J Math Chem (2014) 52:2442–2458 DOI 10.1007/s10910-014-0392-3 ORIGINAL PAPER Analysis of the fractional modification of the monocyclic enzyme cascades, defined in an alternative way involving the two forms of the modified protein Maria D. Masia · Manuela Garcia-Moreno · Francisco Garcia-Sevilla · Julio Escribano · Milagros Molina-Alarcon · Maria L. Amo-Saus · Enrique Arribas · Ramon Varon Received: 30 May 2014 / Accepted: 17 July 2014 / Published online: 3 August 2014 © Springer International Publishing Switzerland 2014 Abstract This paper presents the discussion of a more complete definition of the frac- tional modification of an monocyclic enzyme cascade suggested, but not discussed, by Varon et al. (Bull Math Biol 68(7):1461-1493, 2006) as the quotient of the sum of all forms of the modified protein, i.e. the free one and the intermediate complex converter enzyme of the original protein-modified protein, between the initial concentration of the target protein. From this general equation, obtained under three assumptions nec- essary to linearize the set of differential equations describing the kinetic of the system, we derive, as particular cases, other simpler expressions, by applying additional sim- M. D. Masia University Hospital of San Juan, Alicante, Spain M. Garcia-Moreno · M. L. Amo-Saus · R. Varon Department of Physical Chemistry, Technical School of Industrial Engineering, University of Castilla-La Mancha, Albacete, Spain F. Garcia-Sevilla Department of Electrical Engineering, Electronics, Automation and Communications, Technical School of Industrial Engineering, University of Castilla-La Mancha, Albacete, Spain J. Escribano Department of Agricultural Technology and Science and Genetics, Faculty of Medicine, University of Castilla-La Mancha, Albacete, Spain M. Molina-Alarcon Nursing Department, School of Nursing, University of Castilla-La Mancha, Albacete, Spain E. Arribas Applied Physics Department, High School of Informatics Engineering, University of Castilla-La Mancha, Albacete, Spain R. Varon (B ) Escuela de Ingenieros Industriales, Universidad de Castilla-La Mancha, Av. España s/n, Campus Universitario, 02071 Albacete, Spain e-mail: ramon.varon@uclm.es 123