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Today: Diffusion Why x2 = #Dt (from Equipartition Function) When directed motion (v ≈ constant, x = vt) is better/worse than diffusion (v not constant) depends on how far you have to move. short distances, diffusion of small molecules very good. Biological examples Bacterial vs. Eukaryotic Cells Oxygen transport: how close cells need to be to Oxygen in blood in Lungs Stopping time of Bacteria. Diffusion Move with small kinetic energy because have Edeg. freedom= ½ kBT. http://en.wikipedia.org/wiki/Diffusio n#mediaviewer/File:Blausen_0315 _Diffusion.png http://commons.wikim edia.org/wiki/File:Che mical_surface_diffusio n_slow.gif#mediaview er/File:Chemical_surfa ce_diffusion_slow.gif Edeg. freedom= ½ kBT = ½ kx2 = ½H Omv2 H2O m 2 Diffusion For “small” things, diffusion is a great way to get around. For somewhat larger things, need directed motors. Inertia does not matter for bacteria or anything that is small / microscopic levels. What is velocity of water molecule at room temperature? tcollision = ?? Diffusion: x2 = # Dt Diffusion as a Random Walk 1-D case (first) Particle at x = 0 at t = 0 1.Assume equally likely to step to right as step to left. 2. Takes steps of length L every t seconds i.e. moving with velocity between collisions ±v (L = ±vt) R steps/sec; total of N steps [For now take v, t as constants : they actually depend on size of particle, nature of fluid, temp…] Where <x2> is the average distance squared moved, D: Diffusion constant (measured in length2/ time) L: size of step in time In reality, there is a distribution of step sizes, but this model works amazingly well. In m-dimensions: <DrN2> = 2mDt (H.W.) If in 1 sec, it has gone q distance, then in 2 sec it’s gone? As a function of time, width increases as the √t What values for D? Diffusion Coefficient & Brownian Noise Einstein–one of three 1905 papers, each which should have received a Nobel Prize Stokes-Einstein Equation D = kbT/6phr = kbT/f h = viscosity (1 centipoise for water) r = radius of bead f = frictional coefficient True where “Reynolds number” is low; Flow is sufficiently slow that don’t have eddies, vortexes…is characterized by smooth, constant fluid motion…flow is laminar, where viscous forces are dominant. The Reynolds number is defined as the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions. In contrast turbulent flow occurs at high Reynolds numbers; Where flow is dominated by inertial forces, which tend to produce chaotic eddies, vortices and other flow instabilities D = 250 um2/sec for small molecule in water Life at low Reynold’s Number (Where Einstein’s Equation for D holds) Caramel: no turbulence (High) Reynold’s number Neurons: Signals transmitted via synapses. Your brain: 100 billion neurons, 100 trillion synapses Information flow Pre-synaptic Bouton Axon Synapse (30-100 nm) Post-synaptic Spine Axon Valtschanoff & Weinberg, 2003 Dendrite How long to cross a synapse? D = 250 mm2/sec Nerve synapse: 0.1 mm <x2> = 2Dt 0.01 mm2 = (2)(250 mm2/sec)t t = 20 msec (fast!) Diffusion is fast enough to go across narrow synapse D: diffusion constant, <x2> = 6 Dt If molecule gets bigger, x and D or D small molecule , e.g. O2 = 1000 um2/sec: (D=1/f) Dsucrose = 300 um2/sec (D=1/f) How long does it take for O2 to go from the edge of a cell to the middle? How big cell? 20 um <x2> = 6Dt t = <x2>/6D = 16 msec Ultimately limits the maximum metabolic rate. Bacterial cell ~ 1-3um in size, eukaryotic 10-50 um Metabolism of bacteria much higher than eukaryotes. Size of eukaryotes limited by size (diffusion time of O2). As size gets bigger, everything happens more slowly. Large cell: frog oocytes– basically everything happens slowly. Every cell needs to be within 50-100 mm of blood supply! Oocyte:1-2 mm! If capillary is small enough (and lining relatively free to move O2 and CO2) diffusion works well. How Bacteria move Inertia doesn’t matter for microscopic world Life at low Reynold’s number Why study? 1.Simple Example of F= ma 2.Doesn’t need much biology 3.Results are broadly applicable to microscopic level. Go ~ 25 um/sec: # body lengths/sec? 10 body length/sec Compared to you walking? 4 miles/hr = 6ft/sec = 1bl/sec Compared to you swimming? 50 m/s ~ min ~ m/min ~ ½ bl/s Bacteria are good swimmers! If turn off “propeller,” how far Bacteria coast? F = ma What forces are left on bacteria? ma = mdv/dt – Ffriction (drag) = -gv g = drag coefficient v linear in v (low Reynold’s #) What is drag coefficient? What does it depend on? a) Goopiness of fluid – h = viscosity b) Dimension of object – bigger object, harder to move Fdrag = chrv = 6phrv : g = 6phr r = radius, v = velocity c= constant Remember Stokes-Einstein Equation D = kbT/6phr = kbT/f h = viscosity (1 centipoise for water) r = radius of bead f = frictional coefficient Fdrag = fv Solve eq’n of motion: m dv/dt = -gv Units of g? (m/v) (v/t) = m/t (good) What is mass of bacteria (can you estimate?) 4/3 pr3r = 4 x 10-15kg What units of g = 6phr R= 10-6 meters; h = 0.001 g = 20x10-9 N-s/m = 20 nN-s/m Plugging in the #’s m = 4 x 10-15kg g = 20 nN-s/m t = m/g = 0.2 msec So bacteria stops in 200 nsec—very fast! Once forces are turned off, bacteria forgets about history very quickly! History doesn’t matter to bacteria. Inertia is completely irrelevant for bacteria How far does bacteria coast in 0.2 usec? Inertia is irrelevant to bacteria. Once force is over, no forward motion! Scaling up: What about a person swimming? A good swimmer coasts about 1 body length Inertia is much more important for bigger organism Size of drag force on bacteria? (compared to it’s weight? Bacteria swim as if dragging 10x their own weight! Class evaluation 1. What was the most interesting thing you learned in class today? 2. What are you confused about? 3. Related to today’s subject, what would you like to know more about? 4. Any helpful comments. Answer, and turn in at the end of class.