04_-_Diffusion - Quantitative Physiology I Molecular and...

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Quantitative Physiology I / Molecular and Cellular Systems, BMEN E4001x Notes 04 - Diffusion Chapter 4 of Nelson Diffusion dominates at molecular and cellular scales: Reynold’s number η ρ vL = Re v=average velocity L=characteristic length ρ =media density η =dynamic viscosity (1 poise = 1g*cm -1 *sec -1 ) For a fish (10 cm long) swimming in water (100 cm/sec) Re=10 5 For a bacterium (10 -4 cm long) in water (10 -3 cm/sec) Re=10 -5 So, at molecular and cellular scales, turbulence does not significantly assist diffusion (conduction) in transport. Similarly, convection and radiation are not too important. Key ideas: Cover two views of diffusion, a discrete, step model and a continuum model. Understand two different regimes; motion of a single molecule and how far it goes and understand transport. We will specifically focus on steady state solutions, given their ability to yield simple solutions. Two models of 1D diffusion: This is based on Chapter 4 of Nelson, Biological Physics. I have a few copies that can be lent out in my office, and it should be on reserve at the Engineering Library. Discrete model of diffusion – 1D random walk This is based on the idea that particles have some kinetic energy, represented as thermal energy. They thus have a certain speed, and, based on the number of particles on average in a given volume, a certain free path length. 0 - L - 2 L L 2 L x Following a few particles, we get a larger distribution that starts in one position, and spreads symmetrically. How can be get at that quantitatively? For any given timestep N, the molecule can move either left or right. Encode this as k N = ± 1. x N =x N-1 +k N * L First off, <x N >=0. L * k x x N 1 N N + = - So, the average molecule doesn’t move at all.
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The rest of the distribution looks like a binomial distribution, which reflects the idea that to reach far extents, the molecules had to hop left or right with every step. There are multiple ways to reach intermediate points. As an important measure of diffusion-based spreading, look at <x 2 >: ( 29 ( 29 ( 29 ( 29 ( 29 2 N * N 1 N 2 1 N 2 N 1 N 2 N L k L k * x 2 x L k x x + + = + = - - - last term = L 2 , as <k N 2 >=1 middle term = 0 so... ( 29 ( 29 ( 29 2 N 2 1 N 2 N L k x x + = - or .... ( 29 2 2 N L * N x = Interpret as the RMS distance increases linearly with time. Together with <x N >=0, really, truly points out that these parameters correspond to how a population of molecules, and not individual molecules, behaves. Rewrite as: ( 29 t N t t L D Dt L N x N = = = = * ; 2 ; 2 * 2 2 2 From this, we don’t need to know t or L; the parameter D can be measured from experiment. Continuous model of diffusion – 1D Fick’s law Break the space up into slabs of width L, sides Y and Z, each centered on x=0,L,2L,3L, .... For a given time interval t, all the particles in the slab move either left or right, with equal probability (free diffusion). Choose t such that we only need to examine one crossing, and no particles cross over two boundaries. In the diagram, more particles move from x=0 to
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x=L than from x=L to x=0, simply because there are more particles in x=0 than in x=Lf.
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