Kinetic limits on sensors imposed by rates of diffusion, convection (delivery by flow), and reaction (binding to capture molecule) Basic idea – most sensors flow sample by sensing surface If flow is slow compared to diffusion and binding, region near sensor surface gets depleted of analyte, which slows rate of detection. What would analyte distribution look like at equilibrium? We focus on transient steady-state before most receptors bind analyte

If flow is fast, portions of the sample are never “seen” by the detector. How can one match these rates for best operation in particular applications? How can one estimate analyte conc. near surface? These issues are crucial in real-time sensing (SPR) and when dealing with very low concentration analyte.

Main points to cover 1.Relationships between flow rate Q, fluid velocity U, channel dimensions H and W, pressure P, viscosity  2.How big is depletion region at low flow rates 3.How big is depletion region at high flow rates 4.How much does depletion slow approach to equil.

1.Relations between Q, W, H, U, P, , etc. Common sensor geometry Q [vol/s] = area x average velocity = W H U Total flux [molec/s] = Q x c x area Fluid velocity profile is parabolic with z: u(z) = 6 U z(H-z)/H 2 How would you expect Q to vary with P, W, H, L,  ~ P W H / L  Q= (H 2 /12)PWH/ L  z L

2. How big is depletion region d (roughly) at low flow rates? If flow slows, how does d change? If flow increases, how does d change? At equilibrium, total flux from convection Q c 0 = total flux from diffusion (D (c 0 – 0)/d ) H W => d = H W D / Q d/H = nice dimensionless quantity (for fluid mech!) = 1/Pe H, when d >> H, Pe H <<1 If d << H, does this way of measuring flux make sense? d c0c0

3. How big is depletion region  at high flow rates (e)? Steady-state: time to diffuse  = time to flow over L  2 /D = L/u(  ) u(  ) = 6Q  /WH 2 for  <<H ->  /L = (DH 2 W/6QL 2 ) 1/3 = (1/Pe s ) 1/3

4. How much does depletion slow approach to equil. Estimate c S = conc at which rate molecules diffuse across  from c 0 to c s = rate at which they bind to receptor J D = area x D x (c 0 – c S )/ d S = LW S D (c 0 – c S ) (Pe S ) 1/3 / L J R = k on x c S x # free receptors on surface = k on c S (b m -b) LW S initially, all receptors are free (b=0) c S /c 0 = 1/(1 + k on b m L/D(Pe S ) 1/3 ) = 1/(1+Da) “Damkohler” #, Da More simply, when Da >> 1, Da = c 0 /c S

Usefulness of Damkohler # If binding kinetics are limiting,  eq =  rxn = k off -1 /(1+c 0 /K D ) If transport is limiting (Da>1),  eq = Da  rxn (slower) You can derive this by estimating  eq = time it takes for b m [c 0 /K D /(1+ c 0 /K D )] Area molecules to bind analyte when they bind at rate k on c s b m Area Caveat: formula for  rxn is underestimate when c 0 is so low that at equil. less than 1 receptor molecule binds analyte Da >>1 means that when c 0 is low, c s = c 0 /Da much lower and you enter the regime where t rxn needs correction

Ways to get around some of these limits: Decrease k off, so sensor never “releases” a captured tgt e.g. bind analyte with more than 1 receptor Keep mixing sample to reduce depletion zone; hard to do on micro scale where all flow laminar Use some force to concentrate analyte at sensor surface (magnetic, electrophoretic, laser trap force), i.e. move it faster than diffusion

Summary of useful formulas for this course x rms = (6Dt) 1/2 D [m 2 /s] D = k B T/6  r (Stokes-Einstein) k B T = 4x J = 4pNnm at room temp  viscosity) = Ns/m 2 for water j D [#/(area s)] = D (  c/  x)  (Fick) Pe H = Q/W C Dd/H = 1/Pe H when Pe H < 1 Pe S = 6 2 Pe H  = L/H d S /L = 1/(Pe S ) 1/3 b(t)/b m -> [c 0 /K D /(1 + c 0 /K D )](at steady state)  rxn = k off -1 /(1+c 0 /K D ) K D = k off /k on [M] Da = k on b m L/D(Pe S ) 1/3  eq = Da  rxn when Da>1 k on typically  10 6 /Ms for proteins F drag = 6  r*velocity flow between parallel plates: Q=H 3 W C P/12  L velocity near surface = 6Qz/W C H 2

Next class – microscale cantilever – measure mass of captured analyte Basic idea – put flow cell inside cantilever and operate the cantilever in air or vacuum to minimize drag Will use Manalis formulae to estimate mass transport characteristics! Read: Burg et al (Manalis lab!) Weighing biomolecules… Nature 446:1066 (2007)