Optimal-Sensitivity Analysis of Ion Channel Gating Kinetics

We propose a new approach to analysis of kinetic models for ion channel gating, based on application of fluctuating voltages through a voltage clamp, in addition to conventional techniques. We show that the channel kinetics can be probed in a much more sensitive way, leading to more efficient model selection and more reliable estimates of model parameters. We use wavelet transform as an analytic tool for fluctuating currents and parametric dispersion plots as a measure of model compatibility with experimental data.This revised version was published online in August 2005 with a corrected cover date.     

Nonequilibrium response spectroscopy of voltage-sensitive ion channel gating.

We describe a new electrophysiological technique called nonequilibrium response spectroscopy, which involves application of rapidly fluctuating (as high as 14 kHz) large-amplitude voltage clamp waveforms to ion channels. As a consequence of the irreversible (in the sense of Carnot) exchange of energy between the fluctuating field and the channel protein, the gating response is exquisitely sensitive to features of the kinetics that are difficult or impossible to adequately resolve by means of traditional stepped potential protocols. Here we focus on the application of dichotomous (telegraph) noise voltage fluctuations, a broadband Markovian colored noise that fluctuates between two values. Because Markov kinetic models of channel gating can be embedded within higher-dimensional Markov models that take into account the effects of the voltage fluctuations, many features of the response of the channels can be calculated algebraically. This makes dichotomous noise and its generalizations uniquely suitable for model selection and kinetic analysis. Although we describe its application to macroscopic ionic current measurements, the nonequilibrium response method can also be applied to gating and single channel current recording techniques. We show how data from the human cardiac isoform (hH1a) of the Na+ channel expressed in mammalian cells can be acquired and analyzed, and how these data reveal hidden aspects of the molecular kinetics that are not revealed by conventional methods.     

The use of automated parameter searches to improve ion channel kinetics for neural modeling

The voltage and time dependence of ion channels can be regulated, notably by phosphorylation, interaction with phospholipids, and binding to auxiliary subunits. Many parameter variation studies have set conductance densities free while leaving kinetic channel properties fixed as the experimental constraints on the latter are usually better than on the former. Because individual cells can tightly regulate their ion channel properties, we suggest that kinetic parameters may be profitably set free during model optimization in order to both improve matches to data and refine kinetic parameters. To this end, we analyzed the parameter optimization of reduced models of three electrophysiologically characterized and morphologically reconstructed globus pallidus neurons. We performed two automated searches with different types of free parameters. First, conductance density parameters were set free. Even the best resulting models exhibited unavoidable problems which were due to limitations in our channel kinetics. We next set channel kinetics free for the optimized density matches and obtained significantly improved model performance. Some kinetic parameters consistently shifted to similar new values in multiple runs across three models, suggesting the possibility for tailored improvements to channel models. These results suggest that optimized channel kinetics can improve model matches to experimental voltage traces, particularly for channels characterized under different experimental conditions than recorded data to be matched by a model. The resulting shifts in channel kinetics from the original template provide valuable guidance for future experimental efforts to determine the detailed kinetics of channel isoforms and possible modulated states in particular types of neurons.     

Global parameter optimization for cardiac potassium channel gating models.

Quantitative ion channel model evaluation requires the estimation of voltage dependent rate constants. We have tested whether a unique set of rate constants can be reliably extracted from nonstationary macroscopic voltage clamp potassium current data. For many models, the rate constants derived independently at different membrane potentials are not unique. Therefore, our approach has been to use the exponential voltage dependence predicted from reaction rate theory (Stevens, C. F. 1978. Biophys. J. 22:295-306; Eyring, H., S. H. Lin, and S. M. Lin. 1980. Basic Chemical Kinetics. Wiley and Sons, New York) to couple the rate constants derived at different membrane potentials. This constrained the solution set of rate constants to only those that also obeyed this additional set of equations, which was sufficient to obtain a unique solution. We have tested this approach with data obtained from macroscopic delayed rectifier potassium channel currents in voltage-clamped guinea pig ventricular myocyte membranes. This potassium channel has relatively simple kinetics without an inactivation process and provided a convenient system to determine a globally optimized set of voltage-dependent rate constants for a Markov kinetic model. The ability of the fitting algorithm to extract rate constants from the macroscopic current data was tested using "data" synthesized from known rate constants. The simulated data sets were analyzed with the global fitting procedure and the fitted rate constants were compared with the rate constants used to generate the data. Monte Carlo methods were used to examine the accuracy of the estimated kinetic parameters. This global fitting approach provided a useful and convenient method for reliably extracting Markov rate constants from macroscopic voltage clamp data over a broad range of membrane potentials. The limitations of the method and the dependence on initial guesses are described.     

Metabolic control analysis under uncertainty: framework development and case studies

Information about the enzyme kinetics in a metabolic network will enable understanding of the function of the network and quantitative prediction of the network responses to genetic and environmental perturbations. Despite recent advances in experimental techniques, such information is limited and existing experimental data show extensive variation and they are based on in vitro experiments. In this article, we present a computational framework based on the well-established (log)linear formalism of metabolic control analysis. The framework employs a Monte Carlo sampling procedure to simulate the uncertainty in the kinetic data and applies statistical tools for the identification of the rate-limiting steps in metabolic networks. We applied the proposed framework to a branched biosynthetic pathway and the yeast glycolysis pathway. Analysis of the results allowed us to interpret and predict the responses of metabolic networks to genetic and environmental changes, and to gain insights on how uncertainty in the kinetic mechanisms and kinetic parameters propagate into the uncertainty in predicting network responses. Some of the practical applications of the proposed approach include the identification of drug targets for metabolic diseases and the guidance for design strategies in metabolic engineering for the purposeful manipulation of the metabolism of industrial organisms.     

Isothermal titration calorimetry — A new method for the quantification of microbial degradation of trace pollutants

The environmental fate and, in particular, biodegradation rates of hydrophobic organic compounds (HOC) are of high interest due to the ubiquity, persistence, and potential health effects of these compounds. HOC tend to interact with bioreactor materials and sampling devices and are frequently volatile, so that conventionally derived degradation parameters are often biased. We report on the development and validation of a novel calorimetric approach that serves to gain real time information on the kinetics and the physiology of HOC bioconversion in aqueous systems while overcoming weaknesses of conventional biodegradation experiments. Soil bacteria Mycobacterium frederiksbergense LB501T, Rhodococcus erythropolis K2-3 and Pseudomonas putida G7 were exposed to pulsed titrations of dissolved anthracene, 4-(2,4-dichlorophenoxy)butyric acid or naphthalene, respectively, and the thermal responses were monitored. The combinations of strains and pollutants were selected as examples for complete and partial biodegradation and complete degradation with storage product formation, respectively. Heat production signals were interpreted thermodynamically and in terms of Michaelis-Menten kinetics. The half-saturation constant k_D and the degradation rate r_D^Max were derived. Comparison with conventional methods shows the suitability to extract kinetic degradation parameters of organic trace pollutants from simple ITC experiments, while thermodynamic interpretation provided further information about the metabolic fate of HOC compounds.     

Multiple-state model of ionic channel in reproducing the time course of electrophysiological events.

BACKGROUND: The predictions of the Hodgkin-Huxley model do not accurately fit all the measurements of voltage-clamp currents, gating charge and single-channel currents. There are many quantitative differences between the predicted and measured characteristics of the sodium and potassium channels. For example, the two-state gate model has exponential onset kinetics, whereas the sodium and potassium conductances show S-shaped activation and the sodium conductance shows an exponential inactivation. In this paper we shall examine a more general channel model that can more faithfully represent the measured properties of ionic channels in the membrane of the excitable cell. METHODS: The model is based on the generalisation of the notion of a channel with a discrete set of states. Each state has state attributes such as the state conductance, state ionic current and state gating charge. These variables can have quite different waveforms in time, in contrast with a two-state gate channel model, in which all have the same waveforms. RESULTS: The kinetics of all variables are equivalent: gating and ionic currents give equivalent information about channel kinetics; both the equilibrium values of the current and the time constants are functions of membrane potential. The results are in almost perfect concordance with the experimental data regarding the characteristics of nerve impulse. CONCLUSIONS: The expected values of the gating charge and the ionic conductance are weighted sums of the state occupancy probabilities, but the weights differ: for the expected value of the gating charge the weights are the state gating charges and for the expected value of the ionic conductance the weights are the state conductances. Since these weights are different, the expected values of the gating charge and the ionic conductance will differ.     

Periodic forcing of a K+ channel at various temperatures.

The random sequence of openings and closings of single ion channels and the channel conductances have been the object of intense study over the past two decades with a view toward illuminating the underlying kinetics of the channel protein molecules. Channels that are sensitive to voltage, such as many K(+)-selective channels, have been particularly useful, because the kinetic rates can be manipulated by changing the membrane voltage. Most such studies have been performed under stationary conditions and usually at a single temperature. Here we report the results of experiments with sinusoidal modulation of the membrane potential performed at several temperatures. Dwell time and cycle histograms, objects not normally associated with ion channel experiments, are herein reported. From the last, the transition probability densities for channel opening and closing events are obtained. A new and unusual phase anticipation is observed in the cycle histograms, and its temperature dependence is measured.     

Simulations of voltage clamping poorly space-clamped voltage-dependent conductances in a uniform cylindrical neurite

Significant error is made by using a point voltage clamp to measure active ionic current properties in poorly space-clamped cells. This can even occur when there are no obvious signs of poor spatial control. We evaluated this error for experiments that employ an isochronal I(V) approach to analyzing clamp currents. Simulated voltage clamp experiments were run on a model neuron having a uniform distribution of a single voltage-gated inactivating ionic current channel along an elongate, but electrotonically compact, process. Isochronal Boltzmann I(V) and kinetic parameter values obtained by fitting the Hodgkin-Huxley equations to the clamp currents were compared with the values originally set in the model. Good fits were obtained for both inward and outward currents for moderate channel densities. Most parameter errors increased with conductance density. The activation rate parameters were more sensitive to poor space clamp than the I(V) parameters. Large errors can occur despite normal-looking clamp curves.     

A model for conductance changes in the squid giant axon. II. Channel kinetics

The kinetics of the sodium and potassium channels in voltage clamped squid giant axon following a relaxation of the membrane subunits are examined and compared with the Hodgkin-Huxley equations. Mechanisms are suggested for the turn-off of the sodium conductance and a set of kinetic states are proposed for the potassium channel which are consistent with the experimental observations. Determination of the rate constants for relaxation of the surface subunits which triggers the subsequent changes within the independent channels provide information on the equilibrium constant and free energy for this process. The free energy is observed to approach zero as the depolarizing voltage of the clamp approaches $\text{E}{}_{\mathrm{<mtext>Na</mtext>}}$, the voltage for zero sodium current in voltage clamped axons. Analysis of the final rate constants in the kinetic sequence for potassium indicates a symmetry of the channel when it is in its steady-state configuration during clamp in the absence of external gradients.     

Actionable insights with less data: guiding early building design decisions with streamlined probabilistic life cycle assessment

Purpose

Two obstacles that impede wider use of life cycle assessment (LCA) are its time- and data-intensiveness and the credibility surrounding its results—challenges that grow with the complexity of the product being analyzed. To guide the critical early-design stages of a complicated product like a building, it is important to be able to rapidly estimate environmental impacts with limited information, quantify the resulting uncertainty, and identify critical parameters where more detail is needed.

Methods

The authors have developed the Building Attribute to Impact Algorithm (BAIA) to demonstrate the use of streamlined (not scope-limiting), probabilistic LCA for guiding the design of a building from early stages of the design process when many aspects of the design are unknown or undecided. Early-design uncertainty is accommodated through under-specification—characterizing the design using the available level of detail—and capturing the resulting variability in predicted impacts through Monte Carlo simulations. Probabilistic triage with sensitivity analyses identifies which uncertain attributes should be specified further to increase the precision of the results. The speed of the analyses allows for sequentially refining key attributes and re-running the analyses until the predicted impacts are precise enough to inform decision-making, such as choosing a preferable design alternative.

Results and discussion

Twelve design variants for a hypothetical single-family residential building are analyzed. As information is sequentially added to each variant, the significance of the difference in performance between each variant pair is calculated to determine when enough information has been added to resolve the designs (identify which design is preferable) with high confidence. At the sixth step in the analysis, all variant pairs whose mean impacts differ by at least 4% are resolvable with 90% confidence, even though only six attributes are specified and dozens of attributes remain under-specified. Furthermore, the comparative results for each variant pair are validated against a set of conventional LCA results, showing that BAIA identifies the correct preferable design among each resolvable pair at this step.

Conclusions

Iterative specification guided by probabilistic triage can help identify promising early-design alternatives even when details are only provided for key attributes. The analysis of hypothetical design variants demonstrates that BAIA is both efficient (arrives at statistically defensible conclusions from design variant comparisons based on few pieces of information) and effective (identifies the same preferable design variants as conventional LCAs).

     

Bimodal Kinetics of a Chloride Channel from Human Fibroblasts

Excised patches were used to study the kinetics of a Cl channel newly identified in cultured human fibroblasts (L132). The conductance of ca. 70 pS in 150 mm symmetrical Cl, and the marked outward rectification ascribe this channel to the ICOR family. Long single-channel recordings (>30 min) revealed that the channel spontaneously switches from a kinetic mode characterized by high voltage dependence (with activity increasing with depolarization; mode 1), into a second mode (mode 2) insensitive to voltage, and characterized by a high activity in the voltage range ±120 mV. On patch excision the channel always appeared in mode 1, which was maintained for a variable time (5–20 min). In most instances the channels then switched into mode 2, and never were seen to switch back, in spite of the eight patches that cumulatively dwelled in this mode 2.33-fold as compared to mode 1. Stability plots of long recordings showed that the channel was kinetically stable in both modes, allowing standard analysis of steady-state kinetics to be performed. Open and closed time distributions of mode 1 and mode 2 revealed that the apparent number of kinetic states of the channel was the same in the two modes. The transition from mode 1 into mode 2 was not instantaneous, but required a variable time in the range 5–60 sec. During the transition the channel mean open time was intermediate between mode 1 and mode 2. The intermediate duration in the stability plot however is not to be interpreted as if the channel, during the transition, rapidly switches between mode 1 and mode 2, but represents a distinct kinetic feature of the transitional channel. Received: 31 December 1998/Revised: 13 April 1999     

Validation of Bayesian analysis of compartmental kinetic models in medical imaging

IntroductionKinetic compartmental analysis is frequently used to compute physiologically relevant quantitative values from time series of images. In this paper, a new approach based on Bayesian analysis to obtain information about these parameters is presented and validated.Materials and methodsThe closed-form of the posterior distribution of kinetic parameters is derived with a hierarchical prior to model the standard deviation of normally distributed noise. Markov chain Monte Carlo methods are used for numerical estimation of the posterior distribution. Computer simulations of the kinetics of F18-fluorodeoxyglucose (FDG) are used to demonstrate drawing statistical inferences about kinetic parameters and to validate the theory and implementation. Additionally, point estimates of kinetic parameters and covariance of those estimates are determined using the classical non-linear least squares approach.Results and discussionPosteriors obtained using methods proposed in this work are accurate as no significant deviation from the expected shape of the posterior was found (one-sided P > 0.08). It is demonstrated that the results obtained by the standard non-linear least-square methods fail to provide accurate estimation of uncertainty for the same data set (P < 0.0001).ConclusionsThe results of this work validate new methods for a computer simulations of FDG kinetics. Results show that in situations where the classical approach fails in accurate estimation of uncertainty, Bayesian estimation provides an accurate information about the uncertainties in the parameters. Although a particular example of FDG kinetics was used in the paper, the methods can be extended for different pharmaceuticals and imaging modalities.     

Wavelet analysis of nonequilibrium ionic currents in human heart sodium channel (hH1a)

Nonequilibrium response spectroscopy (NRS), the technique of using rapidly fluctuating voltage pulses in the study of ion channels, is applied here. NRS is known to drive an ensemble of ion channels far from equilibrium where, it has been argued, new details of ion channel kinetics can be studied under nonequilibrium conditions. In this paper, a single-pulse NRS technique with custom-designed waveforms built from wavelets is used. The pulses are designed to produce different responses from two competing models of a human heart isoform of the sodium channel (hH1a). Experimental data using this new type of pulses are obtained through whole-cell recordings from mammalian cells (HEK 293). Wavelet analysis of the model response and the experimental data is introduced to show how these NRS pulses can aid in distinguishing the better of the two models and thus introduces another important application of this new technique.     

Electrical properties of ionic channels formed by Helix pomatia hemocyanin in planar lipid bilayers

Helix pomatia hemocyanin forms ion-conducting channels in planar lipid bilayer membranes when added at mg/ml concentration. These channels have several original features. They fluctuate between one conducting and some poorly conducting states and fluctuations can be grouped in bursts. Different channels can have widely different conductance amplitudes. Both channel conductance and burst lifetime are dependent on the applied voltage. Fluctuations within a burst show a complex kinetic behaviour which has been explained developing a multistate model. The model calls for one single open state and six different closed states. Transitions are allowed only between one of the closed states and the open one and obey first order kinetics. This model is able to fit all our experimental curves obtained in single channel experiments.     

Rapid ligand-regulated gating kinetics of single inositol 1,4,5-trisphosphate receptor Ca2+ release channels

The ubiquitous inositol 1,4,5-trisphosphate receptor (InsP(3)R) intracellular Ca(2+) release channel is engaged by thousands of plasma membrane receptors to generate Ca(2+) signals in all cells. Understanding how complex Ca(2+) signals are generated has been hindered by a lack of information on the kinetic responses of the channel to its primary ligands, InsP(3) and Ca(2+), which activate and inhibit channel gating. Here, we describe the kinetic responses of single InsP(3)R channels in native endoplasmic reticulum membrane to rapid ligand concentration changes with millisecond resolution, using a new patch-clamp configuration. The kinetics of channel activation and deactivation showed novel Ca(2+) regulation and unexpected ligand cooperativity. The kinetics of Ca(2+)-mediated channel inhibition showed the single-channel bases for fundamental Ca(2+) release events and Ca(2+) release refractory periods. These results provide new insights into the channel regulatory mechanisms that contribute to complex spatial and temporal features of intracellular Ca(2+) signals.     

Gating kinetics of potassium channel in rat dorsal root ganglion neurons analyzed with fractal model

The kinetics of ion channels have been widely modeled as a Markov process. In these models it is assumed that the channel protein has a small number of discrete conformational states and kinetic rate constants connecting these states are constant. To study the gating kinetics of voltage-dependent K(+) channel in rat dorsal root ganglion neurons, K(+) channel current were recorded using cell-attached patch-clamp technique. The K(+) channel characteristic of kinetics were found to be statistically self-similar at different time scales as predicted by the fractal model. The fractal dimension D for the closed times and for the open times depend on the pipette potential. For the open and closed times of kinetic setpoint, it was found dependent on the applied pipette potential, which indicated that the ion channel gating kinetics had nonlinear kinetic properties. Thus, the open and closed durations, which had the voltage dependence of the gating of this ion channel, were well described by the fractal model.     

The Xenopus Oocyte Cut-open Vaseline Gap Voltage-clamp Technique With Fluorometry

The cut-open oocyte Vaseline gap (COVG) voltage clamp technique allows for analysis of electrophysiological and kinetic properties of heterologous ion channels in oocytes. Recordings from the cut-open setup are particularly useful for resolving low magnitude gating currents, rapid ionic current activation, and deactivation. The main benefits over the two-electrode voltage clamp (TEVC) technique include increased clamp speed, improved signal-to-noise ratio, and the ability to modulate the intracellular and extracellular milieu.Here, we employ the human cardiac sodium channel (hNa_V1.5), expressed in Xenopus oocytes, to demonstrate the cut-open setup and protocol as well as modifications that are required to add voltage clamp fluorometry capability.The properties of fast activating ion channels, such as hNa_V1.5, cannot be fully resolved near room temperature using TEVC, in which the entirety of the oocyte membrane is clamped, making voltage control difficult. However, in the cut-open technique, isolation of only a small portion of the cell membrane allows for the rapid clamping required to accurately record fast kinetics while preventing channel run-down associated with patch clamp techniques.In conjunction with the COVG technique, ion channel kinetics and electrophysiological properties can be further assayed by using voltage clamp fluorometry, where protein motion is tracked via cysteine conjugation of extracellularly applied fluorophores, insertion of genetically encoded fluorescent proteins, or the incorporation of unnatural amino acids into the region of interest¹. This additional data yields kinetic information about voltage-dependent conformational rearrangements of the protein via changes in the microenvironment surrounding the fluorescent molecule.     

Optimal estimation of ion-channel kinetics from macroscopic currents

Markov modeling provides an effective approach for modeling ion channel kinetics. There are several search algorithms for global fitting of macroscopic or single-channel currents across different experimental conditions. Here we present a particle swarm optimization(PSO)-based approach which, when used in combination with golden section search (GSS), can fit macroscopic voltage responses with a high degree of accuracy (errors within 1%) and reasonable amount of calculation time (less than 10 hours for 20 free parameters) on a desktop computer. We also describe a method for initial value estimation of the model parameters, which appears to favor identification of global optimum and can further reduce the computational cost. The PSO-GSS algorithm is applicable for kinetic models of arbitrary topology and size and compatible with common stimulation protocols, which provides a convenient approach for establishing kinetic models at the macroscopic level.