Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Reaction Kinetics Need to get C i in terms of B j . ∂ B 1 ∂ t = 1 k a ( 1 − B Σ ) C 1 + 1 2 k d B 12 − 1 k d B 1 − 1 2 k a B 1 C 2 , ∂ B 12 = 1 2 k a B 1 C 2 + 2 1 k a B 2 C 1 − 1 2 k d B 12 − 2 1 k d B 12 , ∂ t ∂ B 2 ∂ t = 2 1 k d B 12 + 2 k a ( 1 − B Σ ) C 2 − 2 1 k a B 2 C 1 − 2 k d B 2 .
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Unstirred Layer Consider the set of PDE’s for C 1 ∂ 2 C 1 ∂η 2 = η ∂ C 1 (14) D r ∂ x � ∂ B 1 � ∂ C 1 ∂ t + ∂ B 12 ∂η ( x , 0 , t ) = Da , (15) D r ∂ t Introduce a Laplace transform in x in (14) and use (15): � x � ∂ B 1 � Da + ∂ B 12 d ν C 1 ( x , 0 , t ) = 1 − ( ν, t ) ( x − ν ) 2 / 3 . D 2 / 3 1 ∂ t ∂ t 3 Γ( 2 3 3 ) 0 r
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Ligand Depletion Convolution integral represents upstream ligand depletion � x � ∂ B 1 � Da + ∂ B 12 d ν C 1 ( x , 0 , t ) = 1 − ( ν, t ) D 2 / 3 ∂ t ∂ t ( x − ν ) 2 / 3 3 Γ( 2 1 3 3 ) 0 r Ligand concentration a perturbation away from the outer concentration. Defined as � x d ν J α f ( x ) = f ( ν ) ( x − ν ) 1 − α , (16) 0 one may recognize the integral term in C 1 as a fractional integral, with α = 1 / 3.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Bound State System The bound state system is then: ∂ B 1 ∂ t = ( 1 − B Σ ) C 1 − 1 K d B 1 − 1 2 K a B 1 C 2 + 1 2 K d B 12 ∂ B 12 = 1 2 K a B 1 C 2 − 1 2 K d B 12 + 2 1 K a B 2 C 1 − 2 1 K d B 12 ∂ t ∂ B 2 ∂ t = 2 1 K d B 12 − 2 1 K a B 2 C 1 + 2 K a ( 1 − B Σ ) C 2 − 2 K d B 2 with � x � ∂ B 1 � Da + ∂ B 12 d ν C 1 ( x , 0 , t ) = 1 − ( ν, t ) D 2 / 3 ( x − ν ) 2 / 3 1 ∂ t ∂ t 3 Γ( 2 3 3 ) 0 r � x � ∂ B 1 � Da + ∂ B 12 d ν C 2 ( x , 0 , t ) = 1 − ( ν, t ) 1 ( x − ν ) 2 / 3 3 Γ( 2 ∂ t ∂ t 3 3 ) 0
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Perturbation Approximation Da ≪ 1, so we can search for a perturbation expansion of the form B = 0 B + Da 1 B + O ( Da 2 ) . (17) Leading order: d 0 B = − A 0 B + e 1 + 2 K a e 3 (18) d t 0 B ( t ) = ( I − e − At )[ A − 1 ( e 1 + 2 K a e 3 )] , (19) e.g. well mixed approximation. The spatial dependence in 1 B ( x , t ) ∼ x 1 / 3 .
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Perturbation Approximation Thus we may write B ( x , t ) = ( I − e − At )[ A − 1 ( e 1 + 2 K a + x 1 / 3 Da 1 B ( t )] + O ( Da 2 ) . (20) Problem: 1 B contains a secular term of the form te − λ t in one of its components. A multiple scale expansion would be unweildy, and would have to be manipulated again to obtain an expression of physical relavance.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Another Approximation We are really interested in B . What if we could derive a set of equations for B , and solve them numerically using a standard ODE Package?
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Averaged Bound State System To do this we would integrate both sides of ∂ B 1 = ( 1 − B Σ ) C 1 − 1 K d B 1 − 1 2 K a B 1 C 2 + 1 2 K d B 12 ∂ t ∂ B 12 = 1 2 K a B 1 C 2 − 1 2 K d B 12 + 2 1 K a B 2 C 1 − 2 1 K d B 12 ∂ t ∂ B 2 = 2 1 K d B 12 − 2 1 K a B 2 C 1 + 2 K a ( 1 − B Σ ) C 2 − 2 K d B 2 ∂ t using � x � ∂ B 1 � Da + ∂ B 12 d ν C 1 ( x , 0 , t ) = 1 − ( ν, t ) ( x − ν ) 2 / 3 , D 2 / 3 1 3 Γ( 2 ∂ t ∂ t 3 3 ) r 0 � x � ∂ B 1 � Da + ∂ B 12 d ν C 2 ( x , 0 , t ) = 1 − ( ν, t ) ( x − ν ) 2 / 3 . 1 ∂ t ∂ t 3 Γ( 2 3 3 ) 0
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis How to Deal With Convolution Integral We may exploit the fact that to leading order B is independent of space � x � ∂ B 1 � Da + ∂ B 12 d ν C 1 ( x , 0 , t ) = 1 − ( ν, t ) D 2 / 3 ( x − ν ) 2 / 3 1 ∂ t ∂ t 3 Γ( 2 3 3 ) 0 r B 1 ( x , t ) = 0 B 1 ( t ) + Da 1 B 1 ( x , t ) + O ( Da 2 ) By substituting our expansion into C 1 we arrive at � x � � d 0 B 1 + d 0 B 12 Da d ν ( x − ν ) 2 / 3 + O ( Da 2 ) C 1 ( x , 0 , t ) = 1 − ( t ) 1 D 2 / 3 d t d t 3 Γ( 2 3 3 ) 0 r
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis How to Deal With Convolution Integral Since time dependence factors out of the integral, we may write � � d 0 B 1 + d 0 B 12 + O ( Da 2 ) C 1 ( x , 0 , t ) = 1 − Da h ( x ) (21) d t d t where, � x ( x − ν ) − 2 / 3 d ν = 3 2 / 3 x 1 / 3 1 h ( x ) = Γ( 2 / 3 ) . (22) 3 1 / 3 Γ( 2 / 3 ) 0
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis ERC Equations Using these manipulations and some algebra we can derive a set of nonlinear ODE’s, Effective Rate Constant Equations , for B : d B d t = ( I + Da N ( B )) − 1 ( − A B + e 1 + 2 K a e 3 ) + O ( Da 2 ) .
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis ERC Equation Solution B 1 B 12 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 y 0.5 y 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 1 2 3 4 5 0 1 2 3 4 5 t t B 2 1 0.9 0.8 0.7 0.6 y 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 t
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis ERC Equation Solution Sensogram Signal 1 0.9 0.8 0.7 0.6 y 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 t Da = . 01. Reaction rates equal to one.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Numerics Used a finite difference algorithm ∂ B 1 i , n + 1 = ( 1 − B Σ i , n ) C 1 i , n + 1 − 1 K d B 1 i , n − 1 2 K a B 1 i , n C 2 i , n + 1 + 1 2 K d B 12 i , n , ∂ t ∂ B 12 i , n + 1 = 1 i , n + 1 − 1 i , n + 2 i , n + 1 − 2 2 K a B 2 i , n C 2 2 K d B 12 1 K a B 2 i , n C 1 1 K d B 12 i , n , ∂ t ∂ B 2 i , n + 1 = 2 i , n − 2 i , n + 1 + 2 K a ( 1 − B Σ 1 K d B 12 1 K a B 2 i , n C 1 i , n ) C 2 i , n + 1 − 2 K d B 2 i , n . ∂ t
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Multiple-Component Reactions n i
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Multiple-Component Reactions n i
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Multiple-Component Reactions n i
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Multiple-Component Reactions n i
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Multiple-Component Reactions n i
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Multiple-Component Reactions n i
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Multiple-Component Reactions n i
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Singular Convolution Integral Difficult to deal with singularity � x i � � Da ∂ B 1 ∂ t ( x i − ξ, t n + 1 ) + ∂ B 12 d ξ C 1 i , n + 1 = 1 − ∂ t ( x i − ξ, t n + 1 ) ( ν, t ) 1 D 2 / 3 ξ − 2 / 3 3 Γ( 2 3 3 ) 0 r Use trapezoidal rule to discretize the integral
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Convolution Integral Subtract out the singularity � � x i � ∂ t ( x i − ξ, t n + 1 ) − ∂ B 1 Da ∂ B 1 i , n + 1 C 1 i , n + 1 = 1 − 1 D 2 / 3 ∂ t 3 Γ( 2 3 3 ) r 0 � � �� ∂ t ( x i − ξ, t n + 1 ) − ∂ B 12 ∂ B 1 + ∂ B 12 + ∂ B 12 d ξ 1 i , n + 1 i , n + 1 i , n + 1 + 3 x 3 i ξ − 2 / 3 ∂ t ∂ t ∂ t Even when singularity is subtracted out, convergence is only O (∆ x 2 / 3 ) due to functional form. Temporal convergence O (∆ t 2 ) , from AB2 time-stepping scheme.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Results B 1 0.4 0.3 0.2 0.1 0 0 1 0.5 0.5 1 0 t x Figure : Left: B 1 after 1 second. Right: B 1 after 5 seconds Da = 2 . All reaction rate constants taken to be 1
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Results
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Error in ERC Equations B 1 B 12 1 x 10 −6 6 x 10 −7 0.9 5 0.8 0.7 4 0.6 y 0.5 y 3 0.4 2 0.3 0.2 1 0.1 0 0 0 1 2 3 4 5 0 1 2 3 4 5 t t B 2 1 x 10 −6 0.9 0.8 0.7 0.6 y 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 t
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Error in ERC Equations B 1 | s Error B 12 | s Error 1 x 10 −3 1.2 x 10 −3 0.9 1 0.8 0.7 0.8 Abs Error Abs Error 0.6 0.5 0.6 0.4 0.4 0.3 0.2 0.2 0.1 0 0 0 2 4 6 8 0 2 4 6 8 t t B 2 | s Error 1.5 x 10 −3 Abs Error 1 0.5 0 0 2 4 6 8 t
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis ERC Error vs Da Error B 1 | s Error B 12 | s −4 −4 −6 −6 log(err) log(err) −8 −8 −10 −10 y = 1 . 9150 x − 5 . 2190 y = 1 . 8844 x − 5 . 1452 −12 −12 R 2 = . 9983 R 2 = . 9986 −14 −14 −4 −2 0 2 4 6 −4 −2 0 2 4 6 log(Da) log(Da) Error B 2 | s −4 −6 log(err) −8 −10 y = 1 . 8494 x − 4 . 8553 −12 R 2 = . 9993 −14 −4 −2 0 2 4 6 log(Da)
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Wash Phase We have derived similar results for the wash phase. Recall in the wash phase, only the buffer fluid is flowing through the biosensor.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Wash Phase In this case, we still have the same kinetics system at the boundary, ∂ B 1 ∂ t = ( 1 − B Σ ) C 1 − 1 K d B 1 − 1 2 K a B 1 C 2 + 1 2 K d B 12 , ∂ B 12 = 1 2 K a B 1 C 2 − 1 2 K d B 12 + 2 1 K a B 2 C 1 − 2 1 K d B 12 , ∂ t ∂ B 2 ∂ t = 2 1 K d B 12 − 2 1 K a B 2 C 1 + 2 K a ( 1 − B Σ ) C 2 − 2 K d B 2 . Unbound ligand concentration at the surface will be different, i.e. only trace amounts.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Wash Phase Therefore instead of � x � ∂ B 1 � Da + ∂ B 12 d ν C 1 ( x , 0 , t ) = 1 − ( ν, t ) ( x − ν ) 2 / 3 , D 2 / 3 1 ∂ t ∂ t 3 Γ( 2 3 3 ) 0 r we have � x � ∂ B 1 � Da + ∂ B 12 d ν C 1 ( x , 0 , t ) = − ( ν, t ) ( x − ν ) 2 / 3 , D 2 / 3 1 ∂ t ∂ t 3 Γ( 2 3 3 ) 0 r In this case ∂ B i ∂ t < 0, and C 1 = O ( Da ) .
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Wash Phase Results ERC equations in this case are d B d t = ( I + Da N ( B )) − 1 ( − D B ) + O ( Da 2 ) (23)
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Wash Phase Results: FD Solution
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Wash Phase Results: FD Solution
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Wash Phase Results: ERC Error vs. Da Error B 1 | s Error B 12 | s −4 −4 −6 −6 log(err) log(err) −8 −8 −10 −10 y = 1 . 7702 x − 6 . 4990 y = 1 . 8679 x − 5 . 7978 −12 −12 R 2 = . 9983 R 2 = . 9993 −14 −14 −4 −2 0 2 4 6 −4 −2 0 2 4 6 log(Da) log(Da) Error B 2 | s −4 −6 log(err) −8 −10 y = 1 . 8354 x − 5 . 3414 −12 R 2 = . 9990 −14 −4 −2 0 2 4 6 log(Da)
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Recovering Reaction Rates Overall Goal: What are the reaction rates? Can we actually find cases where different rate constants give the same signal ? Can we develop a curve fitting algorithm?
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Recovering Reaction Rates Take Da = 0 and study the linear set of ODE’s d B d t = − A B + f , B ( 0 ) = 0 . (24) Use (24) as our data.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Recovering Reaction Rates Here ( 1 + 1 K d + 1 1 − 1 2 K a ) 2 K d 1 − 1 ( 1 2 K d + 2 − 2 A = 1 K d ) 2 K a 1 K a 2 K a − 2 ( 2 K a + 2 K d + 2 2 K a 1 K d 1 K a ) and f = e 1 or f = 2 K a e 3
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Methodology First inject ligand one until the system reaches an equilibrium, then inject ligand two. Broke the problem up into cases based on the size of 1 K d , 2 K a , 2 K d .
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Recovering Reaction Rates B 1 ( P 1 a ) 1 1 K d 2 K d ( P 1 b ) 1 2 K a B 12 E 2 2 K d 1 K d 2 (P2) 1 K a ( P 1 c ) 2 K a B 2
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Recovering Reaction Rates 1 K d = 100
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Recovering Reaction Rates B 1 ( P 1 a ) 1 1 K d 2 K d ( P 1 b ) 1 2 K a B 12 E 2 2 K d 1 K d 2 (P2) 1 K a ( P 1 c ) 2 K a B 2
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Recovering Reaction Rates 1 1 K d = 100 , 2 K d = 100
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Recovering Reaction Rates B 1 ( P 1 a ) 1 1 K d 2 k d ( P 1 b ) 1 2 K a B 12 E 2 2 K d 1 K d 2 (P2) 1 k a ( P 1 c ) 2 k a B 2
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Recovering Reaction Rates
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Ambiguous Sensogram Left: 2 K d = 100 , 1 1 2 K a = 100 . Right: 2 K a = 100. Both: C 1 = 1.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Ambiguous Sensogram: Case 1 B 1 ( P 1 a ) 1 1 K d 2 K d ( P 1 b ) 1 2 K a B 12 E 2 2 K d 1 K d 2 (P2) 1 K a ( P 1 c ) 2 K a B 2
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Ambiguous Sensogram: Case 2 B 1 ( P 1 a ) 1 1 K d 2 K d ( P 1 b ) 1 2 K a B 12 E 2 2 K d 1 K d 2 (P2) 1 K a ( P 1 c ) 2 K a B 2
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Clarified Sensogram 100 , 1 1 Left: 2 K d = 2 K a = 100. Right: 2 K a = 100. Both: C 1 = . 1.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Single Ligand Analysis When studying the single ligand process, there is only one type of reaction at the boundary. In this case the reacting species concentration obeys they equation � x � � ∂ B Da ∂ B d ν ∂ t = ( 1 − B ) 1 − ∂ t ( ν, t ) − KB 3 1 / 3 Γ( 2 / 3 ) ( x − ν ) 2 / 3 0 Can we find an analytic expression for B or B when Da = O ( 1 ) ?
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis A Homotopy Method Homotopy : a continuous deformation of one curve into another. H ( t , s ) = ( 1 − s ) γ 0 ( t ) + s γ 1 ( t ) , s ∈ [ 0 , 1 ] Can we try the same thing with differential operators?
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis A Homotopy of Differential Operators Many differential operators A can be composed into a linear part L , and nonlinear part N L ( B ) + N ( B ) = F . (25) � �� � A ( B ) We can draw a homotopy between L and A H ( B , p ) = ( 1 − p ) L ( B ) + p A ( B ) , p ∈ [ 0 , 1 ] . (26)
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Nuts and Bolts Therefore we can propose a series solution to H ( B , p ) = 1 ⇔ ( 1 − p ) L ( B ) + p A ( B ) = F , p ∈ [ 0 , 1 ] . of the form B ( x , t ) = B 0 ( x , t ) + pB 1 ( x , t ) + p 2 B 2 ( x , t ) + · · · .
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Nuts and Bolts Thus when examing the p th coefficient of our series in the equation ⇔ ( 1 − p ) L ( B ) + p A ( B ) = F , p ∈ [ 0 , 1 ] . (27) we will find that the nonlinearity is higher order. That is we will have an equation of the form L ( B i ) = −N ( B 1 , . . . , B i − 1 ) . (28)
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Single Ligand Analysis The equation governing the bound state in the single ligand case is � x � � ∂ B Da ∂ B d ν ∂ t = ( 1 − B ) 1 − ∂ t ( ν, t ) − KB 3 1 / 3 Γ( 2 / 3 ) ( x − ν ) 2 / 3 0 First we obtain an expression for B by averaging each side, and rearranging some terms: � x d B Da ∂ B d ν d t + ( 1 + K ) B + 3 1 / 3 Γ( 2 / 3 )( B − 1 ) ∂ t ( ν, t ) = 1 ( x − ν ) − 2 / 3 0 � �� � � �� � L N
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Series Solution Propose and substitute a series solution. B ( x , t ) = B 0 ( x , t ) + pB 1 ( x , t ) + p 2 B 12 ( x , t ) + · · · (29) H ( B , p ) = 1 . (30) Get linear ODE’s for B 0 ( t ) , B 1 ( t ) , B 2 ( t ) , . . . An approximation to B is then given by B ( t ) = B 0 + B 1 ( t ) + B ( t ) + · · ·
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Two Terms Doing this the first two terms are: B 0 ( t ) = α − 1 ( 1 − e − α t ) B 1 ( t ) = − Da he − 2 t α ( − 1 + e t α − e t α t α + e t α t α 2 ) α 2 α = ( K + 1 ) x 1 / 3 h = 3 1 / 3 Γ( 2 / 3 )
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Issues Convergence of our series. When Da = O ( 1 ) or Da ≫ 1, what guaruntees that our series will converge? Secular term of the form te − α t This is not bad enough make our series converge, but still throws off the accuracy.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Convergence A standard technique is to embed a convergence control paramter q into our homotopy ( 1 − p )( L ( B ) − L ( b 0 )) + qp A ( B ) = 1 , p ∈ [ 0 , 1 ] . (31) Choose q that minimizes ||A ( B ) − 1 || 2 (32) 2 Done numerically in Mathematica.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Time Scale We can fix convergence, but the time scale is still off. B s vs Approximation B s vs Approximation y y 0.6 0.6 0.4 0.4 0.2 0.2 14 t 14 t 2 4 6 8 10 12 2 4 6 8 10 12 B s B s � 0.2 � 0.2 3 Terms 3 Terms Da = 3; This is a 3 term approximation with and without the convergence parameter q
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Time Scale The propose a strained time scale of the form: τ = ( 1 + p ω 1 + p 2 ω 2 + · · · ) t , (33) where the ω i are choosen to eliminate secular terms.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Expansion The first two terms are: B 0 ( τ ) = α − 1 ( 1 − e − ατ ) , B 1 ( τ ; q ) = q Da he − 2 ατ ( e ατ − 1 ) , α 2 τ = ( 1 + ω 1 + ω 2 + · · · ) t , where α = ( 1 + K ) , and ω 1 = − q Da h ( 1 − α − 1 ) , ω 2 = Da hq α − 2 ( α − 1 )( − Da hq + α − q α + Da hq α ) , x 1 / 3 h = 3 1 / 3 Γ( 2 / 3 ) .
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Two Term Expansion, Da = 1 / 2 Absolute Error B s vs Approximation y y 0.030 0.6 0.025 0.020 0.4 0.015 0.2 0.010 0.005 t t 2 4 6 8 0 2 4 6 8 B s � 0.2 2 Terms Two term expansion, Da = 1 / 2.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Three Term Expansion, Da = 2 Absolute Error B s vs Approximation y y 0.030 0.6 0.025 0.020 0.4 0.015 0.2 0.010 0.005 14 t 14 t 2 4 6 8 10 12 0 2 4 6 8 10 12 B s � 0.2 3 Terms Three term Approximation, Da = 2.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Five Term Expansion, Da = 10 Absolute Error B s vs Approximation y y 0.030 0.6 0.025 0.020 0.4 0.015 0.2 0.010 0.005 t t 10 20 30 0 10 20 30 B s � 0.2 5 Terms Five term expansion, Da = 10.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Five Term Expansion, Da = 100 Absolute Error B s vs Approximation y y 0.030 0.6 0.025 0.020 0.4 0.015 0.2 0.010 0.005 t t 50 100 150 200 250 0 50 100 150 200 250 B s � 0.2 5 Terms Five term expansion, Da = 100.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Dissociation Phase The expansion in the dissociation phase is: B 0 ( t ) = e − k τ α , Da e − 2 k τ � − 1 + e k τ � hq B 1 ( t ) = − α 2 τ = ( 1 + ω 1 + ω 2 + · · · ) t ω 1 = − Da hq , ω 2 = Da hq ( − 1 + q + Da hq ) .
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Two Term Expansion, Da = 1 / 2 Absolute Error B s vs Approximation y 0.030 y 0.6 0.025 0.020 0.4 0.015 0.010 0.2 0.005 t t 5 10 15 0 5 10 15 B s � 0.2 2 Terms Two term expansion, Da = 1 / 2.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Three Term Expansion, Da = 2 Absolute Error B s vs Approximation y 0.030 y 0.6 0.025 0.020 0.4 0.015 0.010 0.2 0.005 35 t 35 t 5 10 15 20 25 30 0 5 10 15 20 25 30 B s � 0.2 2 Terms Three term Approximation, Da = 2.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Five Term Expansion, Da = 10 Absolute Error B s vs Approximation y 0.030 y 0.6 0.025 0.020 0.4 0.015 0.010 0.2 0.005 t t 20 40 60 80 100 0 20 40 60 80 100 B s � 0.2 3 Terms Five term expansion, Da = 10.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Five Term Expansion, Da = 100 Absolute Error B s vs Approximation y 0.030 y 0.6 0.025 0.020 0.4 0.015 0.010 0.2 0.005 800 t t 200 400 600 0 20 40 60 80 100 B s � 0.2 4 Terms Five term expansion, Da = 100.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Matches Up With ERC Approximation Absolute Error Absolute Error y y 0.010 0.010 0.008 0.008 0.006 0.006 0.004 0.004 0.002 0.002 t t 2 4 6 8 2 4 6 8 2 Terms 5 Terms � 0.002 � 0.002 ERC ERC Left: Two term approximation vs ERC Approximation. Right: Five-Term approximation vs ERC Approximation.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Conclusions Modeling Multiple-Components in Optical Biosensors. We must consider transport. Full model simplifes to a coupled system of integrodiffential equations. These equations further reduce to a set of nonlinear ODE’s. Formally holds for Da ≪ 1, numerically everywhere. Sensogram Issues Multiple reacting species make interpreting Sensogram data difficult. Can fix this in certain cases by varying of C 1 .
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Conclusions Single-Component Reactions Strongly nonlinear problem when Da = O ( 1 ) . Can find analytic approximations to B by applying a homotopy method. Must used a strained time scale. Matches up with ERC approximations.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis Future Work Tie together multiple-receptor and multiple-ligand model. Nonuniform initial receptor concentration. Helical geometries.
Optical Biosensors Multi-Component Model Recovering Reaction Rates Single Ligand Analysis References C. Bertucci, A. Piccoli, and M. Pistolozzi Optical biosensors as a tool for 1 early determination of absorption of lead candidates and drugs. comb. chem. high throughput screen , 10:433:440, 2007 D. A. Edwards. Transport effects on surface reaction arrays: biosensor 2 applications. Mathematical Biosciences . 12-22,2007 E.F.Grabowski, L.I. Friedman, and E.F. Leonard. Effects of shear rate on 3 diffusion and adhesion of blood platelets to a foreign surface. Ind. Eng. Chem. Fund , 11:224-232, 1972 J. He. The homotopy perturbation technique. Computer Methods in 4 Applied Mechanics and Engineering , 178 (3):257–262, 1999 R.L. Rich, D. G. Myszka, Survey of the Year 2009 Commercial optical 5 biosensor literature. J. Mol. Recognit , 24:892-914,2011
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