Hysteresis Phenomena in Biology

This chapter provides a brief introduction to hysteresis phenomena in biology and reveals the main intention of this book to provide the basics of Preisach models of hysteresis for interdisciplinary researchers and students. From switches in protein-DNA interactions (Chatterjee et al. 2008), microscopic cellular signaling pathways with bistable molecular cascades (Angeli et al. 2004; Qiao et al. 2007), cell division, differentiation, cancer onset and apoptosis (Eissing et al. 2004; Kim et al. 2007; Wilhelm 2009), protein folding (Andrews et al. 2013) and purinergic neuron-astrocyte interactions in the brain (Noori 2011) up to macroscopic biomechanics of cornea (Congdon et al. 2006) and lung deformations (Escolar and Escolar 2004), hysteresis phenomena are ubiquitous in biology. Since the focus of attention towards hysteresis has mostly been the physical phenomena, most textbooks dealing with this concept have adapted the terminology and ideas of related fields of research such as control theory and magnetism, which often exacerbate the access to this theory for computational biologists and other researchers that are not familiar with the utilized terms. Following Krasnosel’skii and Pokrovskii (1989), this monograph is intended to provide an introductionary material purely expressed in mathematical terminology.

In this chapter, we will recapitulate the essential concepts, definitions and theorems of the Lyapunov and Andronov stability theories of dynamical systems. The global aim is to prepare the reader for the mathematical abstraction of biological switches and hysteresis phenomena. Furthermore, the expected readership, which are students and researchers interested in mathematical modeling of biological processes, will benefit in general from this chapter since it provides the essence of stability of dynamical systems in a brief and precise way.

This chapter intends to familiarize the readers with the Preisach model of hysteresis. Since its publication in the 1930s of the last century (Preisach 1935), the model has been further developed and improved and many valuable facts have been accumulated (Everett and Whitton 1952; Everett 1954, 1955; Enderby 1956; Biorci and Pescetti 1958, 1959, 1966; Brown 1962; Bate 1962; Woodward and Della Torre 1960; Della Torre 1965; Damlanian and Visintin 1983; Visintin 1984; Barker et al. 1985; Brokate and Visintin 1989; Krasnosel’skii and Pokrovskii 1989). Here, no attempt of a complete presentation of the theory is made. Instead, we focus on a systematic introduction of the basic but essential concepts of Preisach model of hysteresis, which will help the reader to easily access this complex mathematical theory and prepare him/her for mathematical modeling of biological processes expressing non-linearities of hysteresis type.

Hysteresis may occur in different spatiotemporal scales of consideration. From switches in protein-DNA interactions (Chatterjee et al. 2008), microscopic cellular signaling pathways with bistable molecular cascades (Angeli et al. 2004; Qiao 2007), cell division, differentiation, cancer onset and apoptosis (Sha et al. 2003; Eissing et al. 2004; kim et al. 2007; Wilhelm 2009), protein folding (Andrews et al. 2013) and purinergic neuron-astrocyte interactions in the brain (Noori 2011) up to macroscopic biomechanics of cornea (Congdon et al. 2006) and lung deformations (Escolar and Escolar 2004), hysteresis phenomena are ubiquitous in biology. In this chapter, we will briefly discuss a few examples of hysteresis phenomena in biology, which may qualitatively resemble many other biological processes.