Mathematical modeling of amperometric and potentiometric biosensors and system of non-linear equations – Homotopy perturbation approach A. Meena, L. Rajendran * Department of Mathematics, The Madura College, Madurai 625 011, Tamilnadu, India article info Article history: Received 19 February 2010 Received in revised form 17 March 2010 Accepted 24 March 2010 Available online 27 March 2010 Keywords: Amperometric biosensor Potentiometric biosensor Non-linear reaction–diffusion Non-steady-state Homotopy perturbation method abstract A mathematical model of amperometric and potentiometric biosensor is developed. The model is based on system of reaction–diffusion equations containing a non-linear term related to Michaelis–Menten kinetics of the enzymatic reaction. This paper presents an approximate analytical method (He’s Homotopy pertur- bation method) to solve the non-linear differential equations that describe the diffusion coupled with a Michaelis–Menten kinetics law. Approximate analytical expressions for substrate concentration, product concentration and corresponding current response have been derived for all values of parameter r using perturbation method. These results are compared with available limiting case results and are found to be in good agreement. The obtained results are valid for the whole solution domain. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction Biosensors are analytical devices converting a biochemical rec- ognition reaction into a measurable effect [1–3]. Amperometric biosensors measure the changes in the output current on the work- ing electrode due to the direct oxidation or reduction of products of a biochemical reaction [4]. They, being compact and having rela- tively short response time, are widely applied to monitor chemical substances in the medicine, food technology and the environmen- tal industry [5,6]. The understanding of the kinetic peculiarities of biosensors is of crucial importance for their design. The mathemat- ical modeling is rather widely used to improve the efficiency of the biosensors design and to optimize their configuration [7–10]. Starting from seventies various mathematical models of biosensors have been developed and successfully used to study and optimize analytical characteristics of biosensors [11–16]. A comprehensive review on the modeling of amperometric biosensors has been pre- sented by Schulmeister [17]. As it is evident from Table 1, a sub- stantial amount of work has been done on the modelling and simulation of amperometric and potentiometric biosensors. The amperometric biosensors are known to be reliable, cheaper and highly sensitive for environment, clinical and industrial pur- poses. Starting from the publication of Clark and Lyons [18], the amperometric biosensors became one of the popular and perspec- tive trends to biosensorics [19]. The general features of ampero- metric response were analyzed in the publications of Mell and Maloy [11,20]. However, due to limiting calculation possibilities and unperfected mathematics, the calculations were restricted by two critical concentrations of substrate when enzyme acted at the first and zero-order reaction conditions. Some later reports were also devoted to calculate steady-state and non-stationary kinetics of amperometric biosensor response [20–24]. The devel- opment of the numerical methods of solving of partial differential equations and facilities of modern computers open possibilities to make calculation at all interval of substrates concentration and at different diffusion and enzymatic reaction rate. The developed model is based on non-stationary diffusion equations [25], contain- ing a non-linear term related to the enzymatic reaction. Ames [26] carried out the digital simulation of the biosensor response using the implicit difference scheme. In potentiometric biosensors, the analytical information is ob- tained by converting the recognition process into a potential, which is proportional (in a logarithmic fashion) to the concentra- tion of the reaction product. These devices have been widely used in environmental, medical and industrial applications because of their high selectivity, simplicity and low cost [5,27–29]. It should be pointed out that, complete solutions have not yet been obtained for non-steady-state behaviour because of the non-linearity inher- ent in Michaelis–Menten kinetics [1]. To our knowledge no rigor- ous analytical solutions for non-steady-state concentration and current for all values of r (see Eq. (14) for its definition) have been reported. In this paper, we have derived analytical expressions of concentration and current in order to describe and evaluate the performances of amperometric and potentiometric biosensors using Homotopy perturbation method [30–37]. 1572-6657/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2010.03.027 * Corresponding author. Tel.: +91 9442228951; fax: +91 452 2675238. E-mail address: raj_sms@rediffmail.com (L. Rajendran). Journal of Electroanalytical Chemistry 644 (2010) 50–59 Contents lists available at ScienceDirect Journal of Electroanalytical Chemistry journal homepage: www.elsevier.com/locate/jelechem