Brownian Motion in the Brain: Looking at the Sodium Flow During an Action Potential

By: Chelsea Gottschalk

In class, we touched on Brownian motion, and how the flow of diffusion can be modeled by the equation:

Q = Dm/Dt = DA ((C2-C1)/L), where

Q= flow rate (Kg/m³)

D= coefficient of diffusion (m²/s)

A= cross-sectional area of substance (m²)

C₂= concentration of substance on the outside of diffusion barrier (Kg/ m³)

C₁= concentration of substance on inside of diffusion barrier (Kg/ m³)

            I thought it would be interesting to determine the mass of sodium that moves into a neuron during one action potential. It is no doubt a very small number, but it still represents an action that keeps us alive.

            Nerve tissue in the brain is one of the few excitable tissues in the body, a property that lends itself to fast communication within the vast network of neurons in the CNS as well as structures in the PNS. Signals are propagated throughout this network via action potentials, which start when a signal binds to a receptor on the dendritic surface, allowing some positive ions to flow into the neuron. This brings the negative resting membrane potential (-70 mV) up to its threshold of about -55 mV, which causes voltage-sensitive Na channels to open, leading to a massive influx of sodium (moving down its electrochemical gradient) into the neuron. This influx reaches a maximum of about 35 mV before potassium ion channels open, and positive charge moves out of the cell, repolarizing the neuron and preparing the cell for more signals.

      When sodium moves down its gradient, a large amount of ions flow into the neuron very rapidly. Since this is passive facilitated diffusion (no input from ATP), we can use the equation above to calculate the flow of Na into the cell. Interestingly enough, I found the diffusion coefficient for sodium in a rat brain, which is probably very close to that in humans. This value is D= 1.15 mm²/ms. The average thickness of a neuronal plasma membrane is 7 nm, and the atomic radius of Na is 180 pm. Additionally, the extracellular concentration of Na is 14mM, the intracellular concentration is 12mM, and action potentials last only about 1 ms. We can assume that extracellular sodium is right next to the plasma membrane, so the only distance that it must diffuse is across the membrane. First, let’s convert our units into SI units.

D= 1.15 mm²/ms : 1.0 x 10^-12/mm² : 1000 ms/1 sec = 1.15 x 10^-9 m²/s

L= 7 nm : 1.0 x 10^-9 m/ 1 nm = 7 x 10^-9 m

R= 180 pm : 1.0 x 10^-12/ 1 pm : 1.8 x 10^-10 m

C: 1 mM : 1 M/1000mM : moles/ L : 1 L/ .001 m³ : 22.99 g/ 1 mol Na : 1 Kg/1000 g

C _{extracellular}= 140 mM = 3.22 Kg/ m³

C _{intracellular}= 12 mM = 0.276 Kg/ m³

Now for the calculation:

Q= (1.15 x 10^-9 m²/s)(p(1.8 x 10^-10 m)²)(( 3.22 Kg/ m³ - 0.276 Kg/ m³)/ 7 x 10^-9 m)

Q= 4.9 x 10^-20 Kg/s

Since Q = Dm/Dt, and we know that an action potential lasts approximately 1 ms, or .001 s, then we can solve for the mass of sodium that moves across the dendritic membrane during an action potential.

Dm/.001s= 4.9 x 10^-20 Kg/s

Dm= 4.9 x10^-23 Kg of Na moved during one action potential!

This value may seem incredibly small, and it is. It only takes a very small percentage of the total extracellular concentration of sodium to flow into the neuron for the membrane potential to reach threshold. That is why a neuron could fire for several hundred or a thousand times (without a working Na/K pump) before a noticeable change in concentration would occur.

Although Q= A x v, this is not a closed system, so I do not think we can accurately determine the area over which this diffusion occurs (especially since the surface area of dendrites is so great), meaning that we can not determine the velocity of the sodium ions.

Reference:

Goodman, J. A., Kroenke, C. D., Bretthorst, G. L., Ackerman, J. J. H. and Neil, J. J. (2005),                   Sodium ion apparent diffusion coefficient in living rat brain. Magn Reson Med, 53: 1040–1045. doi: 10.1002/mrm.20444