Computational Test of a Model for Na Channel Clustering

The computational model consists of an axon in contact with an initially nonmyelinating Schwann cell. The idea is to calculate the rate of formation and shape of axonal Na+ channel clusters that form at the edges of the Schwann cell after it begins remyelination and compare these with results from immunofluorescence. As discussed earlier, the Na+ channel density at adult nodes is about 1,000-1,500/|m2. The internodal density is much less (40-60/||m2), but the total number of internodal channels is high (~95% of the axonal population). As mentioned previously, because there is evidence that the nodal channels are derived primarily from an existing axonal pool, we can conclude that the nodal density has a maximum value that cannot be exceeded despite the availability of other channels. From the preceding numbers, we have taken the mean value of 1,250 chan-nels/||m2 for the normal adult node, and the internodal density is then set at 4% of this value, or 50 channels/|m2. We set U as the density of unbound channels, B the density of bound or anchored channels, and D the diffusion coefficient. With the chemical reaction (equation 1) approximated by a first order process with rate constants a and P, the system is then described by equation (2) in one dimension (Crank, 1975).

Each of the rate constants has two possible values, one in regions covered by a Schwann cell (aSC, PSC) and another in "naked" zones with no Schwann cell (aN, PN). Only unbound channels are free to diffuse laterally in the membrane. We took as a starting point for the kinetics of the chemical reaction rate constants derived from data for tyrosine phos-phorylation and dephosphorylation of membrane proteins that were available in the literature. The model is illustrated by the sketch in Fig. 6A. Initially, Na+ channels are uniformly distributed in the axon at a density of 50/|m2 and a = aN, P = PN everywhere. At t = 0 the Schwann cell commits to myelination, and the rate constants in the region of contact (bounded by the MAG-positive regions shown in red) change abruptly to aSC, PSC. As the Schwann cell grows longitudinally, the rate constants in the newly covered zones also change to aSC, PSC. Just outside the MAG-positive zones (destined to become paranodes) nascent microvilli appear, and in this region and beyond, rate constants remain aN and PN.

The axon was 163 |im long, divided initially into a central zone of 83 |im covered by a Schwann cell and two 40-|im uncovered zones on either side of the Schwann cell. The diffusion coefficient of mobile Na+ channels in axon membranes has been measured in fluorescence photobleach recovery experiments to be about 10-10 cm2/sec (see Custer et al., 2003, for details), and this is assigned to all unbound channels, regardless of location. Because data on Na+ channels are not available, we used kinetic data on another membrane protein, the insulin receptor, for plausible starting values of reaction kinetics. The half-time of tyrosine dephosphorylation of the latter protein has been reported in different experiments at 21 sec (Mooney and Anderson, 1989) and 37 sec (Mooney and Bordwell, 1992) at 37°C. Using 30 sec as an intermediate value, we assigned rate constants outside the Schwann cell region of aN = 1.4 min-1 and PN = 0.001 min-1. (The half-time gives the sum of the rate constants, and we made the assumption that the reverse reaction rate was much smaller than the forward rate.) Phosphorylation kinetics were similar (Mooney and Anderson, 1989), and we chose for values under the Schwann cell, aSC = 0.001 min-1 and PSC = 1.0 min-1. The longitudinal growth of Schwann cells was approximated by fitting a single exponential function to the data in Fig. 4 (initial length, 83 |m; final length, 274 |m; t = 17 days). The equations were solved numerically using the method of Euler (Simon, 1986). Accuracy was determined first by varying the distance resolution and the time step. Solutions were stable with x-divisions of 0.2 |m and time steps of 0.15 sec. Final calculations were done with x-divisions of 0.1 |m and time steps of 0.015 sec, but these varied by less than 1% from the former. Further, if the chemical reaction were eliminated, solutions of the diffusion equation alone agreed with the analytical solution (Crank, 1975) within 0.1%.

In Fig. 6B the profile of Na+ channel density at the right edge of the Schwann cell is plotted for several times ranging from 1 to 24 hours. An example of typical new clusters of Na+ channels forming during remyelination at 13 dpi is shown in panel F. It is not possible to know with certainty the time that has elapsed between the differentiation of a particular Schwann cell to a myelinating phenotype and the appearance of adjacent Na+ channel clusters. However, almost all MAG-positive Schwann cells are associated with clusters detected by immunofluorescence during both development and remyelination, and channels must therefore accumulate within hours (Vabnick et al., 1996). Cluster formation is predicted by the model to be correspondingly fast, with significant gradients in density appearing rapidly. However, the shapes of the calculated density profiles do not match the observed immunolabeled sites, especially at shorter times. While some zones of high Na+ channel immunofluorescence appear to have a gradient in density, the great majority are more sharply defined, as in Fig. 6F. The model, on the other hand, predicts for short times a

118 122 126 130 134 138 122 124 126 128 130 126 128 130 132 134 Distance (.Lim) Distance (^m) Distance ((j/n)

£ 24 hours, 1.4 min"1 24 hours, «= 5.6 min-1

£ 24 hours, 1.4 min"1 24 hours, «= 5.6 min-1

Figure 6 Computational model of Na+ channel clustering. The diagram in (A) illustrates the configuration. The x-axis below the sketch gives the distances at time zero. The curves in graph (B) plot Na+ channel density vs. distance along the axon at 1, 2, 4, 6, 12, and 24 hours after the Schwann cell reaches a myelinating state. Since the Schwann cell grows longitudinally according to the curve in Fig. 4, the position of the right edge changes with time. (C and D) Result of varying the rate constant aN at 6 and 24 hours, respectively. The black curves repeat the calculations in (B) on an expanded x-axis. The blue and red curves show results with aN = 2.8 and 5.6 min-1, respectively. (E) Representations of the 24-hour curves with aN = 1.4 min-1 (left) and 5.6 min-1 (right), converted to a gray scale of 15 to 230, which mimics the green immunofluorescence intensity range in the cluster in (F). (F) A cluster of Na+ channels (green) at the tip of a remyelinating Schwann cell process (MAG, red) 13 days after injection of lysolecithin. (C, D, E, and F) are all scaled to the same distance calibration.

Figure 6 Computational model of Na+ channel clustering. The diagram in (A) illustrates the configuration. The x-axis below the sketch gives the distances at time zero. The curves in graph (B) plot Na+ channel density vs. distance along the axon at 1, 2, 4, 6, 12, and 24 hours after the Schwann cell reaches a myelinating state. Since the Schwann cell grows longitudinally according to the curve in Fig. 4, the position of the right edge changes with time. (C and D) Result of varying the rate constant aN at 6 and 24 hours, respectively. The black curves repeat the calculations in (B) on an expanded x-axis. The blue and red curves show results with aN = 2.8 and 5.6 min-1, respectively. (E) Representations of the 24-hour curves with aN = 1.4 min-1 (left) and 5.6 min-1 (right), converted to a gray scale of 15 to 230, which mimics the green immunofluorescence intensity range in the cluster in (F). (F) A cluster of Na+ channels (green) at the tip of a remyelinating Schwann cell process (MAG, red) 13 days after injection of lysolecithin. (C, D, E, and F) are all scaled to the same distance calibration.

sharp peak followed by a gradual decrease in density with distance from the glial edge (for example, the 6-hour curve in Fig. 6B), and even at 24 hours the shapes do not match well. This can be seen in Fig. 6E (left) in which the curve at 24 hours in B has been converted to a gray scale plot for comparison with the immunolabeled image (the distance scales in Fig. 6C, D, E, and F are identical).

To judge which property of this system might be responsible for the deficiencies in the fit, we tested the sensitivity of the model to several parameters. NrCAM and NF186 seem to precede Na+ channels at nascent nodes (Lambert et al., 1997; Custer et al., 2003). At E21 there are relatively few identifiable nodal sites, but at 60% of these, NrCAM was found in the absence of Na+ channels (Custer et al., 2003). Further, in NrCAM null mutant mice, Na+ channel clustering at nodes is delayed by several days, but reaches normal levels by P10. Binding of NrCAM to ankyrinG would be dependent on similar phosphorylation/dephos-phorylation reactions, and clustering mechanisms could thus be similar. Since this protein is smaller than the Na+ channel a-subunit, it might diffuse more rapidly and thus cluster sooner. Could the early presence of these ankyrinG

binding proteins then speed the immobilization reaction of Na+ channels either by concentrating an essential phosphatase or by association with ankyrinG, which has multiple membrane-binding domains? In either case, the rate constant for Na+ channels to bind at the Schwann cell edge (aN) would rise. Results of two- and fourfold increases in aN are shown in the blue and red curves, respectively, in Fig. 6C and D. At 6 hours (Fig. 6C), Na+ channel clustering is predicted to be significantly stronger and more focal. At 24 hours (Fig. 6D), the major change is in the cluster shape. Both provide better fits to the immunofluorescence, and the fourfold 24-hour curve is translated to a gray scale plot in Fig. 6E (right). In contrast, results were only minimally sensitive to fivefold changes (up or down) in PN, and were virtually unaffected by fivefold changes in aSC or PSC because diffusion is much slower than the rate at which channels are released from their cytoskeletal link after conversion of the Schwann cell to a myelinating state. We also modeled the fusion of two clusters to form a node by including two Schwann cells separated by an initial gap of 12 |im. The density profiles matched immunostaining observations quite well (not shown). It should be emphasized that the idea behind development of a computational model is not to prove that a suggested mechanism is correct, but rather the opposite: Should the hypothesis be rejected for lack of consistency with the available data? In this case, the proposed mechanism seems not to be inconsistent with the calculated predictions.

Diabetes Sustenance

Diabetes Sustenance

Get All The Support And Guidance You Need To Be A Success At Dealing With Diabetes The Healthy Way. This Book Is One Of The Most Valuable Resources In The World When It Comes To Learning How Nutritional Supplements Can Control Sugar Levels.

Get My Free Ebook


Post a comment