Bulletin of Mathematical Biology

, Volume 76, Issue 7, pp 1783–1808

Multiple Surface Reactions in Arrays with Applications to Optical Biosensors

Original Article

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

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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