The Estimation of Rapid Rate Constants from Current-Amplitude Frequency Distributions of Single-Channel Recordings

A method is described for estimating rapid rate constants from the distributions of current amplitude observed in single-channel electrical recordings. It has the advantages over previous, similar approaches that it can accommodate both multistate kinetic models and adjustable filtering of the data using an 8-pole Bessel filter. The method is conceptually straightforward: the observed distributions of current amplitude are compared with theoretical distributions derived by combining several simplifying assumptions about the underlying stochastic process with a model of the filter and electrical noise. Parameters are estimated by approximate maximum likelihood. The method was used successfully to estimate rate constants for both a simple two-state kinetic model (the transitions between open and closed states during the rapid gating of an outward-rectifying K⁺-selective channel in the plasma membrane of Acetabularia) and a complex multistate kinetic model (the blockade of the maxi cation channel in the plasma membrane of rye roots by verapamil). For the two-state model, parameters were estimated well, provided that they were not too fast or too slow in relation to the sampling rate. In the three-state model the precision of estimates depended in a complex way on the values of all rate parameters in the model. Received: 4 October 1996/Revised: 2 September 1997     

Coding characters from different life stages for phylogenetic reconstruction: a case study on dragonfly adults and larvae,including a description of the larval head anatomy of Epiophlebia superstes (Odonata: Epiophlebiidae)

The exclusive use of characters coding for specific life stages may bias tree reconstruction. If characters from several life stages are coded, the type of coding becomes important. Here, we simulate the influence on tree reconstruction of morphological characters of Odonata larvae incorporated into a data matrix based on the adult body under different coding schemes. For testing purposes, our analysis is focused on a well‐supported hypothesis: the relationships of the suborders Zygoptera, ‘Anisozygoptera’, and Anisoptera. We studied the cephalic morphology of Epiophlebia, a key taxon among Odonata, and compared it with representatives of Zygoptera and Anisoptera in order to complement the data matrix. Odonate larvae are characterized by a peculiar morphology, such as the specific head form, mouthpart configuration, ridge configuration, cephalic musculature, and leg and gill morphology. Four coding strategies were used to incorporate the larval data: artificial coding (AC), treating larvae as independent terminal taxa; non‐multistate coding (NMC), preferring the adult life stage; multistate coding (MC); and coding larval and adult characters separately (SC) within the same taxon. As expected, larvae are ‘monophyletic’ in the AC strategy, but with anisopteran and zygopteran larvae as sister groups. Excluding larvae in the NMC approach leads to strong support for both monophyletic Odonata and Epiprocta, whereas MC erodes phylogenetic signal completely. This is an obvious result of the larval morphology leading to many multistate characters. SC results in the strongest support for Odonata, and Epiprocta receives the same support as with NMC. Our results show the deleterious effects of larval morphology on tree reconstruction when multistate coding is applied. Coding larval characters separately is still the best approach in a phylogenetic framework. © 2015 The Linnean Society of London     

Water alignment, dipolar interactions, and multiple proton occupancy during water-wire proton transport

A discrete multistate kinetic model for water-wire proton transport is constructed and analyzed using Monte Carlo simulations. In the model, each water molecule can be in one of three states: oxygen lone-pairs pointing leftward, pointing rightward, or protonated (H(3)O(+)). Specific rules for transitions among these states are defined as protons hop across successive water oxygens. Our model also includes water-channel interactions that preferentially align the water dipoles, nearest-neighbor dipolar coupling interactions, and Coulombic repulsion. Extensive Monte Carlo simulations were performed and the observed qualitative physical behaviors discussed. We find the parameters that allow the model to exhibit superlinear and sublinear current-voltage relationships, and show why alignment fields, whether generated by interactions with the pore interior or by membrane potentials, always decrease the proton current. The simulations also reveal a "lubrication" mechanism that suppresses water dipole interactions when the channel is multiply occupied by protons. This effect can account for an observed sublinear-to-superlinear transition in the current-voltage relationship.     

Subthreshold Voltage Noise Due to Channel Fluctuations in Active Neuronal Membranes

Voltage-gated ion channels in neuronal membranes fluctuate randomly between different conformational states due to thermal agitation. Fluctuations between conducting and nonconducting states give rise to noisy membrane currents and subthreshold voltage fluctuations and may contribute to variability in spike timing. Here we study subthreshold voltage fluctuations due to active voltage-gated Na⁺ and K⁺ channels as predicted by two commonly used kinetic schemes: the Mainen et al. (1995) (MJHS) kinetic scheme, which has been used to model dendritic channels in cortical neurons, and the classical Hodgkin-Huxley (1952) (HH) kinetic scheme for the squid giant axon. We compute the magnitudes, amplitude distributions, and power spectral densities of the voltage noise in isopotential membrane patches predicted by these kinetic schemes. For both schemes, noise magnitudes increase rapidly with depolarization from rest. Noise is larger for smaller patch areas but is smaller for increased model temperatures. We contrast the results from Monte Carlo simulations of the stochastic nonlinear kinetic schemes with analytical, closed-form expressions derived using passive and quasi-active linear approximations to the kinetic schemes. For all subthreshold voltage ranges, the quasi-active linearized approximation is accurate within 8% and may thus be used in large-scale simulations of realistic neuronal geometries.     

Fluxes,first passage times,and the reduction of Hill diagrams

The steady-state flux resulting from the coupling of two multistate systems is considered. The dynamics of these systems are described (a) as diffusion along a continuous one-dimensional free-energy profile specified by a conformational coordinate or (b) in terms of transitions between a discrete but arbitrary number of substates. If these multistate systems are connected in a simple way, it is shown that the steady-state flux can be obtained analytically. For both the continuous and discrete cases, the exact flux is shown to be identical to that calculated from a simple kinetic scheme involving only four states, if the effective rate constants of this reduced scheme are appropriately defined in terms of the mean first passage times for moving between various points along the multistate cycles. These results clarify and quantify the manner in which the internal conformational dynamics of two multistate systems influences the steady-state flux.     

Markovian models of low and high activity levels of cardiac ryanodine receptors

The modal gating behavior of single sheep cardiac sarcoplasmic reticulum (SR) Ca2+-release/ryanodine receptor (RyR) channels was assessed. We find that the gating of RyR channels spontaneously shifts between high (H) and low (L) levels of activity and inactive periods where no channel openings are detected (I). Moreover, we find that there is evidence for multiple gating modes within H activity, which we term H1 and H2 mode. Our results demonstrate that the underlying mechanisms regulating gating are similar in native and purified channels. Dwell-time distributions of L activity were best fitted by three open and five closed significant exponential components whereas dwell-time distributions of H1 activity were best fitted by two to three open and four closed significant exponential components. Increases in cytosolic [Ca2+] cause an increase in open probability (Po) within L activity and an increase in the probability of occurrence of H activity. Open lifetime distributions within L activity were Ca2+ independent whereas open lifetime distributions within H activity were Ca2+ dependent. This study is the first attempt to estimate RyR single-channel kinetic parameters from sequences of idealized dwell-times and to develop kinetic models of RyR gating using the criterion of maximum likelihood. We propose distinct kinetic schemes for L, H1, and H2 activity that describe the major features of sheep cardiac RyR channel gating at these levels of activity.     

Trends in morphological evolution in homobasidiomycetes inferred using maximum likelihood: a comparison of binary and multistate approaches

The homobasidiomycetes is a diverse group of macrofungi that includes mushrooms, puffballs, coral fungi, and other forms. This study used maximum likelihood methods to determine if there are general trends (evolutionary tendencies) in the evolution of fruiting body forms in homobasidiomycetes, and to estimate the ancestral forms of the homobasidiomycetes and euagarics clade. Character evolution was modeled using a published 481-species phylogeny under two character-coding regimes: additive binary coding, using DISCRETE, and multistate (five-state) coding, using MULTISTATE. Inferences regarding trends in character evolution made under binary coding were often in conflict with those made under multistate coding, suggesting that the additive binary coding approach cannot serve as a surrogate for multistate methods. MULTISTATE was used to develop a"minimal"model of fruiting body evolution, in which the 20 parameters that specify rates of transformations among character states were grouped into the fewest possible rate categories. The minimal model required only four rate categories, one of which is approaching zero, and suggests the following conclusions regarding trends in evolution of homobasidiomycete fruiting bodies: (1) there is an active trend favoring the evolution of pileate-stipitate forms (those with a cap and stalk); (2) the hypothesis that the evolution of gasteroid forms (those with internal spore production, such as puffballs) is irreversible cannot be rejected; and (3) crustlike resupinate forms are not a particularly labile morphology. The latter finding contradicts the conclusions of a previous study that used binary character coding. Ancestral state reconstructions under binary coding suggest that the ancestor of the homobasidiomycetes was resupinate and the ancestor of the euagarics clade was pileate-stipitate, but ancestral state reconstructions under multistate coding did not resolve the ancestral form of either node. The results of this study illustrate the potential sensitivity of comparative analyses to character state definitions.     

A bayesian random-effects markov model for tumor progression in women with a family history of breast cancer

SUMMARY: It makes intuitive sense to model the natural history of breast cancer, tumor progression from preclinical screen-detectable phase (PCDP) to clinical disease, as a multistate process, but the multilevel structure of the available data, which generally comes from cluster (family)-based service screening programs, makes the required parameter estimation intractable because there is a correlation between screening rounds in the same individual, and between subjects within clusters (families). There is also residual heterogeneity after adjusting for covariates. We therefore proposed a Bayesian hierarchical multistate Markov model with fixed and random effects and applied it to data from a high-risk group (women with a family history of breast cancer) participating in a family-based screening program for breast cancer. A total of 4867 women attended (representing 4464 families) and by the end of 2002, a total of 130 breast cancer cases were identified. Parameter estimation was carried out using Markov chain Monte Carlo (MCMC) simulation and the Bayesian software package WinBUGS. Models with and without random effects were considered. Our preferred model included a random-effect term for the transition rate from preclinical to clinical disease (sigma(2)(2f)), which was estimated to be 0.50 (95% credible interval = 0.22-1.49). Failure to account for this random effect was shown to lead to bias. The incorporation of covariates into multistate models with random effect not only reduced residual heterogeneity but also improved the convergence of stationary distribution. Our proposed Bayesian hierarchical multistate model is a valuable tool for estimating the rate of transitions between disease states in the natural history of breast cancer (and possibly other conditions). Unlike existing models, it can cope with the correlation structure of multilevel screening data, covariates, and residual (unexplained) variation.     

How the folding rates of two‐ and multistate proteins depend on the amino acid properties

Proteins fold by either two‐state or multistate kinetic mechanism. We observe that amino acids play different roles in different mechanism. Many residues that are easy to form regular secondary structures (α helices, β sheets and turns) can promote the two‐state folding reactions of small proteins. Most of hydrophilic residues can speed up the multistate folding reactions of large proteins. Folding rates of large proteins are equally responsive to the flexibility of partial amino acids. Other properties of amino acids (including volume, polarity, accessible surface, exposure degree, isoelectric point, and phase transfer energy) have contributed little to folding kinetics of the proteins. Cysteine is a special residue, it triggers two‐state folding reaction and but inhibits multistate folding reaction. These findings not only provide a new insight into protein structure prediction, but also could be used to direct the point mutations that can change folding rate. Proteins 2014; 82:2375–2382. © 2014 Wiley Periodicals, Inc.     

Analytical description of the activation of multi-state receptors by continuous neurotransmitter signals at brain synapses.

Chemical synaptic transmission is a fundamental component of interneuronal communications in the central nervous system (CNS). Discharge of a presynaptic vesicle containing a few thousand molecules (a quantum) of neurotransmitter into the synaptic cleft generates a transmitter concentration signal that drives postsynaptic ion-channel receptors. These receptors exhibit multiple states, with state transition kinetics dependent on neurotransmitter concentration. Here, a novel and simple analytical approach for describing gating of multi-state receptors by signals with complex continuous time courses is used to describe the generation of glutamate-mediated quantal postsynaptic responses at brain synapses. The neurotransmitter signal, experienced by multi-state N-methyl-D-aspartate (NMDA)- and L-alpha-amino-3-hydroxy-5-methyl-4-isoxazolepropionate (AMPA)-type glutamate receptors at specific points in a synaptic cleft, is approximated by a series of step functions of different intensity and duration and used to drive a Markovian, multi-state kinetic scheme that describes receptor gating. Occupancy vectors at any point in time can be computed interatively from the occupancy vectors at the times of steps in transmitter concentration. Multi-state kinetic schemes for both the low-affinity AMPA subtype of glutamate receptor and for the high-affinity NMDA subtype are considered, and expected NMDA and AMPA components of synaptic currents are calculated. The amplitude of quantal responses mediated by postsynaptic receptor clusters having specific spatial distributions relative to foci of quantal neurotransmitter release is then calculated and related to the displacement between the center of the postsynaptic receptor cluster and the focus of synaptic vesicle discharge. Using this approach we show that the spatial relation between the focus of release and the center of the postsynaptic receptor cluster affects synaptic efficacy. We also show how variation in this relation contributes to variation in synaptic current amplitudes.     

Multistate Models Reveal Long-Term Trends of Northern Spotted Owls in the Absence of a Novel Competitor

Quantifying spatial and temporal variability in population trends is a critical aspect of successful management of imperiled species. We evaluated territory occupancy dynamics of northern spotted owls (Strix occidentalis caurina), California, USA, 1990–2014. The study area possessed two unique aspects. First, timber management has occurred for over 100 years, resulting in dramatically different forest successional and structural conditions compared to other areas. Second, the barred owl (Strix varia), an exotic congener known to exert significant negative effects on spotted owls, has not colonized the study area. We used a Bayesian dynamic multistate model to evaluate if territory occupancy of reproductive spotted owls has declined as in other study areas. The state-space approach for dynamic multistate modeling imputes the number of territories for each nesting state and allows for the estimation of longer-term trends in occupied or reproductive territories from longitudinal studies. The multistate approach accounts for different detection probabilities by nesting state (to account for either inherent differences in detection or for the use of different survey methods for different occupancy states) and reduces bias in state assignment. Estimated linear trends in the number of reproductive territories suggested an average loss of approximately one half territory per year (-0.55, 90% CRI: -0.76, -0.33), in one management block and a loss of 0.15 per year (-0.15, 90% CRI: -0.24, -0.07), in another management block during the 25 year observation period. Estimated trends in the third management block were also negative, but substantial uncertainty existed in the estimate (-0.09, 90% CRI: -0.35, 0.17). Our results indicate that the number of territories occupied by northern spotted owl pairs remained relatively constant over a 25 year period (-0.07, 90% CRI: -0.20, 0.05; -0.01, 90% CRI: -0.19, 0.16; -0.16, 90% CRI: -0.40, 0.06). However, we cannot exclude small-to-moderate declines or increases in paired territory numbers due to uncertainty in our estimates. Collectively, we conclude spotted owl pair populations on this landscape managed for commercial timber production appear to be more stable and do not show sharp year-over-year declines seen in both managed and unmanaged landscapes with substantial barred owl colonization and persistence. Continued monitoring of reproductive territories can determine whether recent declines continue or whether trends reverse as they have on four previous occasions. Experimental investigations to evaluate changes to spotted owl occupancy dynamics when barred owl populations are reduced or removed entirely can confirm the generality of this conclusion.     

Inferential biases linked to unobservable states in complex occupancy models

Modeling of species distributions has undergone a shift from relying on equilibrium assumptions to recognizing transient system dynamics explicitly. This shift has necessitated more complex modeling techniques, but the performance of these dynamic models has not yet been assessed for systems where unobservable states exist. Our work is motivated by the impacts of the emerging infectious disease chytridiomycosis, a disease of amphibians that is associated with declines of many species worldwide. Using this host‐pathogen system as a general example, we first illustrate how misleading inferences can result from failing to incorporate pathogen dynamics into the modeling process, especially when the pathogen is difficult or impossible to survey in the absence of a host species. We found that traditional modeling techniques can underestimate the effect of a pathogen on host species occurrence and dynamics when the pathogen can only be detected in the host, and pathogen information is treated as a covariate. We propose a dynamic multistate modeling approach that is flexible enough to account for the detection structures that may be present in complex multistate systems, especially when the sampling design is limited by a species’ natural history or sampling technology. When multistate occupancy models are used and an unobservable state is present, parameter estimation can be influenced by model complexity, data sparseness, and the underlying dynamics of the system. We show that, even with large sample sizes, many models incorporating seasonal variation in vital rates may not generate reasonable estimates, indicating parameter redundancy. We found that certain types of missing data can greatly hinder inference, and we make study design recommendations to avoid these issues. Additionally, we advocate the use of time‐varying covariates to explain temporal trends in the data, and the development of sampling techniques that match the biology of the system to eliminate unobservable states when possible.     

Analysis of the kinetic isotope effects on initial rates in transient kinetics

A method was described recently for circumventing the difficulties in determining intrinsic kinetic isotope effects from eigenvalues obtained in transient kinetic experiments (Maniscalco, Tally, and Fisher (2004) Arch. Biochem. Biophys. 425, 165-172). The method, based on the isotope effects on initial rates of formation of intermediates, was validated by numerical simulation for only a few linear reaction sequences. A general analytical proof of the validity of the method is given in this work. The mathematical approach, using rate laws and L'H?pital's rule, allows more complex reaction schemes to be analyzed. Several are analyzed in this article, illustrating the broad applicability and possible complications in this approach for determining intrinsic isotope effects. Some possible applications are noted, with particular attention being paid to nonlinear reaction schemes, the effect of measuring signals rather than concentration, and the ability to distinguish stepwise from concerted reactions.     

Direct measurement of equilibrium constants for high-affinity hemoglobins

The biological functions of heme proteins are linked to their rate and affinity constants for ligand binding. Kinetic experiments are commonly used to measure equilibrium constants for traditional hemoglobins comprised of pentacoordinate ligand binding sites and simple bimolecular reaction schemes. However, kinetic methods do not always yield reliable equilibrium constants with more complex hemoglobins for which reaction mechanisms are not clearly understood. Furthermore, even where reaction mechanisms are clearly understood, it is very difficult to directly measure equilibrium constants for oxygen and carbon monoxide binding to high-affinity (K(D) < 1 micro M) hemoglobins. This work presents a method for direct measurement of equilibrium constants for high-affinity hemoglobins that utilizes a competition for ligands between the "target" protein and an array of "scavenger" hemoglobins with known affinities. This method is described for oxygen and carbon monoxide binding to two hexacoordinate hemoglobins: rice nonsymbiotic hemoglobin and Synechocystis hemoglobin. Our results demonstrate that although these proteins have different mechanisms for ligand binding, their affinities for oxygen and carbon monoxide are similar. Their large affinity constants for oxygen, 285 and approximately 100 micro M(-1) respectively, indicate that they are not capable of facilitating oxygen transport.     

Monotonic and non-monotonic single channel open time distributions with two sequential open states

Some conditions under which kinetic schemes including two sequential open states of identical conductance will display a non-monotonic (i.e. with a deficit of short open times and a maximum at t>0) distribution of single channel open times are described theoretically. Neither a closed cyclic scheme nor exclusively irreversible transitions between states are required for non-monotonic distributions. A required condition for the schemes considered here is that all openings are to a state from which closing is not possible. It is the presence of a precursor process to channel closing that produces the non-monotonic distribution. Following each channel opening some time is required for a transition into the second open state from which all closings proceed. Simple schemes of this sort cannot provide the basis of any experimental reports of non-monotonic distributions.     

Attributable risk estimation for adjusted disability multistate models: Application to nosocomial infections

Attributable risk has become an important concept in clinical epidemiology. In this paper, we suggest to estimate the attributable risk of nosocomial infections using a multistate approach. Recently, a multistate model (called progressive disability model in the literature) has been developed in order to take into consideration both the time‐dependency of the risk factor (e.g., nosocomial infections) and the presence of competing risks (e.g., death and discharge) at each time point. However, this approach does not take into account the possible heterogeneity of the study population. In this paper, we investigate an extension of this model and suggest an adjusted disability multistate model including covariates in each transition. This new multistate model has led us to define the concepts of overall and profiled attributable risk. We use a classical semiparametric approach to estimate the model and the new attributable risk. A simulation study is investigated and we show, in particular, that neglecting the presence of covariates when estimating the model can lead to an important bias. The methodology developed in this paper is applied to data on ventilator‐associated pneumonia in 12 French intensive care units.     

Integrated allosteric model of voltage gating of HCN channels

Hyperpolarization-activated (pacemaker) channels are dually gated by negative voltage and intracellular cAMP. Kinetics of native cardiac f-channels are not compatible with HH gating, and require closed/open multistate models. We verified that members of the HCN channel family (mHCN1, hHCN2, hHCN4) also have properties not complying with HH gating, such as sigmoidal activation and deactivation, activation deviating from fixed power of an exponential, removal of activation "delay" by preconditioning hyperpolarization. Previous work on native channels has indicated that the shifting action of cAMP on the open probability (Po) curve can be accounted for by an allosteric model, whereby cAMP binds more favorably to open than closed channels. We therefore asked whether not only cAMP-dependent, but also voltage-dependent gating of hyperpolarization-activated channels could be explained by an allosteric model. We hypothesized that HCN channels are tetramers and that each subunit comprises a voltage sensor moving between "reluctant" and "willing" states, whereas voltage sensors are independently gated by voltage, channel closed/open transitions occur allosterically. These hypotheses led to a multistate scheme comprising five open and five closed channel states. We estimated model rate constants by fitting first activation delay curves and single exponential time constant curves, and then individual activation/deactivation traces. By simply using different sets of rate constants, the model accounts for qualitative and quantitative aspects of voltage gating of all three HCN isoforms investigated, and allows an interpretation of the different kinetic properties of different isoforms. For example, faster kinetics of HCN1 relative to HCN2/HCN4 are attributable to higher HCN1 voltage sensors' rates and looser voltage-independent interactions between subunits in closed/open transitions. It also accounts for experimental evidence that reduction of sensors' positive charge leads to negative voltage shifts of Po curve, with little change of curve slope. HCN voltage gating thus involves two processes: voltage sensor gating and allosteric opening/closing.