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Not Just Diffusion Bio435 Diffusion with Dri3 Mean and Variance - - PowerPoint PPT Presentation
Not Just Diffusion Bio435 Diffusion with Dri3 Mean and Variance - - PowerPoint PPT Presentation
Not Just Diffusion Bio435 Diffusion with Dri3 Mean and Variance Mean Total steps N=t/ t Displacement x i , i=1,2,N Total displacement x tot = x 1 + x 2 + x N < x>=a*k + t + (a)*k t + (0)*(1k +
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Mean and Variance
Mean Total steps N=t/Δt Displacement Δxi , i=1,2,…N Total displacement Δxtot=Δx1+Δx2+…ΔxN <Δx>=a*k+Δt + (‐a)*k‐Δt + (0)*(1‐k+Δtk‐Δt) = a(k+ ‐k‐)Δt Variance (avg. sq. displacement) var<Δx2>=a2*k+Δt + (‐a)2*k‐Δt ‐ <Δx2>= 2a2(k++k‐)Δt Since k±Δt<<1 and <Δx2><<<
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Velocity
For low Reynold’s Numbers F = γv γ = FricVonal coefficient p(x, t+Δt) = …
v = Δx Δt = a k+ − k−
( )
∂p(x,t) ∂t = −v ∂p(x,t) ∂x + D∂ 2p(x,t) ∂x 2
Dri3 velocity v = a(k+ ‐ k‐) Diffusion coefficient D= (k+ + k‐)a2/2
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Einstein RelaVon: Macroscopic
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Einstein‐Smoluchowski RelaVon
Einstein, A. (1905), "Über die von der molekularkineVschen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen", Annalen der Physik 17: 549–560 Smoluchowski, M. (1906), "Zur kineVschen Theorie der Brownschen Molekularbewegung und der Suspensionen", Annalen der Physik 21: 756–780
D = kBT 6πηr
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Monomer Diffusion to Capture by Polymer
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ChemorecepVon as Diffusion to Capture
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Ligand ConcentraVon Profile
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Uptake Rate of Ligand of Perfect Receptor
a=radius of spherical cell c0=far field concentraVon D=diffusion coefficient
dn dt = 4πDac0
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Uptake Rate of Ligand with Imperfect Receptor
a=radius of spherical cell c0=far field concentraVon D=diffusion coefficient kon = finite absorpVon rate of receptor M = no. of surface bound receptors
dn dt = Mkonc(a)
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For Imperfect Receptors
At the cell surface c(a) =
c0 1+ Mkon 4πDa
( )
Mkon 4πDa >>1 Mkon 4πDa <<1
When c(a)=0 When c(a)=c0
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ImplicaVons
When receptor number increased, nothing added to absorb ligands What is minimal receptor number to mimic perfectly absorbing surface? Solving
dn dt = M c0 1+ Mkon 4πDa
( )
M = 4πDa kon
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FRAP: Measuring GFP mobility in E. coli
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Principle
- Prebleach image
- Bleach
- Postbleach image series
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PreparaVons
- Collect 10 ml sterile LB tubes from CA
- Bacteria E. coli K12 with pGFP ampicillin resistant
- Grow in 50 ug/ml Ampicillin (stock 100 ug/ml)
- Monitor O.D.600nm
- At OD=0.6, add final conc. 100 uM IPTG to induce
- verexpression of GFP (stock 100 mM)
- Use cephalexin to lengthen cells 20 min before
removing culture
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- E. Coli K12 with pGFP
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Analysis
Alignment of the images (only necessary if the regions of interest moved over Vme). 1. Fluorescence intensity quanVficaVon (obtaining the raw data) 2. Background subtracVon 3. CorrecVons due to laser fluctuaVons, photobleaching during acquisiVon (postbleach) and total fluorescence loss caused by the bleaching step 4. NormalizaVon 5. Mobile/immobile fracVon 6. T½ hal3ime of the equilibraVon of bleached and unbleached molecules 7. TheoreVcal models to addiVonally determine binding characterisVcs of the analysed molecule
IN = IT II
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Recovery Profile
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Measure
- Bleached area
- Unbleached area
- Neighbouring cell
- Background (no cell)
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Data Fiwng
C0=depth of bleach at Vme t=0 R0=IniVal half‐width of bleach D=diffusion coefficient t=Vme point
C = C0 ⋅ R0 R0
2 + 8Dt
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ObservaVons and Results
- Diffusion coefficient
- Mobile fracVon
- TheoreVcal diffusion coefficient of GFP
- Error bars
- Intrinsic bleach rate
- Other model fits to experiment
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NEXT
- Assignment 3: diffusion to capture
- FRAP experiment 16‐March
– 17th March culture E. coli GFP – 18th March microscope in 6 batches – 18th March onwards: data analysis – 24th March discuss results
- Crowding
- Cytoskeletal dynamics
- Molecular Motors and their dynamics
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