Lecture 7

Data processing in SPRData processing in SPRSurface Reaction Kinetics on a

Bi hiBiochip

Surface plasmon sensor• Principle of affinity SP biosensor

Data processing

• Processing steps:ocess g s eps– zero response to the base line before analyte

injectioninjection– align all responses so that injection starts at the

same pointsame point– subtract the reference cell response – subtract the averaged response to buffer injection

Data processingmeasured response:measured response: receptorreceptor

receptor and reference aligned base

receptor and reference aligned basereferencereference aligned, base line subtractedaligned, base line subtracted

referencereference 4 injections4 injectionsreference subtractedreference subtracted

4 injections averaged + buffer injection

4 injections averaged + buffer injection

data, ready to be analyzeddata, ready to be analyzed

buffer injection subtractedbuffer injection subtracted

Data processing• Software:

– Scrubber2, (Biosensor Tools http://www.cores.utah.edu/Interaction/scrubber.html)ttp // co es uta edu/ te act o /sc ubbe t )

– BiaEvaluation (Biacore AB, Sweden)

General biosensor modelKinetics

Reaction Mass Reactionkinetics

Mass transport

Rates of chemical reactions

Consider reaction:A+2B -> 3C+DTh tThe rates are:

dtBd

dtAd

dtCd

dtDd ][

21][][

31][

−=−==

We can define rate of reaction: ξνξ ddndt

dv jj == ,][

0 -> 3C+D-A-2Bdt

jν

Rate laws

• Rate of reaction is often proportional to the p pconcentration raised to some power, e.g.

ba BAkv ][][=

• Overall order of the reaction: a+b+...order of the reaction with respect to A: a

BAkv ][][

order of the reaction with respect to A: a• Reactions of zero order

kv =

First order reaction close to equilibrium

• Consider reactions:

BA→ABBA

→→

• At equilibrium forward and reverse rates are the same

BdAd ,][][=

eqBkKBkAk

dtdt][

][][

,

' ===eq

eqeq AkKBkAk

][,][][ ' ===

Equilibrium constant

Modeling Molecular Interaction in SPR• The aim: design a model kinetic equation that

describes how amount of ligand-analyte depends on time concentration of anal te and amo nt of freetime, concentration of analyte and amount of free binding sites left.

The signal is proportional to the mass (per unit area) of bound analyte:

mass of the analyte

surface concentration of a complex

Aconst Mξ γ= ⋅ ⋅ 1:1 binding

The maximal signal:

S Aconst Mξ β= ⋅ ⋅

/ /S A

S

ξ βξ ξ γ β=

surface concentration of a ligand

Pseudo-first order Kinetics• For an analyte A (in solution) and a receptor R (immobilized)

ka

d

k

kA R AR⎯⎯→+ ←⎯⎯

• Association rate:

( ) [ ] [ ]0[ ]; ;a

ad k A R RAdγ α β γ α β γ= − = = =( ) [ ] [ ]0

[ ]adtβ γ β γ

• Dissociation rate:

ad

d kdtγ γ= −

• Summing:

( )a dd k kdtγ α β γ γ= − −

• Summing:

( )dt

Pseudo-first order Kinetics• Let’s consider a situation when a concentration of analyte a0 is injected into

the volume V with the surface S.

0V S V constα γ α+ = =

d Sγ ⎛ ⎞( )0a dd Sk kdt Vγ α γ β γ γ⎛ ⎞= − − −⎜ ⎟

⎝ ⎠

• At the equilibrium (when time passed)

kd γγ

( )0

0 eqa

deq eq

kd KSdt kV

γγ

α γ β γ= = =

⎛ ⎞− −⎜ ⎟⎝ ⎠V⎝ ⎠

ka2<ka1K tK=const

Pseudo-first order Kinetics( )0a d

d Sk kdt Vγ α γ β γ γ⎛ ⎞= − − −⎜ ⎟

⎝ ⎠• if the concentration of analyte is high:

( ) ( )0 ; eqaa d

kd k k Kγγ α β γ γ= − − = =( ) ( )0

0

;a dd eqdt k

β γ γα β γ−

Pseudo-first order Kinetics• Equilibrium analysis (not affected by mass transport)

0

(1 )EQ eq K

Kξ γ αξ β

= =+ 0(1 )S Kξ β α+

Other kinetic models• Zero-order reaction following initial binding (conformational

change in AR complex that blocks dissociation)

1 2 *a ak k

k kA R AR AR⎯⎯→ ⎯⎯→+ ←⎯⎯ ←⎯⎯

1 2d dk k← ←

22 1 2 2a d

d k kdtγ γ γ= −

11 0 1 2 1 1 2 1 2 2( )a d a d

dtd k k k kdtγ α β γ γ γ γ γ= − − − − +

• As the total mass is measured by the sensor

1 2/ ( ) /Sξ ξ γ γ β= +

Other kinetic models• Parallel pseudo first-order reactions (e.g. two different

receptors or two different analytes)

1

1 1ak

A R AR⎯⎯→+ ←⎯⎯1

2

2

1 1

2 2

d

a

d

k

k

kA R AR

←

⎯⎯→+ ←⎯⎯2d

[ ] [ ] (1 )R p R pβ β β β= = = = −1 1 2 2

21 0 1 1 1 1

[ ] [ ] (1 )

( )a d

R p R pd k kdt

β β β βγ α β γ γ

= = = =

= − −

12 0 2 2 2 2( )

/ ( ) /

a dd k kdtγ α β γ γ

ξ ξ β

= − −

1 2/ ( ) /Sξ ξ γ γ β= +

Other kinetic models• Multivalent receptor binding:

single receptor binds more than one molecule (e.g. streptavidin, ( g p ,antibodies, triplex formation)

1akA R AR→1

1

2

a

d

a

k

k

A R AR

AR A ARA

⎯⎯→+ ←⎯⎯

⎯⎯→+ ←2dk

AR A ARA+ ←⎯⎯

[ ] [ ] (1 )R Rβ β β β1 1 2 2

22 0 1 2 2

[ ] [ ] (1 )

a d

R p R pd k kdt

β β β βγ α γ γ

= = = = −

= −

11 0 1 2 1 1 2 0 1 2 2( )a d a d

dtd k k k kdtγ α β γ γ γ α γ γ= − − − − +

1 2/ ( 2 ) /Sξ ξ γ γ β= +

Thermodynamics in SPR• change in Gibbs energy can be found from equilibrium

constant:0 lnG RT KΔ = −

• Enthalpy and entropy of the reaction can be found fromEnthalpy and entropy of the reaction can be found from temperature dependence (van’t Hoff equation)

G H T SΔ = Δ − ΔG H T SΔ Δ Δ

• Activation energy for association and dissociation can be found from Arrhenius equation:found from Arrhenius equation:

exp( / )actk P E RT= −

Association/Dissociation in an experiment

Plateau + Span

Plateau

Dissociation:

Response = Span * e-kd* t + PlateauAssociation:

Response = Req * (1 - e -kobs * t)p q ( )

Mass transport effects• In a flow cell:

– flow is laminarvolume flow rate

Reh

ρηΦ

=Reynolds number:

2Re ( / ) 0.998 /h mm sη

= Φ ⋅For water at 20ºC:

flow is laminar for Re<2100.

– velocity profile is parabolic

max32

vhwΦ

=

in a case of no diffusion the transport

( ) ( )x y t y y x y tα α∂ ∂⎛ ⎞

– in a case of no diffusion the transport is by convection

max( , , ) ( , , )4 1x y t y y x y tv

t h h xα α∂ ∂⎛ ⎞= − −⎜ ⎟∂ ∂⎝ ⎠

Mass transport effects• Let’s take into account diffusion

( , , )x y tα∂

2 2

( , , )

( ) ( )

x y tt

x y t x y t

α

α α

∂=

∂⎛ ⎞∂ ∂

2 2

( , , ) ( , , )

( )

x y t x y tDx y

y y x y t

α α

α

⎛ ⎞∂ ∂= + −⎜ ⎟∂ ∂⎝ ⎠

∂⎛ ⎞max

( , , )4 1y y x y tvh h x

α∂⎛ ⎞− −⎜ ⎟ ∂⎝ ⎠

Mass transport effects• Complete model can be solved numerically like

followingg– Navier-Stokes and Convection-Diffusion in the volume, e.g.

2 2( ) ( ) ( ) ( )x y t x y t x y t y y x y tα α α α⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞max2 2

( , , ) ( , , ) ( , , ) ( , , )4 1x y t x y t x y t y y x y tD vt x y h h x

α α α α⎛ ⎞∂ ∂ ∂ ∂⎛ ⎞= + − −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

– Reaction kinetics on a biochip

[ ]( , ) ( 0 ) ( ) ( )d x t k t t k tγ β[ ]( , ) ( ,0, ) ( , ) ( , )a dk x t x t k x tdtγ α β γ γ= − −

Boundary condition at the biochip– Boundary condition at the biochip( ,0, ) ( , )x t x tD

tα γ∂ ∂

=∂ ∂y t∂ ∂

SPR case: Mass transfer+Reaction

Simplified model of mass transport A i• Assumptions:– analyte transport in x direction is convective (no diffusion)

2max 1v hPeDl

= >

l it d d ( ) id d li ( l

2( ) ( ) ( )x y t x y t y x y tα α α⎛ ⎞∂ ∂ ∂

– velocity dependence v(y) considered linear (as only area near surface is important)

max2

( , , ) ( , , ) ( , , )4x y t x y t y x y tD vt y h x

α α α⎛ ⎞∂ ∂ ∂= −⎜ ⎟∂ ∂ ∂⎝ ⎠

“two compartment model”: bulk of the channel with– two compartment model : bulk of the channel with convection transport and constant α and layer of thickness hlnext to the sensornext to the sensor.

( )01

Md dkdt h dtα γα α⎡ ⎤= − −⎢ ⎥⎣ ⎦

1/32max1.282M

v Dkhl

⎛ ⎞≈ ⎜ ⎟

⎝ ⎠ldt h dt⎢ ⎥⎣ ⎦ hl⎝ ⎠

Data processing• Software:

– Scrubber2, (Biosensor Tools http://www.cores.utah.edu/Interaction/scrubber.html)ttp // co es uta edu/ te act o /sc ubbe t )

– BiaEvaluation (Biacore AB, Sweden)

Data processing

• Global analysis:Ak

A B AB→

All responses within the data set are fitted to the same

A

DkA B AB⎯⎯→+ ←⎯⎯

– All responses within the data set are fitted to the same values of kA and kD.

– Chi-squared is calculated for all curvesChi squared is calculated for all curves

m A

m D

k k

k kA A B AB⎯⎯→ ⎯⎯→+←⎯⎯ ←⎯⎯mass transfer limited

– conditions to reduce mass transfer effect: low ligand density, high flow rate.

Data processing• Protein-antibody interaction

Best fit usingBest fit using bimolecular model

Best fit using mass-transport model

Thermodynamics for drug discovery

even contribution

entropy driven

enthalpy driven

G H T SΔ = Δ − Δ

large entropy: affinity increases w. temperature

Shuman et al, J.Biomol.Rec.17, p.106 (2004)

Problem

• Derive equations for parallel pseudo first-order e e equa o s o pa a e pseudo s o dereaction kinetics with a single receptor and two different analytesdifferent analytes.