Fractalization in biology

This book is a collection of contributions by seven authors in the field of mathematical biology. Its usefulness lies primarily in the exposure of some current, advanced topics, rather than a systematic coverage of the field. In this multi-author book, tighter editorial control would have eliminated a number of grammatical errors and mis-spellings.

Many biological systems are governed by fluctuations and, thus, techniques of statistical physics seem to provide a suitable mechanism for helping to understand them. There are six chapters in the book, spanning topics such as fluctuations, fractal geometry and stochastic scaling, self-organized criticality, patterns and correlations, fluctuation-driven transport and collective motion.

In spite of being named the Basic concepts, the first chapter provides essentially an executive summary of the rest of the book. An introductory chapter dealing with the concepts that underpin the rest of the book would have broadened the readership.

This book proposes several models of reality and examines their correctness. Selection of models is influenced by realization that the system under investigation consists of many similar interacting units. Many parameters are involved, and determination of the most significant ones is rarely possible.

Critical state is normally reached by changing one of the control parameters. Reaching the critical state without any parameter modification relates to the important concept of self-organized criticality. Applying this technique to the evolution model, through the mutation and selection processes, it turns out that the evolution/extinction rate is not constant, but displays fluctuations. These can be linked to major extinction events in the past. One may then hypothesize that there is no need for external events, such as meteorite collisions, to explain the intermittent nature of the extinctions.

A large part of the book deals with bacterial colonies. Topics such as proliferation, motility and chemotaxis are examined in detail. In systematic studies aimed at determining the influence of environmental factors, different morphology diagrams were obtained. Applying statistical measures, it was found that the boundary of the colony is best represented by a self-affine curve with long-range correlations. The independence of the Hurst exponent on the bacterial colony yields strong support for the ‘universality’ feature, implying that the macroscopic patterns are not influenced by many microscopic features.

It is known that growing bacteria on nutrient-poor agar substrates leads to complex branching, with tip-splitting patterns. Many of the characteristic forms can be explained in terms of diffusion-limited processes, reflecting the fact that bacterial growth depends on locally available nutrients. One of the first models reproducing the fractal characteristics is the classic Diffusion Limited Aggregation (DLA). Other models are developed modelling motile and non-motile bacteria, by taking into account the microscopic ‘rules’ and macroscopic morphology.

The complex nature of DNA is addressed from the standpoint of statistical physics. It is demonstrated that the idea of the long-range correlations is appropriate for characterizing the highly heterogeneous nature of DNA. The location of coding and non-coding parts in the genome is still not well established. Application of the Detrended Fluctuation Analysis (DFA) facilitates the differentiation between these parts.

Next, the dimensional analysis of the brain's electrical activity is examined. It is disconcerting that this section uses words in place of Greek letters and disregards subscripts. Following a brief description of linear and non-linear methods for the analysis of the EEG, the section concentrates on the correlation, D₂, and point-correlation, PD₂, dimensions. It is argued that the accuracy of the latter algorithm is superior, especially for finite data samples, and several clinical applications are considered.

Many different mechanisms of biological motion exist and these are based on the individual motor proteins. These motors move in discrete steps and usually convert chemical energy into mechanical energy. The dynamical protein network lies at the basis of intracellular transport. The structure of linear and rotary motor families is examined in great detail. Fluctuation-driven transport serves as a basis for modelling biological systems. Mechanical properties of muscle contraction are derived from biochemical parameters, and an analytical formula for the force-velocity is obtained. It appears that rather simple assumptions suffice to describe the complex process of muscle contraction.

The last chapter deals with collective motion, from the universality standpoint, whereby the same pattern may develop in completely different environments. The collective features of organisms are studied using the self-propelled models in one, two and three dimensions. The book ends by investigating the social aspect of self-organization and examines the dynamics of pedestrians. Some of the empirical findings, such as trail formation, can be realistically modelled.

Overall, a book such as this is valuable for researchers, as it brings together physicists, mathematicians and biologists in an attempt to quantify some of the most complex real-life situations.

M.M. Novak School of Mathematics Kingston University U.K.