Modeling mammary organogenesis from biological first principles: Cells and their physical constraints

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.

Constraints Shape Cell Function and Morphology by Canalizing the Developmental Path along the Waddington's Landscape

“…More precisely, constraints modulate the effect triggered by physical forces, thus determining when (and how) biological activities are allowed to instantiate. [ 40 ] Here, the critical factor is the capability of the system to act on its own behalf to use energy (work) for recreating those conditions (constraints) that make work possible (work cycle), [ 41 ] thus compensating the (physiological) decay. The capacity to generate proper constraints out of its own processes is unknown in classical physics, and confers the organism a history—as the dynamical relationships between forces, parameters and functions changes over times and those changes are direct consequences of the previous modifications [ 42 ] —which may have a role in determining future organismal dynamics.…”

An Integrative Approach Toward Biology, Organisms, and Cancer

“…Objectively, this concept of constraints opens a point of entry for the mathematization of biology. In fact, we modeled mammary gland morpho-genesis using the notion of default state and its constraints [17]. …”

“…What does require an explanation is the identification and mode of action of the constraints that limit the instantiation of the default state both in unicellular and in multicellular organisms. In other words, what requires an explanation is the departure from the default state, namely, proliferative quiescence, lack of motility, and restrained variation [17]. …”

“…Given the interdependence of the parts in an organism, it is insufficient to analyze a single constraint or a given set of constraints in isolation. Nonetheless, we obtained an insightful explanation of glandular morphogenesis by analyzing constraints on the default state in a 3D model of the breast [17]. Admittedly, additional constraints at the tissue level and organismal regulation acting via hormones should be studied for an increasingly comprehensive biological analysis.…”

“…Of note, such an approach is very different from the applied mathematics transplanted directly from Physics that is routinely used to model biological phenomena [9, 18]. We favor, instead, to model biological phenomena using biological principles ([17] Montévil, this book).…”