2- Bacteria - Plasma Membrane - Diffusion

2- Bacteria - Plasma Membrane - Diffusion - BME 418...

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BME 418, Quantitative Cell Biology Alan J. Hunt Diffusion in one dimension For simplicity assume δ is a constant increment of movement that occurs in a time interval τ . The probability of a step in the negative direction is the same as in the positive: P(- δ ) = P(+ δ ) = ½ And the expected mean position after n steps ( t = n τ ) is: 0 ) ( = n x What about the absolute displacement, that is: 2 1 2 ) ( n x ( 29 2 2 2 2 ) 1 ( 2 ) 1 ( ) 1 ( ) ( δ δ δ + - ± - = ± - = n x n x n x n x Now this can have many outcomes depending on the value of first and second terms. We can compute the average over these possible outcomes as: [ ] = + - ± - = N i i i n x n x N n x 1 2 2 2 ) 1 ( 2 ) 1 ( 1 ) ( δ δ Since it is equally probable that the particle will move either direction, the second (cross) term will average out to zero giving us: [ ] = + - = N i i n x N n x 1 2 2 2 ) 1 ( 1 ) ( δ Or 2 2 2 ) 1 ( ) ( δ + - = n x n x i We can evaluate this by inductive reasoning. Since, x (0) = 0, then 0 ) 0 ( 2 = x . Thus: 2 2 ) 1 ( δ = x , 2 2 2 ) 2 ( δ = x , …….
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  • Winter '08
  • HUNT
  • Quantitative Cell Biology, Alan J. Hunt

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