Transport Effects on Multiple-Component Reactions in Optical Biosensors

0.59 MB·

by  Ryan M. Evans

NonamemanuscriptNo.
(willbeinsertedbytheeditor)
Transport Effects on Multiple-Component Reactions in
Optical Biosensors
RyanM.Evans · DavidA.Edwards
Received:date/Accepted:date
7
1
0
Abstract Manybiochemicalreactionsinvolveastreamofchemicalreactants(ligand
2
molecules)flowingoverasurfacetowhichotherreactants(receptors)areconfined.
n
Scientistsmeasurerateconstantsassociatedwiththesereactionsinanopticalbiosen-
a
J sor:aninstrumentinwhichligandmoleculesareconvectedthroughaflowcell,over

6 a surface on which receptors are immobilized. In applications such as DNA dam-
2 agerepairmultiplesimultaneousreactionsoccuronthesurfaceofthebiosensor.We
quantify transport effects on such multiple-component reactions, which result in a
]
N nonlinearsetofintegrodifferentialequationsforthereactingspeciesconcentrations.
In physically relevant parameter regimes, these integrodifferential equations further
M
reducetoanonlinearsetofordinarydifferentialequations,whichmaybeusedtoes-
. timaterateconstantsfrombiosensordata.Weverifyourresultswithasemi-implicit
o
i finitedifferencealgorithm.
b
- Keywords Biochemistry·Opticalbiosensors·Rateconstants·Partialdifferential
q
equations·Integrodifferentialequations·Numericalmethods
[

1
v
1 Introduction
9
8
4 Many biochemical reactions in nature involve a stream of chemical reactants (lig-
0 and molecules) flowing through a fluid-filled volume, and another reactant (recep-
0
.
2 ThisworkwasdonewiththesupportoftheNationalScienceFoundationunderawardnumbernsf-dms
0 1312529,andwasalsopartiallysupportedbytheNationalResearchCouncilthroughanNRCpostdoctoral
7 fellowship.
1
R.M.Evans
:
v AppliedandComputationalMathematicsDivision,InformationandTechnologyLab,
i NationalInstituteforStandardsandTechnology,Gaithersburg,MD20899,USA
X E-mail:[email protected]
r D.A.Edwards
a
DepartmentofMathematicalSciences,UniversityofDelaware,Newark,DE19716,USA
E-mail:[email protected]
2 RyanM.Evans,DavidA.Edwards
tors)confinedtoasurface.Suchsurface-volumereactionsoccurduringplateletad-
hesion (Austin, 2009), drug absorption (Bertucci et al., 2007), antigen-antibody in-
teractions (Raghaven et al., 1994), and DNA damage repair (Zhuang et al., 2008).
Fundamental to understanding these reactions is getting accurate quantitative mea-
surements of the underlying reaction rate constants. To measure rate constants as-
sociatedwithsurface-volumereactions,scientistsuseopticalbiosensors:seeFigure
1.1foraschematicofonesuchinstrument;here,x=0correspondstotheinletand
(cid:101)
y=0correspondstothesensorsurface.Biosensortechnologyhasbecomeextremely
(cid:101)
popularinrecentyears;10,000papershavecitedtheuseofanopticalbiosensorasof
2009alone(RichandMyszka,2011).
Forourpurposes,biosensorexperimentsarepartitionedintotwophases:thein-
jectionphase,andthewashphase.Duringtheinjectionphase,ligandmoleculesare
injectedintothebiosensorviaabufferfluidinastandardtwo-dimensionalPoiseuille
(cid:0) (cid:1)
flowprofilefromtheinletatx(cid:101)=0atconcentrationC(cid:101) x(cid:101),y(cid:101),(cid:101)t ;thetildevariablesde-
note dimensional quantities throughout. The ligand molecules then diffuse through
thebufferontothechannelfloortobindwithreceptorsimmobilizedonthesurface.
Mass changes on the floor due to ligand binding are averaged over a portion of the
channelfloor[x ,x ]toproducemeasurementsoftheform
(cid:101)min (cid:101)max
1 (cid:90) x(cid:101)max
B(cid:101)(t)= x −x B(cid:101)(x(cid:101),(cid:101)t)dx(cid:101), (1.1)
(cid:101)max (cid:101)min x(cid:101)min
(cid:0) (cid:1)
whereB(cid:101) x(cid:101),(cid:101)t istheconcentrationofboundligandmolecules.Aftertheboundstate
reachesachemicalequilibrium,scientiststhenpreparethedeviceforanotherexper-
iment by washing it with the buffer fluid–this is the wash phase of the experiment.
Onlybufferisflowingthroughthebiosensorduringthewashphase,notbuffercon-
tainingtheligandmolecules.Thishastheeffectofcausingallboundligandmolecules
togetsweptoutofthebiosensor,therebycleaningthedeviceforanotherexperiment.
Measuringrateconstantswithbiosensordatareliesonhavinganaccuratemathe-
maticalmodelofthisprocess;mathematicalmodelshavebeensuccessfullyproposed
and progressively refined throughout the years: (Edwards, 1999, 2000, 2001, 2006,
2011; Edwards et al., 1999; Lebedev et al., 2006; Zumbrum and Edwards, 2014,
2015).Althoughsuchmathematicalmodelsaretypicallylimitedtoreactionsinvolv-
ing only a single molecule or a single step, chemists are currently using biosensor
technology to measure rate constants associated with reactions involving multiple
components. For example, chemists are now using biosensor experiments to eluci-
date how cells cope with DNA damage. Harmful DNA lesions can impair a cell’s
abilitytoreplicateDNA,anditsabilitytosurvive.Onewayacellmayrespondtoa
DNAlesionisthroughDNAtranslesionsynthesis(Friedberg,2005;Lehmannetal.,
2007; Plosky and Woodgate, 2004). For a description of this process we refer the
interested reader to the references included herein; however, for our purposes it is
sufficienttoknowthatDNAtranslesionsynthesisinvolvesmultipleinteractingcom-
ponents:aProliferatingCellNuclearAntigen(PCNA)molecule,polymeraseδ,and
polymerase η. Further, in order for a successful DNA translesion synthesis event
to occur polymerase δ must bind with a PCNA molecule. A central question sur-
roundingDNAtranslesionsynthesisiswhetherpolymeraseδ directlybindswiththe
TransportEffectsonMultiple-ComponentReactionsinOpticalBiosensors 3
bound
complex
unbound
unbound ligand
receptor
magnified view of
area in small
circle
H
parabolic flow
ỹ
reacting zone

  evanescent
wave
̃xmin ̃xmax  ̃x

L  

Fig. 1.1: Cross-sectional schematic of an optical biosensor experiment. The instru-
menthaslengthL(cid:101)andheightH(cid:101).Ligandmoleculesareconvectedintoinstrumentin
a Poiseuille flow profile and transported to the surface to bind with receptors. The
receptorsarelimitedtothereactingzonex=x tox=x .
(cid:101) (cid:101)min (cid:101) (cid:101)max
PCNA,orwhetherthepolymeraseδ andPCNAcomplexformsasaresultofaligand
switchingprocess(Zhuangetal.,2008).
TheformerscenarioisdepictedinFigure1.2,wherewehaveshownpolymerase
δ directlybindingwithaPCNAmolecule,i.e.thereaction:
P1:E+L2−(cid:41)2−−(cid:101)k(cid:42)−a EL2. (1.2a)
2(cid:101)kd
Here, we have denoted the PCNA molecule and polymerase δ as E and L respec-
2
tively.Additionally,2(cid:101)kadenotestherateatwhichL2bindswithaPCNAmoleculeE,
and2(cid:101)kd denotestherateatwhichL2 dissociatesfromaPCNAmoleculeE.Wewill
refertothisaspathwayone,orsimplyP asin(1.2a).
1
TheligandswitchingprocessisshowninFigure1.3andstatedpreciselyas:
P2:E+L1−(cid:41)1−−(cid:101)k(cid:42)−a EL1, EL1+L2(cid:41)−12−−(cid:101)k(cid:42)−a EL1L2−(cid:41)21−−(cid:101)k(cid:42)−d EL2+L1. (1.2b)
1(cid:101)kd 12(cid:101)kd 21(cid:101)ka
First, polymerase η (denoted L ) binds with a PCNA molecule; next polymerase δ
1
associateswiththepolymeraseη andPCNAcomplex;finallypolymeraseη switches
out,therebyleavinguswiththepolymeraseδ andPCNAcomplex.Eachoneofthe
steps in thismultiple-component process is reversible. In (1.2b) and Figure1.3, the
rateconstants1(cid:101)ka and1(cid:101)kd denotetheratesatwhichpolymeraseη bindsandunbinds
with a PCNA molecule (respectively), ij(cid:101)ka denotes the rate at which ligand Li binds
withtheproductELj,andij(cid:101)kddenotestherateatwhichLidissociatesfromtheproduct
EL L .Inthelattertwoexpressionstheindicesiand jcanequaloneortwo.Weshall
1 2
refertothispathwaytwo,orsimplyP asin(1.2b).
2
4 RyanM.Evans,DavidA.Edwards
L2
2(cid:101)ka
L2
2(cid:101)kd
EL2 E
Fig. 1.2: Direct binding of polymerase δ with a PCNA molecule. We have labeled
polymeraseδ andthePCNAmoleculeasL andE respectively.
2
L2
L1 L1 L2
1(cid:101)ka 21(cid:101)ka 12(cid:101)kd
1(cid:101)kd 21(cid:101)kd 12(cid:101)ka
E EL1 EL1L2 EL2
Fig.1.3:Schematicoftheligandswitchingprocess.First,polymeraseη bindswith
the PCNA molecule; next, polymerase δ binds with the polymerase η and PCNA
complex; finally, polymerase η dissociates, leaving us with the polymerse δ and
PCNA complex. We have labeled polymerase η, polymerase δ, and the PCNA
moleculeasL ,L ,andE respectively.
1 2
By measuring the rate constants associated with this multiple-component pro-
cess,onecouldthereforedeterminewhetherEL formsasaresultofligandbinding
2
or the ligand switching process described above. To measure the kinetic rate con-
stants, scientists first immobilize PCNA molecules on the surface of the biosensor.
Afterreceptorimmobilization,scientistsinjectL andL intotheinstrumentatcon-
1 2
centrationsC(cid:101)1(x(cid:101),y(cid:101),(cid:101)t) andC(cid:101)2(x(cid:101),y(cid:101),(cid:101)t). The ligands L1 and L2 are then transported to
the surface to bind with available receptor sites E, creating the three products EL ,
1
EL1L2,andEL2atconcentrationsB(cid:101)1(x(cid:101),(cid:101)t), B(cid:101)12(x(cid:101),(cid:101)t),andB(cid:101)2(x(cid:101),(cid:101)t).
Thepresenceofmultiplereactingspecieschangestheformofbiosensorfeedback
(i.e.,thesensogramreading).Mostopticalbiosensorsmeasureonlymasschangesat
thesurfaceduetoligandbinding,andproducelumpedmeasurementsoftheform
S(cid:102)((cid:101)t)=s(cid:101)1B(cid:101)1((cid:101)t)+(s(cid:101)1+s(cid:101)2)B(cid:101)12((cid:101)t)+s(cid:101)2B(cid:101)2((cid:101)t), (1.3)
whereB(cid:101)i((cid:101)t)isdefinedanalogouslyto(1.1).Measuringrateconstantsassociatedwith
this multiple-component process involves calculating the sensogram signal (1.3),
whichreliesonamathematicalmodel.Thestraightforwardwell-stirredkineticsap-
proximationisinappropriatesinceitdoesnotaccountformechanismswhichtrans-
portligandmoleculestothereactingsurface.Thoughtransporteffectsonbimolecular
TransportEffectsonMultiple-ComponentReactionsinOpticalBiosensors 5
kineticsarewellstudied,fewifanyattemptshavebeenmadetoanalyticallyquantify
sucheffectsonmultiple-componentreactions.Hence,wemodelthefullconvection-
diffusionsystemwithmultiplecoupledreactionsattheboundary.HighPe´cletnumber
flowimpliesthattransporteffectsarerelevantonlyinathinunstirredregionnearthe
surface, and our Partial Differential Equation (PDE) system reduces asymptotically
toacouplednonlinearsetofIntegrodifferentialEquations(IDEs).Inphysicallyrel-
evantparameterregimes,thissetofIDEsfurthersimplifiestoasetofOrdinaryDif-
ferentialEquations(ODEs),whichcanbeusedtodeterminetherateconstantsfrom
biosensordata.BycomparingthesolutionofourODEsystemwithasemi-implicit
finite-difference solution to the IDE system, we show that this reduction holds for
a wide parameter range, thus providing experimentalists with a tool to measure the
rate constants in (1.2) and reveal the underlying pathway of polymerase δ in DNA
translesionsynthesis.
Weremarkthattheabovediscussionmayraiseuniquenessconcerns,sincemore
thanonesetofrateconstantsmaypossiblycorrespondtothesamesignal(1.3).Fortu-
nately,throughvaryingtheuniformin-flowconcentrationsoftheligands,C(cid:101)1(0,y(cid:101),(cid:101)t)=
C(cid:101)1,uandC(cid:101)2(0,y(cid:101),(cid:101)t)=C(cid:101)2,u,onemayresolvethisill-posednessincertainphysicallyrel-
evantscenarios(Evans,R.M.andEdwards,D.A.andLi,W.,inpreparation).This
approachtoidentifyingthecorrectsetofrateconstantsinthepresenceofambiguous
dataisrelatedtothe“globalanalysis”techniqueinbiologicalliterature(Karlssonand
Fa¨lt,1997;Mortonetal.,1995).
2 GoverningEquations
2.1 InjectionPhase
Wenowpresenttheequationsgoverningreactionkineticsduringtheinjectionphase;
tothisendwelet
B(cid:101)(x(cid:101),(cid:101)t)=(B(cid:101)1(x(cid:101),(cid:101)t),B(cid:101)12(x(cid:101),(cid:101)t),B(cid:101)2(x(cid:101),(cid:101)t))T
denote a vector in R3 whose components are the three bound state concentrations;
B(cid:101)Σ =B(cid:101)1+B(cid:101)12+B(cid:101)2; and R(cid:101)T denote the initial empty receptor concentration. Note,
thetotalconcentrationofemptyreceptorsatanypointduringtheexperimentisgiven
(cid:0) (cid:1)
by [E](x(cid:101),(cid:101)t)=R(cid:101)T−B(cid:101)Σ x(cid:101),(cid:101)t . Applying the law of mass action to (1.2a) and (1.2b)
givesthefollowingsetofpartialdifferentialequationsforB(cid:101)i(x(cid:101),(cid:101)t):
∂∂B(cid:101)(cid:101)t1 =1(cid:101)ka(R(cid:101)T−B(cid:101)Σ)C(cid:101)1(x(cid:101),0,(cid:101)t)+12(cid:101)kdB(cid:101)12−1(cid:101)kdB(cid:101)1−12(cid:101)kaB(cid:101)1C(cid:101)2(x(cid:101),0,(cid:101)t),
∂∂B(cid:101)(cid:101)t12 =12(cid:101)kaB(cid:101)1C(cid:101)2(x(cid:101),0,(cid:101)t)+21(cid:101)kaB(cid:101)2C(cid:101)1(x(cid:101),0,(cid:101)t)−12(cid:101)kdB(cid:101)12−21(cid:101)kdB(cid:101)12, (2.1)
∂∂B(cid:101)(cid:101)t2 =2(cid:101)ka(R(cid:101)T−B(cid:101)Σ)C(cid:101)2(x(cid:101),0,(cid:101)t)+21(cid:101)kdB(cid:101)12−2(cid:101)kdB(cid:101)2−21(cid:101)kaB(cid:101)2C(cid:101)1(x(cid:101),0,(cid:101)t),
B(cid:101)(x(cid:101),0)=0;
6 RyanM.Evans,DavidA.Edwards
these equations hold on the reacting surface when y(cid:101)=0 and x(cid:101)∈[0,L(cid:101)]. The instru-
ment’s height H(cid:101) is much less than its length L(cid:101), hence its aspect ratio ε =H(cid:101)/L(cid:101) is
small.
Onthereactingsurfaceaty˜=0,thediffusivefluxofligandisusedupinbinding
toformthevariousboundstates:
−D(cid:101)1∂∂C(cid:101)y1(x(cid:101),0,(cid:101)t)=−1(cid:101)ka(R(cid:101)T−B(cid:101)Σ)C(cid:101)1(x(cid:101),0,(cid:101)t)+1(cid:101)kdB(cid:101)1 (2.2a)
(cid:101)
−21(cid:101)kaB(cid:101)2C(cid:101)1(x(cid:101),0,(cid:101)t)+21(cid:101)kdB(cid:101)12,
−D(cid:101)2∂∂C(cid:101)y2(x(cid:101),0,(cid:101)t)=−2(cid:101)ka(R(cid:101)T−B(cid:101)Σ)C(cid:101)2(x(cid:101),0,(cid:101)t)+2(cid:101)kdB(cid:101)2 (2.2b)
(cid:101)
−12(cid:101)kaB(cid:101)1C(cid:101)2(x(cid:101),0,(cid:101)t)+12(cid:101)kdB(cid:101)12.
Here D(cid:101)1 and D(cid:101)2 (respectively) denote the diffusion rates of L1 and L2 through the
buffer.Thefourtermsontheright-handsideof(2.2a)are(inorder):lossofC(cid:101)1 toan
emptyreceptortoformB(cid:101)1;dissociationofB(cid:101)1,whichaddstoC(cid:101)1;lossofC(cid:101)1 toB(cid:101)2 to
formB(cid:101)12;anddissociationofB(cid:101)12intoB(cid:101)2andC(cid:101)1.Thetermsontheright-handsideof
(2.2b)havesimilarinterpretations.Butthenusing(2.1),wehave
D(cid:101)1∂∂C(cid:101)y(cid:101)1(x(cid:101),0,(cid:101)t)= ∂B(cid:101)1∂((cid:101)tx(cid:101),(cid:101)t)+∂B(cid:101)1∂2((cid:101)tx(cid:101),(cid:101)t), (2.3a)
D(cid:101)2∂∂C(cid:101)y(cid:101)2(x(cid:101),0,(cid:101)t)= ∂B(cid:101)1∂2((cid:101)tx(cid:101),(cid:101)t)+∂B(cid:101)2∂((cid:101)tx(cid:101),(cid:101)t). (2.3b)
WenowpresenttheevolutionequationsfortheunboundconcentrationsC(cid:101)1(x(cid:101),y(cid:101),(cid:101)t)
and C(cid:101)2(x(cid:101),y(cid:101),(cid:101)t). Due to the geometry of the device, a unidirectional flow model is
appropriate(ZumbrumandEdwards,2014).ThisPoiseuillechannelflowleadstoa
parabolicvelocityflowprofile;thus,theconvection-diffusionequationsforC(cid:101)i(x(cid:101),y(cid:101),(cid:101)t)
taketheform:
(cid:32) (cid:18) (cid:19) (cid:33)
∂C(cid:101)i =D(cid:101)i∇(cid:101)2C(cid:101)i−(cid:101)v·∇(cid:101)C(cid:101)i, (cid:101)v= V(cid:101)y(cid:101) 1− y(cid:101) ,0 , (2.4)
∂(cid:101)t H(cid:101) H(cid:101)
wherei=1, 2;theaboveequationsholdwhen(x(cid:101),y(cid:101))∈(0,L(cid:101))×(0,H(cid:101))and(cid:101)t>0;and
V(cid:101) isthecharacteristicvelocityassociatedwithourflow.Furthermore:initiallythere
isnounboundligandinthechannel,ligandisconvectedinatuniformratesC(cid:101)i,uafter
the initial time(cid:101)t =0, and no-flux conditions hold downstream at x(cid:101)=L(cid:101) and on the
ceilingaty(cid:101)=H(cid:101).
DuetohighPe´cletnumberflow,Edwardshasdemonstrated(Edwards,1999)that
transporteffectsarerelevantonlyinathinunstirredlayernearthesurface,andthat
theappropriatedimensionlessvariablesare:
x= x(cid:101), y= y(cid:101), η =Pe1/3y, Pe=V(cid:101)H(cid:101)2, t=1(cid:101)kaC(cid:101)1,u(cid:101)t,
L(cid:101) L(cid:101) L(cid:101)D(cid:101)2
(2.5)
B(x,t)= B(cid:101)i(x,t), C(x,η,t)=C(cid:101)i(x(cid:101),y(cid:101),(cid:101)t).
i i
R(cid:101)T C(cid:101)i,u
TransportEffectsonMultiple-ComponentReactionsinOpticalBiosensors 7
These are the natural scalings associated with the reaction-limited and transport-
dominant parameter regime, in which ligand molecules are transported to the un-
stirred layer and gradually bind with receptor sites at the surface. Introducing the
scalings(2.5)into(2.1)resultsinthedimensionlessboundstatesystem:
∂B
1 =(1−B )C +1K B − K B −1K B C , (2.6a)
∂t Σ 1 2 d 12 1 d 1 2 a 1 2
∂B
12 =1K B C +2K B C −1K B − K B , (2.6b)
∂t 2 a 1 2 1 a 2 1 2 d 12 1 d 12
∂B
2 = K (1−B )C + K B − K B −2K B C . (2.6c)
∂t 2 a Σ 2 1 d 12 2 d 2 1 a 2 1
B(x,0)=0, (2.6d)
onthesurfaceatη =0.Thedimensionlessrateconstantsin(2.6)aredefinedas:
jK = C(cid:101)i,u·ij(cid:101)ka, jK = (cid:101)kd ,
i a i d
C(cid:101)1,u·1(cid:101)ka C(cid:101)1,u·1(cid:101)ka
where i=1, 2, and j =1, 2, or can be blank (K and K are the dimensionless
i a i d
analogsofi(cid:101)ka,ori(cid:101)kd).Wenon-dimensionalizethesensogramreading(1.3)bysetting
S(t)= S(cid:102)(t) =B (t)+(cid:18)1+s(cid:101)2(cid:19)B (t)+s(cid:101)2B (t). (2.7)
1 12 2
R(cid:101)T·s(cid:101)1 s(cid:101)1 s(cid:101)1
Thediffusivefluxconditions(2.3)become:
(cid:18) (cid:19)
∂C Da ∂B ∂B
1(x,0,t)= 1 + 12 , (2.8a)
∂η F ∂t ∂t
r
(cid:18) (cid:19)
∂C ∂B ∂B
2(x,0,t)=Da 12 + 2 , (2.8b)
∂η ∂t ∂t
where
Da= 1(cid:101)kaR(cid:101)T(H(cid:101)L(cid:101))1/3, F =CD, C =C(cid:101)1,u, D = D(cid:101)1. (2.9)
r r r r r
(V(cid:101)D(cid:101)2)1/3 C(cid:101)2,u D(cid:101)2
TheDamko¨hlernumber,denotedDa,isakeydimensionlessgroupwhichrepresents
the relative strength of reaction to diffusion. In practice, it is desirable to design
biosensorexperimentssuchthatDa(cid:28)1.WhenDa(cid:28)1,diffusionandreactionoccur
ondifferenttimescales,andoneisinthebestpositiontomeasurereaction.Typically
thisisthecase,althoughattimesexperimentalistscanonlyincreasevelocitiesenough
tomakeDa=O(1)(Edwards,1999).Further,F measuresthediffusionstrengthof
r
eachreactingspecies,ascharacterizedbytheproductoftheinputconcentrationsand
thediffusioncoefficients.
It remains to derive the equations for the unbound concentrations C(x,η,t) in
i
the unstirred layer. To this end, we substitute the dimensionless variables (2.5) into
8 RyanM.Evans,DavidA.Edwards
(2.4) and expand the result in a perturbation series for large Pe. The leading order
equationsare:
∂2C ∂C
D 1 =η 1, (2.10a)
r ∂η2 ∂x
∂2C ∂C
2 =η 2, (2.10b)
∂η2 ∂x
withtheboundarydata:
C(0,η,t)=1, (2.11a)
i
limC(x,η,t)=1, (2.11b)
i
η→∞
andequations(2.8).Equation(2.11a)tellsusthatatx=0,theconcentrationisgiven
bythenormalizedinletvalue.Equation(2.11b)expressestherequirementthatasone
exitstheunstirredlayer,i.e.asη→∞,theconcentrationmustmatchtheundisturbed
valuein thebulk flow.A moredetailed descriptionof thisprocess maybe foundin
(Edwards,1999).
To study (2.6) we need the value of C only on the reacting surface at η =0.
i
Byanalogywith(Edwards,1999)wecantransform(2.10a)and(2.10b)intoAiry’s
equationsthroughaLaplacetransforminx.Thederivativeofthetransformedsolution
atη =0isknown,soonecanshow
D1/3Da (cid:90) x(cid:18)∂B ∂B (cid:19) dν
C (x,0,t)=1− r 1 + 12 (ν,t) , (2.12a)
1 FrΓ(2/3)31/3 0 ∂t ∂t (x−ν)2/3
Da (cid:90) x(cid:18)∂B ∂B (cid:19) dν
C (x,0,t)=1− 12 + 2 (ν,t) . (2.12b)
2 Γ(2/3)31/3 0 ∂t ∂t (x−ν)2/3
Earlyintheexperiment,ligandmoleculesdiffusetothesurfacetobindwithreceptor
sitesupstream,beforetheydosodownstream.Thisphenomenonofupstreamligand
depletionisreflectedin(2.12),andreadilyseeninFigure2.1;herewehavedepicted
results of our numerical method described in Section 4. Thus, during the injection
phasetheboundstateevolutionisgovernedbytheIDEsystem(2.6),with(2.12).
2.2 WashPhase
Wenowpresenttherelevantdimensionlessequationsforthewashphase.Inpractice,
the injection phase is run until the bound state concentration reaches a steady state
(RichandMyszka,2009).Sincetheboundligandconcentrationevolvesonamuch
slower time scale than the unbound ligand concentration (Edwards, 1999), the un-
boundligandwillhavealsoreachedsteadystatebythetimethewashphasebegins;
i.e.,C(x,y,0)=1for(x,y)∈[0,1]2,atthestartofthewashphase.Thereactionkinet-
i
icsarethereforegovernedby(2.6),with(2.6d)replacedbythesteadystatesolution
TransportEffectsonMultiple-ComponentReactionsinOpticalBiosensors 9
Fig.2.1:Injectionphaseofbiosensorexperiment,uptot=5,obtainedthroughsolv-
ing(2.6),(2.12)withthenumericalmethoddescribedinSection4.Allrateconstants
were taken equal to one, and Da taken equal to two to visualize upstream ligand
depletion,whichisespeciallyevidentinB .
12
to(2.6)duringtheinjectionphase:
B(x,0)=A−1f, (2.13)
(1+ K +1K ) 1−1K 1 
1 d 2 a 2 d
A= −12Ka (12Kd+21Kd) −21Ka , (2.14)
K K −2K ( K + K +2K )
2 a 2 a 1 d 2 a 2 d 1 a
 
1
f= 0 . (2.15)
K
2 a
Equationssimilarto(2.11b)hold:
C(0,η,t)=0, (2.16a)
i
limC(x,η,t)=0. (2.16b)
i
η→∞
10 RyanM.Evans,DavidA.Edwards
Equation(2.16a)istheinflowcondition,and(2.16b)expressestherequirementthat
theconcentrationinthelayermustmatchtheconcentrationC(x,y,t)=0intheouter
i
layer. In a similar manner to the injection phase, one can use (2.10) together with
(2.16)toshow
D1/3Da (cid:90) x(cid:18)∂B ∂B (cid:19) dν
C (x,0,t)=− r 1 + 12 (ν,t) , (2.17a)
1 FrΓ(2/3)31/3 0 ∂t ∂t (x−ν)2/3
Da (cid:90) x(cid:18)∂B ∂B (cid:19) dν
C (x,0,t)=− 12 + 2 (ν,t) . (2.17b)
2 Γ(2/3)31/3 0 ∂t ∂t (x−ν)2/3
Thus,duringthewashphase,theboundstateevolutionisgovernedbytheIDEsystem
(2.6a)–(2.6c),(2.13),and(2.17).
3 EffectiveRateConstantApproximation
Duringbothphasesoftheexperiment,theboundstateconcentrationobeysanonlin-
earsetofIDEswhichishopelesstosolveinclosedform.Moreover,weareultimately
interested in the average concentration B(t)–not B(x,t)–since from B(t) we can re-
constructthesensogramsignal(2.7)(i.e.,thequantityofphysicalrelevance).Thus,
we seek to find an approximation to B(t), and begin by finding one during the in-
jection phase. We first average each side of (2.6) and (2.12) in the sense of (1.1).
Immediately,weareconfrontedwithtermssuchas
(cid:18) Da (cid:90) x(cid:18)∂B ∂B (cid:19) dν (cid:19)
B C =B 1− 12 + 2 , (3.1)
1 2 1 31/3Γ(2/3) 0 ∂t ∂t (x−ν)2/3
ontherighthandsideof(2.6a).IntheexperimentallyrelevantcaseofsmallDa,we
aremotivatedtoexpandB(x,t)inaperturbationseries:
B(x,t)=0B(x,t)+O(Da). (3.2)
Inthislimit,theleadingorderof(2.12)isjustC =1.Usingthisresultin(2.6),we
i
havethatthegoverningequationfor0Bisindependentofx:
d0B
=−A0B+f,
dt
whereAisgivenby(2.14)andfby(2.15).Hencetheleading-orderapproximation
0B(t)=A−1(I−e−At)f (3.3)
isindependentofspace.Substituting(3.3)into(2.12),thetime-dependenttermsmay
befactoredoutoftheintegrand,leavingthespatialdependenceofC varyingasx1/3.
j
Thisistheonlyspatialvariationin(2.6)atO(Da);hencewemaywrite
B(x,t)=0B(t)+Dax1/3·1B(t)+O(Da2). (3.4)
Asaresultof(3.4)wehavetherelation
DaB(x,t)=Da0B(t)+O(Da2), (3.5)
i i