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Multiple Surface Reactions in Arrays with Applications to Optical Biosensors


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.

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\(A\) :

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


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


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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Correspondence to Matthew E. Zumbrum.

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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).

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  • Surface reactions
  • Perturbation methods