# American Institute of Mathematical Sciences

2016, 13(6): 1159-1168. doi: 10.3934/mbe.2016036

## Modelling random antibody adsorption and immunoassay activity

 1 School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland, Ireland 2 Biomedical Diagnostics Institute, Dublin City University, Glasnevin, Dublin 9, Ireland

Received  December 2015 Revised  April 2016 Published  August 2016

One of the primary considerations in immunoassay design is optimizing the concentration of capture antibody in order to achieve maximal antigen binding and, subsequently, improved sensitivity and limit of detection. Many immunoassay technologies involve immobilization of the antibody to solid surfaces. Antibodies are large molecules in which the position and accessibility of the antigen-binding site depend on their orientation and packing density.
In this paper we propose a simple mathematical model, based on the theory known as random sequential adsorption (RSA), in order to calculate how the concentration of correctly oriented antibodies (active site exposed for subsequent reactions) evolves during the deposition process. It has been suggested by experimental studies that high concentrations will decrease assay performance, due to molecule denaturation and obstruction of active binding sites. However, crowding of antibodies can also have the opposite effect by favouring upright orientations. A specific aim of our model is to predict which of these competing effects prevails under different experimental conditions and study the existence of an optimal coverage, which yields the maximum expected concentration of active particles (and hence the highest signal).
Citation: D. Mackey, E. Kelly, R. Nooney. Modelling random antibody adsorption and immunoassay activity. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1159-1168. doi: 10.3934/mbe.2016036
##### References:
 [1] D. A. Edwards, Steric hindrance effects on surface reactions: Applications to BIAcore, J. Math. Biol., 55 (2007), 517-539. doi: 10.1007/s00285-007-0093-7. [2] J. W. Evans, Random and cooperative sequential adsorption, Rev. Mod. Phys., 65 (1993), 1281. doi: 10.1103/RevModPhys.65.1281. [3] V. Gubala, C. Crean, R. Nooney, S. Hearty, B. McDonnell, K. Heydon, R. O'Kennedy, B. D. MacCraith and D. E. Williams, Kinetics of immunoassays with particles as labels: Effect of antibody coupling using dendrimers as linkers, Analyst, 136 (2011), 2533-2541. doi: 10.1039/c1an15017k. [4] M. K. Hassan, J. Schmidt, B. Blasius and J. Kurths, Jamming coverage in competitive random sequential adsorption of a binary mixture, Phys. Rev. E, 65 (2002), 045103. doi: 10.1103/PhysRevE.65.045103. [5] P. L. Krapivsky, Kinetics of random sequential parking on a line, J. Stat. Phys., 69 (1992), 135-150. doi: 10.1007/BF01053786. [6] A. Rényi, On a one-dimensional problem concerning random space-filling (In Hungarian), Publ. Math. Inst. Hung. Acad. Sci, 3 (1958), 109-127. [7] J. C. Roach, V. Thorsson and A. F. Siegel, Parking strategies for genome sequencing, Genome Research, 10 (2000), 1020-1030. doi: 10.1101/gr.10.7.1020. [8] B. Saha. T. H. Evers and M. W. J. Prins, How antibody surface coverage on nanoparticles determines the activity and kinetics of antigen capturing for biosensing, Anal. Chem., 86 (2014), 8158-8166. [9] W. Schramm and S. Paek, Antibody-antigen complex formation with immobilized immunoglobulins, Anal. Biochem., 205 (1992), 47-56. doi: 10.1016/0003-2697(92)90577-T. [10] J. Talbot, G. Tarjus, P. R. Van Tassel and P. Viot, From car parking to protein adsorption: An overview of sequential adsorption processes, Colloids Surf. A, 165 (2000), 287-324. doi: 10.1016/S0927-7757(99)00409-4. [11] M. L. M. Vareiro, J. Liu, W. Knoll, K. Zak, D. Williams and A. T. A. Jenkins, Surface plasmon fluorescence measurements of human chorionic gonadotrophin: role of antibody orientation in obtaining enhanced sensitivity and limit of detection, Anal. Chem., 77 (2005), 2426-2431. doi: 10.1021/ac0482460. [12] D. Wild (editor), The Immunoassay Handbook, $3^{rd}$ edition, Elsevier, 2005. [13] M. E. Wiseman and C. W. Frank, Antibody adsorption and orientation on hydrophobic surfaces, Langmuir, 28 (2012), 1765-1774. doi: 10.1021/la203095p. [14] H. Xu, J. R. Lu and D. E. Williams, Effect of surface packing density of interfacially adsorbed monoclonal antibody on the binding of hormonal antigen human chorionic gonadotrophin, J. Phys. Chem. B, 110 (2006), 1907-1914. doi: 10.1021/jp0538161. [15] H. Xu, X. Zhao, C. Grant, J. R. Lu, D. E. Williams and J. Penfold, Orientation of a monoclonal antibody adsorbed at the solid/solution interface: A combined study using atomic force microscopy and neutron reflectivity, Langmuir, 22 (2006), 6313-6320. doi: 10.1021/la0532454. [16] X. Zhao, F. Pan, B. Cowsill, J. R. Lu, L. Garcia-Gancedo, A. J. Flewitt, G. M. Ashley and J. Luo, Interfacial immobilization of monoclonal antibody and detection of human prostate-specific antigen, Langmuir, 27 (2011), 7654-7662. doi: 10.1021/la201245q.

show all references

##### References:
 [1] D. A. Edwards, Steric hindrance effects on surface reactions: Applications to BIAcore, J. Math. Biol., 55 (2007), 517-539. doi: 10.1007/s00285-007-0093-7. [2] J. W. Evans, Random and cooperative sequential adsorption, Rev. Mod. Phys., 65 (1993), 1281. doi: 10.1103/RevModPhys.65.1281. [3] V. Gubala, C. Crean, R. Nooney, S. Hearty, B. McDonnell, K. Heydon, R. O'Kennedy, B. D. MacCraith and D. E. Williams, Kinetics of immunoassays with particles as labels: Effect of antibody coupling using dendrimers as linkers, Analyst, 136 (2011), 2533-2541. doi: 10.1039/c1an15017k. [4] M. K. Hassan, J. Schmidt, B. Blasius and J. Kurths, Jamming coverage in competitive random sequential adsorption of a binary mixture, Phys. Rev. E, 65 (2002), 045103. doi: 10.1103/PhysRevE.65.045103. [5] P. L. Krapivsky, Kinetics of random sequential parking on a line, J. Stat. Phys., 69 (1992), 135-150. doi: 10.1007/BF01053786. [6] A. Rényi, On a one-dimensional problem concerning random space-filling (In Hungarian), Publ. Math. Inst. Hung. Acad. Sci, 3 (1958), 109-127. [7] J. C. Roach, V. Thorsson and A. F. Siegel, Parking strategies for genome sequencing, Genome Research, 10 (2000), 1020-1030. doi: 10.1101/gr.10.7.1020. [8] B. Saha. T. H. Evers and M. W. J. Prins, How antibody surface coverage on nanoparticles determines the activity and kinetics of antigen capturing for biosensing, Anal. Chem., 86 (2014), 8158-8166. [9] W. Schramm and S. Paek, Antibody-antigen complex formation with immobilized immunoglobulins, Anal. Biochem., 205 (1992), 47-56. doi: 10.1016/0003-2697(92)90577-T. [10] J. Talbot, G. Tarjus, P. R. Van Tassel and P. Viot, From car parking to protein adsorption: An overview of sequential adsorption processes, Colloids Surf. A, 165 (2000), 287-324. doi: 10.1016/S0927-7757(99)00409-4. [11] M. L. M. Vareiro, J. Liu, W. Knoll, K. Zak, D. Williams and A. T. A. Jenkins, Surface plasmon fluorescence measurements of human chorionic gonadotrophin: role of antibody orientation in obtaining enhanced sensitivity and limit of detection, Anal. Chem., 77 (2005), 2426-2431. doi: 10.1021/ac0482460. [12] D. Wild (editor), The Immunoassay Handbook, $3^{rd}$ edition, Elsevier, 2005. [13] M. E. Wiseman and C. W. Frank, Antibody adsorption and orientation on hydrophobic surfaces, Langmuir, 28 (2012), 1765-1774. doi: 10.1021/la203095p. [14] H. Xu, J. R. Lu and D. E. Williams, Effect of surface packing density of interfacially adsorbed monoclonal antibody on the binding of hormonal antigen human chorionic gonadotrophin, J. Phys. Chem. B, 110 (2006), 1907-1914. doi: 10.1021/jp0538161. [15] H. Xu, X. Zhao, C. Grant, J. R. Lu, D. E. Williams and J. Penfold, Orientation of a monoclonal antibody adsorbed at the solid/solution interface: A combined study using atomic force microscopy and neutron reflectivity, Langmuir, 22 (2006), 6313-6320. doi: 10.1021/la0532454. [16] X. Zhao, F. Pan, B. Cowsill, J. R. Lu, L. Garcia-Gancedo, A. J. Flewitt, G. M. Ashley and J. Luo, Interfacial immobilization of monoclonal antibody and detection of human prostate-specific antigen, Langmuir, 27 (2011), 7654-7662. doi: 10.1021/la201245q.
 [1] Don A. Jones, Hal L. Smith, Horst R. Thieme. Spread of viral infection of immobilized bacteria. Networks and Heterogeneous Media, 2013, 8 (1) : 327-342. doi: 10.3934/nhm.2013.8.327 [2] Naoki Sato, Toyohiko Aiki, Yusuke Murase, Ken Shirakawa. A one dimensional free boundary problem for adsorption phenomena. Networks and Heterogeneous Media, 2014, 9 (4) : 655-668. doi: 10.3934/nhm.2014.9.655 [3] Daniela Calvetti, Erkki Somersalo. Microlocal sequential regularization in imaging. Inverse Problems and Imaging, 2007, 1 (1) : 1-11. doi: 10.3934/ipi.2007.1.1 [4] Ye Chen, Keith W. Hipel, D. Marc Kilgour. A multiple criteria sequential sorting procedure. Journal of Industrial and Management Optimization, 2008, 4 (3) : 407-423. doi: 10.3934/jimo.2008.4.407 [5] Karam Allali, Sanaa Harroudi, Delfim F. M. Torres. Optimal control of an HIV model with a trilinear antibody growth function. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 501-518. doi: 10.3934/dcdss.2021148 [6] Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366 [7] Benedict Geihe, Martin Rumpf. A posteriori error estimates for sequential laminates in shape optimization. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1377-1392. doi: 10.3934/dcdss.2016055 [8] Dawan Mustafa, Bernt Wennberg. Chaotic distributions for relativistic particles. Kinetic and Related Models, 2016, 9 (4) : 749-766. doi: 10.3934/krm.2016014 [9] Amina Mecherbet. Sedimentation of particles in Stokes flow. Kinetic and Related Models, 2019, 12 (5) : 995-1044. doi: 10.3934/krm.2019038 [10] Karl Grill, Christian Tutschka. Ergodicity of two particles with attractive interaction. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4831-4838. doi: 10.3934/dcds.2015.35.4831 [11] David Cowan. A billiard model for a gas of particles with rotation. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 101-109. doi: 10.3934/dcds.2008.22.101 [12] Wilfrid Gangbo, Andrzej Świech. Optimal transport and large number of particles. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1397-1441. doi: 10.3934/dcds.2014.34.1397 [13] Alexander Rezounenko. Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1547-1563. doi: 10.3934/dcdsb.2017074 [14] Jian Xiong, Yingwu Chen, Zhongbao Zhou. Resilience analysis for project scheduling with renewable resource constraint and uncertain activity durations. Journal of Industrial and Management Optimization, 2016, 12 (2) : 719-737. doi: 10.3934/jimo.2016.12.719 [15] Marie Levakova. Effect of spontaneous activity on stimulus detection in a simple neuronal model. Mathematical Biosciences & Engineering, 2016, 13 (3) : 551-568. doi: 10.3934/mbe.2016007 [16] Maria Francesca Carfora, Enrica Pirozzi. Stochastic modeling of the firing activity of coupled neurons periodically driven. Conference Publications, 2015, 2015 (special) : 195-203. doi: 10.3934/proc.2015.0195 [17] S. M. Crook, M. Dur-e-Ahmad, S. M. Baer. A model of activity-dependent changes in dendritic spine density and spine structure. Mathematical Biosciences & Engineering, 2007, 4 (4) : 617-631. doi: 10.3934/mbe.2007.4.617 [18] Irada Dzhalladova, Miroslava Růžičková. Simplification of weakly nonlinear systems and analysis of cardiac activity using them. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3435-3453. doi: 10.3934/dcdsb.2021191 [19] Nikolaos Kazantzis, Vasiliki Kazantzi. Characterization of the dynamic behavior of nonlinear biosystems in the presence of model uncertainty using singular invariance PDEs: Application to immobilized enzyme and cell bioreactors. Mathematical Biosciences & Engineering, 2010, 7 (2) : 401-419. doi: 10.3934/mbe.2010.7.401 [20] Songqiang Qiu, Zhongwen Chen. An adaptively regularized sequential quadratic programming method for equality constrained optimization. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2675-2701. doi: 10.3934/jimo.2019075

2018 Impact Factor: 1.313