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MAE6291 Class 2 ELISA (Enzyme-linked immune sandwich assay) Binding kinetics and equilibrium Brownian motion and diffusion Surface plasmon resonance (SPR) sensors. monoclonal or polyclonal binds diff. epitope than capture ab may be directly conj. to enzyme

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MAE6291 Class 2

ELISA (Enzyme-linked immune sandwich assay)

Binding kinetics and equilibrium

Brownian motion and diffusion

Surface plasmon resonance (SPR) sensors


monoclonal or polyclonal

binds diff. epitope than capture ab

may be directly conj. to enzyme

or enzyme introduced via 3rdAb

that binds detection Ab

or enzyme-avidin conjugate

if detection Abbiotinylated


Typical ELISA format

monoclonal or polyclonal

usually immobilized on plate

by commercial supplier

could be microbial antigen if

goal is to detect antibodies to it

then detection ab is anti-IgG


Typical protocol

Add sample in ~200ml, incubate ~1.5h (why so long?), wash

Add 20 Ab coupled to enzyme (e.g HRP).incubate 1.5h, wash

Add enz. substrate (e.g. tetramethylbenzene)

Incubate 30min (in dark)

Add stop solution (H2SO4) (why?), read OD (within 30min)

Analyte with know concentration serially diluted in some

wells to compare intensities

to that of sample

Result: analyte conc. in sample


Reaction (receptor binding) kinetics

Let bm = total receptor conc. on sensor surface [moles/area]

b(t) = conc of receptors that have bound analyte at time t

Assume analytebinds receptor at rate ~

freeanalyteconc., c0,* free receptor conc., [bm – b(t)]

and dissociates from receptor at rate ~b(t)

db(t)/dt= konc0 [bm – b(t)] – koffb(t)

kon and koff are constants


db(t)/dt= konc0 [bm – b(t)] – koffb(t)

Interpretation of binding constants

kon = av. # “binding” collisions/s each receptor molecule

makes with an analyte molecule when analyteconc = 1

in whatever units you use, e.g. #/m3 or “molar”, M, moles/l

Units of konare #/conc.*time, e.g. M-1s-1

koff = rate each receptor-analyte complex dissociates in #/s

Define KD=koff/konUnits of KD are conc., e.g. M


db(t)/dt= konc0 [bm – b(t)] – koffb(t)

Solution: b(t)/bm= fraction of receptors with analyte= A(1-e-Bt)

where A = [c0/KD /(1 + c0/KD)] and B = konc0 + koff

A = c0/KD /(1 + c0/KD)



t = 1/B = koff-1/(1+c0/KD)


c0/KD /(1 + c0/KD)


natural to measure concentrations in units of KD

when c0 = KD, half of receptors have bound analyte

c0 >> KD, fraction of receptors with analyte -> 1

c0 << KD, most receptors are free

typical values kon~ 106/Ms ( =10-21m3/s) fairly constant

koff~ 1/s to 1/103s (varies a lot)

KD~mM (weak) to nM (tight binding)

Note smaller KD <-> tighter binding (slower koff)


t = koff-1/(1 + c0 /KD)


Receptor and analyte are interchangeable in this model

Excess receptor drives binding of analyte

Excess analyte drives binding of receptor

Model assumes mixing keeps concentration uniform

c0 = freeanalyteconc, lower than initial concci in ELISA

Model describes behavior of large # of molecules;

when only a few analyte molecules/sensor,

need to consider stochastic issues more carefully

More detailed analysis takes into account depletion of analyte in region just above receptor due to its binding and flux if convective flow (next week)


Details of some example ELISA assays


kon= 106/Ms, koff = 10-3/s, KD = 10-9M

c0 ~ 0.1pM = 107/well (100ml) << KD

bm~ 1/(10nm)2= 1011/3mm well

What fraction of receptors bind analyte?

How long to reach equilibrium?

10 nm Assay takes hours bec. of equil. times

Sensitivity limited by # enz. req. to generate signal and “noise” due to non-specific sticking (of detection Ab and subsequent reagents)

Reagent costs <1$/well; ~$100/plate; OD plate reader ~$10-30K


What is sensitivity of high end ELISA?

How many molecules in

100ml sample @ 0.5pg/ml

If KD=nM and 1011 capture

Abs/well, what fraction of

analyte is bound?

Hint: switch roles of anal.

and receptor in formula

for fx bound


Analyte MW = 20kDa (20,000g/mole)


Diffusion and Brownian motion

  • Why do molecules diffuse?
  • How far do they go (on average) in time t?
  • Let <x2(t)> = av. displacement2 of molecule after time t
  • Why talk in terms of <x2(t)>? What is <x(t)>?
  • <x2(t)> =~ t Why not ~t2? They keep changing directions.
  • In 1-d, suppose molecule moves randomly +e every t sec
  • <(xnt)2> = <(x(n-1)t +e)2> = <(x(n-1)t)2 + 2x(n-1)te+ e2> = ne2
  • since n = t/t, <x2(t)> = (e2/t)t =Dtwhere D is diff. const.
  • Note D=e2/thas units of m2/s

Deep connection between D and viscous drag:

both are due to bombardment by adj. molecules

In laminar flow, drag force on sphere radius r

Fdrag = gv

where g = 6phr Stokes law

h=viscosity, 10-3 Ns/m2 in water

v = velocity of sphere

Einstein relation: Dg = kBT = 4x10-21J at R.T.

Utility: If you know r, you know D = kBT/6phr

mol. with r~2nm has D = 4x10-21/.04x 10-9 = 10-10m2/s

virus with r~200nm has D = 10-12m2/s


About how long does it take a 2nm molecule to

diffuse 3mm? (approx. distance to

surface in microtiter plate well)?

t = x2/6D in 3dimensions = 9x10-6/6x10-10 = 104s (hours!)

twice dimension # why ELISA takes so long

Bulk flow of molecules down a concentration gradient

is due to diffusion

Fick’s Law:

flux j D (# molecules crossing unit area per s) = D dc/dx

check that units agree!

this becomes important for SPR sensors


Surface plasmon resonance (SPR) sensors

Like real-time immune

capture assay without

need for second Ab

Sensing principle – light incident on glass surface

> critical angle is totally reflected; if glass has

thin metallic coating, evanescent wave in metal

polarized in plane of incidence canexcite coherent

movement of electrons on metal surface when

resonance condition kx = ksp is met


Surface Plasmon Resonance – prism configuration



Since kx = (w/c) ng sin q = ksp,

surface plasmon excited

at particular incidence angle qr

-> decrease in reflection intensity

ksp very sensitive to n2~ mass bound to metal surface

To first order, Dqr ~ Dm


How sensitive is Dqrto Dm?

Typical sensitivity limit is ~1pg (~107 molecules for MW 105)

in sensor area 1mm2 (roughly comparable to ELISA);

this is equiv. to covering ~ 1/1000th of surface

Less sensitive per bound molecule when tgt is small

(signal ~mass bound) but still can be used to study

protein-drug interactions

Major advantages cf to ELISA

real time results, don’t need label, more automatable

get kinetic parameters kon, koff, KD, and bm as well as c0

Major disadvantage – cost ~$100K/machine + exp. chips


Lots of info available on web and from vendors, e.g.

Biacore, now part of GE

Note SPR facilitates depositing your own receptors





Characteristic SPR binding curves

Conc that

-> half max

binding = KD

Flow in


Flow in buffer

(analyte elutes)

charact. off

time = 1/koff

1RU=10-4 degrees




Can you estimate 1/koff from these curves?

Can you estimate conc. that gives half max binding?

Do they wait long enough to reach binding equilibrium?


Are binding kinetics influenced by flow rate, channel

dimensions, sensor dimensions?

E.g. as molecules adsorb to surface, are they depleted from

region just above surface? What does this do to

concentration near surface? To binding rate?

# molecules that bind/s = koncs [bm-b(t)]

Does incoming flow keep cs = c0?

Need more detailed model of mass transport to extract

KD, kon from raw data = next week’s topic


Example – kinetic constants extracted from SPR data

for binding of various engineered, antibody-like

molecules to their targets

Note ~106/Ms ~1/1000s ~nM


SPR is work-horse for analysis of protein interactions

in biochemistry labs/ pharmaceutical industry

Does it need to be so expensive?

Texas Instruments’ SPR chip - SPREETA

Sensors and

Actuators B 91

(2003) 266–274

Could you put

one in cell phone?


Other questions

What limits sensitivity of SPR?

What determines width of resonance?

What is effect of spatial inhomogeneity of

bound analyte on molecular scale?

What is effect of distance of analyte from

metal surface?


Summary –

most assays we’ll consider involve analyte binding

to surface,

with direct detection (“label-free”) due

to effect on some physical phenomenon – e.g. surface

plasmon resonance frequency,

or detection via attachment of

a “label” that provides a signal,

e.g. enzyme that generates dye



Specificity for particular analyte (essential in

biosensors because almost always dealing with

very complex mixtures of related molecules) comes

from capture molecules (e.g. antibody, compl. DNA)

This means interactions with capture molecules are crucial

b(t)/bm = [c0/KD /(1 + c0/KD)][1-exp-(konc0 + koff)t]

(c0/KD)/(1 + c0/KD)



t = 1/(konc0 + koff)


Sensitivity usually not better than 107 molecules [pM]

for conventional sensors, but coming down to 1

molecule for new technologies (major subject of course)

Assay times affected by mass transport kinetics plus diffusion plus kinetics of binding to capture molecules – topic for next week


Intro to next week’s paper – mass transport limits

for standard format biosensors

sensor surface

Sensor kinetics depends on 3 main processes:

sample flow rate, diffusion of molecules to sensor

surface, binding kinetics


Good matching of rates of these processes

important for sensor function

Binding of target molecules to sensor surface depletes

region near surface of targets (lowers cs), slows response

Color shows

depletion of

targets (blue)

in regions of

flow cell as

function of

time if no flow

analyteconc scale


Flow restores target molecules to depleted regions

but very rapid flow (in e) -> many molecules whiz

by before they can diffuse to surface



Depletion region dimensions depend on D, L, H, flow rate

Simple ratios of these parameters, PeH and PeS, determine

if sensor is operating in regime that looks like c versus e

What does concentration profile look like if you

wait for equilibrium?


Another ratio, Da, determines if time to reach

steady-state is determined mainly by chemical reaction

trxn = koff-1/(1 + c0 /KD)

or strongly affected by diffusion, geometry and receptor density

teq = Datrxnwhere Da is function of D, L, H, kon and bm

This analysis especially important when pushing sensitivity

to detect very low conc. targets


Divide segments of Manalis paper for student presentations

Fig. 1 + summarize relations between flow rate, speed

as function of height, pressure, from rest of paper

Fig. 2 – depletion region dimensions as function of

time in absence of flow

Fig. 3. – intitial (quasi-steady state) depletion region dimensions given flow, Peclet numbers

Fig. 4. and Box 3 – 2 particluar sensors

Fig. 5. and Damkohler # Box 2 – low conc.