Chapter 1. Introduction to a random walk simulation for amperometric recordings of quantal neurotransmitter release

This chapter is modified from our review article on presynaptic mechanisms that regulate quantal size (Sulzer and Pothos, 2000).

The diffusion of molecules is surprisingly non-intuitive. Crank's classic volume on diffusion (Crank, 1975) is useful (and challenging reading) for examples of how to derive formulae to describe diffusion under different conditions. One such formula was adapted by Robert Chow to estimate the relationship between distance and amperometric spike shape (Chow and von Ruden, 1995). Nevertheless, the complexity of deriving the appropriate formulae as conditions are altered is a drawback to their use.

In contrast, the random walk, discussed by H.C. Berg (Berg, 1983), provides the same answers but is simpler to manipulate. This approach was used by Schroeder et al. from Mark Wightman's group in a study on amperometric spike shape and fractional recovery (Schroeder et al., 1992). An advantage of a random walk simulation we have adapted to be easily performed on spreadsheets (even by biologists!), as shown below. We have some sample Excel files attached you can use to begin with.

The following chapters show how to extend this to amperometry and cyclic voltammetry recordings in brain slice or in vivo. What follows immediately is a first lesson on using a random walk simulation, using a "numerical analysis" for quantal recordings with amperometry, which helps in the more complex analyses to follow.

A spreadsheet modeling diffusion in one dimension over time (t) can be made with columns representing distance and rows representing time intervals (see example below). Since the vectors of diffusion are random, for a population of molecules diffusing in one dimension, one half will move to the right and one half will move to the left. Therefore, the cells in a row t (x+1) equal the average of the cells in from the two neighboring columns in row t (x).

The release site in this example is modeled as a reflecting surface(column a). In the spread sheet below, a reflecting surface on the left-hand side receives half of the molecules from its neighboring column and donates half of its molecules to its neighboring column. This is simulated by making a cell at t (x+1) the average of itself and its neighbor at t (x).

For modeling the electrode surface in amperometric recordings, we assume a consuming surface, as catecholamines are oxidized and do not return to a reduced state. A consuming surface (column g) destroys all the molecules that encounter the surface at t (x), and at t (x+1) receives half of the molecules from the cell in its neighboring column (column f) at t (x). The cell in the column next to the electrode (column f) in row t (x+1) receives only half of the molecules from the cell in its other neighboring column (column e) in row t (x), and no molecules from cells in the electrode column (column g). "avg" is an averaging function, the columns represent distances, and the rows iterations (time). The dimensions of the distance axis can be set arbitrarily. More columns representing smaller distances will yield greater accuracy and smoother shapes.( In this case we smooth the values in column g by averaging 4 iterations.)

In the following example, 7000 molecules are released from the reflecting column a at t = 0. The diffusion of the molecules is shown for 0.5 µm bins and 20 iterations: The relationship between the iterations on the spread sheet and time can be determined from the diffusion coefficient (D) for the respective molecule and medium. The diffusion coefficients for classical transmitters in water are available from standard references and most small hydrophilic molecules like classical neurotransmitters are pretty similar.

The relationship between time (t) and distance (x) is: t = x^2/2D

For bins of 0.5 µm and the apparent diffusion coefficient D for dopamine in water at 34 degrees Celsius (6.9 x 10^-6 cm^2/s, Nicholson, 1996) the time t represented by each row is

t = (0.5 x 10^-5 cm)^2 / (2 x 6.9 x 10^-6 cm^2/sec) = 0.182 ms

The amperometric current can be converted into 'number of molecules' by applying Faraday's law, which for dopamine results in 1 pCoul (=pA s)= 3.1212 x 10^6 molecules (if one assumes that two electrons are donated per dopamine molecule. This appears accurate at physiological pH for events shorter than ~ 1 sec. For longer lasting events 4 electrons can be donated. See our review for further information.)

For 182 µs time intervals as in our example, this would correspond to

3.1212 x 10^6 molecules/pA s x 182 x 10^-6 s = 568 molecules / pA

Now the iterations can be expressed as time intervals and the number of molecules as current, so that genuine amperometric recordings can be compared with simulations.

References

Berg HC. (1983). Random Walks in Biology (Princeton, NJ: Princeton University Press).

Chow RH, von Ruden L. (1995). Electrochemical detection of secretion from single cells. In Single-channel recording, B. Sakmann and E. Neher, eds. (New York: Plenum Press), pp. 245-276.

Crank J. (1975) The Mathematics of Diffusion (Oxford, UK: Oxford University Press).

Nicholson (1996) Diffusion of albumins in rat cortical slices and relevance to volume transmision. Neurosience 75:839-847.

Schroeder TJ, Jankowski JA, Kawagoe KT, Wightman RM, Lefrou C, Amatore C (1992). Analysis of diffusional broadening of vesicular packets of catecholamines released from biological cells during exocytosis. Analytical Chemistry 64: 3077-3083.

Sulzer D, Pothos EN (2000). Presynaptic mechanisms that regulate quantal size. Reviews in the Neurosciences 11: 159-212.