Are rate constants constant?

ModFossa: A library for modeling ion channels using Python

The creation and simulation of ion channel models using continuous-time Markov processes is a powerful and well-used tool in the field of electrophysiology and ion channel research. While several software packages exist for the purpose of ion channel modeling, most are GUI based, and none are available as a Python library. In an attempt to provide an easy-to-use, yet powerful Markov model-based ion channel simulator, we have developed ModFossa, a Python library supporting easy model creation and stimulus definition, complete with a fast numerical solver, and attractive vector graphics plotting.

The what and where of adding channel noise to the Hodgkin-Huxley equations

Conductance-based equations for electrically active cells form one of the most widely studied mathematical frameworks in computational biology. This framework, as expressed through a set of differential equations by Hodgkin and Huxley, synthesizes the impact of ionic currents on a cell's voltage--and the highly nonlinear impact of that voltage back on the currents themselves--into the rapid push and pull of the action potential. Later studies confirmed that these cellular dynamics are orchestrated by individual ion channels, whose conformational changes regulate the conductance of each ionic current. Thus, kinetic equations familiar from physical chemistry are the natural setting for describing conductances; for small-to-moderate numbers of channels, these will predict fluctuations in conductances and stochasticity in the resulting action potentials. At first glance, the kinetic equations provide a far more complex (and higher-dimensional) description than the original Hodgkin-Huxley equations or their counterparts. This has prompted more than a decade of efforts to capture channel fluctuations with noise terms added to the equations of Hodgkin-Huxley type. Many of these approaches, while intuitively appealing, produce quantitative errors when compared to kinetic equations; others, as only very recently demonstrated, are both accurate and relatively simple. We review what works, what doesn't, and why, seeking to build a bridge to well-established results for the deterministic equations of Hodgkin-Huxley type as well as to more modern models of ion channel dynamics. As such, we hope that this review will speed emerging studies of how channel noise modulates electrophysiological dynamics and function. We supply user-friendly MATLAB simulation code of these stochastic versions of the Hodgkin-Huxley equations on the ModelDB website (accession number 138950) and http://www.amath.washington.edu/~etsb/tutorials.html.

Analysis of proton exchange kinetics with time-dependent exchange rate

Mass spectrometry is used to probe the kinetics of hydrogen-deuterium exchange in lysozyme in pH 5, 6 and 7.4. An analysis based on a Verhulst growth model is proposed and effectively applied to the kinetics of the hydrogen exchange. The data are described by a power-like function which is based on a time-dependence of the exchange rate. Experimental data ranging over many time scales is considered and accurate fits of a power-like function are obtained. Results of fittings show correlation between faster hydrogen-deuterium exchange and increase of pH. Furthermore a model is presented that discriminates between easily exchangeable hydrogens (located in close proximity to the protein surface) and those protected from the exchange (located in the protein interior). A possible interpretation of the model and its biological significance are discussed.

Adaptive transition rates in excitable membranes

Adaptation of activity in excitable membranes occurs over a wide range of timescales. Standard computational approaches handle this wide temporal range in terms of multiple states and related reaction rates emanating from the complexity of ionic channels. The study described here takes a different (perhaps complementary) approach, by interpreting ion channel kinetics in terms of population dynamics. I show that adaptation in excitable membranes is reducible to a simple Logistic-like equation in which the essential non-linearity is replaced by a feedback loop between the history of activation and an adaptive transition rate that is sensitive to a single dimension of the space of inactive states. This physiologically measurable dimension contributes to the stability of the system and serves as a powerful modulator of input–output relations that depends on the patterns of prior activity; an intrinsic scale free mechanism for cellular adaptation that emerges from the microscopic biophysical properties of ion channels of excitable membranes.

Order-based representation in random networks of cortical neurons

The wide range of time scales involved in neural excitability and synaptic transmission might lead to ongoing change in the temporal structure of responses to recurring stimulus presentations on a trial-to-trial basis. This is probably the most severe biophysical constraint on putative time-based primitives of stimulus representation in neuronal networks. Here we show that in spontaneously developing large-scale random networks of cortical neurons in vitro the order in which neurons are recruited following each stimulus is a naturally emerging representation primitive that is invariant to significant temporal changes in spike times. With a relatively small number of randomly sampled neurons, the information about stimulus position is fully retrievable from the recruitment order. The effective connectivity that makes order-based representation invariant to time warping is characterized by the existence of stations through which activity is required to pass in order to propagate further into the network. This study uncovers a simple invariant in a noisy biological network in vitro; its applicability under in vivo constraints remains to be seen.

On the possible methods for the mathematical description of the ball and chain model of ion channel inactivation

Ion channels are large transmembrane proteins that are able to conduct small inorganic ions. They are characterized by high selectivity and the ability to gate, i.e. to modify their conductance in response to different stimuli. One of the types of gating follows the ball and chain model, according to which a part of the channel’s protein forms a ball connected with the intracellular side of the channel by a polypeptide chain. The ball is able to modify the conductance of the channel by properly binding to and plugging the channel pore. In this study, the polypeptide ball is treated as a Brownian particle, the movements of which are limited by the length of the chain. The probability density of the ball’s position is resolved by different diffusional operators — parabolic (including the case with drift), hyperbolic, and fractional. We show how those different approaches shed light on different aspects of the movement. We also comment on some features of the survival probabilities (which are ready to be compared with electrophysiological measurements) for issues based on the above operators.