Skip to main content

Transport Effects on Multiple-Component Reactions in Optical Biosensors


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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9


  • BIAcore T200 data file (2013) General electric life sciences, GE Healthcare bio-sciences AB, Björkgatan 30, 751 84 Uppsala, Sweden, April 2013

  • de la Torre JG, Huertas ML, Carrasco B (2000) Calculation of hydrodynamic properties of globular proteins from their atomic-level structure. Biophys J 78(2):719–730

    Article  Google Scholar 

  • Edwards DA (1999) Estimating rate constants in a convection-diffusion system with a boundary reaction. IMA J Appl Math 63(1):89–112

    Article  MathSciNet  MATH  Google Scholar 

  • Edwards DA (2000) Biochemical reactions on helical structures. SIAM J Appl Math 42(4):1425–1446

    Article  MathSciNet  MATH  Google Scholar 

  • Edwards DA (2001) The effect of a receptor layer on the measurement of rate constants. Bull Math Biol 63(2):301–327

    Article  MathSciNet  MATH  Google Scholar 

  • Edwards DA (2006) Convection effects in the BIAcore dextran layer: surface reaction model. Bull Math Biol 68:627–654

    Article  MathSciNet  MATH  Google Scholar 

  • Edwards DA (2011) Transport effects on surface reaction arrays: biosensor applications. Math Biosci 230(1):12–22

    Article  MathSciNet  MATH  Google Scholar 

  • Edwards DA, Evans RM, Li W (2017) Measuring kinetic rate constants of multiple-component reactions with optical biosensors. Anal Biochem 533:41–47

  • Edwards DA, Jackson S (2002) Testing the validity of the effective rate constant approximation for surface reaction with transport. Appl Math Lett 15(5):547–552

    Article  MathSciNet  MATH  Google Scholar 

  • Edwards DA, Goldstein B, Cohen DS (1999) Transport effects on surface–volume biological reactions. J Math Biol 39(6):533–561

    MATH  Google Scholar 

  • Friedberg EC (2005) Suffering in silence: the tolerance of DNA damage. Nat Rev Mol Cell Biol 6(12):943–953

    Article  Google Scholar 

  • Karlsson R, Fält A (1997) Experimental design for kinetic analysis of protein-protein interactions with surface plasmon resonance biosensors. J Immunol Methods 200(1):121–133

    Article  Google Scholar 

  • Lebedev K, Mafé S, Stroeve P (2006) Convection, diffusion, and reaction in a surface-based biosensor: modeling of cooperativity and binding site competition and in the hydrogel. J Colloid Interface Sci 296:527–537

    Article  Google Scholar 

  • Lehmann AR, Niimi A, Ogi T, Brown S, Sabbioneda S, Wing JF, Kannouche PL, Green CM (2007) Translesion synthesis: Y-family polymerases and the polymerase switch. DNA Repair 6(7):891–899

    Article  Google Scholar 

  • Morton TA, Myszka DG, Chaiken IM (1995) Interpreting complex binding kinetics from optical biosensors: a comparison of analysis by linearization, the integrated rate equation, and numerical integration. Anal Biochem 227(1):176–185

    Article  Google Scholar 

  • Nie Q, Zhang Y-T, Zhao R (2006) Efficient semi-implicit schemes for stiff systems. J Comput Phys 214(2):521–537

    Article  MathSciNet  MATH  Google Scholar 

  • Plosky BS, Woodgate R (2004) Switching from high-fidelity replicases to low-fidelity lesion-bypass polymerases. Curr Opin Genet Dev 14(2):113–119

    Article  Google Scholar 

  • Rich RL, Myszka DG (2009) Extracting kinetic rate constants from binding responses. In: Cooper MA (ed) Label-free bioesnsors. Cambridge University Press, Cambridge

    Google Scholar 

  • Rich RL, Cannon MJ, Jenkins J, Pandian P, Sundaram S, Magyar R, Brockman J, Lambert J, Myszka DG (2008) Extracting kinetic rate constants from surface plasmon resonance array systems. Anal Biochem 373(1):112–120

    Article  Google Scholar 

  • Yarmush ML, Patankar DB, Yarmush DM (1996) An analysis of transport resistances in the operation of BIAcore; implications for kinetic studies of biospecific interactions. Mol Immunol 33(15):1203–1214

    Article  Google Scholar 

  • Zhuang Z, Johnson RE, Haracska L, Prakash L, Prakash S, Benkovic SJ (2008) Regulation of polymerase exchange between pol\(\eta \) and pol\(\delta \) by monoubiquitination of pcna and the movement of dna polymerase holoenzyme. Proc Natl Acad Sci 105(14):5361–5366

    Article  Google Scholar 

  • Zumbrum M (2013) Extensions for a surface–volume reaction model with application to optical biosensors. Ph.D. thesis, University of Delaware

  • Zumbrum M, Edwards DA (2014) Multiple surface reactions in arrays with applications to optical biosensors. Bull Math Biol 76(7):1783–1808

    Article  MathSciNet  MATH  Google Scholar 

  • Zumbrum M, Edwards DA (2015) Conformal mapping in optical biosensor applications. J Math Biol 71(3):533–550

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Ryan M. Evans.

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)

$$\begin{aligned} V=\frac{6Q}{WH}. \end{aligned}$$

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

Table 4 Dimensional parameter ranges, taken from references de la Torre et al. (2000), Gen (2013), Rich et al. (2008), and Yarmush et al. (1996)
Table 5 Dimensionless parameters

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Evans, R.M., Edwards, D.A. Transport Effects on Multiple-Component Reactions in Optical Biosensors. Bull Math Biol 79, 2215–2241 (2017).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


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