Cell shapes in tissues are affected by the biophysical interaction between cells. Tissue forces can influence specific cell features such as cell geometry and cell surface area. Here, we examined the 2-dimensional shape, size, and perimeter of pleural epithelial cells at various lung volumes. We demonstrated a 1.53-fold increase in 2-dimensional cell surface area and a 1.43-fold increase in cell perimeter at total lung capacity compared to residual lung volume. Consistent with previous results, close inspection of the pleura demonstrated wavy folds between pleural epithelial cells at all lung volumes. To investigate a potential explanation for the wavy folds, we developed a physical simulacrum suggested by D\'Arcy Thompson in On Growth and Form. The simulacrum suggested that the wavy folds were the result of redundant cell membranes unable to contract. To test this hypothesis, we developed a numerical simulation to evaluate the impact of an increase in 2-dimensional cell surface area and cell perimeter on the shape of the cell-cell interface. Our simulation demonstrated that an increase in cell perimeter, rather than an increase in 2-dimensional cell surface area, had the most direct impact on the presence of wavy folds. We conclude that wavy folds between pleural epithelial cells reflects buckling forces arising from the excess cell perimeter necessary to accommodate visceral organ expansion.

The celebrated 1917 work \"On Growth and Form\" of D\'Arcy W. Thompson has established a landmark for mathematical biology, introducing new perspectives of study and research in biology, providing mathematical methods to morphology of biological systems. In this brief historical essay, we recall the novelties and relevance of the work from a retrospective stance, above all pointing out the crucial role played by it in the dawning of epigenetic standpoint. The role of underlying epigenetic processes in generation of biological forms via similarity transformations is analyzed within the framework of D\'Arcy Thompson. The significance of D\'Arcy Thompson as a predecessor of the relational biology and of the epigenetic concepts of evolution is discussed.

Mathematical models have been widespread in biology since its emergence as a modern experimental science in the 19th century. Focusing on models in developmental biology and heredity, this article (1) presents the properties and epistemological basis of pertinent mathematical models in biology from Mendel\'s model of heredity in the 19th century to Eric Davidson\'s model of developmental gene regulatory networks in the 21st; (2) shows that the models differ not only in their epistemologies but also in regard to explicitly or implicitly taking into account basic biological principles, in particular those of biological specificity (that became, in part, replaced by genetic information) and genetic causality. The article claims that models disregarding these principles did not impact the direction of biological research in a lasting way, although some of them, such as D\'Arcy Thompson\'s models of biological form, were widely read and admired and others, such as Turing\'s models of development, stimulated research in other fields. Moreover, it suggests that successful models were not purely mathematical descriptions or simulations of biological phenomena but were based on inductive, as well as hypothetico-deductive, methodology. The recent availability of large amounts of sequencing data and new computational methodology tremendously facilitates system approaches and pattern recognition in many fields of research. Although these new technologies have given rise to claims that correlation is replacing experimentation and causal analysis, the article argues that the inductive and hypothetico-deductive experimental methodologies have remained fundamentally important as long as causal-mechanistic explanations of complex systems are pursued.

In 1917, the publication of On Growth and Form by D\'Arcy Wentworth Thompson challenged both mathematicians and naturalists to think about biological shapes and diversity as more than a confusion of chaotic forms generated at random, but rather as geometric shapes that could be described by principles of physics and mathematics. Thompson\'s work was based on the ideas of Galileo and Goethe on morphology and of Russell on functionalism, but he was first to postulate that physical forces and internal growth parameters regulate biological forms and could be revealed via geometric transformations in morphological space. Such precise mathematical structure suggested a unifying generative process, as reflected in the title of the book. To Thompson it was growth that could explain the generation of any particular biological form, and changes in ontogeny, rather than natural selection, could then explain the diversity of biological shapes. Whereas adaptationism, widely accepted in evolutionary biology, gives primacy to extrinsic factors in producing morphological variation, Thompson\'s \'laws of growth\' provide intrinsic directives and constraints for the generation of individual shapes, helping to explain the \'profusion of forms, colours, and other modifications\' observed in the living world.

The mechanisms by which organisms acquire their sizes and shapes through growth was a major focus of D\'Arcy Thompson\'s book On Growth and Form. By applying mathematical and physical principles to a range of biological forms, Thompson achieved fresh insights, such as the notion that diverse biological shapes could be related through simple deformations of a coordinate system. However, Thompson considered genetics to lie outside the scope of his work, even though genetics was a growing discipline at the time the book was published. Here, we review how recent advances in cell, developmental, evolutionary and computational biology allow Thompson\'s ideas to be integrated with genes and the processes they influence to provide a deeper understanding of growth and morphogenesis. We consider how genes interact with subcellular-, cellular- and tissue-level processes in plants to yield patterns of growth that underlie the developmental and evolutionary shape transformations Thompson so eloquently described.

D\'Arcy Thompson wrote his most celebrated book, On Growth and Form, at University College, Dundee in the 1910s. This paper explores some of the influences on Thompson\'s ideas and the context in which the book was written. It examines the controversial nature of some of the book\'s content and the connections between science and art that it embodies. The book\'s reception is discussed and the gradual take-up of his ideas within biology, as well as looking at the many other fields that have been influenced by Thompson\'s ideas.