Class 2.

Review major points from class 1

New MaterialReaction (receptor binding) kineticsDiffusionEinstein relationFick’s LawSPR sensors

Review of last week

Biological molecules are often complex polymerse.g. proteins, DNA

Interact via fundamental forces (e.g. electrostatics)but these are often shielded by ions in solution-> complex weak interactions ~kBT over short distances (e.g hydrogen bonds,

hydrophobic effects, base pairing)Lots of molecule-specific interactionsBoltzman distribution – chance that molecule is in

state with energy E ~ exp(-E/kBT)

Did not review derivation of Boltzman distribution

But could for those with mathematical bent…

(if time and interest at end of class)

New topic – binding kinetics

Volume of liquid sample = V

“free” ligand molecule L

“free” antibody molecule Ab

antibody that has bound ligand AbL

Consider a microtiter well sensor coated with antibodiesfilled with sample containing ligand molecules

Let [AbT] = total # antibody molecules on sensor surface/sensor vol V (in some sense the Ab concentration)

[LT] = total # of ligand molecules in the sensor/V

note [ ] indicates concentration

[Ab(t)] = concentration of antibody molecules that havenot bound ligand at time t, i.e. that are free to bind L

[L(t)] = concentration of ligand molecules that have notbound antibody at time t, i.e. that are free to bind Ab

Ab + L <-> AbL

By conservation of molecules

[AbT] = [Ab] + [AbL] (1)

[LT] = [L] + [AbL] (2)

Simple model for Ab + L AbL

d/dt [AbL] = kon [Ab] [L] – koff [AbL]

at equilibrium, d/dt [AbL] = 0 => kon [Ab] [L] = koff [AbL]

[Ab] [L]/[AbL] = koff/kon = KD (3)

kon

koff

-><-

Note (1), (2) and (3) are 3 eqns w/ 6 “variables”:([AbT], [LT], [Ab], [L], [AbL], KD

Frequently only interested in fraction of Abs thathave bound L, i.e. [AbL]/[AbT]

(or fraction of L that have bound Ab, i.e. [AbL]/[LT])

Then (1-3) can be rewritten (you should check this!):

[AbL]/[AbT] = [L]/KD / (1 + [L]/KD) (4)

[AbL]/[LT] = [Ab]/KD / (1 + [Ab]/KD) (5)

The “signal” from L binding (e.g. dye color in some ELISAs)is often proportional to the amount of ligand that binds, i.e. [AbL] or the rhs of 4 or 5

Then if you plot signal vs free Ab or L concentration, you get sigmoid curve, half max at [L] = KD (or [Ab]=KD)

This can be used to determine the KD

[L]/KD

signa

l

[Ab]/KD

signa

l

1 1

½ max + ½ max +

Units

d/dt [AbL] = kon [Ab] [L] – koff [AbL]

conversion factor CFchemistry SI units (SI = chem x CF)

[X] M = moles/liter #/m3 6*1026

koff s-1 s-1 1kon M-1s-1 = l/mole-s (#/m3)-1s-1 (6*1026)-1

KD M m3/#

KD is a measure of the strength of an interaction; the lowerthe KD, the tighter the binding; [X]/KD is unit-less

kon is # reactive collisions/sec each molecule of Ab or L makes when it’s partner is at “unit” concentration

[L]/KD

signa

l

[AbL]/[AbT] = [L]/KD / (1 + [L]/KD) (4)

[AbL]/[LT] = [Ab]/KD / (1 + [Ab]/KD) (5)

Note when [L] << KD, fraction of Ab “bound” -> [L]/KD

when [L]>> KD, fraction of Ab “bound” -> 1

High conc. of either free species, [L] or [Ab],“drives” its partner into a complex

Eqns (1) – (3) are symmetric in Ab and L, so symmetry of 4 and 5 is expected

Caveat: (4) or (5) are useful when AbT (or LT) is in excess so that complex formation does not deplete the species in excess, i.e. [Ab] @ [AbT]) (or [L] @ [LT]) because you often know [AbT] or [LT] but not [Ab] or [L]

[AbL]/[AbT] = [L]/KD / (1 + [L]/KD) (4)

[AbL]/[LT] = [Ab]/KD / (1 + [Ab]/KD) (5)

Example: If you want to measure ligand at pM conc. by ELISA, is (4) or (5) relevant? Assume KD is nM,you can pack antibodies on the plastic surface at1012/cm2 (do you want high or low [AbT]?), a wellhas an area of ~0.1cm2, and sample volume is 100ml

If ligand is likely to be at nM conc, is (4) or (5) useful tomeasure KD?

You don’t need to know [Ab] or [L] if you don’t careabout KD and just want [LT] for an unknown bycomparing the signal it gives to that from known concentrations [LT] of a standard

Note [Ab] and [L] are not really symmetric becauseAb is fixed and L is free to diffuse; the abovetreats the Ab as if it were free in solution atconc = # surface molecules/ vol; dicey!

If neither Ab nor L is in excess, it’s better to use (1) and (2) to eliminate [Ab] and [L] from (3), giving aquadratic eqn for [AbL]:

[AbL]2 – [AbL] (KD + [LT] + [AbT]) + [AbT][LT] = 0

=> [AbL] = -b/2 + (b2/4-c)1/2

Is the + or – solution non-physical? Hint: if [LT] = 0, what is c, and what must [AbL] be?

You could use this to get [LT] if you measure signal (~{AbL])and know KD and [AbT]

b c

What is sensitivity (lowest detectable conc) in a good ELISA?

How many molecules in100ml sample @ 0.5pg/mlIf MW = 2*104g/mole?

If KD=nM and 1011 captureAbs/well, what fraction ofligand is bound?Hint: is Ab or L in excess?How does [Ab] cf to KD?

*

Sensitivity usually limited by noise from non-specific sticking of enzyme -> color

The simple model also gives kinetic information:

d [AbL(t)]/dt = kon [AbT-AbL(t)] [LT – AbL(t)] – koff [AbL(t)]

Suppose LT >> Ab so that LT – AbL(t) @ LT.

Divide all terms by [AbT]Let f(t) = fraction of Ab bound at time t, f(t) = AbL(t)/AbT

This gives simple diff eqn for f(t):

f’(t) = -f(t) {(LTkon + koff)} + konLT

f’(t) = -f(t) {(LTkon + koff)} + konLT

f(t) has simple solution: f(t) = A(1-e-Bt)

where A = [LT/KD /(1 + LT/KD)] and B = konLT + koff

(if you are unfamiliar with this, check by substitution)

Plot of f(t) vs tfraction of Ab that bind L = f(t)

time

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

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

Note f(t) exponentiallyapproaches equil. valuewith characteristictime t

f(t)

time

LT/KD /(1 + LT/KD)

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

If LT << KD, t -> 1/koff If LT >> KD, t -> koff

-1 KD/LT = 1/(konLT)

typical values kon ~ 104 - 106/Ms ( =10-23 – 10-21 m3/s)koff ~ 1/s to 10-3s KD ~ mM (weak) to nM (tight binding)

If LT is not in excess, the diff. eqn. is not simple and requiresnumerical solution

What determines kon?

Need digression to discuss Brownian motionand diffusion

Diffusion and Brownian motion

Why do molecules diffuse?How far away do they go (on average) in time t?

Let <x2(t)> = av. displacement2 of molecule after time tWhy talk in terms of <x2(t)> instead of <x(t)>?

<x2(t)> = ~ t Why not ~t2? They keep changing directions. In 1-d, suppose molecule moves randomly +d every t sec <(xnt)2> = <(x(n-1)t + d)2> = <(x(n-1)t)2 + 2x(n-1)td + d2> = nd2

since n = t/t, <x2(t)> = (d2/t)t = 2Dt where D is diff. const.

Note D= d2/2t has units of m2/s. Time to diffuse x = x2/2DHow can we estimate D?

Deep connection between D and viscous drag:both are due to bombardment by adj. molecules

Why do objects pushed at constant F move with constant velocity in viscous media rather thanaccelerate according to F = ma?

Einstein suggested they do accelerate with a=F/m for short times, but then are struck sufficiently hard byother molecules that their velocity is randomized

Let average time between velocity-randomizing collisions = t

Terminal velocity (before next collision) = (F/m) t

Does this microscopic terminal velocity=macroscopic vdrift

of body subject to force F in medium with viscosity h?

In laminar flow, drag force on sphere radius rFdrag = gv v = Fdrag/gwhere g = 6phr Stokes law

h=viscosity, 10-3 Ns/m2 in water(viscosity = shear force/velocity gradient ^ to shear)

If the two are the same, F/6phr=(F/m)t => t/m = 1/6phr

Finally, all objects in thermal equilibrium have averagekinetic energy mvx

2 = kBT => vx = d/t = (kBT/m)1/2

This gives 3 relations between “microscopic” quantities d, t, m and “macroscopic”quantitites g (= 6phr), kBT, and D:

d2/2t =D (4)t/m = 1/6phr (5)d/t = (kBT/m)1/2 (6)

Use (4) and (5) to eliminate d and m in (6) ->

D = kBT/6phr Einstein relation (lost factor of 2)

This gives you D for a molecule in medium withviscosity h, if you can estimate its radius r (!)

Example:

What is the diffusion constant in water for an antibodymolecule whose radius is 3nm?

viscosity of water h = 10-3 Ns/m2

(check units = F/area/gradient of velocity)

kBT = 4*10-21J at room temperature (T~300oK)(useful to remember this for this course!)

D = kBT/6phr = 4*10-21 J/(6*3.14*10-3*3*10-9Js/m2) = 6*10-11m2/s Note units!100x larger virus has 100x smaller D since D~1/r

About how long does it take a 2nm molecule to diffuse 3mm? (approx. distance tosurface in microtiter plate well)?

t = x2/6D in 3dimensions = 9x10-6/6x10-10 = 104s (hours!) twice dimension # that’s why ELISA takes so long!

Diffusion explains bulk flow of molecules down a concentration gradient c(x) | c(x+d)total flux ½ c(x) A d/t -> <- ½ c(x+d) A d/tnet flux j D (# molecules crossing unit area per s)

= {(c(x) – c(x+d))/d} (d2/2t) = -D dc/dxcheck that units agree!

This is Fick’s Law

How does this relate to kon?

Diffusion limited molecular collision rate is tricky to derive but consider that molecule of radius r diffusing distance r in time t = r2/6D collides with all ligand molecules in volume ~ pr3

# collisions/s @ pr3[L]/(r2/6D) = 6D pr[L] = 6prkBT[L]/6phr@ kBT[L]/h = kon [L] =>

kon = kBT/h = 4*10-21/10-3m3/s @ 2*109M-1s-1

Observed kon’s are ~ 104 – 106 M-1s-1, suggesting thatonly 1 in 103-105 collisions results in binding

How are kon’s measured?

SPR (surface plasmon resonance) sensors is 1 way

Quantitative details of SPR not important for thiscourse, but try to get qualitative understanding

Surface plasmon resonance (SPR) sensors

Like real-time immune capture assay

Transduction methodSensing principle – light incident on glass surface

> critical angle is totally reflected; if glass hasthin metallic coating, evanescent wave in metalpolarized in plane of incidence can excite coherent movement of electrons on metal surface when resonance condition kx = ksp is met

Surface Plasmon Resonance – prism configuration

q

Since kx = (w/c) n sin q = ksp, surface plasmon excitedat particular incidence angle qr

-> decrease in reflection intensity

ksp very sensitive to n2 ~ mass bound to metal surfaceTo first order, Dqr ~ Dm

2

How sensitive is Dqr to 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 ligand is small (signal ~mass bound) but still can be used to study protein-drug interactions

Advantages cf to ELISAreal time results, don’t need label, more automatableget kinetic parameters kon, koff, KD, and LT as well as L

Disadvantages – cost ~$100K/machine, expensive flow cells

Lots of info available on web and from vendors, e.g.Biacore, now part of GE

http://www.biacore.com/lifesciences/technology/introduction/following_interaction/index.htmlhttp://www.biacore.com/lifesciences/technology/introduction/data_interaction/index.html

Note SPR facilitates depositing your own receptors

http://www.protein.iastate.edu/seminar/BIACore/TechnologyNotes/TechnologyNote1.pdf

http://www.bama.ua.edu/~chem/seminars/student_seminars/fall08/f08-papers/bokatzian-johnson-sem.pdf

http://www.biosensingusa.com/Application101.html

Flow inanalyte

Flow in buffer(analyte elutes)

Conc that-> half maxbinding = KD

charact. offtime = 1/koff

Characteristic SPR binding curve data

analyte conc.nM1R

U=1

0-4 d

egre

es

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?

Example – kinetic constants extracted from SPR datafor binding of various engineered, antibody-likemolecules to their targets

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

Note SPR provides “real-time” binding assay, so close to a reporter of events modeled in binding kinetics theory

More complex phenomena revealed by SPR sensors

E.g. as molecules adsorb to surface, they are depleted fromregion just above surface. What does this do toconcentration near surface? To binding rate?# molecules that bind/s = kon Ls [Ab(t)]

Does incoming flow keep Ls = LT? koff may be underestimated because of rapid rebinding

Need more detailed model of mass transport to extractKD, kon from raw data = topic in 2 weeks

SPR is work-horse for analysis of protein interactionsin 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 putone in cell phone?

References/reading

Random Walks in Biology, Howard Berg, 1993a great paperback, p. 5-11, 17-42

Chapter on diffusion in Nelson

http://www.fas.harvard.edu/~scphys/nsta/viscosity_brownian.pdf

Short chapter from Harvard class on Einstein relation

Next week

Do homework problems on binding kinetics, diffusionRead general article on single-molecule ELISA to get

basic idea, then article on theory of single-molassay (in J of Immunol) for review ofbinding kinetics and state –of-the-art application

See questions on theory article (on Blackboard) to help you digest itpick a figure from this paper to discuss in class