# Multiple Surface Reactions in Arrays with Applications to Optical Biosensors

## Abstract

We analyze surface-volume reactions in the context of optical biosensors with arrays of reacting zones. For arrays having zones with the same rate constants, we consider a two-dimensional reacting zone boundary definition and quantify ligand depletion with the effective Damköhler number. We use asymptotics to obtain ligand depletion results for the one-dimensional case, and also compute results for the circular reacting zone case. For arrays having zones with different rate constants, depletion effects cannot be expressed as the product of time-dependent and space-dependent terms, and we propose two effective rate constant equations for this case.

### Keywords

Surface reactions Perturbation methods### List of Symbols

### Variables and Parameters

- \(A\)
Area of reacting zone (17)

- \(a\)
Constant in \(\text {Da}\) bound

- \(\tilde{B}(\tilde{x}, \tilde{z}, \tilde{t})\)
Bound ligand concentration, units \(N/L^2\)

- \(b\)
Constant in \(\text {Da}\) bound

- \(\tilde{C}(\tilde{x}, \tilde{y}, \tilde{z}, \tilde{t})\)
Ligand concentration, units \(N/L^3\) (1)

- \(\tilde{C}_\text {u}\)
Uniform feed ligand concentration, units \(N/L^3\) (1)

- \(c\)
Constant in \(\text {Da}\) bound

- \(\tilde{D}\)
Molecular diffusion coefficient, units \(L^2/T\)

- \(\text {Da}, {}_i\text {Da}\)
Damköhler number (7)

- \(\text {Da}_i\)
Effective Damköhler number for \(i^{\text {th}}\) reacting zone (18)

- \(d\)
Constant in \(\text {Da}\) bound

- \(f_1, f_2\)
General functions in discussion of the boundedness of \(\text {Da}_i(t)\)

- \(g\)
Constant in average ligand depletion

- \(\tilde{H}\)
Height of biosensor channel, units \(L\) (1)

- \(H\)
Harmonic number

- \(h\)
Spatial function for ligand concentration (14)

- \(\overline{h}\)
Constant in average ligand concentration

- \(I(x, z)\)
Indicator function for reacting zone (8)

- \(i\)
Row variable

- \(j\)
Column variable

- \(K\)
Scaled affinity constant (5)

- \(\tilde{k}_{\text {on}}, \tilde{k}_{\text {off}}\)
Interaction rate constants, units \(L^3/NT\) and \(1/T\)

- \(\tilde{L}\)
Length of biosensor channel, units \(L\)

- \(\tilde{L}_\text {r}\)
Diameter of a circular reacting zone, units \(L\) (1)

- \(m\)
Parameter for reacting zone boundary definition

- \(n\)
Indexing variable

- Pe
Peclét number

- \(\tilde{R}\)
Receptor concentration on reacting surface, units \(N/L^2\)

- \(\mathcal{R}_\mathrm{r}\)
Reacting surface (5)

- \(r\)
Root function (21)

- \(Re\)
Reynolds number

- \(S{[\cdot ]}\)
Sensogram (17)

- \(\tilde{t}\)
Reaction time scale, units \(T\) (1)

- \(\tilde{V}\)
Characteristic velocity, units \(L/T\)

- \(\tilde{W}\)
Width of biosensor channel, units \(L\)

- \(x(z; j)\)
Boundary for reacting zone (21)

- \(\tilde{x}, \tilde{y}, \tilde{z}\)
Spatial variables, units \(L\) (1)

- \(\varGamma \)
Gamma function

- \(\eta \)
Boundary layer variable (2)

- \(\kappa _{\text {on}}\)
Ratio of association rate constant to the first reacting zone association rate constant (28)

- \(\nu \)
Convolution integral variable

### Other Notation

- 0
as a subscript, used to indicate leading-order perturbation expansion

- \(-\)
as a subscript, used to indicate smaller quadratic root

- \(-\)
as a superscript, used to indicate the beginning of a reacting zone

- +
as a subscript, used to indicate larger quadratic root

- +
as a superscript, used to indicate the end of a reacting zone

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