Pore Dimensions of Ion Channels: methods

2. Methods

HOLE Monte Carlo Simulated Annealing

The HOLE method has been described in detail elsewhere 6, 11. The program requires the user to supply the coordinates of the ion channel of interest in Brookhaven Protein Data Bank (pdb) format. An initial point p which lies anywhere within the central channel is also needed. In addition the user specifies a vector v that is approximately in the direction of the channel (normally referred to as the channel direction vector). The program reads atoms from within the pdb file and sets up a van der Waals radius for each (various sets are available). It is normal to exclude solvent and ions during the read: the program allows this by the specification of a keyword stating the residue types to ignore.

It is easy to find the maximum radius R(p) of a sphere centred at a point p without overlap with the van der Waals surface of any atom:

R(p) = min_i=1^Natom [|X_i-p|- vdW_i] where X_i is the position of atom number i, vdW_i its van der Waals radius and Natom the total number of atoms. The radius R(p) can be regarded as an objective function of the point p. By using Monte Carlo simulated annealing, adjusting p, the radius of the sphere can be maximized within the pore of the channel. In all cases p is kept on a plane normal to the channel direction vector v:

Note that all distance searches are conducted in three dimensions. Once the highest radius sphere centre on a particular plane has been established a new search is initiated by taking a step of length s in the direction of the channel direction vector v. This results in searches being conducted for a series of parallel planes with a sphere of maximum radius being found for each:

The net result can be thought of as producing the locus of a flexible sphere "squeezing" through the centre of the ion channel. The use of Monte Carlo Simulated Annealing reduces the possibility of the routine getting stuck in a local minimum. In addition it allows the mapping out of complex internal topologies (such as annexin V¹²) using multiple runs.

A flow chart giving a detailed description of the basic HOLE method is available.

Visualization methods

There are a number of ways to analyze and visualize the results of a HOLE run. One of the most useful is plot a graph of pore radius against coordinate z along the channel direction vector v:

A graph can be used to examine issues such as how tight the central constriction of the channel is, whether water molecules could fit within the channel and how large the mouth of the pore is. The HOLE package does not explicitly produce graphs but instead provides easily accessible numerical information so that the user can use the graph plotting program of his/her choice.

Graphical presentation can be complimented by examination of HOLE objects using a molecular graphics program. Two objects can be produced and displayed in conjunction with the ion channel structural model.

The centre line of the channel can be shown by plotting the locus of the centre of the sphere as it proceeds through the channel.
The locus of the surface of the spheres define the pore lining surface. It is often useful to colour code the surface. The standard scheme is to use red to indicate parts of the pore where there is insufficient room to accommodate a water molecule, green where a single water molecule could be placed and blue for places sufficiently large for two water molecules to be placed side by side:
(click to see picture full size)

The HOLE package is designed to be used primarily with the quanta molecular graphics program. However, conversion routines are supplied to allow use with:

Sybyl
InsightII
O - Alwyn Jones crystallographic package. For further details see the O home page.
mage: Kinemage is a protein display and viewing language developed by David Richardson. Kinemages can be viewed by the mage program which is available in versions for MS-windows, mac and various unix machines. Mage is available by ftp from suna.biochem.duke.edu. (If you have mage then you can use this example).
Work which is still in progress allows the use of Virtual Reality Modeling Language (VRML) which may become standard way of representing 3D objects on the world wide web. If you have a suitable viewer (see http://www.ch.ic.ac.uk/VRML/ for details) then try looking at the standard gramicidin example in vrml format.

A recent advance allows an alternative view of the internal surface of a pore - from the inside. In this case we set up a cylindrical co-ordinate system working from the pore centreline:

(click to see picture full size)

This coordinate system can then be used to display properties of the pore lining in two dimensions. A simple way of imagining how the maps are constructed is to imagine that the internal surface of the pore is cut down a line and then rolled flat. A number of properties of the internal surface of the pore can then be displayed, for instance, whether the surface is lined by oxygen or nitrogen atoms. The versatile contour plotting program surfer is used to process data produced by HOLE. This provides a very different way of looking at the gramicidin channel:

(click to see picture full size)

Analyzing anisotropy

The use of a spherical probe to analyse the pore dimensions of a channel has proved to be extremely useful^6,
11. However, the pores of many larger ion channels are clearly anisotropic as exemplified by porins^15,16. To be able to analyse and consistently measure such structural aspects an extension of the original HOLE method is introduced.

Instead of using a sphere to probe the internal surface of the channel in question a spherocylinder (capsule) is considered. This object can be considered to be a simple extension of a sphere where the centre is spread from a point onto a line section. It can be defined in terms of three properties: two centres and a radius (in comparison to a sphere which has a single centre and a radius):

(click to see picture full size)

In the HOLE method the axis of the capsule is held at right angles to the channel direction vector v. Only small adaptations are required to implement this: instead of a single centre being considered in the optimization, two independent centres are introduced. Each of these is constrained to move on a plane normal to the channel direction vector v. Instead of optimizing the radius of the sphere as in the original routine the area of the capsule on the plane normal to v is maximized (although this is converted in the program into an effective radius to ease comparisons). As the number of independent variables is doubled from two to four, more steps of simulated annealing are required to achieve stability in results. Despite this increased cpu requirement most channels can be analyzed in well under one hour of cpu time on a modern workstation. The result of optimizing a capsule inside the pore of gramicidin is shown below:

(click to see picture full size) The use of the capsule option allows the measurement of the anisotropy of a channel and properties such as the rotation of the capsule vector as it proceeds through the channel. Examples of this can be found in the Applications section. The more accurate delimitation of the cross-sectional area of the channel as a function of distance along the channel vector is also useful when making predictions of the conductance properties of channels as discussed below.

Predicting conductance properties using HOLE

The difficulty in experimentally determining the three-dimensional structures of ion channels often make it necessary to resort to modelling techniques. Although modelling can lead to insights into the functional and structural properties of channels there is always the problem of knowing what level of confidence to place in the model. For this reason, tractable methods of validating a given model of a channel are required. HOLE has been adapted so that a reasonable prediction of the conductance of a channel can be rapidly made on the basis of its structure¹¹.

The method is based on simple Ohmic considerations. The HOLE program can be thought of as measuring the cross section area A(z) of a pore as a function of distance along the channel direction vector z. Consider the pore to be filled with an electrolytic solution of resisitivity Þ. A reasonable approximation of the resistance of the channel is given by:

G^-1_macro = Þ s/A(z) where s is the width between parallel planes used in HOLE (see above). Note that in ion channel patch clamping studies it is normal to consider conductance (G) measured in Siemens (S) rather than resistance. This type of approach was pioneered by Hille² and introduced into the HOLE methodology by Sansom and Kerr ¹⁷.

The above equation assumes that the conductivity of an ionic solution within a channel is equal to that of bulk solution. This would be true if ion channels had macroscopic dimensions (much larger than a water molecule). In practice it is found that real channels have a conductance which is around five times lower than that expected from the above equation giving the macroscopic limit G_macro. In order to make a reasonable estimate of the conductance of a channel, an empirically-based correction factor is used, which is chosen to be dependent on the minimum radius of the channel:

G_pred = G_macro/C(R_min)

The correction factor was parameterized 11 on the results obtained for the gramicidin channel¹⁴ and the E. coli OmpF porin¹⁵. The prediction routine was tested by "predicting" the conductance found for all channel-forming proteins and peptides where an experimental structure is available¹¹. Results are summarized in the following graph:

(click to see picture full size)

Overall, the algorithm yield good results with predictions accurate to within a average factor of 1.8 to the experimental values. This accuracy is sufficient to make the method a useful part in validating model structures. Further work has been undertaken to reparameterize the correction function on the basis of all well defined structures which were considered¹¹. Future work will involve include using molecular dynamics simulation procedures to find quantities such as the variation of the self-diffusion co-efficient of water molecules within the channel and including such effects within the prediction methodology.

It is also possible to adapt the procedure to predict the effect of adding non-electrolytes such as PEG to conductance measurements. Such experiments can be interpreted in terms of a radius profile for a channel. Encouraging preliminary results¹¹ have been obtained comparing the expected profile calculated by HOLE from the X-ray structure of cholera toxin B subunit¹⁹ with the experimental result²¹.

Oliver S. Smart (last modified 20/12/96)