We present here a maximal likelihood algorithm for estimating single-channel kinetic parameters from idealized patch-clamp data. The algorithm takes into account missed events caused by limited time resolution of the recording system. Assuming a fixed dead time, we derive an explicit expression for the corrected transition rate matrix by generalizing the theory of Roux and Sauve (1985, Biophys. J. 48:149-158) to the case of multiple conductance levels. We use a variable metric optimizer with analytical derivatives for rapidly maximizing the likelihood. The algorithm is applicable to data containing substates and multiple identical or nonidentical channels. It allows multiple data sets obtained under different experimental conditions, e.g., concentration, voltage, and force, to be fit simultaneously. It also permits a variety of constraints on rate constants and provides standard errors for all estimates of model parameters. The algorithm has been tested extensively on a variety of kinetic models with both simulated and experimental data. It is very efficient and robust; rate constants for a multistate model can often be extracted in a processing time of approximately 1 min, largely independent of the starting values.

Allosteric transitions allow fast regulation of protein function in living systems. Even though the end points of such conformational changes are known for many proteins, the characteristics of the paths connecting these states remain largely unexplored. Rate-equilibrium linear free-energy relationships (LFERs) provide information about such pathways by relating changes in the free energy of the transition state to those of the ground states upon systematic perturbation of the system. Here we present an LFER analysis of the gating reaction pathway of the muscle acetylcholine receptor. We studied the closed <==> open conformational change at the single-molecule level following perturbation by series of single-site mutations, agonists and membrane voltages. This method provided a snapshot of several regions of the receptor at the transition state in terms of their approximate positions along the reaction coordinate, on a scale from 0 (closed-like) to 1 (open-like). The resulting map reveals a spatial gradient of positional values, which suggests that the conformational change proceeds in a wave-like manner, with the low-to-high affinity change at the transmitter-binding sites preceding the complete opening of the pore.

SUMMARYWe present a maximum likelihood method for the modelling of aggregated Markov processes. The method utilizes the joint probability density of the observed dwell time sequence as likelihood. A forward-backward recursive procedure is developed for efficient computation of the likelihood function and its derivatives with respect to the model parameters. Based on the calculated forward and backward vectors, analytical formulae for the derivatives of the likelihood function are derived. The method exploits the variable metric optimizer for search of the likelihood space. It converges rapidly and is numerically stable. Numerical examples are given to show the effectiveness of the method.

N-methyl-D-aspartate receptor (NMDAR) channels mediate the slow component of excitatory potentials at glutamatergic synapses. They have complex kinetic behavior, and much remains to be understood about NMDAR gating mechanisms and the molecular events that shape the synaptic current. Here we show that an individual NMDAR produces at least three stable patterns of activity. For all modes, channels gate by the same mechanism and can occupy either of two open states. The relative stability of the open states differs across modes because of a common perturbation to the NMDAR structure that may be subject to cellular control. Simulations indicate that native NMDAR-mediated synaptic responses arise mainly from the most common mode, and that the slow rise and decay of the current can be attributed to multiple transitions between fully liganded open and closed states rather than to agonist dissociation.

In five members of a family and another unrelated person affected by a slow-channel congenital myasthenic syndrome (SCCMS), molecular genetic analysis of acetylcholine receptor (AChR) subunit genes revealed a heterozygous G to A mutation at nucleotide 457 of the alpha subunit, converting codon 153 from glycine to serine (alpha G153S). Electrophysiologic analysis of SCCMS end plates revealed prolonged decay of miniature end plate currents and prolonged activation episodes of single AChR channels. Engineered mutant AChR expressed in HEK fibroblasts exhibited prolonged activation episodes strikingly similar to those observed at the SCCMS end plates. Single-channel kinetic analysis of engineered alpha G153S AChR revealed a markedly decreased rate of ACh dissociation, which causes the mutant AChR to open repeatedly during ACh occupancy. In addition, ACh binding measurements combined with the kinetic analysis indicated increased desensitization of the mutant AChR. Thus, ACh binding affinity can dictate the time course of the synaptic response, and alpha G153 contributes to the low binding affinity for ACh needed to speed the decay of the synaptic response.

At central excitatory synapses, N-methyl-D-aspartate (NMDA) receptors, which have a high affinity for glutamate, produce a slowly rising synaptic current in response to a single transmitter pulse and an additional current after a second, closely timed stimulus. Here we show, by examining the kinetics of transmitter binding and channel gating in single-channel currents from recombinant NR1/NR2A receptors, that the synaptic response to trains of impulses is determined by the molecular reaction mechanism of the receptor. The rate constants estimated for the activation reaction predict that, after binding neurotransmitter, receptors hesitate for approximately 4 ms in a closed high-affinity conformation before they either proceed towards opening or release neurotransmitter, with about equal probabilities. Because only about half of the initially fully occupied receptors become active, repetitive stimulation elicits currents with distinct waveforms depending on pulse frequency. This high-affinity/low-efficiency activation mechanism might serve as a link between stimulation frequency and the directionality of the ensuing synaptic plasticity.

Hidden Markov modeling (HMM) provides an effective approach for modeling single channel kinetics. Standard HMM is based on Baum's reestimation. As applied to single channel currents, the algorithm has the inability to optimize the rate constants directly. We present here an alternative approach by considering the problem as a general optimization problem. The quasi-Newton method is used for searching the likelihood surface. The analytical derivatives of the likelihood function are derived, thereby maximizing the efficiency of the optimization. Because the rate constants are optimized directly, the approach has advantages such as the allowance for model constraints and the ability to simultaneously fit multiple data sets obtained at different experimental conditions. Numerical examples are presented to illustrate the performance of the algorithm. Comparisons with Baum's reestimation suggest that the approach has a superior convergence speed when the likelihood surface is poorly defined due to, for example, a low signal-to-noise ratio or the aggregation of multiple states having identical conductances.

We used single-channel recording and model-based kinetic analyses to quantify the effects of mutations in the extracellular domain (ECD) of the α-subunit of mouse muscle–type acetylcholine receptors (AChRs). The crystal structure of an acetylcholine binding protein (AChBP) suggests that the ECD is comprised of a β-sandwich core that is surrounded by loops. Here we focus on loops 2 and 7, which lie at the interface of the AChR extracellular and transmembrane domains. Side chain substitutions in these loops primarily affect channel gating by either decreasing or increasing the gating equilibrium constant. Many of the mutations to the β-core prevent the expression of functional AChRs, but of the mutants that did express almost all had wild-type behavior. Rate-equilibrium free energy relationship analyses reveal the presence of two contiguous, distinct synchronously-gating domains in the α-subunit ECD that move sequentially during the AChR gating reaction. The transmitter-binding site/loop 5 domain moves first (Φ = 0.93) and is followed by the loop 2/loop 7 domain (Φ = 0.80). These movements precede that of the extracellular linker (Φ = 0.69). We hypothesize that AChR gating occurs as the stepwise movements of such domains that link the low-to-high affinity conformational change in the TBS with the low-to-high conductance conformational change in the pore.