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 analytedistribution look likeat equilibrium?

We focus on transientsteady-state before mostreceptors bind analyte

If flow is fast, portions of the sample are never “seen” bythe 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 h

2. How big is depletion region at low flow rates3. How big is depletion region at high flow rates4. How much does depletion slow approach to equil.

1. Relations between Q, W, H, U, P, h, etc. Common sensor geometry

Q [vol/s] = area x average velocity = W H UTotal flux [molec/s] = Q x c x areaFluid velocity profile is parabolic with z: u(z) = 6 U z(H-z)/H2

How would you expect Q to vary with P, W, H, L, ?h

~ P W H / L h Q= (H2/12)PWH/L h

z

L

2. How big is depletion region d (roughly) at low flow rates?

If flow slows, howdoes d change?

If flow increases,how does d change?

At equilibrium, total flux from convection Q c0 = total flux from diffusion (D (c0 – 0)/d ) H W => d = H W D / Qd/H = nice dimensionless quantity (for fluid mech!)

= 1/PeH, when d >> H, PeH <<1If d << H, does this way of measuring flux make sense?

dc0

3. How big is depletion region d at high flow rates (e)?

Steady-state:time to diffuse d = time to flow over L

d2/D = L/u(d)u(d) = 6Qd/WH2 for

d<<H -> d/L = (DH2W/6QL2)1/3

= (1/Pes)1/3

4. How much does depletion slow approach to equil.

Estimate cS = conc at which rate molecules diffuse across d from c0 to cs = rate at which they bind to receptor

JD = area x D x (c0 – cS)/dS = LWSD(c0 – cS)(PeS)1/3/L JR = kon x cS x # free receptors on surface = kon cS (bm-b) LWS

initially, all receptors are free (b=0)

cS /c0 = 1/(1 + konbmL/D(PeS)1/3) = 1/(1+Da)

“Damkohler” #, Da

More simply, when Da >> 1, Da = c0 /cS

Usefulness of Damkohler #

If binding kinetics are limiting, teq = trxn = koff-1/(1+c0/KD)

If transport is limiting (Da>1), teq = Da trxn (slower)

You can derive this by estimating teq = time it takes forbm [c0/KD/(1+ c0/KD)] Area molecules tobind analyte when they bind at rate kon cs bm Area

Caveat: formula for trxn is underestimate when c0 is so low that at equil. less than 1 receptor molecule binds analyte

Da >>1 means that when c0 is low, cs = c0/Da much lowerand you enter the regime where trxn needs correction

Ways to get around some of these limits:

Decrease koff, so sensor never “releases” a captured tgte.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 sensorsurface (magnetic, electrophoretic, laser trap force), i.e. move it faster than diffusion

Summary of useful formulas for this course

xrms = (6Dt)1/2 D [m2/s]D = kBT/6phr (Stokes-Einstein)kBT = 4x10-21J = 4pNnm at room temp(h viscosity) = 10-3Ns/m2 for water

jD [#/(area s)] = D (Dc/Dx) (Fick)PeH = Q/WCD d/H = 1/PeH when PeH < 1PeS = 6l2PeH l = L/H dS/L = 1/(PeS)1/3

b(t)/bm -> [c0/KD /(1 + c0/KD)] (at steady state)trxn = koff

-1/(1+c0/KD) KD = koff/kon [M]Da = konbmL/D(PeS)1/3 teq = Da trxn when Da>1kon typically ~106/Ms for proteinsFdrag = 6phr*velocity flow between parallel plates:Q=H3WCP/12hL velocity near surface = 6Qz/WCH2

Next class – microscale cantilever – measure mass of captured analyte

Basic idea – put flow cell inside cantilever and operatethe cantilever in air or vacuum to minimize drag

Will use Manalis formulae to estimate mass transportcharacteristics!

Read: Burg et al (Manalis lab!) Weighing biomolecules… Nature 446:1066 (2007)