Lecture 7. Surface Reaction Kinetics on a

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Aldous Marsh
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1 Lecture 7 Data processing in SPR Surface Reaction Kinetics on a Biochip

2 Surface plasmon sensor Principle of affinity SP biosensor

3 Data processing Processing steps: zero response to the base line before analyte injection align all responses so that injection starts at the same point subtract the reference cell response subtract the averaged response to buffer injection

reference subtracted 4 injections averaged + buffer

4 Data processing measured response: reference receptor receptor and reference aligned, base line subtracted reference subtracted 4 injections averaged + buffer injection buffer injection subtracted data, ready to be analyzed

5 Data processing Software: Scrubber2, (Biosensor Tools edu/ te act o /sc ubbe t BiaEvaluation (Biacore AB, Sweden)

6 General biosensor model Kinetics Reaction inetics Mass transport

7 Rates of chemical reactions Consider reaction: A+2B -> 3C+D The rates are: d [ D] 1 d[ C] d[ A] 1 d[ B] = = = dt 3 dt dt 2 dt We can define rate of reaction: 0 -> 3C+D-A-2B v d[ ξ ] =, dn dt j = ν dξ j ν j

8 Rate laws Rate of reaction is often proportional p to the concentration raised to some power, e.g. a b v = [ A ] [ B ] Overall order of the reaction: a+b+... order of the reaction with respect to A: a Reactions of zero order v =

9 First order reaction close to equilibrium Consider reactions: A B B At equilibrium forward and reverse rates are the same A d[ A] d[ B] =, dt dt ' [ A ] eq = [ B ] eq, K = = ' [ B] [ A] eq eq Equilibrium constant

Modeling Molecular Interaction in SPR The aim: design a model inetic equation that describes how amount of ligand-analyte depends on time, concentration of analyte and amount of free binding sites

10 Modeling Molecular Interaction in SPR The aim: design a model inetic equation that describes how amount of ligand-analyte depends on time, 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 ξ = const M A γ 1:1 binding ξ The maximal signal: S = const M β ξ / ξ = γ / β S A surface concentration of a ligand

11 Pseudo-first order Kinetics For an analyte A (in solution) and a receptor R (immobilized) a A+ R AR d Association rate: dγ a = aα β γ α = [ A]; β = R ; γ = RA 0 dt Dissociation rate: dγ a dt Summing: = ( ) [ ] [ ] d γ dγ = aα ( β γ) dγ dt

12 Pseudo-first order Kinetics Let s consider a situation when a concentration of analyte a 0 is injected into the volume V with the surface S. αv + γs = α V = const 0 d γ S = a α0 γ ( β γ) dγ dt V At the equilibrium (when time passed) d γ γ a eq = 0 K = = dt S d α0 γeq β γ V ( ) eq a2 < a1 K=const

13 Pseudo-first order Kinetics dγ S = a α0 γ ( β γ) dγ dt V if the concentration of analyte is high: dγ γ a eq = aα 0 ( β γ ) dγ ; K = = dt d α0 β γeq ( )

14 Pseudo-first order Kinetics Equilibrium analysis (not affected by mass transport) ξeq γ eq Kα0 = = ξ β (1 + Kα ) S + 0

15 Other inetic models Zero-order reaction following initial binding (conformational change in AR complex that blocs dissociation) A a1 a2 + R AR AR d1 d2 dγ 2 dt dγ1 dt = γ γ a2 1 d2 2 = α ( β γ γ ) γ γ + γ a d1 1 a2 1 d2 2 As the total mass is measured by the sensor ξ / ξ = ( γ + γ )/ β S 1 2 *

16 Other inetic models Parallel pseudo first-order reactions (e.g. two different receptors or two different analytes) a 1 A + R AR 1 1 d1 2 2 a 2 A + R AR d 2 β = [ R ] = p β β = [ R ] = (1 p ) β dγ 2 = a 1α0( β1 γ1) d1γ1 dt dγ1 = a2α0( β2 γ2) d2γ2 dt ξ / ξ = ( γ + γ )/ β S 1 2

17 Other inetic models Multivalent receptor binding: single receptor binds more than one molecule (e.g. streptavidin, antibodies, triplex formation) a1 1 A+ R AR d 1 a 2 AR + A ARA d 2 β = [ R ] = p β β = [ R ] = (1 p ) β dγ 2 = a2αγ 0 1 d2γ2 dt dγ1 = a 1α 0( β γ1 γ2) d1γ1 a2α0γ1+ d2γ2 dt ξ / ξ = ( γ + 2 γ )/ β S 1 2

18 Thermodynamics in SPR change in Gibbs energy can be found from equilibrium constant: 0 Δ G = RTln K Enthalpy and entropy of the reaction can be found from temperature dependence (van t Hoff equation) Δ G = Δ H T Δ S Activation energy for association and dissociation can be found from Arrhenius equation: act = Pexp( E / RT)

19 Association/Dissociation in an experiment Plateau + Span Plateau Association: Response = Req * (1 - e -obs * t ) Dissociation: Response = Span * e -d* t + Plateau

20 Mass transport effects In a flow cell: flow is laminar Reynolds number: For water at 20ºC: ρφ Re = ηh volume flow rate 2 Re = ( Φ/ ) / h mm s flow is laminar for Re<2100. velocity profile is parabolic v max = 3 2 Φ hw in a case of no diffusion the transport is by convection α ( xyt,, ) y y α ( xyt,, ) = 4vmax 1 t h h x

21 Mass transport effects Let s tae into account diffusion α ( xyt,, ) = t 2 2 α ( xyt,, ) α( xyt,, ) = D x y y y α ( x, y, t ) 4vmax 1 h h x

22 Mass transport effects Complete model can be solved numerically lie following Navier-Stoes and Convection-Diffusion in the volume, e.g. α α α α t x y h h x 2 2 ( xyt,, ) ( xyt,, ) ( xyt,, ) (,, ) 4 y D v 2 2 max 1 y = + xyt Reaction inetics on a biochip d γ ( x, t ) dt [ ] = α ( x,0, t ) β γ( x, t ) γ ( x, t ) a d Boundary condition at the biochip α ( x,0, t) γ ( x, t) D = y t

23 SPR case: Mass transfer+reaction

24 Assumptions: Simplified model of mass transport analyte transport in x direction is convective (no diffusion) 2 vmaxh Pe = > 1 Dl velocity dependence d v(y) considered d linear (as only area near surface is important) 2 α ( xyt,, ) α ( xyt,, ) y α ( xyt,, ) = D 4v 2 max t y h x two compartment model : bul of the channel with convection transport and constant α and layer of thicness h l next to the sensor. 2 dα 1 dγ vmaxd = M ( α0 α) M dt h l dt hl 1/3

25 Data processing Software: Scrubber2, (Biosensor Tools edu/ te act o /sc ubbe t BiaEvaluation (Biacore AB, Sweden)

26 Data processing Global analysis: A A+ B AB D All responses within the data set are fitted to the same values of A and D. Chi-squared is calculated for all curves mass transfer limited A m A A + B AB m D conditions to reduce mass transfer effect: low ligand density, high flow rate.

27 Data processing Protein-antibody interaction Best fit using bimolecular model Best fit using mass-transport model

28 Thermodynamics for drug discovery even contribution entropy driven enthalpy driven Δ G =ΔH TΔS large entropy: affinity increases w. temperature Shuman et al, J.Biomol.Rec.17, p.106 (2004)

29 Problem Derive e equations for parallel a pseudo first-order ode reaction inetics with a single receptor and two different analytes.

Lecture 4. SPR: immobilization strategies Kinetics

Lecture 4. SPR: immobilization strategies Kinetics Lecture 4 SPR: immobilization strategies Kinetics Surface plasmon sensor Principle of affinity SP biosensor The Aim of the Lecture How to make our SPR sensor selective? What type of a signal we expect