2nd Edition, Encyclopedic Dictionary of Modern Mathematical Sciences, Paperbound Edition, Volume 2: Mathematical BiologyBook Review
Authors: Heisuke Hironaka (Editorial Committee Representative), Yuzuru Iwasa (Editorial Secretary)
The 2nd Edition of the Encyclopedic Dictionary of Modern Mathematical Sciences (published in 2009) divided into separate volumes (8 volumes in total)
Publication Date: January 29, 2026 Price: 3,740 yen (tax included) Publisher: Maruzen Publishing
From the late 20th to the 21st century, the life sciences underwent a dramatic transformation. With advances in molecular biology, the actual nature of genes and proteins was successively revealed, and the "components" that make up life phenomena came to be described with unprecedented precision. On the other hand, the accumulation of this vast knowledge gave rise to a new question: "How does life behave as a dynamic system?" "The 2nd Edition Mathematical Sciences Encyclopedia: Popularized Edition, Vol. 2: Mathematics of Life" (Maruzen Publishing) is a book compiled precisely during this transitional period.
The defining feature of this book lies not merely in organizing a specific field as a textbook of "mathematical biology," but in reconstructing the interface between the life sciences and mathematical sciences from an exceptionally broad perspective. The structure—ranging across ecology, behavior and society, evolution, development and morphogenesis, neuroscience, medicine, bioinformatics, and systems biology—reflects an editorial philosophy that seeks to reframe biological phenomena as cross-hierarchical dynamical systems. Of particular interest is how classical morphogenesis research using reaction-diffusion systems or cellular automata is positioned in a continuous flow alongside systems biology, which deals with gene-protein networks and fluctuations. This well reflects the zeitgeist of the life sciences as they shifted from the "description of structures" to the "understanding of state transitions."
In the preface, Heisuke Hironaka states that the purpose of this book is to answer the questions, "What is mathematical science?" and "What is the potential of mathematical science?" He then introduces a striking episode from the editorial process: "This book is becoming increasingly complex" and "That is precisely what mathematical science is." This reveals an important implication: the stance that mathematical science is not an endeavor to simplify reality and confine it within beautiful theories, but rather a methodology for confronting the complexity of reality head-on.
This perspective is also extremely important when considering recent developments in systems biology. For example, a series of studies by Ferrell and colleagues on ultrasensitivity, bistability, and oscillation clearly demonstrated that biochemical reaction networks do not merely act as continuous response systems, but behave as nonlinear dynamical systems involving "thresholds," "irreversibility," "memory," and "decision-making." In the cell cycle, MAPK cascades, calcium signaling, and others, cells do not simply respond proportionally to inputs; rather, they undergo sharp state transitions across certain thresholds and sometimes even exhibit hysteresis. This understanding represents a major shift from viewing biological phenomena as a "collection of molecular parts" to viewing them as "systems transitioning within a state space."
Having been personally engaged in reproductive and developmental biology research using *Xenopus* oocytes for many years, I deeply resonate with the concerns raised in this book. Many of the phenomena observed in oocytes—such as the intracellular calcium wave at fertilization, egg activation, release from and re-arrest of meiotic arrest, and the irreversible progression of the cell cycle—are quintessential "state transitions." In these processes, cells do not linger in intermediate states; rather, at a certain moment, they behave collectively in a phase-transition-like manner. The ultrasensitive responses and bistable switches mathematically described by Ferrell and colleagues were precisely the language needed to understand such phenomena. As one reads through this book, it becomes abundantly clear that, already around 2009, the Japanese mathematical science community had acutely sensed this direction.
More than a decade after its publication, the life sciences have grown even larger, expanding rapidly into single-cell analysis, omics analysis, AI, and data-driven research. Yet, at the same time, the importance of understanding biological phenomena as "dynamic systems" seems to be increasing all the more. This is because merely acquiring massive amounts of data cannot reach the fundamental question of "why cells change their state at that specific timing." Reading this book again today, one realizes that its sense of awareness regarding these issues has by no means aged. Rather, one is struck by how it anticipated the challenges facing modern life sciences today.
Rather than a specialized textbook heavy on mathematical formulas, this book is an intellectual map that illustrates from various perspectives what it means to think about life through mathematics. It offers numerous insights not only for researchers engaged in the life sciences, but also for readers interested in complex systems science and information science. Above all, it conveys to the present day the fervor of an era when the life sciences were shifting from a "science of parts" to a "science of dynamics."
Kenichi Sato (Kyoto Sangyo University, Faculty of Life Sciences）
