Mathematical Modeling in the Study of Organisms and Their Parts.

Mathematical modeling is a very powerful tool to understand natural phenomena. Such a tool carries its own assumptions and should always be used critically. In this chapter we highlight the key ingredients and steps of modeling and focus on their biological interpretation. Particularly, we discuss the role of theoretical principles in writing models. We also highlight the meaning and interpretation of equations. The main aim of this chapter is to facilitate the interaction between biologists and mathematical modelers. We focus on the case of cell proliferation and motility in the context of multicellular organisms.

Historicity at the heart of biology.

Most mathematical modeling in biology relies either implicitly or explicitly on the epistemology of physics. The underlying conception is that the historicity of biological objects would not matter to understand a situation here and now, or, at least, historicity would not impact the method of modeling. We analyze that it is not the case with concrete examples. Historicity forces a conceptual reconfiguration where equations no longer play a central role. We argue that all observations depend on objects defined by their historical origin instead of their relations as in physics. Therefore, we propose that biological variations and historicity come first, and regularities are constraints with limited validity in biology. Their proper theoretical and empirical use requires specific rationales.

The Identity of Organisms in Scientific Practice: Integrating Historical and Relational Conceptions.

We address the identity of biological organisms at play in experimental and modeling practices. We first examine the central tenets of two general conceptions, and we assess their respective strengths and weaknesses. The historical conception, on the one hand, characterizes organisms' identity by looking at their past, and specifically at their genealogical connection with a common ancestor. The relational conception, on the other hand, interprets organisms' identity by referring to a set of distinctive relations between their parts, and between the organism and its environment. While the historical and relational conceptions are understood as opposed and conflicting, we submit that they are also fundamentally complementary. Accordingly, we put forward a hybrid conception, in which historical and relational (and more specifically, organizational) aspects of organisms' identity sustain and justify each other. Moreover, we argue that organisms' identity is not only hybrid but also bounded, insofar as the compliance with specific identity criteria tends to vanish as time passes, especially across generations. We spell out the core conceptual framework of this conception, and we outline an original formal representation. We contend that the hybrid and bounded conception of organisms' identity suits the epistemological needs of biological practices, particularly with regards to the generalization and reproducibility of experimental results, and the integration of mathematical models with experiments.

A Combined Morphometric and Statistical Approach to Assess Nonmonotonicity in the Developing Mammary Gland of Rats in the CLARITY-BPA Study.

BACKGROUND: The Consortium Linking Academic and Regulatory Insights on Bisphenol-A (CLARITY-BPA) is a rare collaboration of guideline-compliant (core) studies and academic hypothesis-based studies to assess the effects of bisphenol A (BPA). OBJECTIVES: We aimed to a) determine whether BPA showed effects on the developing rat mammary gland using new quantitative and established semiquantitative methods in two laboratories, b) develop a software tool for automatic evaluation of quantifiable aspects of the mammary ductal tree, and c) compare those methods. METHODS: Sprague-Dawley rats were exposed to BPA, vehicle, or positive control [ethinyl estradiol (EE2)] by oral gavage beginning on gestational day (GD)6 and continuing with direct dosing of the pups after birth. There were two studies: subchronic and chronic. The latter used two exposure regimes, one stopping at postnatal day (PND)21 (stop-dose) the other continuing until tissue harvest (continuous). Glands were harvested at multiple time points; whole mounts and histological specimens were analyzed blinded to treatment. RESULTS: The subchronic study's semiquantitative analysis revealed no significant differences between control and BPA dose groups at PND21, whereas at PND90 there were significant differences between control and the lowest BPA dose and between control and the lowest EE2 dose in animals in estrus. Quantitative, automatized analysis of the chronic PND21 specimens displayed nonmonotonic BPA effects, with a breaking point between the 25 and 250µg/kg body weight (BW) per day doses. This breaking point was confirmed by a global statistical analysis of chronic study animals at PND90 and 6 months analyzed by the quantitative method. The BPA response was different from the EE2 effect for many features. CONCLUSIONS: Both the semiquantitative and the quantitative methods revealed nonmonotonic effects of BPA. The quantitative unsupervised analysis used 91 measurements and produced the most striking nonmonotonic dose-response curves. At all time points, lower doses resulted in larger effects, consistent with the core study, which revealed a significant increase of mammary adenocarcinoma incidence in the stop-dose animals at the lowest BPA dose tested. https://doi.org/10.1289/EHP6301.

A Primer on Mathematical Modeling in the Study of Organisms and Their Parts.

Mathematical modeling is a very powerful tool for understanding natural phenomena. Such a tool carries its own assumptions and should always be used critically. In this chapter, we highlight the key ingredients and steps of modeling and focus on their biological interpretation. In particular, we discuss the role of theoretical principles in writing models. We also highlight the meaning and interpretation of equations. The main aim of this chapter is to facilitate the interaction between biologists and mathematical modelers. We focus on the case of cell proliferation and motility in the context of multicellular organisms.

Toward a theory of organisms: Three founding principles in search of a useful integration.

Organisms, be they uni- or multi-cellular, are agents capable of creating their own norms; they are continuously harmonizing their ability to create novelty and stability, that is, they combine plasticity with robustness. Here we articulate the three principles for a theory of organisms, namely: the default state of proliferation with variation and motility, the principle of variation and the principle of organization. These principles profoundly change both biological observables and their determination with respect to the theoretical framework of physical theories. This radical change opens up the possibility of anchoring mathematical modeling in biologically proper principles.

Theoretical principles for biology: Variation.

Darwin introduced the concept that random variation generates new living forms. In this paper, we elaborate on Darwin's notion of random variation to propose that biological variation should be given the status of a fundamental theoretical principle in biology. We state that biological objects such as organisms are specific objects. Specific objects are special in that they are qualitatively different from each other. They can undergo unpredictable qualitative changes, some of which are not defined before they happen. We express the principle of variation in terms of symmetry changes, where symmetries underlie the theoretical determination of the object. We contrast the biological situation with the physical situation, where objects are generic (that is, different objects can be assumed to be identical) and evolve in well-defined state spaces. We derive several implications of the principle of variation, in particular, biological objects show randomness, historicity and contextuality. We elaborate on the articulation between this principle and the two other principles proposed in this special issue: the principle of default state and the principle of organization.

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

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.

Theoretical principles for biology: Organization.

In the search of a theory of biological organisms, we propose to adopt organization as a theoretical principle. Organization constitutes an overarching hypothesis that frames the intelligibility of biological objects, by characterizing their relevant aspects. After a succinct historical survey on the understanding of organization in the organicist tradition, we offer a specific characterization in terms of closure of constraints. We then discuss some implications of the adoption of organization as a principle and, in particular, we focus on how it fosters an original approach to biological stability, as well as and its interplay with variation.

The biological default state of cell proliferation with variation and motility, a fundamental principle for a theory of organisms.

The principle of inertia is central to the modern scientific revolution. By postulating this principle Galileo at once identified a pertinent physical observable (momentum) and a conservation law (momentum conservation). He then could scientifically analyze what modifies inertial movement: gravitation and friction. Inertia, the default state in mechanics, represented a major theoretical commitment: there is no need to explain uniform rectilinear motion, rather, there is a need to explain departures from it. By analogy, we propose a biological default state of proliferation with variation and motility. From this theoretical commitment, what requires explanation is proliferative quiescence, lack of variation, lack of movement. That proliferation is the default state is axiomatic for biologists studying unicellular organisms. Moreover, it is implied in Darwin's "descent with modification". Although a "default state" is a theoretical construct and a limit case that does not need to be instantiated, conditions that closely resemble unrestrained cell proliferation are readily obtained experimentally. We will illustrate theoretical and experimental consequences of applying and of ignoring this principle.

SAMA: A Method for 3D Morphological Analysis.

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.

In search of principles for a Theory of Organisms.

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.

Biological organisation as closure of constraints.

We propose a conceptual and formal characterisation of biological organisation as a closure of constraints. We first establish a distinction between two causal regimes at work in biological systems: processes, which refer to the whole set of changes occurring in non-equilibrium open thermodynamic conditions; and constraints, those entities which, while acting upon the processes, exhibit some form of conservation (symmetry) at the relevant time scales. We then argue that, in biological systems, constraints realise closure, i.e. mutual dependence such that they both depend on and contribute to maintaining each other. With this characterisation in hand, we discuss how organisational closure can provide an operational tool for marking the boundaries between interacting biological systems. We conclude by focusing on the original conception of the relationship between stability and variation which emerges from this framework.

From single cells to tissues: interactions between the matrix and human breast cells in real time.

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.

From bottom-up approaches to levels of organization and extended critical transitions.

Biological thinking is structured by the notion of level of organization. We will show that this notion acquires a precise meaning in critical phenomena: they disrupt, by the appearance of infinite quantities, the mathematical (possibly equational) determination at a given level, when moving at an "higher" one. As a result, their analysis cannot be called genuinely bottom-up, even though it remains upward in a restricted sense. At the same time, criticality and related phenomena are very common in biology. Because of this, we claim that bottom-up approaches are not sufficient, in principle, to capture biological phenomena. In the second part of this paper, following (Bailly, 1991b), we discuss a strong criterium of level transition. The core idea of the criterium is to start from the breaking of the symmetries and determination at a "first" level in order to "move" at the others. If biological phenomena have multiple, sustained levels of organization in this sense, then they should be interpreted as extended critical transitions.

The Inert vs. the Living State of Matter: Extended Criticality, Time Geometry, Anti-Entropy - An Overview.

The physical singularity of life phenomena is analyzed by means of comparison with the driving concepts of theories of the inert. We outline conceptual analogies, transferals of methodologies and theoretical instruments between physics and biology, in addition to indicating significant differences and sometimes logical dualities. In order to make biological phenomenalities intelligible, we introduce theoretical extensions to certain physical theories. In this synthetic paper, we summarize and propose a unified conceptual framework for the main conclusions drawn from work spanning a book and several articles, quoted throughout.

A 2-dimensional geometry for biological time.

This paper proposes an abstract mathematical frame for describing some features of biological time. The key point is that usual physical (linear) representation of time is insufficient, in our view, for the understanding key phenomena of life, such as rhythms, both physical (circadian, seasonal...) and properly biological (heart beating, respiration, metabolic...). In particular, the role of biological rhythms do not seem to have any counterpart in mathematical formalization of physical clocks, which are based on frequencies along the usual (possibly thermodynamical, thus oriented) time. We then suggest a functional representation of biological time by a 2-dimensional manifold as a mathematical frame for accommodating autonomous biological rhythms. The "visual" representation of rhythms so obtained, in particular heart beatings, will provide, by a few examples, hints towards possible applications of our approach to the understanding of interspecific differences or intraspecific pathologies. The 3-dimensional embedding space, needed for purely mathematical reasons, allows to introduce a suitable extra-dimension for "representation time", with a cognitive significance.

Protention and retention in biological systems.

This article proposes an abstract mathematical frame for describing some features of cognitive and biological time. We focus here on the so called "extended present" as a result of protentional and retentional activities (memory and anticipation). Memory, as retention, is treated in some physical theories (relaxation phenomena, which will inspire our approach), while protention (or anticipation) seems outside the scope of physics. We then suggest a simple functional representation of biological protention. This allows us to introduce the abstract notion of "biological inertia".