence reminiscent of heterotypic gap junctions. These gating properties could be fit using a combination of homotypic and heterotypic channel properties. Thus, whereas our biochemical evidence suggests that Cx40 and Cx43 form heteromeric connexons, we conclude that they are functionally insignificant with regard to voltage-dependent gating.     

Identification of a potential regulator of the gap junction protein pannexin1

Recent studies have revealed a second class of gap-junction-forming proteins in vertebrates. These genes are termed pannexins, and it has been suggested that they perform similar functions as connexins. Pannexin1 is expressed in diverse tissues including the central nervous system and seems to form gap junction channels in the Xenopus oocyte expression system. Since protein interacting partners have frequently been described for connexins, the most prominent family of gap junction forming proteins, we thus started to search for candidate genes of pannexin interacting partners. Kvbeta3, a protein belonging to the family of regulatory beta-subunits of the voltage-dependent potassium channels, was identified as a binding partner of pannexin1 in an E. coli two-hybrid system. This result was verified by confocal laser scanning microscopy using double transfected Neuro2A cells. The colocalization of both proteins at the plasma membrane is suggestive of functional interaction.     

Biochemical studies of mammalian oogenesis: Metabolic cooperativity between granulosa cells and growing mouse oocytes

Freeze fracture and lanthanum tracer experiments have shown that gap junctions exist throughout folliculogenesis between granulosa cells and growing mouse oocytes (Anderson and Albertini, J. Cell Biol.71, 680–686, 1976). The following lines of experimentation in the present study suggest that metabolic cooperativity exists between granulosa cells and their enclosed oocytes, i.e., gap junctions are functional, and that in most cases examined, greater than 85% of the metabolites present in follicle-enclosed oocytes were originally taken up by the granulosa cells and transferred to the oocyte via gap junctions: (1) When incubated with various radiolabeled compounds, follicle-enclosed oocytes contained more intracellular radioactivity than did oocytes with no attached granulosa cells (denuded oocytes); (2) for two radiolabeled ribonucleosides examined, the distribution of phosphorylated metabolites in follicle-enclosed oocytes resembled that of granulosa cells and differed significantly from that in denuded oocytes; (3) pulse-chase experiments with radiolabeled ribonucleosides revealed that during the chase period more radioactivity became associated with the follicle-enclosed oocyte; (4) treatments known to disrupt gap junctions in other cell types were effective in reversibly uncoupling metabolic cooperativity between granulosa cells and oocytes; and (5) a series of control experiments using (a) medium conditioned by granulosa cells and (b) cocultures of denuded oocytes and granulosa cells in which physical contact between the two cell types was not permitted demonstrated that contact between follicle cells and oocytes was necessary for observing metabolic cooperativity. Metabolic cooperativity was also found between follicle cells and oocytes in the two culture systems which support growth of mouse oocytes in vitro. The fact that oocytes do not grow well, if at all, in the absence of follicle cells and the large contribution of nutrients apparently furnished to the oocyte by the granulosa cells is consistent with the concept that gap junction mediated metabolic cooperativity between follicle cells and their enclosed oocytes is vital for mammalian oocyte growth.     

Differential scanning calorimetric study of the effect of sterol side chain length and structure on dipalmitoylphosphatidylcholine thermotropic phase behavior.

We have investigated the thermotropic phase behavior of dipalmitoylphosphatidylcholine (DPPC) bilayers containing a series of cholesterol analogues varying in the length and structure of their alkyl side chains. We find that upon the incorporation of up to approximately 25 mol % of any of the side chain analogues, the DPPC main transition endotherm consists of superimposed sharp and broad components representing the hydrocarbon chain melting of sterol-poor and sterol-rich phospholipid domains, respectively. Moreover, the behavior of these components is dependent on sterol side chain length. Specifically, for all sterol/DPPC mixtures, the sharp component enthalpy decreases linearly to zero by 25 mol % sterol while the cooperativity is only moderately reduced from that observed in the pure phospholipid. In addition, the sharp component transition temperature decreases for all sterol/DPPC mixtures; however, the magnitude of the decrease is dependent on the sterol side chain length. With respect to the broad component, the enthalpy initially increases to a maximum around 25 mol % sterol, thereafter decreasing toward zero by 50 mol % sterol with the exception of the sterols with very short alkyl side chains. Both the transition temperature and cooperativity of the broad component clearly exhibit alkyl chain length-dependent effects, with both the transition temperature and cooperativity decreasing more dramatically for sterols with progressively shorter side chains. We ascribe the chain length-dependent effects on transition temperature and cooperativity to the hydrophobic mismatch between the sterol and the host DPPC bilayer (see McMullen, T. P. W., Lewis, R. N. A. H., and McElhaney, R. N. (1993) Biochemistry 32:516-522).(ABSTRACT TRUNCATED AT 250 WORDS)     

生态网络中物质、能量流动的时间链分析

本文以Ｍａｒｋｏｖ过程理论为基础,利用转移矩阵对生态网络中物质、能量流动和随机行为进行的描述;将输入的物质、能量在生态网络中宏观分布随时间的变化定义为物质、能量流动的时间链,并给一般生态网络中物质、能量流动时间链的分析方法。两个稳态生态网络中物质、能量流动的时间链分析表明,时间链直观地反映了物质、能量在流动中流失或耗散的宏观行为,由于物质再循机普遍存在,使得物质流动的时间链与能量流动的时间链有着质     

Hidden Markov model analysis of intermediate gating steps associated with the pore gate of shaker potassium channels.

Cooperativity among the four subunits helps give rise to the remarkable voltage sensitivity of Shaker potassium channels, whose open probability changes tenfold for a 5-mV change in membrane potential. The cooperativity in these channels is thought to arise from a concerted structural transition as the final step in opening the channel. Recordings of single-channel ionic currents from certain other channel types, as well as our previous recordings from T442S mutant Shaker channels, however, display intermediate conductance levels in addition to the fully open and closed states. These sublevels might represent stepwise, rather than concerted, transitions in the final steps of channel activation. Here, we report a similar fine structure in the closing transitions of Shaker channels lacking the mutation. Describing the deactivation time course with hidden Markov models, we find that two subconductance levels are rapidly traversed during most closing transitions of chimeric, high conductance Shaker channels. The lifetimes of these levels are voltage-dependent, with maximal values of 52 and 22 micros at -100 mV, and the voltage dependences of transitions among these states suggest that they arise from equivalent conformational changes occurring in individual subunits. At least one subconductance level is found to be traversed in normal conductance Shaker channels. We speculate that voltage-dependent conformational changes in the subunits give rise to changes in a "pore gate" associated with the selectivity filter region of the channel, producing the subconductance states. As a control for the hidden Markov analysis, we applied the same procedures to recordings of the recovery from N-type inactivation in Shaker channels. These transitions are found to be instantaneous in comparison.     

Covariate Adjustment of Event Histories Estimated from Markov Chains: The Additive Approach

Markov chain models are frequently used for studying event histories that include transitions between several states. An empirical transition matrix for nonhomogeneous Markov chains has previously been developed, including a detailed statistical theory based on counting processes and martingales. In this article, we show how to estimate transition probabilities dependent on covariates. This technique may, e.g., be used for making estimates of individual prognosis in epidemiological or clinical studies. The covariates are included through nonparametric additive models on the transition intensities of the Markov chain. The additive model allows for estimation of covariate-dependent transition intensities, and again a detailed theory exists based on counting processes. The martingale setting now allows for a very natural combination of the empirical transition matrix and the additive model, resulting in estimates that can be expressed as stochastic integrals, and hence their properties are easily evaluated. Two medical examples will be given. In the first example, we study how the lung cancer mortality of uranium miners depends on smoking and radon exposure. In the second example, we study how the probability of being in response depends on patient group and prophylactic treatment for leukemia patients who have had a bone marrow transplantation. A program in R and S-PLUS that can carry out the analyses described here has been developed and is freely available on the Internet.