Bulletin of Mathematical Biology

, Volume 76, Issue 7, pp 1783–1808 | Cite as

Multiple Surface Reactions in Arrays with Applications to Optical Biosensors

  • Matthew E. Zumbrum
  • David A. Edwards
Original Article


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.


Surface reactions Perturbation methods 

List of Symbols

Variables and Parameters


Area of reacting zone (17)


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

\(\tilde{B}(\tilde{x}, \tilde{z}, \tilde{t})\)

Bound ligand concentration, units \(N/L^2\)


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)


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


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)


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

\(f_1, f_2\)

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


Constant in average ligand depletion


Height of biosensor channel, units \(L\) (1)


Harmonic number


Spatial function for ligand concentration (14)


Constant in average ligand concentration

\(I(x, z)\)

Indicator function for reacting zone (8)


Row variable


Column variable


Scaled affinity constant (5)

\(\tilde{k}_{\text {on}}, \tilde{k}_{\text {off}}\)

Interaction rate constants, units \(L^3/NT\) and \(1/T\)


Length of biosensor channel, units \(L\)

\(\tilde{L}_\text {r}\)

Diameter of a circular reacting zone, units \(L\) (1)


Parameter for reacting zone boundary definition


Indexing variable


Peclét number


Receptor concentration on reacting surface, units \(N/L^2\)


Reacting surface (5)


Root function (21)


Reynolds number

\(S{[\cdot ]}\)

Sensogram (17)


Reaction time scale, units \(T\) (1)


Characteristic velocity, units \(L/T\)


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


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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Copyright information

© Society for Mathematical Biology 2014

Authors and Affiliations

  1. 1.Department of MathematicsTemple UniversityPhiladelphiaUSA
  2. 2.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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