Here we discuss the application and articulation of biological principles for mathematical modeling of morphogenesis in the case of mammary ductal morphogenesis, with an emphasis on the default state.
Abstract
Unlike inert objects, organisms and their cells have the ability to initiate activity by themselves, and thus change their properties or states even in the absence of an external cause. This crucial difference led us to search for principles suitable for the study organisms. We propose that cells follow the default state of proliferation with variation and motility, a principle of biological inertia. This means that in the presence of sufficient nutrients, cells will express their default state. We also propose a principle of variation that addresses two central features of organisms, variation and historicity. To address interdependence between parts, we use a third principle, the principle of organization: more specifically, the notion of the closure of constraints. Within this theoretical framework, constraints are specific theoretical entities defined by their relative stability with respect to the processes they constrain. Constraints are mutually dependent in an organized system and act on the default state. Here we discuss the application and articulation of these principles for mathematical modeling of morphogenesis in a specific case, that of mammary ductal morphogenesis, with an emphasis on the default state. Our model has both a biological component, the cells, and a physical component, the matrix that contains collagen fibers. Cells are agents that move and proliferate unless constrained; they exert mechanical forces that i) act on collagen fibers and ii) on other cells. As fibers are organized, they constrain the cells’ ability to move and to proliferate. This model exhibits a circularity that can be interpreted in terms of the closure of constraints. Implementing our mathematical model shows that constraints to the default state are sufficient to explain the formation of mammary epithelial structures. Finally, the success of this modeling effort suggests a step-wise approach whereby additional constraints imposed by the tissue and the organism can be examined in silico and rigorously tested by in vitro and in vivo experiments, in accordance with the organicist perspective we embrace.
En biologie, le hasard est une notion essentielle pour comprendre les variations ; cependant, cette notion n'est généralement pas conceptualisée avec précision. Nous apportons ici quelques éléments allant dans ce sens.
Abstract
La physique possède plusieurs concepts de hasard qui reposent néanmoins tous sur l’idée que les possibilités sont données d’avance. En revanche, un nombre croissant de biologistes théoriciens cherchent à introduire la notion de nouvelles possibilités, c’est-à-dire des modifications de l’espace des possibles - une idée déjà discutée par Bergson et qui n’a pas été véritablement poursuivie scientifiquement jusqu’à récemment (sauf, en un sens, dans la systématique, c’est-à-dire la méthode de classification des êtres vivants). Alors, le hasard opère au niveau des possibilités elles-mêmes et est à la base de l’historicité des objets biologiques. Nous soulignons que ce concept de hasard n’est pas seulement pertinent lorsqu’on cherche à prédire l’avenir. Au contraire, il façonne les organisations biologiques et les écosystèmes. À titre d’illustration, nous soutenons qu’une question cruciale de l’Anthropocène est la disruption des organisations biologiques que l’histoire naturelle a structurées, conduisant à un effondrement des possibilités biologiques.
In biology, randomness is a critical notion to understand variations; however this notion is typically not conceptualized precisely. Here we provide some elements in that direction.
Abstract
Physics has several concepts of randomness that build on the idea that the possibilities are pre-given. By contrast, an increasing number of theoretical biologists attempt to introduce new possibilities, that is to say, changes of possibility space – an idea already discussed by Bergson and that was not genuinely pursued scientifically until recently (except, in a sense, in systematics, i.e, the method to classify living beings). Then, randomness operates at the level of possibilities themselves and is the basis of the historicity of biological objects. We emphasize that this concept of randomness is not only relevant when aiming to predict the future. Instead, it shapes biological organizations and ecosystems. As an illustration, we argue that a critical issue of the Anthropocene is the disruption of the biological organizations that natural history has shaped, leading to a collapse of biological possibilities.
Qu'est qu'une forme en biologie? Au delà des définitions mathématiques, quel est le statut théorique des formes biologiques?
Abstract
Dans l’interface entre biologie et mathématiques, les formes et les processus de morphogenèse sont souvent étudiées pour eux-mêmes. Nous pensons que cette manière de procéder est insuffisante pour capturer le sens biologique de ces formes. La biologie comporte des spécificités qui se manifestent tant sur le plan philosophique que sur celui des principes théoriques : en particulier, tout processus biologique tel qu’un processus de morphogenèse ou une régulation physiologique (i) s’inscrit dans l’évolution et dans une histoire naturelle et (ii) s’intègre dans un organisme dont il dépend et auquel il participe. Nous aborderons alors le sens des formes biologiques à l’aune de ces principes, tant au niveau de la théorie qu’au niveau de la compréhension de l’accès expérimental aux objets biologiques.
Turing distingue l’imitation d’un phénomène de sa modélisation. En biologie, il n'y a cependant pas encore de cadre théorique pour encadrer la pratique de modélisation.
Abstract
Turing distingue soigneusement l’imitation de la modélisation d’un phénomène. Cette dernière vise à saisir la structure causale du phénomène étudié. En biologie, il n’y a cependant pas de cadre théorique bien établi pour encadrer la pratique de modélisation. Nous partons de l’articulation entre la compréhension du vivant et la thermodynamique, en particulier le second principe. Ceci nous conduira à expliciter les défis théoriques et épistémologiques pour la compréhension mathématique du vivant. En particulier, l’historicité du vivant est un défi rarement abordé explicitement dans ce domaine. Nous pensons que ce défi nécessite un renversement complet de l’épistémologie de la physique afin d’aborder de manière théoriquement précise les organismes vivants. Ce changement épistémologique est pertinent tant pour la pratique théorique que pour l’interprétation des protocoles et résultats expérimentaux.
L’invention et le développement des ordinateurs a ouvert de nouvelles possibilités pour la modélisation. En physique, l’existence de théories mathématisées
Abstract
L’invention et le développement des ordinateurs a ouvert de nouvelles possibilités pour la modélisation. En physique, l’existence de théories mathématisées permet d’utiliser l’ordinateur pour calculer des solutions approchées à des problèmes déjà bien circonscrits théoriquement et épistémologiquement. En biologie, par contre, il n’existe pas de théorie jouant ce rôle épistémologique, et l’informatique a permis l’émergence de pratiques de modélisation combinant plusieurs cadres mathématiques. L’ouvrage de Franck Varenne porte sur ces pratiques novatrices, leur histoire et leur épistémologie, à travers le cas des simulations de morphogenèse d’arbres et plus généralement de plantes.
We propose a theoretical framework to model the behavior of cells in tissues and develop an application in the case of duct morphogenesis in mammary glands.
Abstract
We developed 3D culture methods that reproduce in vitro mammary gland ductal morphogenesis. We are proposing a conceptual framework to understand morphogenetic events based on epistemologically sound biological principles instead of the common practice of using only physical principles. More specifically, our theoretical framework is based on the principle that the default state of cells is proliferation with variation and motility. We emphasize the role played by the agency of cells embedded in a gel and the circularity that is relevant for the intended process, whereby cells act upon other cells and on matrix elements, and are subject to the agentivity of neighboring cells. This circularity strongly differs from classical linear causality. Finally, our approach opens up the study of causal determination to multilevel explanations rather than to reductive ones involving only molecules in general and genes in particular.
Keywords: Morphogenesis, extracellular matrix, theoretical principles, default state of cells, modelization.
This focused issue of Progress in Biophysics and Molecular Biology is entitled "From the century of the genome to the century of the organism: New theoretical approaches." It was developed during Ana M. Soto’s tenure as Blaise Pascal Chair of Biology 2013-15 at the Ecole Normale Supérieure (ENS,...
Abstract
This focused issue of Progress in Biophysics and Molecular Biology is entitled "From the century of the genome to the century of the organism: New theoretical approaches." It was developed during Ana M. Soto’s tenure as Blaise Pascal Chair of Biology 2013-15 at the Ecole Normale Supérieure (ENS, Paris, France). Giuseppe Longo was the Pascal Chair host at the ENS. This ongoing theoretical was also used as the content of a 10 session course attended by graduate students and post-graduates, which took place at the National Museum of Natural History and at the ENS. The attendants of course encouraged the guest editors to make this material easily available, hence the origin of PBMB issue.
We developed a mathematical model of mammary gland based on proper biological principles: the default state of cells and the principle of organization.
Abstract
Abstract In multicellular organisms, relations among parts and between parts and the whole are contextual and interdependent. These organisms and their cells are ontogenetically linked: an organism starts as a cell that divides producing non-identical cells, which organize in tri-dimensional patterns. These association patterns and cells types change as tissues and organs are formed. This contextuality and circularity makes it difficult to establish detailed cause and effect relationships. Here we propose an approach to overcome these intrinsic difficulties by combining the use of two models; 1) an experimental one that employs 3D culture technology to obtain the structures of the mammary gland, namely, ducts and acini, and 2) a mathematical model based on biological principles. The typical approach for mathematical modeling in biology is to apply mathematical tools and concepts developed originally in physics or computer sciences. Instead, we propose to construct a mathematical model based on proper biological principles. Specifically, we use principles identified as fundamental for the elaboration of a theory of organisms, namely i) the default state of cell proliferation with variation and motility and ii) the principle of organization by closure of constraints. This model has a biological component, the cells, and a physical component, a matrix which contains collagen fibers. Cells display agency and move and proliferate unless constrained; they exert mechanical forces that i) act on collagen fibers and ii) on other cells. As fibers organize, they constrain the cells on their ability to move and to proliferate. The model exhibits a circularity that can be interpreted in terms of closure of constraints. Implementing the mathematical model shows that constraints to the default state are sufficient to explain ductal and acinar formation, and points to a target of future research, namely, to inhibitors of cell proliferation and motility generated by the epithelial cells. The success of this model suggests a step-wise approach whereby additional constraints imposed by the tissue and the organism could be examined in silico and rigorously tested by in vitro and in vivo experiments, in accordance with the organicist perspective we embrace.
Software for Automated Morphological Analysis is a method by which epithelial structures grown in 3D cultures can be imaged, reconstructed and analyzed.
Abstract
Three-dimensional (3D) culture models are critical tools for understanding tissue morphogenesis. A key requirement for their analysis is the ability to reconstruct the tissue into computational models that allow quantitative evaluation of the formed structures. Here, we present Software for Automated Morphological Analysis (SAMA), a method by which epithelial structures grown in 3D cultures can be imaged, reconstructed and analyzed with minimum human intervention. SAMA allows quantitative analysis of key features of epithelial morphogenesis such as ductal elongation, branching and lumen formation that distinguish different hormonal treatments. SAMA is a user-friendly set of customized macros operated via FIJI (http://fiji.sc/Fiji), an open-source image analysis platform in combination with a set of functions in R (http://www.r-project.org/), an open-source program for statistical analysis. SAMA enables a rapid, exhaustive and quantitative 3D analysis of the shape of a population of structures in a 3D image. SAMA is cross-platform, licensed under the GPLv3 and available at http://montevil.theobio.org/content/sama.
Documentation for Software for Automated Morphological Analysis, a method by which epithelial structures grown in 3D cultures can be imaged, reconstructed and analyzed.
Abstract
Three-dimensional (3D) culture models are critical tools for understanding tissue morphogenesis. A key requirement for their analysis is the ability to reconstruct the tissue into computational models that allow quantitative evaluation of the formed structures. Here, we present Software for Automated Morphological Analysis (SAMA), a method by which epithelial structures grown in 3D cultures can be imaged, reconstructed and analyzed with minimum human intervention. SAMA allows quantitative analysis of key features of epithelial morphogenesis such as ductal elongation, branching and lumen formation that distinguish different hormonal treatments. SAMA is a user-friendly set of customized macros operated via FIJI (http://fiji.sc/Fiji), an open-source image analysis platform in combination with a set of functions in R (http://www.r-project.org/), an open-source program for statistical analysis. SAMA enables a rapid, exhaustive and quantitative 3D analysis of the shape of a population of structures in a 3D image. SAMA is cross-platform, licensed under the GPLv3 and available at http://montevil.theobio.org/content/sama.
Lacking an operational theory to understand life cycles hinders progress in biology. We discuss elements towards such a theory, such as inertia and thermodynamics.
Abstract
Lacking an operational theory to explain the organization and behaviour of matter in unicellular and multicellular organisms hinders progress in biology. Such a theory should address life cycles from ontogenesis to death. This theory would complement the theory of evolution that addresses phylogenesis, and would posit theoretical extensions to accepted physical principles and default states in order to grasp the living state of matter and define proper biological observables. Thus, we favour adopting the default state implicit in Darwin’s theory, namely, cell proliferation with variation plus motility, and a framing principle, namely, life phenomena manifest themselves as non-identical iterations of morphogenetic processes. From this perspective, organisms become a consequence of the inherent variability generated by proliferation, motility and self-organization. Morphogenesis would then be the result of the default state plus physical constraints, like gravity, and those present in living organisms, like muscular tension.
Bulk properties do not determine shape; however, localized regions of collagen fiber alignment are required for ductal elongation and branching.
Abstract
Background: Mammary gland morphogenesis involves ductal elongation, branching, and budding. All of these processes are mediated by stroma - epithelium interactions. Biomechanical factors, such as matrix stiffness, have been established as important factors in these interactions. For example, epithelial cells fail to form normal acinar structures in vitro in 3D gels that exceed the stiffness of a normal mammary gland. Additionally, heterogeneity in the spatial distribution of acini and ducts within individual collagen gels suggests that local organization of the matrix may guide morphogenesis. Here, we quantified the effects of both bulk material stiffness and local collagen fiber arrangement on epithelial morphogenesis. Results: The formation of ducts and acini from single cells and the reorganization of the collagen fiber network were quantified using time-lapse confocal microscopy. MCF10A cells organized the surrounding collagen fibers during the first twelve hours after seeding. Collagen fiber density and alignment relative to the epithelial surface significantly increased within the first twelve hours and were a major influence in the shaping of the mammary epithelium. The addition of Matrigel to the collagen fiber network impaired cell-mediated reorganization of the matrix and increased the probability of spheroidal acini rather than branching ducts. The mechanical anisotropy created by regions of highly aligned collagen fibers facilitated elongation and branching, which was significantly correlated with fiber organization. In contrast, changes in bulk stiffness were not a strong predictor of this epithelial morphology. Conclusions: Localized regions of collagen fiber alignment are required for ductal elongation and branching suggesting the importance of local mechanical anisotropy in mammary epithelial morphogenesis. Similar principles may govern the morphology of branching and budding in other tissues and organs.
Keywords: Collagens, Morphogenesis, Extracellular matrix, Gels, Anisotropy, Stiffness, Scanning electron microscopy, Mammary gland development
