Abstract
We analyze surfacevolume reactions in the context of optical biosensors with arrays of reacting zones. For arrays having zones with the same rate constants, we consider a twodimensional reacting zone boundary definition and quantify ligand depletion with the effective Damköhler number. We use asymptotics to obtain ligand depletion results for the onedimensional 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 timedependent and spacedependent terms, and we propose two effective rate constant equations for this case.
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Abbreviations
 \(A\) :

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
 0:

as a subscript, used to indicate leadingorder 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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If the same letter appears with and without a tilde, the letter with a tilde has dimension and the letter without a tilde is dimensionless. Units are listed in terms of length \((L)\), mass \((M)\), moles \((N)\), or time \((T)\).
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Zumbrum, M.E., Edwards, D.A. Multiple Surface Reactions in Arrays with Applications to Optical Biosensors. Bull Math Biol 76, 1783–1808 (2014). https://doi.org/10.1007/s115380149977z
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DOI: https://doi.org/10.1007/s115380149977z
Keywords
 Surface reactions
 Perturbation methods