Bulletin of Mathematical Biology

, Volume 79, Issue 10, pp 2215–2241 | Cite as

Transport Effects on Multiple-Component Reactions in Optical Biosensors

  • Ryan M. EvansEmail author
  • David A. Edwards
Original Article


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.


Biochemistry Optical biosensors Rate constants Integrodifferential equations Numerical methods 


  1. BIAcore T200 data file (2013) General electric life sciences, GE Healthcare bio-sciences AB, Björkgatan 30, 751 84 Uppsala, Sweden, April 2013Google Scholar
  2. 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):719–730CrossRefGoogle Scholar
  3. Edwards DA (1999) Estimating rate constants in a convection-diffusion system with a boundary reaction. IMA J Appl Math 63(1):89–112MathSciNetCrossRefzbMATHGoogle Scholar
  4. Edwards DA (2000) Biochemical reactions on helical structures. SIAM J Appl Math 42(4):1425–1446MathSciNetCrossRefzbMATHGoogle Scholar
  5. Edwards DA (2001) The effect of a receptor layer on the measurement of rate constants. Bull Math Biol 63(2):301–327MathSciNetCrossRefzbMATHGoogle Scholar
  6. Edwards DA (2006) Convection effects in the BIAcore dextran layer: surface reaction model. Bull Math Biol 68:627–654MathSciNetCrossRefzbMATHGoogle Scholar
  7. Edwards DA (2011) Transport effects on surface reaction arrays: biosensor applications. Math Biosci 230(1):12–22MathSciNetCrossRefzbMATHGoogle Scholar
  8. Edwards DA, Evans RM, Li W (2017) Measuring kinetic rate constants of multiple-component reactions with optical biosensors. Anal Biochem 533:41–47Google Scholar
  9. Edwards DA, Jackson S (2002) Testing the validity of the effective rate constant approximation for surface reaction with transport. Appl Math Lett 15(5):547–552MathSciNetCrossRefzbMATHGoogle Scholar
  10. Edwards DA, Goldstein B, Cohen DS (1999) Transport effects on surface–volume biological reactions. J Math Biol 39(6):533–561zbMATHGoogle Scholar
  11. Friedberg EC (2005) Suffering in silence: the tolerance of DNA damage. Nat Rev Mol Cell Biol 6(12):943–953CrossRefGoogle Scholar
  12. Karlsson R, Fält A (1997) Experimental design for kinetic analysis of protein-protein interactions with surface plasmon resonance biosensors. J Immunol Methods 200(1):121–133CrossRefGoogle Scholar
  13. 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 296:527–537CrossRefGoogle Scholar
  14. 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):891–899CrossRefGoogle Scholar
  15. 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):176–185CrossRefGoogle Scholar
  16. Nie Q, Zhang Y-T, Zhao R (2006) Efficient semi-implicit schemes for stiff systems. J Comput Phys 214(2):521–537MathSciNetCrossRefzbMATHGoogle Scholar
  17. Plosky BS, Woodgate R (2004) Switching from high-fidelity replicases to low-fidelity lesion-bypass polymerases. Curr Opin Genet Dev 14(2):113–119CrossRefGoogle Scholar
  18. Rich RL, Myszka DG (2009) Extracting kinetic rate constants from binding responses. In: Cooper MA (ed) Label-free bioesnsors. Cambridge University Press, CambridgeGoogle Scholar
  19. 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):112–120CrossRefGoogle Scholar
  20. 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):1203–1214CrossRefGoogle Scholar
  21. 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):5361–5366CrossRefGoogle Scholar
  22. Zumbrum M (2013) Extensions for a surface–volume reaction model with application to optical biosensors. Ph.D. thesis, University of DelawareGoogle Scholar
  23. Zumbrum M, Edwards DA (2014) Multiple surface reactions in arrays with applications to optical biosensors. Bull Math Biol 76(7):1783–1808MathSciNetCrossRefzbMATHGoogle Scholar
  24. Zumbrum M, Edwards DA (2015) Conformal mapping in optical biosensor applications. J Math Biol 71(3):533–550MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Society for Mathematical Biology (outside the USA) 2017

Authors and Affiliations

  1. 1.Applied and Computational Mathematics Division, Information and Technology LaboratoryNational Institute of Standards and TechnologyGaithersburgUSA
  2. 2.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

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