## Abstract

Optical biosensors are often used to measure kinetic rate constants associated with chemical reactions. Such instruments operate in the *surface–volume* configuration, in which ligand molecules are convected through a fluid-filled volume over a surface to which receptors are confined. Currently, scientists are using optical biosensors to measure the kinetic rate constants associated with DNA translesion synthesis—a process critical to DNA damage repair. Biosensor experiments to study this process involve multiple interacting components on the sensor surface. This multiple-component biosensor experiment is modeled with a set of nonlinear integrodifferential equations (IDEs). It is shown that in physically relevant asymptotic limits these equations reduce to a much simpler set of ordinary differential equations (ODEs). To verify the validity of our ODE approximation, a numerical method for the IDE system is developed and studied. Results from the ODE model agree with simulations of the IDE model, rendering our ODE model useful for parameter estimation.

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## Additional information

This work was done with the support of the National Science Foundation under Award Number NSF-DMS 1312529. The first author was also partially supported by the National Research Council through an NRC postdoctoral fellowship.

## Appendix: Parameter Values

### Appendix: Parameter Values

Parameter values from the literature are tabulated.

The variables \( W,\ Q\) represent the dimensional width and flow rate; the other dimensional variables are as in Sect. 2. The flow rate is related to the velocity through the formula (Edwards 2011)

Using the dimensional values above, we calculated the following *extremal bounds* on the dimensionless variables.

Here \(\epsilon =H/L\) is the aspect ratio, and \(\mathrm {Re}=VH^2/(\nu L)\) is the appropriate Reynolds number associated with our system.

The authors wish to emphasize that the bounds in Table 5 are naïve extremal bounds calculated by using minimum and maximum values for the dimensional parameters in Table 4. In particular, the values for the dimensionless rate constants in Table 5 are not estimates of their true values; they are minimum and maximum values calculated using combinations of *extremal values* for the parameters in Table 5. A large variation in the dimensionless rate constants is highly unlikely, since this scenario corresponds to one in which one of the association rate constants is very large, and another association rate constant very small. We would also like to note that a large variation in some of the parameters, such as the kinetic rate constants or \(\mathrm {Da}\), would necessitate very small values for either or both of \(\Delta t\) and \(\Delta x\) in our numerical method.

Furthermore, one may be concerned about the upper bound on the Reynolds number, the lower bound on the Péclet number, and the upper bound on the Damköhler number. All of these extremal bounds were calculated using a flow rate of \(1\ \upmu \mathrm {L}/\mathrm {min}\)—the slowest flow rate possible on the BIAcore T200 (Gen 2013). Even with the fastest reactions, one can still design experiments to minimize transport effects by increasing the flow rate *Q* (thus the velocity), decreasing the initial empty receptor concentration \(R_{\mathrm {T}}\), and decreasing the ligand inflow concentrations \(C_{1,\mathrm {u}}\) and \(C_{2,\mathrm {u}}\). In the case of the fastest reaction \({}_1k_{\mathrm {a}}=3\times 10^9\ \mathrm {cm}^3/(\mathrm {mol}\, \mathrm {s})\), we can take: \( Q=390\ \upmu \mathrm { L}/\mathrm {min},\ V=.75\ \mathrm {cm}/\mathrm {s},\ R_{\mathrm {T}}=7.76\times 10^{-13}\ \mathrm {mol}/\mathrm {cm}^2,\ C_{1,\mathrm {u}}=C_{2,\mathrm {u}}=2.96\times 10^{-12}\ \mathrm {mol}/\mathrm {cm}^3 .\) These choices yield the dimensionless parameters \(\mathrm {Re}=0.09,\ \mathrm {Pe}=136.26,\ \mathrm {Da}=5.16;\) these values are perfectly in line with our analysis and the validity of our ERC equations.

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Evans, R.M., Edwards, D.A. Transport Effects on Multiple-Component Reactions in Optical Biosensors.
*Bull Math Biol* **79**, 2215–2241 (2017). https://doi.org/10.1007/s11538-017-0327-9

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DOI: https://doi.org/10.1007/s11538-017-0327-9

### Keywords

- Biochemistry
- Optical biosensors
- Rate constants
- Integrodifferential equations
- Numerical methods