Maximum likelihood estimation of ion channel kinetics from macroscopic currents

We describe a maximum likelihood method for direct estimation of rate constants from macroscopic ion channel data for kinetic models of arbitrary size and topology. The number of channels in the preparation, and the mean and standard deviation of the unitary current can be estimated, and a priori constraints can be imposed on rate constants. The method allows for arbitrary stimulation protocols, including stimuli with finite rise time, trains of ligand or voltage steps, and global fitting across different experimental conditions. The initial state occupancies can be optimized from the fit kinetics. Utilizing arbitrary stimulation protocols and using the mean and the variance of the current reduce or eliminate problems of model identifiability (Kienker, 1989). The algorithm is faster than a recent method that uses the full autocovariance matrix (Celentano and Hawkes, 2004), in part due to the analytical calculation of the likelihood gradients. We tested the method with simulated data and with real macroscopic currents from acetylcholine receptors, elicited in response to brief pulses of carbachol. Given appropriate stimulation protocols, our method chose a reasonable model size and topology.     

Estimation of ion channel kinetics from fluctuations of macroscopic currents

For single channel recordings, the maximum likelihood estimation (MLE) of kinetic rates and conductance is well established. A direct extrapolation of this method to macroscopic currents is computationally prohibitive: it scales as a power of the number of channels. An approximated MLE that ignored the local time correlation of the data has been shown to provide estimates of the kinetic parameters. In this article, an improved approximated MLE that takes into account the local time correlation is proposed. This method estimates the channel kinetics using both the time course and the random fluctuations of the macroscopic current generated by a homogeneous population of ion channels under white noise. It allows arbitrary kinetic models and stimulation protocols. The application of the proposed algorithm to simulated data from a simple three-state model on nonstationary conditions showed reliable estimates of all the kinetic constants, the conductance and the number of channels, and reliable values for the standard error of those estimates. Compared to the previous approximated MLE, it reduces by a factor of 10 the amount of data needed to secure a given accuracy and it can even determine the kinetic rates in macroscopic stationary conditions.     

Efficient maximum likelihood estimation of kinetic rate constants from macroscopic currents

A new method is described that accurately estimates kinetic constants, conductance and number of ion channels from macroscopic currents. The method uses both the time course and the strength of correlations between different time points of macroscopic currents and utilizes the property of semiseparability of covariance matrix for computationally efficient estimation of current likelihood and its gradient. The number of calculation steps scales linearly with the number of channel states as opposed to the cubic dependence in a previously described method. Together with the likelihood gradient evaluation, which is almost independent of the number of model parameters, the new approach allows evaluation of kinetic models with very complex topologies. We demonstrate applicability of the method to analysis of synaptic currents by estimating accurately rate constants of a 7-state model used to simulate GABAergic macroscopic currents.     

Fractal model of ion-channel kinetics

Markov models with discrete states, such as closed in equilibrium with closed in equilibrium with open have been widely used to model the kinetics of ion channels in the cell membrane. In these models the transition probabilities per unit time (the kinetic rate constants) are independent of the time scale on which they are measured. However, in many physical systems, a property, L, depends on the scale, epsilon, at which it is measured such that L(epsilon) alpha epsilon 1-D where D is the fractal dimension. Such systems are said to be 'fractal'. Based on the assumption that the kinetic rates are given by k(t) alpha t1-D we derive a fractal model of ion-channel kinetics. This fractal model has fewer adjustable parameters, is more consistent with the dynamics of protein conformations, and fits the single-channel recordings from the corneal endothelium better than the discrete-state Markov model.     

Optimal fed batch experiment design for estimation of monod kinetics of Azospirillum brasilense: from theory to practice

In this paper the problem of reliable and accurate parameter estimation for unstructured models is considered. It is illustrated how a theoretically optimal design can be successfully translated into a practically feasible, robust, and informative experiment. The well-known parameter estimation problem of Monod kinetic parameters is used as a vehicle to illustrate our approach. As known for a long time, noisy batch measurements do not allow for unique and accurate estimation of the kinetic parameters of the Monod model. Techniques of optimal experiment design are, therefore, exploited to design informative experiments and to improve the parameter estimation accuracy. During the design process, practical feasibility has to be kept in mind. The designed experiments are easy to implement in practice and do not require additional monitoring equipment. Both design and experimental validation of informative fed batch experiments are illustrated with a case study, namely, the growth of the nitrogen-fixing bacteria Azospirillum brasilense.     

Discerning ionic currents and their kinetics from input impedance data

The excitable nature of a biological cell is manifested in the many voltage gated ion channels that perforate its membrane. The forms of the associated ionic currents, and in particular the functions that govern their kinetics, permit one to distinguish, electrophysiologically, between various cell types. We show, in the context of FitzHugh-Nagumo and Morris-Lecar models and without recourse to voltage or space clamping, that such currents and kinetics may be stably inferred from a cell’s voltage response to a specified input current.     

Single-channel, macroscopic, and gating currents from sodium channels in the squid giant axon.

Single-channel, macroscopic ionic, and macroscopic gating currents were recorded from the voltage-dependent sodium channel using patch-clamp techniques on the cut-open squid giant axon. To obtain a complete set of physiological measurements of sodium channel gating under identical conditions, and to facilitate comparison with previous work, comparison was made between currents recorded in the absence of extracellular divalent cations and in the presence of physiological concentrations of extracellular Ca2+ (10 mM) and Mg2+ (50 mM). The single-channel currents were well resolved when divalent cations were not included in the extracellular solution, but were decreased in amplitude in the presence of Ca2+ and Mg2+ ions. The instantaneous current-voltage relationship obtained from macroscopic tail current measurements similarly was depressed by divalents, and showed a negative slope-conductance region for inward current at negative potentials. Voltage dependent parameters of channel gating were shifted 9-13 mV towards depolarized potentials by external divalent cations, including the peak fraction of channels open versus voltage, the time constant of tail current decline, the prepulse inactivation versus voltage relationship, and the charge-voltage relationship for gating currents. The effects of divalent cations are consistent with open channel block by Ca2+ and Mg2+ together with divalent screening of membrane charges.     

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

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

Slo3 K+ channels: voltage and pH dependence of macroscopic currents

The mouse Slo3 gene (KCNMA3) encodes a K(+) channel that is regulated by changes in cytosolic pH. Like Slo1 subunits responsible for the Ca(2+) and voltage-activated BK-type channel, the Slo3 alpha subunit contains a pore module with homology to voltage-gated K(+) channels and also an extensive cytosolic C terminus thought to be responsible for ligand dependence. For the Slo3 K(+) channel, increases in cytosolic pH promote channel activation, but very little is known about many fundamental properties of Slo3 currents. Here we define the dependence of macroscopic conductance on voltage and pH and, in particular, examine Slo3 conductance activated at negative potentials. Using this information, the ability of a Horrigan-Aldrich-type of general allosteric model to account for Slo3 gating is examined. Finally, the pH and voltage dependence of Slo3 activation and deactivation kinetics is reported. The results indicate that Slo3 differs from Slo1 in several important ways. The limiting conductance activated at the most positive potentials exhibits a pH-dependent maximum, suggesting differences in the limiting open probability at different pH. Furthermore, over a 600 mV range of voltages (-300 to +300 mV), Slo3 conductance shifts only about two to three orders of magnitude, and the limiting conductance at negative potentials is relatively voltage independent compared to Slo1. Within the context of the Horrigan-Aldrich model, these results indicate that the intrinsic voltage dependence (z(L)) of the Slo3 closed-open equilibrium and the coupling (D) between voltage sensor movement are less than in Slo1. The kinetic behavior of Slo3 currents also differs markedly from Slo1. Both activation and deactivation are best described by two exponential components, both of which are only weakly voltage dependent. Qualitatively, the properties of the two kinetic components in the activation time course suggest that increases in pH increase the fraction of more rapidly opening channels.     

Maximum likelihood estimation of molecular motor kinetics from staircase dwell-time sequences

Molecular motors, such as kinesin, myosin, or dynein, convert chemical energy into mechanical energy by hydrolyzing ATP. The mechanical energy is used for moving in discrete steps along the cytoskeleton and carrying a molecular load. High resolution single molecule recordings of motor steps appear as a stochastic sequence of dwells, resembling a staircase. Staircase data can also be obtained from other molecular machines such as F1 -ATPase, RNA polymerase, or topoisomerase. We developed a maximum likelihood algorithm that estimates the rate constants between different conformational states of the protein, including motor steps. We model the motor with a periodic Markov model that reflects the repetitive chemistry of the motor step. We estimated the kinetics from the idealized dwell-sequence by numerical maximization of the likelihood function for discrete-time Markov models. This approach eliminates the need for missed event correction. The algorithm can fit kinetic models of arbitrary complexity, such as uniform or alternating step chemistry, reversible or irreversible kinetics, ATP concentration and mechanical force-dependent rates, etc. The method allows global fitting across stationary and nonstationary experimental conditions, and user-defined a priori constraints on rate constants. The algorithm was tested with simulated data, and implemented in the free QuB software.     

An alternate method for estimation of cell growth kinetics from batch cultures

An alternative to estimation of cell growth kinetics via continuous culture experiments is proposed in this article. The method employed is based on batch culture experiments with very small inocula (initial cell concentrations being typically less than 5000 cells/mL). Such low initial cell concentrations result in extended exponential cell growth phase during which culture conditions remain unchanged, thereby permitting precise estimation of specific cell growth rates from batch experiments especially for fast-growing microorganisms such as Bacillus species. The effectiveness and utility of this approach are demonstrated via several experiments conducted with a wild-type strain (Bacillus subtilis TN106) and a recombinant strain (B. subtilis TN106[pAT5]). True establishment of exponential growth phase requires insignificant variance of most of the culture conditions during the initial growth phase. Satisfaction of this requirement is demonstrated for microbial systems investigated here. This approach is especially well suited for recombinant microorganisms containing segregationally unstable plasmids, since estimation of growth kinetics of these from continuous cultures is very difficult and highly unreliable due to continual reversion of recombinant ceils to plasmid-free host cells unless some selection pressure is applied at levels sufficient to keep the presence of plasmid-free cells minimal.     

A simple macroscopic model for the diffusion and adsorption kinetics of r-adenovirus

The diffusion of viruses toward cells is a limiting step of the infection process. To be modeled correctly, this step must be evaluated in combination with the adsorption of the virus to the cell surface, which is a rapid but reversible step. In this paper, the recombinant adenovirus (rAd) diffusion and its adsorption to 293S cells in suspension were both measured and modeled. First, equilibrium experiments permitted to determine the number of receptors on the surface of 293S (R(T) = 3,500 cell(-1)) and the association constant (K(A) = 1.9 x 10(11) M(-1)) for rAd on these cells based on a simple monovalent adsorption model. Non-specific binding of the virus to the cell surface was not found to be significant. Second, total virus particle degradation rates between 5.2 x 10(-3) and 4.0 x 10(-2) min(-1) were measured at 37 degrees C in culture medium, but no significant virus degradation was observed at 4 degrees C. Third, free viral particle disappearance rates from a mixed suspension of virus and cells were measured at different virus concentrations. Experimental data were compared to a phenomenological dynamic model comprising both the diffusion and the adsorption steps. The diffusion to adsorption ratio, a fitted parameter, confirmed that the contact process of a virus with a cell is indeed diffusion controlled. However, the characteristic diffusion time constants obtained, based on a reversible adsorption model, were eightfolds smaller than those reported in the literature, based on diffusion models that assume irreversible adsorption.     

Asymmetrical ion-channel model inferred from two-dimensional crystallization of a peptide antibiotic

The structural organization of ion channels formed in lipid membranes by amphiphilic alpha-helical peptides is deduced by applying direct structural methods to different lipid/alamethicin systems. Alamethicin represents a hydrophobic alpha-helical peptide antibiotic forming voltage-gated ion channels in lipid membranes. Here the first direct evidence for the existence of large-scale two-dimensional crystalline domains of alamethicin helices, oriented parallel to the air/water interface, is presented using synchrotron x-ray diffraction, fluorescence microscopy, and surface pressure/area isotherms. Proofs are obtained that the antibiotic peptide injected into the aqueous phase under phospholipid monolayers penetrates these monolayers, phase separates, and forms domains within the lipid environment, keeping the same, parallel orientation of the alpha-helices with respect to the phospholipid/water interface. A new asymmetrical, "lipid-covered ring" model of the voltage-gated ion channel of alamethicin is inferred from the structural results presented, and the mechanism of ion-channel formation is discussed.     

Stochastic Markovian modeling of electrophysiology of ion channels: reconstruction of standard deviations in macroscopic currents

Markovian models of ion channels have proven useful in the reconstruction of experimental data and prediction of cellular electrophysiology. We present the stochastic Galerkin method as an alternative to Monte Carlo and other stochastic methods for assessing the impact of uncertain rate coefficients on the predictions of Markovian ion channel models. We extend and study two different ion channel models: a simple model with only a single open and a closed state and a detailed model of the cardiac rapidly activating delayed rectifier potassium current. We demonstrate the efficacy of stochastic Galerkin methods for computing solutions to systems with random model parameters. Our studies illustrate the characteristic changes in distributions of state transitions and electrical currents through ion channels due to random rate coefficients. Furthermore, the studies indicate the applicability of the stochastic Galerkin technique for uncertainty and sensitivity analysis of bio-mathematical models.     

Relating microscopic charge movement to macroscopic currents: the Ramo-Shockley theorem applied to ion channels

Since the discovery of gating current, electrophysiologists have studied the movement of charged groups within channel proteins by changing potential and measuring the resulting capacitive current. The relation of atomic-scale movements of charged groups to the gating current measured in an external circuit, however, is not obvious. We report here that a general solution to this problem exists in the form of the Ramo-Shockley theorem. For systems with different amounts of atomic detail, we use the theorem to calculate the gating charge produced by movements of protein charges. Even without calculation or simulation, the Ramo-Shockley theorem eliminates a class of interpretations of experimental results. The theorem may also be used at each time step of simulations to compute external current.     

Optimal trait scoring for age estimation

What are the effects that genetics has had on Anthropological research and how can we think anthropologically about Genetics? Just as genetic data have encouraged new hypotheses about human phenotypic variation, evolutionary history, population interaction, and environmental effects, so too has Anthropology offered to genetic studies a new interpretive locus in its history and perspective. This introduction examines how the fields of Anthropology and Genetics have arrived at a crucial moment at which their interaction requires careful examination and critical reflection. The papers discussed here exemplify how we may engage in such a trans-disciplinary conversation. They speak to the future of thoughtful interaction between genetic and anthropological literature and seek a new integration that embodies the holism of the human biological sciences.