Mathematics and biology: a Kantian view on the history of pattern formation theory

Contrasting philosophical and scientific views in the long history of studying the generation of form in development

Focusing on the opposing ways of thinking of philosophers and scientists to explain the generation of form in biological development, I show that today's controversies over explanations of early development bear fundamental similarities to the dichotomy of preformation theory versus epigenesis in Greek antiquity. They are related to the acceptance or rejection of the idea of a physical form of what today would be called information for the generating of the embryo as a necessary pre-requisite for specific development and heredity. As a recent example, I scrutinize the dichotomy of genomic causality versus self-organization in 20th and 21st century theories of the generation of form. On the one hand, the generation of patterns and form, as well as the constant outcome in development, are proposed to be causally related to something that is "preformed" in the germ cells, the nucleus of germ cells, or the genome. On the other hand, it is proposed that there is no pre-existing form or information, and development is seen as a process where genuinely new characters emerge from formless matter, either by immaterial "forces of life," or by physical-chemical processes of self-organization. I also argue that these different ways of thinking and the research practices associated with them are not equivalent, and maintain that it is impossible to explain the generation of form and constant outcome of development without the assumption of the transmission of pre-existing information in the form of DNA sequences in the genome. Only in this framework of "preformed" information can "epigenesis" in the form of physical and chemical processes of self-organization play an important role.

Preparation of surgical meshes using self-regulating technology based on reaction-diffusion processes

While reaction-diffusion processes are utilized in multiple scientific fields, these phenomena have seen limited practical application in the polymer industry. Although self-regulating processes driven by parallel reaction and diffusion can lead to patterned structures, most polymeric products with repeating subunits are still prepared by methods that require complex and expensive instrumentation. A notable, high-added-value example is surgical mesh, which is often manufactured by weaving or knitting. In our present work, we demonstrate how the polymer and the biomedical industry can benefit from the pattern-forming capabilities of reaction-diffusion. We would like to propose a self-regulating method that facilitates the creation of surgical meshes from biocompatible polymers. Since the control of the process assumes a thorough understanding of the underlying phenomena, the theoretical background, as well as a mathematical model that can accurately describe the empirical data, is also introduced and explained. Our method offers the benefits of conventional techniques while introducing additional advantages not attainable with them. Most importantly, the method proposed in this paper enables the rapid creation of meshes with an average pore size that can be adjusted easily and tailored to fit the intended area of application.

Collective behavior and self-organization in neural rosette morphogenesis

Neural rosettes develop from the self-organization of differentiating human pluripotent stem cells. This process mimics the emergence of the embryonic central nervous system primordium, i.e., the neural tube, whose formation is under close investigation as errors during such process result in severe diseases like spina bifida and anencephaly. While neural tube formation is recognized as an example of self-organization, we still do not understand the fundamental mechanisms guiding the process. Here, we discuss the different theoretical frameworks that have been proposed to explain self-organization in morphogenesis. We show that an explanation based exclusively on stem cell differentiation cannot describe the emergence of spatial organization, and an explanation based on patterning models cannot explain how different groups of cells can collectively migrate and produce the mechanical transformations required to generate the neural tube. We conclude that neural rosette development is a relevant experimental 2D in-vitro model of morphogenesis because it is a multi-scale self-organization process that involves both cell differentiation and tissue development. Ultimately, to understand rosette formation, we first need to fully understand the complex interplay between growth, migration, cytoarchitecture organization, and cell type evolution.

Self-Organization and Genomic Causality in Models of Morphogenesis

The debate about what causes the generation of form and structure in embryological development goes back to antiquity. Most recently, it has focused on the divergent views as to whether the generation of patterns and form in development is a largely self-organized process or is mainly determined by the genome, in particular, complex developmental gene regulatory processes. This paper presents and analyzes pertinent models of pattern formation and form generation in a developing organism in the past and the present, with a special emphasis on Alan Turing's 1952 reaction-diffusion model. I first draw attention to the fact that Turing's paper remained, at first, without a noticeable impact on the community of biologists because purely physical-chemical models were unable to explain embryological development and often also simple repetitive patterns. I then show that from the year 2000 and onwards, Turing's 1952 paper was increasingly cited also by biologists. The model was updated to include gene products and now seemed able to account for the generation of biological patterns, though discrepancies between models and biological reality remained. I then point out Eric Davidson's successful theory of early embryogenesis based on gene-regulatory network analysis and its mathematical modeling that not only was able to provide a mechanistic and causal explanation for gene regulatory events controlling developmental cell fate specification but, unlike reaction-diffusion models, also addressed the effects of evolution and organisms' longstanding developmental and species stability. The paper concludes with an outlook on further developments of the gene regulatory network model.

The idea of constancy in development and evolution - Scientific and philosophical perspectives

The ability of developmental systems to produce constant phenotypes, even in a wide range of different environments, and the longstanding stability of species are among the most remarkable phenomena in biology. I argue that understanding the longstanding constancy and stability of species or the constant outcome of development in different environments are also prerequisites for explaining stable change (i.e., change that does not consist of random plasticity). Various approaches to account for stable changes in development are based on the causal role of genes and an organized genome, mathematical-physical-chemical models, or a combination of both. I argue that the constancy of developmental outcome and the longstanding stability of species are associated with organisms' structural and organizational hierarchies, particularly highly organized gene-regulatory networks and genetic causality, which are fundamental principles of life. Mathematical-physical-chemical models that marginalize these principles cannot convincingly account for the observed constancy in development and evolution. However, an integration of physical-chemical processes such as reaction-diffusion mechanisms and genome-based mechanisms of form generation has recently proved fruitful in explaining the development of some periodic structures. Constancy and change were also major topoi in ancient Greek philosophy, in which prominent philosophical schools such as the atomists attempted to bridge the antinomy between them by basing stable change on constant entities. I argue that the idea of change, that is, change without losing complexity or even increasing it, being based on modifications of the otherwise reliable transmission of genomes over long periods of time has a historical parallel in the writings of these ancient speculative thinkers, notwithstanding the fundamental differences between the two thought systems.

French flag gradients and Turing reaction-diffusion versus differentiation waves as models of morphogenesis

The Turing reaction-diffusion model and the French Flag Model are widely accepted in the field of development as the best models for explaining embryogenesis. Virtually all current attempts to understand cell differentiation in embryos begin and end with the assumption that some combination of these two models works. The result may become a bias in embryogenesis in assuming the problem has been solved by these two-chemical substance-based models. Neither model is applied consistently. We review the differences between the French Flag, Turing reaction-diffusion model, and a mechanochemical model called the differentiation wave/cell state splitter model. The cytoskeletal cell state splitter and the embryonic differentiation waves was first proposed in 1987 as a combined physics and chemistry model for cell differentiation in embryos, based on empirical observations on urodele amphibian embryos. We hope that the development of theory can be advanced and observations relevant to distinguishing the embryonic differentiation wave model from the French Flag model and reaction-diffusion equations will be taken up by experimentalists. Experimentalists rely on mathematical biologists for theory, and therefore depend on them for what parameters they choose to measure and ignore. Therefore, mathematical biologists need to fully understand the distinctions between these three models.

Pattern formation mechanisms of self-organizing reaction-diffusion systems

Embryonic development is a largely self-organizing process, in which the adult body plan arises from a ball of cells with initially nearly equal potency. The reaction-diffusion theory first proposed by Alan Turing states that the initial symmetry in embryos can be broken by the interplay between two diffusible molecules, whose interactions lead to the formation of patterns. The reaction-diffusion theory provides a valuable framework for self-organized pattern formation, but it has been difficult to relate simple two-component models to real biological systems with multiple interacting molecular species. Recent studies have addressed this shortcoming and extended the reaction-diffusion theory to realistic multi-component networks. These efforts have challenged the generality of previous central tenets derived from the analysis of simplified systems and guide the way to a new understanding of self-organizing processes. Here, we discuss the challenges in modeling multi-component reaction-diffusion systems and how these have recently been addressed. We present a synthesis of new pattern formation mechanisms derived from these analyses, and we highlight the significance of reaction-diffusion principles for developmental and synthetic pattern formation.

Self-Organization in Pattern Formation

Self-organization is pervasive in development, from symmetry breaking in the early embryo to tissue patterning and morphogenesis. For a few model systems, the underlying molecular and cellular processes are now sufficiently characterized that mathematical models can be confronted with experiments, to explore the dynamics of pattern formation. Here, we review selected systems, ranging from cyanobacteria to mammals, where different forms of cell-cell communication, acting alone or together with positional cues, drive the patterning of cell fates, highlighting the insights that even very simple models can provide as well as the challenges on the path to a predictive understanding of development.

Positional information and reaction-diffusion: two big ideas in developmental biology combine

One of the most fundamental questions in biology is that of biological pattern: how do the structures and shapes of organisms arise? Undoubtedly, the two most influential ideas in this area are those of Alan Turing's 'reaction-diffusion' and Lewis Wolpert's 'positional information'. Much has been written about these two concepts but some confusion still remains, in particular about the relationship between them. Here, we address this relationship and propose a scheme of three distinct ways in which these two ideas work together to shape biological form.

Obituary: Hans Meinhardt (1938-2016)

Patrick Müller and Christiane Nüsslein-Volhard reflect on the life and career of their colleague Hans Meinhardt.

The significance and scope of evolutionary developmental biology: a vision for the 21st century

Evolutionary developmental biology (evo-devo) has undergone dramatic transformations since its emergence as a distinct discipline. This paper aims to highlight the scope, power, and future promise of evo-devo to transform and unify diverse aspects of biology. We articulate key questions at the core of eleven biological disciplines-from Evolution, Development, Paleontology, and Neurobiology to Cellular and Molecular Biology, Quantitative Genetics, Human Diseases, Ecology, Agriculture and Science Education, and lastly, Evolutionary Developmental Biology itself-and discuss why evo-devo is uniquely situated to substantially improve our ability to find meaningful answers to these fundamental questions. We posit that the tools, concepts, and ways of thinking developed by evo-devo have profound potential to advance, integrate, and unify biological sciences as well as inform policy decisions and illuminate science education. We look to the next generation of evolutionary developmental biologists to help shape this process as we confront the scientific challenges of the 21st century.

On the interplay between mathematics and biology: hallmarks toward a new systems biology

This paper proposes a critical analysis of the existing literature on mathematical tools developed toward systems biology approaches and, out of this overview, develops a new approach whose main features can be briefly summarized as follows: derivation of mathematical structures suitable to capture the complexity of biological, hence living, systems, modeling, by appropriate mathematical tools, Darwinian type dynamics, namely mutations followed by selection and evolution. Moreover, multiscale methods to move from genes to cells, and from cells to tissue are analyzed in view of a new systems biology approach.

Modelling and simulating reaction-diffusion systems using coloured Petri nets

Reaction-diffusion systems often play an important role in systems biology when developmental processes are involved. Traditional methods of modelling and simulating such systems require substantial prior knowledge of mathematics and/or simulation algorithms. Such skills may impose a challenge for biologists, when they are not equally well-trained in mathematics and computer science. Coloured Petri nets as a high-level and graphical language offer an attractive alternative, which is easily approachable. In this paper, we investigate a coloured Petri net framework integrating deterministic, stochastic and hybrid modelling formalisms and corresponding simulation algorithms for the modelling and simulation of reaction-diffusion processes that may be closely coupled with signalling pathways, metabolic reactions and/or gene expression. Such systems often manifest multiscaleness in time, space and/or concentration. We introduce our approach by means of some basic diffusion scenarios, and test it against an established case study, the Brusselator model.

Modelling from the experimental developmental biologists viewpoint

In this review we consider Reaction-Diffusion as the archetype of a model in developmental biology. We consider its history in relation to experimental work since it was first proposed in 1952 by Turing and revived in the 1970s by Meinhardt. We then discuss the most recent examples of experiments that address this model, including the challenges that remain in capturing the physico-chemical manifestation of the model mechanism in a real developmental system. Finally we discuss the model's current status and use in the experimental community.

Biology is more theoretical than physics

The word "theory" is used in at least two senses--to denote a body of widely accepted laws or principles, as in "Darwinian theory" or "quantum theory," and to suggest a speculative hypothesis, often relying on mathematical analysis, that has not been experimentally confirmed. It is often said that there is no place for the second kind of theory in biology and that biology is not theoretical but based on interpretation of data. Here, ideas from a previous essay are expanded upon to suggest, to the contrary, that the second kind of theory has always played a critical role and that biology, therefore, is a good deal more theoretical than physics.

Morphogenetic action through flux-limited spreading

A central question in biology is how secreted morphogens act to induce different cellular responses within a group of cells in a concentration-dependent manner. Modeling morphogenetic output in multicellular systems has so far employed linear diffusion, which is the normal type of diffusion associated with Brownian processes. However, there is evidence that at least some morphogens, such as Hedgehog (Hh) molecules, may not freely diffuse. Moreover, the mathematical analysis of such models necessarily implies unrealistic instantaneous spreading of morphogen molecules, which are derived from the assumptions of Brownian motion in its continuous formulation. A strict mathematical model considering Fick's diffusion law predicts morphogen exposure of the whole tissue at the same time. Such a strict model thus does not describe true biological patterns, even if similar and attractive patterns appear as results of applying such simple model. To eliminate non-biological behaviors from diffusion models we introduce flux-limited spreading (FLS), which implies a restricted velocity for morphogen propagation and a nonlinear mechanism of transport. Using FLS and focusing on intercellular Hh-Gli signaling, we model a morphogen gradient and highlight the propagation velocity of morphogen particles as a new key biological parameter. This model is then applied to the formation and action of the Sonic Hh (Shh) gradient in the vertebrate embryonic neural tube using our experimental data on Hh spreading in heterologous systems together with published data. Unlike linear diffusion models, FLS modeling predicts concentration fronts and the evolution of gradient dynamics and responses over time. In addition to spreading restrictions by extracellular binding partners, we suggest that the constraints imposed by direct bridges of information transfer such as nanotubes or cytonemes underlie FLS. Indeed, we detect and measure morphogen particle velocity in such cell extensions in different systems.

Emergence of life from multicomponent mixtures of chemicals: the case for experiments with cycling physicochemical gradients

The emergence of life from planetary multicomponent mixtures of chemicals is arguably the most complicated and least understood natural phenomenon. The fact that living cells are non-equilibrium systems suggests that life can emerge only from non-equilibrium chemical systems. From an astrobiological standpoint, non-equilibrium chemical systems arise naturally when solar irradiation strikes rotating surfaces of habitable planets: the resulting cycling physicochemical gradients persistently drive planetary chemistries toward "embryonic" living systems and an eventual emergence of life. To better understand the factors that lead to the emergence of life, I argue for cycling non-equilibrium experiments with multicomponent chemical systems designed to represent the evolving chemistry of Hadean Earth ("prebiotic soups"). Specifically, I suggest experimentation with chemical engineering simulators of Hadean Earth to observe and analyze (i) the appearances and phase separations of surface active and polymeric materials as precursors of the first "cell envelopes" (membranes) and (ii) the accumulations, commingling, and co-reactivity of chemicals from atmospheric, oceanic, and terrestrial locations.

Computational approaches to developmental patterning

Computational approaches are breaking new ground in understanding how embryos form. Here, we discuss recent studies that couple precise measurements in the embryo with appropriately matched modeling and computational methods to investigate classic embryonic patterning strategies. We include signaling gradients, activator-inhibitor systems, and coupled oscillators, as well as emerging paradigms such as tissue deformation. Parallel progress in theory and experiment will play an increasingly central role in deciphering developmental patterning.

The mechanics behind cell polarity

The generation of cell polarity is one of the most intriguing symmetry-breaking events in biology. It is involved in almost all physiological and developmental processes and, despite the differences between plant and animal cell structures, cell polarity is generated by a similar core mechanism that comprises the extracellular matrix (ECM), Rho GTPase, the cytoskeleton, and the membranes. Several recent articles show that mechanical factors also contribute to the establishment and robustness of cell polarity, and the different molecular actors of cell polarity are now viewed as integrators of both biochemical and mechanical signals. Although cell polarity remains a complex process, some level of functional convergence between plants and animals is revealed. Following comparative presentation of cell polarity in plants and animals, we will discuss the theoretical background behind the role of mechanics in polarity and the relevant experimental tests, focusing on ECM anchorage, cytoskeleton behavior, and membrane tension.