Hybrid Markov-mass action law model for cell activation by rare binding events: Application to calcium induced vesicular release at neuronal synapses

Abstract

Binding of molecules, ions or proteins to small target sites is a generic step of cell activation. This process relies on rare stochastic events where a particle located in a large bulk has to find small and often hidden targets. We present here a hybrid discrete-continuum model that takes into account a stochastic regime governed by rare events and a continuous regime in the bulk. The rare discrete binding events are modeled by a Markov chain for the encounter of small targets by few Brownian particles, for which the arrival time is Poissonian. The large ensemble of particles is described by mass action laws. We use this novel model to predict the time distribution of vesicular release at neuronal synapses. Vesicular release is triggered by the binding of few calcium ions that can originate either from the synaptic bulk or from the entry through calcium channels. We report here that the distribution of release time is bimodal although it is triggered by a single fast action potential. While the first peak follows a stimulation, the second corresponds to the random arrival over much longer time of ions located in the synaptic terminal to small binding vesicular targets. To conclude, the present multiscale stochastic modeling approach allows studying cellular events based on integrating discrete molecular events over several time scales.

Electrodiffusion phenomena in neuroscience: a neglected companion

“…At the same time, the flows of particles crossing the volume boundary are continuously recalculated into the flux, field and other parameters representing boundary conditions for the compartmental or analytical model adopted for the large (or infinite) surrounding volume [46][47][48][49] .…”

The first one hundred nanometers inside the pre-synaptic terminal where calcium diffusion triggers vesicular release

“…Thus, the concentration of buffers that favor a synchronous release should have an optimal value: not too low and not too large. In [23,35], the number of calcium entering through the VGCC vary from 80 to 500, which is also the case in [24], where they open around 12 Ca 2+ channels with a single channel current of 0.15 pA and a duration of 105 µs, leading to ≈ 500 − 600 calcium ions. Some of the 260 buffers with at least two binding sites, could bind calcium ions on their direct way to the calcium sensor underneath the vesicle.…”

“…To conclude, there is not yet a derived formula from physical principle to connect the flux or transient calcium concentration and the release probability, however, stochastic approaches are used to estimate the arrival of ions to the calcium sensors [25,24,23,35]. In addition, the three dimensional vesicular organization should be accounted either directly, by implementing vesicles as obstacles or by computing the Brownian flux to small targets located underneath.…”

“…Following calcium diffusion, once calcium ions are bound to buffers, they can possibly unbind, but often the exact value of the backward rate constant is unknown. In recent mathematical models [28,23,35], vesicles are released when all 5 binding sites at a single sensor are occupied. If less than 5 calcium ions are bound, the vesicle is waiting for the final ions to arrive.…”

“…The Hodgkin-Huxley model [36] can be used [35] to generate a calcium influx current inside the pre-synaptic terminal. Following the opening of VGCC, this current corresponds to an entry per channel during a mean time of ≈1 ms for approximately 80 calcium ions, compatible with a previous estimation of 200 reviewed in [21] or with the 45 ions per channels described in [28]: with a total of 12 channels, this would represent 540 ions entering during ≈ 0.1 ms. After entry, the calcium flux can be split into a ionic component that reaches the small region below the vesicle, and another one reaching the pre-synaptic bulk.…”