1 Multistep Transformation Method for Discrete and Continuous Time Enzyme Kinetics Z. Vosika 1 , G. Lazović 1 1 Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, Belgrade, 11120/35, Serbia zvosika@mas.bg.ac.rs Abstract— In this paper we develop the new physical- mathematical time scale kinetic approach-model applied on organic and non-organic particles motion. Concretely, here, at first, this new research approach is based on enzyme particles dynamics results. At the beginning, a time scale is defined to be an arbitrary closed subset of the real numbers R, with the standard inherited topology. Mathematical examples of time scales include real numbers R, natural numbers N, integers Z, the Cantor set (i.e. fractals), and any finite union of closed intervals of R. Calculus on time scales (TSC) was established in 1988 by Stefan Hilger. TSC, by construction, is used to describe the complex process. This method may utilized for description of physical (classical mechanics), material (crystal growth kinetics, physical chemistry kinetics - for example, kinetics of barium-titanate synthesis), (bio)chemical or similar systems and represents major challenge for contemporary scientists. In this sense, the Michaelis- Menten (MM) mechanism is the one of the best known and simplest nonlinear biochemical network which deserves appropriate attention. Generally speaking, such processes may be described of discrete time scale. Reasonably it could be assumed that such a scenario is possible for MM mechanism. In this work, discrete time MM kinetics (dtMM) with time various step h, is investigated. Instead of the first derivative by time used first backward difference h. Physical basics for new time scale approach is a new statistical thermodynamics, natural generalization of Tsallis non-extensive or similar thermodynamics. A reliable new algorithm of novel difference transformation method, namely multi-step difference transformation method (MSDETM) for solving system of nonlinear ordinary difference equations is proposed. If h tends to zero, MSDETM transformed into multi-step differential transformation method (MSDTM). In the spirit of TSC, MSDETM describes analogously MSDTM. Keywords—kinetics, enzymes, materials, ceramics, BaTiO3, MSDETM . I. INTRODUCTION TSC, as a relatively new physical-mathematical method, is shown in [1], [2], [3]. Therefore,in scientific records we have analyzed that is evident a relatively small number appropriate applications or numerical methods for solving such nonlinear problems. One good choice, in this way, for solving numerical or practical problems, is the MSDTM [4], [5]. As we know, that in nature exists several examples of kinetics: classical mechanics (as a synonym of dynamics), fluid equations, plasma and chemical kinetics, particularly in chemical physics and physical chemistry and biology kinetics [6-13]. The well known is the fact that in medicine and biology holds a special position enzyme kinetics (Figure 1). Figure 1. One example of enzyme: acetylcholinesterase (AChE). It is known that this enzyme catalyzes the breakdown of acetylcholine and of some other choline esters that function as neurotransmitters. Kinetics, therefore, in principle, research many different materials, especially ceramics phenomena. From the other side, in the area of a non-organic materials, we performed the investigations on BaTiO3 - ceramics [14-22] as an complex material example that is often used in different applications. Barium-titanate, as a crystallography, of the grains monocristals is a part of cubic “perovskite family”. The perovskite structure is adopted by many oxides that have the chemical formula ABO3 (Figure 2 (a)).