BB7: Mathematical modelling of ion channels

Project completed January 17, 2012

Background

Ion channels are pore-forming proteins in cell membranes that provide a path to connect the cell interior to the exterior by allowing ions and water molecules to pass through. The transport of ions across ion channels plays a key role in mediating all electrical activities, producing signals in the nervous system and in regulating cardiac contraction. In response to stimuli, ion channels may change between open and closed states (gating). They may also be selective – a specific channel may only allow specific types of ion to permeate the channel.

Molecular dynamics simulations of some potassium channels have shown that ions move in single file through the narrow channel. When an ion enters the channel each ion is observed to fluctuate for a long time around an energy minimum known as a binding site, before it hops to the next binding site in the channel. It is through hopping that the channel conducts ions. Eventually, the ion escapes from the channel. 

Due to their importance in biology and pharmacology, it is necessary to understand gating and selectivity in ion channels, and how the channel conducts ions. However, there are many modelling challenges, notably due to the difference in length scales between the ions, the channel, and the cell.

Many different approaches have been proposed in the literature to study ion transport in a channel. Researchers at the Oxford Centre for Collaborative Applied Mathematics (OCCAM) have derived a model for an ion channel focusing on the ion permeation process starting with Brownian dynamics, with the aim to make a reduced and tractable model. 

Techniques and Challenges

Using the fact that the motion of ions depends only on the current state of the system, the theory reduces a continuous ion diffusion simulated by Brownian dynamics to a discrete Markovian process of ion hopping between binding sites. A hierarchy of Fokker-Planck equations, indexed by channel occupancy, was derived from continuous Brownian dynamics.  From these equations, the mean ion escape times and splitting probabilities (denoting from which side an ion has escaped) can be calculated. By then equating these with the corresponding expressions from the discrete Markov model, the individual escape rates and hopping rates can be determined. 

One binding site: The researchers illustrated the theory using a simple model of a channel with one binding site. The channel was assumed to have a maximum occupancy of two ions (ne at the binding site and one free ion). Alternatively, it could have just one ion or zero ions (the three states are referred to as 0-ion, 1-ion or 2-ion).

Extending to multiple binding sites: Extending the model to have more general channels with multiple binding sites leads to the increased difficulties of computing higher dimensional PDEs and simulating a larger sample of Brownian dynamics. 

Since each binding site can be occupied or empty, as the number of binding sites increases, the number of states in the system grows exponentially and the system becomes too complicated to analyse. However, by using the observation from the two-ion channel system that the ion channel is never in the 0-ion state, the researchers postulated that a channel with n binding sites has only three states: two states of (n+1)-ion occupancy with n ions in the binding sites and one ion at the left or right entry points, and one state of n-ion occupancy with each ion at one binding site. 

Results

One binding site: First, using the simple model, the researchers investigated how the conductivity of the ion channel depends on various parameters. Interestingly, the channel is almost never in the 0-ion state. This observation complies with the knock-on hypothesis which states that ion permeation is triggered by a new ion entering a saturated channel and eventually pushing an ion out the other end (see Figure 1).

Extending to multiple binding sites: Using the simplified multiple binding site model, the researchers were able to calculate the current in the channel.

The novelty in this new approach is that the state of the system is determined by the positions of all the ions simultaneously, thus capturing all the correlations between the positions of different ions.

This new theory is an improvement on the traditional Kramer’s rate theory as it has the notable feature of including the effect of the geometry of the potential energy landscape. The model also showed an intricate coupling between transition rates, mean escape time and splitting probability due to the complex system of Markovian states for multi-ion channels.