Theoretical principles for biology: Variation Ma¨el Mont´evila,b,∗, Matteo Mossiob , Arnaud Pochevillec , Giuseppe Longod,e a Laboratoire”Mati`ere et Syst` emes Complexes” (MSC), UMR 7057 CNRS, Universit´ e Paris 7 Diderot, Paris, France b Institut d’Histoire et de Philosophie des Sciences et des Techniques (IHPST) - UMR 8590, 13, rue du Four, 75006 Paris, France c Department of Philosophy and Charles Perkins Centre, The University of Sydney, Sydney, Australia d Centre Cavaill` es, R´epublique des Savoirs, CNRS USR3608, Coll` ege de France et Ecole Normale Sup´erieure, Paris, France e Department of Integrative Physiology and Pathobiology, Tufts University School of Medicine, Boston, MA USA Abstract 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. Keywords: Variability, Historicity, Genericity, Biological randomness, Organization, Theory of organisms Since the beginning of physics, symmetry 2 Invariance and symmetries: physics as the considerations have provided us with an extremely domain of generic objects 3 powerful and useful tool in our effort to understand nature. Gradually they have become the backbone 3 Variation and symmetry changes: biology as of our theoretical formulation of physical laws. the domain of specific objects 5 Tsung-Dao Lee 3.1 Randomness proper to specific objects . . . 6 The artificial products do not have any molecular 3.2 Constraints and specific objects . . . . . . . 7 dissymmetry; and I could not indicate the existence 3.3 Constraints and randomness . . . . . . . . . 7 of a more profound separation between the 3.4 Historicity . . . . . . . . . . . . . . . . . . . 9 products born under the influence of life and all the 3.5 Contextuality . . . . . . . . . . . . . . . . . 10 others. 3.6 Variability . . . . . . . . . . . . . . . . . . . 11 L. Pasteur 3.7 Modelization and specific objects . . . . . . 12 3.8 Conclusive remarks on the principle of vari- ation . . . . . . . . . . . . . . . . . . . . . . 13 Contents 4 Bringing organization into the picture 13 4.1 Organization grounds constraints in specific 1 Introduction 2 objects . . . . . . . . . . . . . . . . . . . . . 13 4.2 The epistemological status of closure under ∗ Corresponding author variation . . . . . . . . . . . . . . . . . . . . 14 Email addresses: mael.montevil@gmail.com (Ma¨ el Mont´evil), matteo.mossio@univ-paris1.fr (Matteo Mossio), 4.3 Relevant variation with respect to closure . 14 arnaud.pocheville@sydney.edu.au (Arnaud Pocheville), 4.4 Closure remains closure under variation . . 15 longo@ens.fr (Giuseppe Longo) URL: http://montevil.theobio.org (Ma¨ el Mont´ evil), 5 Non-identical iteration of morphogenetic pro- http://www.di.ens.fr/users/longo/ (Giuseppe Longo) cesses 16 • Published as: Ma¨ el Mont´evil, Matteo Mossio, Arnaud Pocheville, Giuseppe Longo, Theoretical principles for biology: Vari- ation, Progress in Biophysics and Molecular Biology, Available on- 6 Conclusions: back to theoretical principles 17 line 13 August 2016, ISSN 0079-6107, http://dx.doi.org/10.1016/ j.pbiomolbio.2016.08.005. Preprint submitted to Progress in biophysics and molecular biology August 22, 2016 1. Introduction Such a theoretical framework does not (yet) exist for the biology of organisms and our proposal aims at contributing A striking feature of living beings is their ability to to the elaboration of the “biological counterpart” of the change. All naturalists know that two individuals of the theoretical frameworks and abstract objects at work in same species usually display important qualitative differ- physics. ences. All experimentalists know that two replicate ex- It is worth emphasizing that, although we will elab- periments can give quite unexpectedly different results – orate on the concept of variation by analogy with and even in the absence of any abnormality in the experimental in contrast to the physico-mathematical perspective, we setup. by no means advocate a physico-mathematical treatment Variation took a central role in biological reasoning in of biological phenomena. Rather, we think that biology Darwin’s book The Origin of the Species (1859) in which in general, and the biology of organisms in particular, it served as a means to explain the current diversity of requires a significant change of perspective with respect life, by virtue of the concept of “descent with modifica- to the physical viewpoints and methodologies. Typically, tion” (Darwin, 1859, pp.119-124): organisms might show physics provide an ahistorical understanding of the phe- some differences from their parents, these differences might nomena studied2 . In contrast to physics and in line with be heritable and, under some proper conditions, accumu- the theory of evolution, we argue that historicity is an es- late to form new lineages. Importantly, to Darwin, some sential feature of biological phenomena and that biological of these variations would be “chance” variations, that is, historicity stem from the principle of variation. changes that would be unrelated to the conditions of exis- The principle of variation is related to the other princi- tence of the organisms, and even unpredictable (Darwin, ples put forward in this special issue: the biological default 1859, p.p 131, 314)1 . In so doing, Darwin introduced con- state (proliferation with variation and motility), and the tingency and historicity into biological thinking: accidents principle of organization. The default state is described would happen along life’s trajectory, which would at the as a primary generator of variation; when a cell divides, it same time be unpredictable, unrepeatable, and have long generates two non-identical cells (Soto et al., 2016). The lasting effects (Gould, 2002, p. 1334). principle of variation specifies the nature of the difference In this paper, we elaborate on the Darwinian idea of between these cells. The principle of organization is a way “chance” variation. We argue that variation should be to interpret biological functions as a property stemming given the status of a principle in biology, and in particu- from the role that parts play in the maintaining of a sys- lar organismal biology. Informally, the principle of varia- tem (Mossio et al., 2016, 2009; Mont´evil & Mossio, 2015). tion states that biological objects (such as organisms) con- According to this principle, a biologically relevant part tinually undergo modifications. Some of these variations (constraint) both depends on and maintains other parts of have functional repercussions, which we discuss with pre- the organism, thus forming a mutual dependence (labeled cise concepts in section 3). Moreover, whatever the math- “closure” for historical reasons). In Mossio et al. (2016), ematical frame used to describe an object, unpredictable variation and organization are discussed as two intertwined variations are nevertheless possible: the principle of varia- principles: organization is a condition for variation and fa- tion thus implies that the existence of exceptions is the rule vors its propagation, whereas variation is a condition for in biology. However, a proper biological theory cannot be the maintenance and adaptation of biological organization a mere catalog of exceptions. Accommodating the changes and for the generation of functional innovations. In sec- biological organisms undergo during their lives (ontogene- tion 4 of this text, we argue that any relevant variation is sis), as well as during evolution (phylogenesis), in a general a variation of an organization. theory is a specific challenge raised by biological systems, Biological variation occurs at all levels of organization, in particular in contrast to physical theorizing. from the molecular level to large scale structures and func- In physics, theoretical definitions enable us to discuss tions (West-Eberhard, 2003; Dueck et al., 2016). Single abstractly and adequately the behavior of objects (such cell observations on one side and high throughput tech- as the trajectory in space of a punctual object of mass m nologies on the other enable biologists to observe both in classical mechanics, or the behavior of quantum objects inter-cellular and inter-individual variations, which have as a vector in a Hilbert space in Quantum Mechanics). received an increasing amount of attention (Elowitz et al., 2002; Collective, 2005; Rivenbark et al., 2013). There 1 This concept of chance variation contrasts sharply with, for in- stance, the concept of variation of Lamarck (1809) another father of 2 As a matter of fact, physical approaches and methodologies are theoretical biology. To Lamarck, variations would be directed by the not confined to the physical and biophysical domain and have, in conditions of existence. This directedness entails that if the condi- part, percolated in biology and even social sciences. Such was the tions of existence re-occur in time, evolution is repeatable and thus, case, for instance, of the proposal of vital forces by some vitalists. ahistorical (Gould, 2002, p. 191). Other 19th century writers would These vital forces were conceived by analogy with Newtonian grav- advocate that variation would be so canalized (by the properties of itation and would entail spontaneous generation as a result of this the organisms) as to direct evolution (when evolution was acknowl- force acting on the right objects (De Klerk, 1979). Vital forces are an edged). See e.g. Bowler (2005; Pocheville & Danchin (2016) for more example of how the physico-mathematical approach typically implies details. an ahistorical understanding of the living, as we stress below. 2 are many generators of variation among which are ran- objects are objects which are all of the same kind from the dom gene expression, instability in morphogenetic pro- point of view of the theory (they typically obey the same cesses and randomness in biological rhythms. In partic- equations). An apple, the Earth, an anvil, for example, ular, cellular proliferation generates variation (Soto et al., are all objects with a given mass and center of gravity 2016). As for temporal scales, living systems undergo vari- and, from the point of view of classical mechanics, they all ation during their lives (ontogenesis), as well as during obey the same equations in the vacuum. Moreover, they evolution (phylogenesis), and these two aspects cannot be continue to obey the same equation during their dynamics analyzed independently (Danchin & Pocheville, 2014). In even though they undergo some changes: this is because, in this paper, we focus on variation as a general feature of physical language, their changes are restricted to changes biological systems without a privileged level of analysis. of state. Equations are not about specific values of the This enables us to discuss general features that are proper parameters or states; instead they jointly describe generic to biology and to stress key differences with respect to relations between parameters, states and the changes of physics. states4 . This is why changes of state of an object do not The central implication of this paper is the distinc- affect the validity of the equation which describe its behav- tion between the objects as conceived in physical theories ior. For example, the mass is an element of the description (generic objects) and the objects as conceived in biology of some generic objects, formalized by a generic variable on the basis of biological variation (specific objects)3 . In m representing jointly and synthetically all the possible what follows, we discuss first shortly what generic objects masses. are, what kind of manipulation they enable, and how their Physical objects, hence, are generic objects. More gen- analysis grounds physical theories (section 2). Then, we erally physical ‘laws’ are about generic objects. Consider contrast generic objects with the variation that biological for example the fundamental principle of dynamics: mass objects exhibit. We propose that biological objects should times acceleration equals the sum of external forces applies be understood as specific (in section 3). Specific objects to the object. Here, the “external forces” are understood are, in particular, fundamentally historical, variable and in a completely generic manner and any kind of forces may contextual. Thus, the specificity of organisms encompasses be involved. biological individuation and diversity. We also discuss the Typically, a physical object is described in a mathe- interplay between specific objects and physical morpho- matical space which is generated by the various quantities genesis. Then, in section 4 we elaborate on the integration required to describe this object. This mathematical space between the principle of variation and the principle of or- is called the ‘phase space’5 . In classical mechanics, the ganization, between the notion of biological specificity for phase space is the space of positions and momenta. This biological objects and the notion of organization and “con- mathematical space is given in advance; it pre-exists the tingent genericity” (Moreno & Mossio, 2015; Mont´evil & description of the object. The behavior of the physical Mossio, 2015; Mossio et al., 2016). Finally, we develop the object is defined as the way in which the object changes idea that biological systems are characterized by the non- in its phase space. The space is also assumed to provide identical iterations of morphogenetic processes (section 5). all the causes of the changes of the object, and thus it specifies the quantities that should be measured experi- mentally. In classical mechanics, positions and momenta, 2. Invariance and symmetries: physics as the do- in combination with properties such as the mass, are the main of generic objects quantities required to understand the changes of positions The principle of variation poses novel challenges with and momenta over time. respect to how mathematics enables us to describe the A phase space, however, is not sufficient to understand world. To better identify these challenges, we first make a the behavior of an object because the quantities it pro- detour by physics and show the role mathematics play in vides need to be articulated together to understand the physical theories. changes of the object6 . In physics, a theoretical framework Physics is based on mathematized theories. Histori- requires equations that depend on the variables symboliz- cally, the development of physical theories has been inter- twined with the development of appropriate mathematics 4 Simulations suffer from a shortcoming in this respect. While a to frame and define their objects: they have “co-evolved”. program does describe generic relationships between the variables, a We submit that mathematized physical theories rely on simulation run only provides one trajectory for specific values of its input. Whether this trajectory is representative of the behavior of the manipulation of generic objects (Bailly & Longo, 2011; the system for other values of the input, that is to say whether the Longo & Mont´evil, 2014a). The notion of generic objects is behavior obtained is generic or not, is a very difficult mathematical abstract, as it lies at the core of physicomathematical rea- issue (Stoer & Bulirsch, 2013). 5 Some physicists restrict the notion of ‘phase space’ to positions soning. However, the intuitive idea is quite simple: generic and momenta. Here, phase space means in general the space of mathematical description of the object. 6 The a priori diversity of possible trajectories in such a space is 3 An introduction to this distinction is given by Soto & Longo unfathomable in the sense that no axiomatic is sufficient to describe (2016). all their possible mathematical features. 3 ing the quantities describing the objects. The behavior of Phase space an object, that is to say its changes, is determined as a ho sts fy specific trajectory by equations that single it out in the ri e at ve phase space. Equations are valid for the phase space (or, ner ge at least, some regions of it) and depend on its quantities. trajectories The behavior of the object is completely determined by Symmetries ve the quantities that define its phase space and the corre- rif e y m in ju sponding equations. Predicting a trajectory corresponds sti ter to making this trajectory mathematically remarkable. To fy de Equations this end, equations typically correspond to optimization principles (for energy, entropy, entropy production, etc.), which enable physicists to single out a trajectory, the op- Figure 1: Articulation of the different components defining a generic timal trajectory that the system follows according to the object in physics. Equations determine the trajectory of a system, theory. Optimization principles and the ability to derive and this trajectory takes place in a mathematical space. Both the equations and the space have a structure that is described by the the- equations are essential for fundamental physical theories oretical symmetries that frame the object and that are valid by hy- and special models to make predictions. pothesis. There is a fundamental feedback that we do not represent For the purposes of this paper, the key question to be here: trajectories are the endpoint which fundamentally justifies the asked at this point is what justifies the use of the spaces whole theoretical construction of the generic object as experimenter can observe them. and equations in the theoretical constructions of physics. In part, these mathematical structures stem from axioms and are justified by their consequences. However, there aspects of an object invariant. For example, cutting an is more to say on the nature of fundamental hypotheses apple into two halves does not change the way it falls in of physics and the way in which they justify the use of the vacuum. Hence, an apple and its two halves are sym- mathematics. metric (they are the same) from the point of view of free Because whole classes of concrete objects are described fall in classical mechanics. Allometric relationships pro- in the same mathematical frame, they are studied as the vide a biological example of symmetry (Longo & Mont´evil, same generic object, and all have the same behavior. As 2014b). In mammals, the average period of rhythms such we evoked above, a piece of lead, an apple, or a planet are as heart rate or respiratory rate is found experimentally all the same objects from the viewpoint of classical grav- to depend on mass with the relation τ ∝ M 1/4 . Measur- itation: they all are point-wise objects with a position, a ing such relationships amounts to assuming that the basic momentum, a mass, and they all are subject to the prin- properties of metabolism are preserved under the trans- ciple of inertia and gravitational forces, described by the formations consisting of changing sizes and species, and same equations. In this respect, there is no relevant dif- thus that mammals of different masses are symmetric as ference between them and they are described jointly and for their internal rhythms (West & Brown, 2005; Longo synthetically as the same generic object. At the core of & Mont´evil, 2014b). Lastly, the assumption that different this approach to natural phenomena lies the identifica- replicates of an experiment enable us to access the same tion of non-identical objects. This identification of non- situation also corresponds to an assumption of symmetry identical objects is made explicit by transformations that between the replicates: they are all supposed to behave in leave these objects invariants (i.e. symetries). Putting fundamentally the same way8 . an emphasis on transformations is a modern approach in Symmetries are the basis of the mathematical struc- mathematics and physics that we build upon in this paper. tures in physics; that is the phase space and the relevant In particular, invariants are best described by the trans- formations that preserve them and which make explicit a biology. For instance, it is possible to deform a balloon into a sphere mathematical structure. or a rod shape without tearing and/or stitching, but it is impossible Generic objects are, for the most part, defined by the to transform it into a donut. Similarly, it is impossible to transform transformations that preserve them, and that enable us to a cell into two cells without tearing and/or stitching the membrane, define stable mathematical structures. We call such trans- where stitching corresponds here to the fusion of the membrane by pinching, and tearing corresponds to the final separation of the cells. formations ‘symmetries’. The notion of symmetry we use In all these cases, continuous deformations are considered as symme- is more general than the concept of geometrical symme- tries, insofar as they preserve topological invariants and, reciprocally, try in a three-dimensional space. Yet, the underlying idea the topological invariants are the ones preserved by continuous de- formations. As a result, one can define different categories of shapes is the same: geometrical symmetries are transformations on the basis of their inter-transformability. Continuous deformations which leave a geometrical figure invariant. Rotating a cir- fall under our concept of symmetry and are characteristic to the field cle around its center, for instance, does not modify the cir- of topology. 8 Notice that such an assumption is required in order to perform cle: it verifies a central symmetry7 . Similarly, symmetries statistical analyses. The most common statistical assumption is that (in general) are transformations which leave the relevant two variables are identically distributed, that is to say that the two considered situations are symmetric as far as their probability dis- 7 Another example comes from topology, a notion very useful in tributions are concerned. 4 equations. Accordingly, they constitute fundamental phys- view, qualitative changes typically imply changes of the ical assumptions which are less anthropomorphic than the relevant mathematical structures and, accordingly, changes notion of law and more meaningful than conservation prin- of symmetries. For example, changes of states of matter in ciples (see for example Van Fraassen, 1989; Bailly & Longo, phase transitions typically correspond to changes of sym- 2011; Longo & Mont´evil, 2014a). metries: a liquid is symmetric by rotation while a crystal For instance, the choice of an origin, three axes and a is not, because of its microscopic structure (see figure 3). metric are mandatory in order to write equations and per- In the biological domain, the organization of any cur- form measurement of positions and velocities (in Galilean, rent organism has been shaped by permanent qualitative special, or general relativity). Although different choices changes, that is, through changes of symmetries. A given are possible, the consistency of the theory depends on the biological organization is determined by an accumulation fact that the trajectories obtained in different reference of changes of symmetries both on the evolutionary and the systems are, in a fundamental sense, the same: in par- ontogenetic times13 . These changes correspond to changes ticular, they are invariant under suitable classical or rel- in the manner in which functions are performed, or even ativistic transformations of the reference system. Thus, to the appearance or loss of functions. the equations of physics are symmetric under these trans- Acknowledging that organisms can vary in this strong, formations9 . In general, the same trajectory should be functional sense, is not trivial: historically, the preforma- obtained before and after transformations which are fun- tionists (as for development), and the fixists (as for evolu- damental symmetries in the theory 10 , and these symme- tion) have held just the opposite view. If the homunculus tries enable us at the same time to formulate and justify is already in the egg, or, in modern terms, if dna already the equations and the phase space 11 . contains a blueprint of the organism, then development is In short, physical objects are understood as generic just the unfolding of an already existing organism (with all objects that follow specific trajectories. Theoretical sym- its relevant properties and functions). Similarly, if species metries ground this approach to natural phenomena. The do not change over geological time, then obviously organ- epistemological structure of generic objects is summarized isms conserve the same functions. in figure 1. In the next section, we discuss the princi- The idea that biological objects genuinely develop and ple of variation and the major challenges that biological evolve over time corresponds to the idea that the mathe- variation raises when one tries to frame biological objects matical structures required to describe them also change theoretically. over time. Thus, stating that development and evolution involve symmetry changes constitutes nothing more than a mathematical interpretation of the departure from the 3. Variation and symmetry changes: biology as the preformationist or fixist stances of development and evo- domain of specific objects lution. Evolution is rarely considered as entirely deter- A central and pervasive property of biological systems mined as the unfolding of historical necessities. Similarly, is their ability to change their organization over time12 . development should not be seen as the unfolding of a pre- These changes are not just quantitative changes, they are constituted organization but instead as a cascade of folding also qualitative. From a physico-mathematical point of leading to the setting up of an organization (figure 2 and 4). The crucial consequence of this view is that, because 9 Similarly, in electromagnetism the choice of assigning negative of their permanent symmetry changes, biological objects or positive charges to electrons is arbitrary; therefore, permuting the sign of charges has to leave the equations invariants (the derived should not be considered as generic objects. Organisms trajectories remain the same). are not well defined as invariant under transformations. 10 In a mathematical model, some symmetries are theoretical sym- When an organism is transformed, and in particular when metries which cannot be violated while others are more pragmatic the flow of time operates on it, the organism may undergo symmetries that correspond to a particular situation. The two things should not be conflated. For example, a theoretical symmetry is the unpredictable qualitative changes. As a result, biologi- assumption that all directions of the empty space are equivalent. cal objects are not well described by the virtuous cycle However, in a particular setting, all directions may not be equiva- described in figure 1. Accordingly, trajectories are not en- lent, for example because of the position and the gravitational field tirely framed by a mathematical framework: they may es- of some planets. Another theoretical symmetry is the symmetry be- tween positive and negative charges in classical electromagnetism. cape such frameworks and require a change in the symme- 11 Such justification of equations by symmetries is, in particular, tries, space of description, and equations used to describe the core of Noether’s theorem, which justifies the conservation of the object (figure 2). energy (resp. momenta) on the basis of a symmetry by time (resp. We propose then to understand biological objects (and space) translation of fundamental equations, among many other con- served quantities (Byers, 1999; Longo & Mont´ evil, 2014c). organisms in particular) as specific objects 14 . Specific ob- 12 While we mean here ‘organization’ in the technical sense dis- jects are constituted by a particular history of relevant and cussed in Mossio et al. (2016), the reader can also interpret the notion in a more informal manner. The different parts of an or- ganism depends on each other and form a coherent whole. This 13 A more detailed presentation of most of these ideas can be found interdependence of the parts and their relation to the whole form in Longo & Mont´ evil (2014a) and Longo & Mont´ evil (2011, 2013). the organization of organisms. 14 Our concept of specificity should not be confused with other 5  Phase space Phase space change ve es rifi es ge re fy ca ne ui ri te es pe ve ra ra req t ne trajectories Symmetry ge Symmetries change ve es re rif fi qu pe y r i ire ve s ca jus fie es tif t i y Equations Equation change us j Figure 2: Scheme of an elementary symmetry change in biology. An initial situation, on the left, is described by analogy with physics (see figure 1). However in biology, variation can escape such a frame. Understanding the object then requires a change of symmetry and of the whole mathematical structure framing the object. Trajectories are at the center of this change, they escape the initial frame and thus require a change of the symmetries describing the object. unpredictable symmetry changes over time, at all time- be defined generally as unpredictability with respect to a scales. Specific objects can be understood as the opposite theory. The notion of randomness which stems from the of generic objects: two instantiations of a specific object principle of variation is not endowed with a probability may always differ by at least one of their relevant qual- measure. itative aspects (in a given theoretical frame), while two Let us first characterize randomness in the case of a ba- instantiations of a generic object do not. For example, sic symmetry breaking, typically encountered in physical two organisms, be they clones, may always differ in one of models. Let us start with a situation which is symmetric, their relevant qualitative properties, for instance because for example a gas (figure 3, top). All directions are equiv- they may have undergone differences in their morphogen- alent for this object: all macroscopic quantities (density esis, i.e., they have been constituted by different develop- of the gas, pressure, etc.) stay the same after rotation. mental histories. When the symmetry is broken, directions are no longer On the basis of the concept of specific objects, we can equivalent; for example, there are privileged directions cor- now state the principle of variation: responding to a crystal structure after a phase transition (figure 3, bottom). The symmetry of the initial situation Principle of variation: means that all directions are initially equivalent and then Biological organisms are specific objects. that it is not possible to deduce the subsequent privileged The principle implies that biological organisms undergo directions in the crystal. As a result, the directions of the changes of symmetry over time and that, as we discuss be- crystal are random in this theoretical account. Moreover, low, some of these changes cannot be stated in advance15 . since all directions are symmetric in the initial conditions, In other words, the mathematical structure required to all directions have the same probability to become one of describe organisms is not stable with respect to the flow the crystal’s privileged directions. of time. Qualitative changes of structures and functions This physical situation exemplifies how symmetry break- occur over time and some of them are unpredictable. ing and randomness are associated and how the initial We now expand on several aspects and implications of symmetries define and justify probabilities (see Longo & the principle of variation. Mont´evil, to appear, for a general analysis of this associa- tion). 3.1. Randomness proper to specific objects Symmetry breaking and the associated randomness are A fundamental feature of the principle of variation is relevant for biology but we submit that they are not suf- that it includes an original notion of randomness: the very ficient. Biological randomness includes a fundamentally fact that biological objects undergo unpredictable symme- different notion. In the above case, the possible outcomes try changes. Generally speaking, the notion of randomness (all the possible directions in three dimensions) are defined is often conflated with the idea that events have some prob- before the symmetry breaking, as it is the mathematical ability of occurrence. However, scientific approaches to space on which symmetries act. Saying that the gas is randomness are richer than the notion of (classical) prob- symmetric by rotation requires us to define rotations and abilities (see for example Longo et al., 2011, for a discus- therefore the set of all possible directions on which rota- sion at the crossroads of different fields). Randomness may tions act. In biology, in contrast, the principle of variation poses that the list of possible outcomes and therefore the concepts of ‘biological’ specificity, such as chemical specificity of en- relevant symmetry changes are not pre-defined. For exam- zymes, or causal and informational specificity (see Griffiths et al., ple, it is not possible to embed all the spaces of description 2015). of current and future organisms within the space of de- 15 We would argue that even the rate of possible symmetry changes scription of the last universal common ancestor (LUCA). cannot be stated in advance. 6  3.2. Constraints and specific objects = The principle of variation does not preclude the pres- ence of elements of stability in biological systems. On the contrary, in order to show experimentally and describe the- oretically a change of symmetry, the preceding and follow- ing situations have to be stable enough to be described. In 6= other words, a set of symmetries has to be at least approx- imately valid long enough before it changes for an observer to discuss it and after the change the new set has to be met for some time too. For example a given geometry Figure 3: Example of a symmetry breaking. The left pictures corre- of bones is conserved during movements of the organism spond to an initial situation and the right ones to the same situations at short time-scales, which corresponds to the conserved after a rotation (represented by the arrows). The above diagrams show a disordered situation such as a gas or a liquid. This situa- symmetry of a solid (the relative positions of points in a tion is statistically symmetric by rotation, there are no privileged solid do not change). However, this geometry is plastic directions. By contrast, the situation below corresponds to a crys- at longer time scales and very important changes can oc- tal such as graphite. It is not symmetric by rotation (except with cur especially during development (West-Eberhard, 2003). an angle of 180◦ ) and it thus has directions which have an intrinsic physical meaning. The transition from the situation above to the The change of two bones geometry at different times thus situation below implies the introduction of new relevant elements: corresponds to a symmetry change, but the symmetries of the directions of the crystals, which are random. these bones are met at short time-scales. We call constraints the relevant stable elements at work in biological systems and their associated symmetries. Con- A part of the relevant symmetry and symmetry changes straints are local stable elements, in the sense that they can only be listed a posteriori, that is, after their real- only concern a particular aspect of a given organism. In ization. These changes only make sense as a result of a addition, constraints are contingent insofar as they, and previous history. Not only lineages, but also individual their associated symmetries, may change over biological organisms, are subject to biological randomness, as their time (which is implied by the principle of variation). development can sometimes take new routes which were In short, we define constraints as symmetries (i.e. sta- not expected in advance (e.g. West-Eberhard, 2003). ble mathematical structure) witch have a restricted range Note that we consider symmetry changes in general of validity and are used to describe a part of a specific and not just symmetry breaking. Symmetry breaking cor- object. responds to symmetry changes which start from a situation that respects a given symmetry to a situation where this 3.3. Constraints and randomness symmetry is no longer valid, as discussed above. Other symmetry changes are possible, for example one can go In this section, we discuss the articulation between two from an asymmetric situation to a symmetric one. In bi- kinds of randomness in specific objects. This discussion is ology, symmetry changes include the appearance of new more technical and may be skipped in a first reading. and unpredictable symmetries corresponding to new rele- A constraint (or a combination of constraints) exerted vant parts and their functioning. For example, the appear- on biological dynamics may lead to a situation in which ance of sexual reproduction in evolution corresponds to a symmetry changes (if any) occur in a generic manner, separation of individuals in two genders in many species, typically as symmetry breaking. In the case of generic where new symmetries (or equivalence) between males on symmetry changes, these ‘random’ changes can be stated one side and females on the other become fundamental in advance, even though their specific outcome cannot. as for their role in reproduction. New associated variables This randomness can be derived from constraints, and it become relevant, for example the sex ratio of a population. is weaker than the randomness proper to specific objects. Because of symmetry changes, the phase spaces of bi- Let us start with morphogenesis as an example. Most ological objects also change in unpredictable ways over (if not all) mathematical models of morphogenesis involve time. Symmetry and phase space changes constitute a a symmetry change, which usually is a symmetry break- specific form of randomness, proper to biological systems ing. Consider for instance Turing’s model of morphogene- (Longo & Mont´evil, 2012; Longo et al., 2012a; Kauffman, sis (Turing, 1952)16 . In this model, the equations describ- 2013; Longo & Mont´evil, 2013). Biological randomness ing reactions and diffusion of chemicals remain invariant, typically manifests itself through the appearance of new so that their properties (rate of reactions, coefficient of relevant quantities, parts, functions, and behaviors over diffusion, etc.) are stable constraints. In turn, these con- time (for example limbs, toes, toenails, all the quantities straints lead together to a symmetry breaking, because of required to describe them and the various functions that they can have). 16 Turing’s model is based on a basic symmetry breaking, where a situation that is initially symmetric by rotation forms a pattern of alternation of concentrations of chemicals (and new quantities are needed to describe where this pattern is located). 7 the sensitivity of the non-linear dynamics to initial con- given space of possibilities which may be given a priori ditions (an instability, says Turing): minor fluctuations probabilities. Biological objects are — by hypothesis — trigger different outcomes. specific, but when we describe a particular change of sym- Another very different example of biological symmetry metry, it is studied a posteriori as a generic aspect of the breaking is the dna recombinations in the maturation of object, and can be added to the past possibilities of a sys- lymphocytes (Thomas-Vaslin et al., 2013). The random tem. Randomness is then not correctly framed by a priori process of recombination in a cell can be seen as a symme- probabilities. Probabilities, if any, are defined a posteri- try breaking from a situation where all the recombinations ori. A specific possibility is accommodated by the space to come are equivalently possible to a situation where only of possibilities, but this space is obtained a posteriori and one recombination is actually realized in each cell. After obviously does not include all future possibilities. recombinations, the description of the system has to in- Let us unpack this idea. A physical symmetry breaking clude which possibility each cell has “chosen”. This sym- is a simple elementary process: a symmetry is met by the metry breaking makes the diversification of the immune system, and after the symmetry breaking event, the sym- repertoire possible under the constraint of enzymes. metry is no longer met. The possible breakings are given Both cases (morphogenesis and dna recombinations) by the initial set of symmetries and make mathematical involve stable constraints, in an extremely sensitive pro- sense when they can be described in a given mathematical cess, which leads to a change of symmetry. These con- space where the symmetry operates. However, if a situ- straints are stable parts of the organization of the consid- ation is and always has been completely symmetric, the ered organisms. As a result, the associated changes are symmetries do not change anything and thus, cannot be robust in the sense that they will occur as a consequence properly evidenced as transformations (because the object of these constraints. In such situations, a generic change of is not changed at all). Thus the logic required to describe symmetry is established, which generates “new” relevant a new symmetry breaking has two steps. First the sym- quantities but in a generic manner, i.e. the change belongs metry that will be broken has to be added to the initial to a set of predefined possibilities. These new quantities definition of the system and accordingly the states that are are new in a weaker sense than the unpredictable new di- initially symmetric have to be added to the phase space mensions of description that specific objects can generate. of the object. They are added because they are required For example, the recombinations in the immune system to accommodate their future breaking. Then, and only can be seen as generic, as a set of possible physico-chemical then, may the symmetry be broken. Such a modeling is recombinations of molecules. The outcome of such recom- retrodictive: the mathematical space, needed for an equa- binations is probably unique because the odds of perform- tional model, can be given only after the change has been ing the same recombinations twice are vanishingly small, observed. In general, then, a biological dynamic must be but this outcome is still generic. The situation is anal- understood as a possible path, out of many established ogous to the physical case of the positions of individual along the biological dynamics, which consists in the com- molecules in a gas which are basically unique, whereas the position of stepwise symmetry changes. gas is still in a generic configuration because the gas is in In a given situation, some symmetry changes can be a configuration of maximum entropy. However the actual spelled out and analyzed in a generic framework because immune repertoire in an adult mammal is not fully deter- they are stabilized by (local) constraints. Let us consider mined by the generic properties of recombinations because such an elementary biological symmetry change, for exam- the recombinations are just a part of the process estab- ple in a morphogenesis model. We can describe it explicitly lishing this repertoire. The immune repertoire strongly with generic constraints but it is also possible to leave it depends on the specific history of the given organism, its implicit and consider that this single symmetry change is environment, non-genetic inheritance (through milk and taken into account by the specificity of the object, among the microbiome), etc. (Thomas-Vaslin et al., 2013). The many other changes. The choice depends on the perspec- immune repertoire has a causal structure that is not deter- tive adopted to understand a given situation, including the mined by pre-existing regularities. The dependency on the scale of description and the phenomena of interest. For ex- organism’s history is functional, it determines the immune ample, the intestine folding are usually kept implicit when response to specific pathogens and contributes to the dy- studying brain morphogenesis. namic relationship with the microbiome. The biologically Even though the boundaries of specific and generic as- relevant properties of the immune repertoire are not the pects of an organism are relative and may change after a generic properties of recombinations, instead they are the new possibility is acknowledged or as a result of a change specific properties which stem from a history. Hence, the of perspective, the accurate description of any biological actual repertoire of the adult contains more meaningful organism will always involve a component of specificity. In novel structures than the initial probabilistic recombina- a given representation of an organism, all changes of sym- tions. metry are then either accommodated by the specificity of Now, every time we describe a symmetry change ac- the object or by generic symmetry changes. The concept cording to current physico-mathematical methodology, it of the specificity of biological objects aims to enable us takes a generic form, that is, a possible change in a pre- to take into account theoretically all symmetry changes 8 without spelling out all of them explicitly. their time is that of a process. Their historicity is embed- ded within a pre-defined phase space. 3.4. Historicity The fact that we can understand such spontaneous ob- Historical objects are objects whose properties are ac- jects on the basis of a stable generic mathematical struc- quired or lost over time, and cannot all be described ahead ture is not fortuitous. Indeed, their spontaneous charac- of time. The fact that biological organisms are specific ter corresponds to the fact that these objects can emerge objects straightforwardly implies that they are historical from homogeneous initial conditions in the mathematical objects and, in particular, contingent objects in Gould’s framework used to describe them. By contrast, specific sense (Beatty, 1995; Gould, 1989). Historicity thus goes objects are not framed by stable mathematical structures: hand in hand with biological randomness, which corre- they cannot be derived from homogeneous initial condi- sponds to the fact that a situation after a random event tions and cannot be obtained spontaneously in practice. cannot be stated with certainty before the event. Thus, Even in the “origin of life” field, the aim is to produce a system showing biological randomness shows historicity: a cell which can evolve and not a cell that is similar to the object takes a particular path among several possible all current cells as they have evolved for billions of years. paths through time. Reciprocally, historical objects nec- Moreover the aim is certainly not to obtain a cell similar essarily show some randomness. to any specific species (Pross & Pascal, 2013). Let us first consider an analogy with dynamic systems. According to the principle of variation, biological ob- We can see a trajectory defined by a differential equation jects are the result of a cascade of unpredictable symmetry as the sum of infinitesimal changes from the initial condi- changes, which implies that they do not follow optimiza- tions to any time point. By analogy, it is conceivable to see tion principles and that they are not spontaneous. To be biological historicity as a sum or a sequence of variations sure, biological objects did appear spontaneously in the since the origin of life. However, this idea does not have history of life, but should one re-run the history of the a well-defined mathematical and theoretical sense, insofar Earth, one could not expect to obtain the same biological as such a history is not entirely accessible. Nevertheless, objects. It is not even possible to state in advance the it is still possible to clarify the present in the light of the mathematical space of possible forms that could be ob- past — and, as a matter of fact, this is precisley one of the tained. The historicity of biological objects is not embed- aims of evolutionary theories. ded within the phase space anymore (as it was in physics): As discussed in Longo et al. (2015), although histor- rather, the principle of variation means that the phase ical objects exist also in physics, they are historical in a space itself is historical (figure 4). weaker sense. Self-organized physical objects, for instance, At first sight, though, the claim that the phase spaces are sometimes described as historical, mostly because they in biology are historical seems too strong: aren’t there depend on a symmetry breaking. For example, the appear- some aspects of biological objects which are ahistorical? ance of convection cells in a fluid corresponds to a qualita- Evolutionary convergences, for instance, seem to be an ex- tive change in the macroscopic dynamics of the fluid. Nev- ample of an ahistorical aspect of the living: convergent ertheless, self-organized objects are spontaneous: they can features seem to be obtained independently of (some as- be obtained de novo. Theoretically, they can be described pects of) the past history of the organism. Let us first as the spontaneous self-organization of flows of energy and point out that evolutionary convergences are not about matter. Even the physical situation of the early history of invariant properties of a given object over time, they are the universe can be obtained experimentally “just” by tun- about mathematical structures that are similar in differ- ing a parameter (by obtaining very high local densities of ent historical paths. Let us consider the case of the cam- energy with particles accelerators)17 . era eye of the vertebrates and of the cephalopods as an Despite these analogies though, physical self-organizing example. These eyes have different evolutionary origins processes have no historical or evolutionary time in a strong but they are nevertheless similar and one could argue that theoretical sense; they may just have the time of a process. they would be instances of the same generic object from a They entirely obey optimality principles from physics and physicomathematical viewpoint, when described in terms past events have not shaped their properties, insofar as the of optical geometry for example. symmetry breakings that self-organizing processes may en- The principle of variation, however, implies that the counter are all pre-defined within the theory. A hurricane convergence is very unlikely to be qualitatively exact. There does have, so to speak, a “birth”, a “life”, and it does even- would always be a relevant biological description which tually “die out”; yet, hurricanes have been the “same” kind would distinguish them sharply by pointing to differences of object for the past four billion years on Earth. Again, in their organization and in their articulation with the rest of the organism. For instance, the retina is inverted in ver- tebrates: the axons of photoreceptors and their connection 17 Incidentally, the idea of spontaneous generation in biology to ganglion cells and the optic nerve are located between stemmed from the same kind of reasoning: (generic) biological ob- the receptors and the light source, creating a blind spot at jects would appear spontaneously by self-organization in the appro- priate milieu (De Klerk, 1979). the level of the optic nerve. In cephalopods, axons are be- hind the photoreceptor which does not create such a blind 9 spot. A close analysis of both the phylogenetic and the on- Let us discuss two examples of internalization of the togenetic paths makes the difference understandable: the context on the developmental and on the evolutionary time high modularity of the cephalopods’ brain derives from an scales, to show how it can lead to unexpected behaviors of early separation of the brain’s modules by an invagination biological objects. of the ectoderm, in contrast to the evagination of the dien- On the developmental time-scale, an example of inter- cephalon, due to the late separation of the eye component nalization of past contexts is provided by the response of of vertebrates’ brains. cells to hormones (Soto & Sonnenschein, 2005). Basically, In short, the principle of variation implies that strict the response of a cell to hormones does not depend only evolutionary (or developmental) convergence never occurs: on the specific receptor and corresponding hormone in- symmetry changes are such that biological objects drift in volved but, rather, on the developmental history of this a burgeoning phase space, and partial convergences always cell. More precisely, precursor erythroid cells are expected embed hidden differences which may be of importance with to differentiate into red blood cells when their erythropoi- regard to the considered behavior of the biological object etin receptors bind with erythropoietin. However, precur- in that phase space. Reciprocally, the similarity between sor erythroid cells which have been engineered to lack ery- the organizations of different organisms stems from com- thropoietin receptors and instead have receptors for pro- mon descent, that is to say from a shared history. lactin do differentiate into red blood cells when they are exposed to prolactin, a hormone associated with lactation 3.5. Contextuality (Socolovsky et al., 1997). Conversely, mammary epithelial Organisms are contextual objects. In our theoretical cells can be engineered to have a hybrid receptor with an framework, the symmetries of organisms depend on its en- extracellular part of a prolactin receptor and an intracel- vironment — both on its immediate environment and the lular part of an erythropoietin receptor. These engineered environments encountered in its past history. cells respond like normal mammary epithelial cells to pro- The fact that the symmetries of an organism depend on lactin (Brisken et al., 2002). These examples show that it its immediate environment constitutes another similarity is not the molecular specificity of a signal binding to a re- with self-organizing physical systems mentioned above, as ceptor that determines the response of a cell to a hormone. the latter strongly depend on their boundary conditions. In contact with a hormone for which it has a receptor, a However, the principle of variation makes the contextuality cell rather responds according to the context of its cellular of biological objects more fundamental than that of phys- lineage during development, that is its trajectory in time ical systems. Contrary to physics, the possible changes of and space (Soto & Sonnenschein, 2005). symmetry due to a change of the context are not all pre- On the evolutionary time-scale, a component of an or- defined. This means that an organism in a new environ- ganism, as a result of a history, may be used for differ- ment may undergo unpredictable reorganizations, which ent purposes in different contexts. The phenomenon of a correspond to different relations between its internal con- character (be it the result of past natural selection or not) straints and the environment, as well as different relations which is coopted for a current use has been named ‘ex- between its internal constraints, tout court. For example, aptation’ by Gould & Vrba (1982). They provide many we do not know a priori the many changes that can occur key examples, for instance: “the jaw arises from the first when bacteria that used to live with many other species gill arch, while an element of the second arch becomes, in their natural and historical environment are grown as in jawed fishes, the hyomandibula (suspending the upper an isolated strain in laboratory conditions. Similarly, it is jaw to the braincase) and later, in tetrapods, the stapes, always difficult to assess whether the behavior of cells cul- or hearing bone” (Gould, 2002, p.1108). An ex-aptation tured in vitro is an artifact of in vitro culture, or whether is a re-interpretation, or re-use, of a trace of the past in it is biologically relevant (meaning that it corresponds to a new context and, therefore, cannot be derived from the a behavior that happens in the context of the multicellular initial function of the parts involved. As a consequence, organism from which they were taken, see Mont´evil et al., the detailed structure of the internal ear can be better 2016). understood by looking into the cumulative history of ex- The contextuality of biological objects is coupled with aptations. their historicity (Miquel & Hwang, 2016): biological or- In light of the principle of variation, the internalization ganizations tend to maintain the effects of former envi- of current and past contexts provides one way (although ronments and may even internalize their relationship with not the only one) in which symmetry changes can occur the environment over time. This holds at the develop- throughout the history of an organism. As an illustration, mental scale (think of how early plastic responses to the the internalization of the context contributes to explaining environment might be ’frozen’ later in development, see the difficulty of replicating biological experiments, insofar also Gilbert & Epel (2009)), at the scale of several gen- as aspects of an experimental situation which can be rel- erations (for example through epigenetics), and at longer evant to the studied behavior may not be measured and evolutionary scales (think, for instance, of the presence of can be traces of an (unknown) past (Begley & Ellis, 2012). lungs and lack of gills in marine mammals, which reflects a past terrestrial life). 10  Symmetries change Symmetries Besides the flow of time, the second set of transforma- (Constraints) (Constraints) tions relevant to variability are the permutations of dif- are a part of are a part of ferent organisms or different populations. Permutations establishes establishes correspond to the interchanging of different objects. They are fundamental symmetries in many physical frames: for determines time example, it is axiomatic that all electrons follow the same History Specific object Specific object randomness equations (but they can be in many different states). In s s ne ne experimental biology, permutations of different animals or mi i rm ter cells are often assumed to be symmetries: when one con- te de de siders different animals of a control group, a common as- Context Context sumption is that they behave in the same way and that the quantitative variation observed stems from a proba- Figure 4: Biological objects and their theoretical structure. Spe- bility distribution that would apply to all of them. This cific objects are not defined by invariants and invariant preserving assumption, in one form or another, is required to apply transformations. Instead, specific objects such as organisms undergo random variation over biological time. Their behaviors are not given theorems of statistical analysis. by a synchronic description. Instead, they depend on a history and a According to the principle of variation, however, the context. Constraints are restricted invariants and symmetries, which permutation between these organisms cannot be taken as may change over time and frame a part of the behavior of specific a symmetry. Of course, organisms are related by a shared objects. Experiments and mathematical models usually investigate constraints and their changes. history, which enables us to determine that they are mice, rats, etc., of a given strain. Yet, the transformation which replaces one organism by another in the same group cor- 3.6. Variability responds to a comparison between the results of divergent The principle of variation underlies biological variabil- paths stemming from a shared history. Here, divergence is ity: the fact that multiple organisms or the same organism taken in a strong sense and implies symmetry changes and or lineage at different times exhibit differences when com- not mere quantitative changes conserving the same sym- pared to each other. metries. For example, qualitative behaviors differ between The flow of time is the most fundamental transforma- different strains of the same species, even in unicellular tion acting on biological objects: as we argued, biological oganisms (Vogel et al., 2015). Now, we illustrate this idea symmetries and accordingly biological organizations are with a historical example. not preserved as time passes. At the end of the 19th century, Sir Francis Galton, one Variability tends to be stronger when considering large of the founders of the notion of heredity, came up with a evolutionary time scales than for shorter time scales. When device, known as the bean box or the quincunx (see figure one follows the succeeding generations from the LUCA to 5). The quincunx facilitated the simulation of a binomial a randomly chosen current organism, for example a rat, distribution (the device would be used to simulate “normal many relevant aspects of the description needed to under- variability”, Galton (1894, pp.63f)). The device consisted stand these organisms appear and disappear through time. in a vertical frame with three parts: a funnel in its upper Variability is also significant at physiological time scales, part, rows of horizontal pins stuck squarely in its middle even at those that are much shorter than the lifespan of part, and a series of vertical compartments in its lower the considered organism. Heart rate, for example, does not part. A charge of small items (say, beans or balls) would obey homeostasis stricto sensu: the beat to beat interval is be thrown through the funnel, travel through the pins, not invariant (in a healthy situation), and it does not even possibly bouncing in any direction, and would be gathered display fluctuations around a stable average value. In- by the vertical compartments at the bottom (where they stead, the beat to beat interval fluctuates in a multiscale would not move anymore). In the end, the distribution of manner (West, 2006; Longo & Mont´evil, 2014b). Typi- the items in the bottom compartment would approximate cally, the heart rate of a healthy subject displays patterns a binomial distribution. of accelerations and deceleration at all time scales during In our terms, the bean box works the following way. wake hours. Note, however, that the typical symmetries The items share a common history when they get into the of multiscale fluctuations (scale symmetries) are not met funnel, and this common history leaves a trace in the re- either. Rather, many factors impact the multiscale feature sult: depending on where the funnel is placed into the of these variations of rhythms. For example, the current device (e.g. in the middle or not), the distribution of the activity of the subject, her age, her life habits (smoking, items in the end varies. When the items exit the funnel, exercising, etc.) and diseases change these multiscale fea- they take divergent paths (by bouncing on the pins) until tures (Longo & Mont´evil, 2014b). These differences in the they reach a vertical compartment. This is, however, di- patterns of the variability of the beat to beat interval can vergence in a weak sense. For the bean box to work, all even be used for diagnostic purposes (West, 2006; Bailly the items have to be supposed to be symmetrical, and all et al., 2011). the realized paths have to be supposedly taken from the same underlying distribution. As a matter of fact, this 11  3.7. Modelization and specific objects Current mathematical modeling practices in biology borrow mostly the epistemology of physics and are based on generic objects following specific trajectories. So far, we have argued that the theorizing of physical phenomena is based on stable mathematical structures and on the corre- sponding analysis of generic objects. We advocate, by con- trast, that biological organisms are specific objects moving along possible phylo-ontogenetic trajectories. Organisms have a historical and contextual nature and change their organization and functions over time. This physicomathematical modeling practice in biology leads to many technical and epistemological problems. For example Boolean networks (see Kauffman, 1993) are used to model gene networks and are defined as random net- works where the existence of an edge between two nodes follows a given probability distribution. Such an assump- tion is a way to model protein or gene networks in an ahis- Figure 5: Galton’s quincux (Galton, 1894, pp.63). A ball falls but torical manner (and for example to generate them de novo obstacles lead it to move randomly to the right or to the left. The in simulations). This disregards the fact that the actual outcome is variability in the position of the balls at the bottom of phenomena are the result of evolution, and thus that ac- the device. This device illustrates variation in a pre-defined set of possibilities. Biological variation, by contrast, sometimes involves tual biological networks depend on the historical interplay the constitution of new possibilities, which would amount for the between living beings and their environment, even at the ball to jump outside of the quincunx. molecular level (Yamada & Bork, 2009). Hence, they are not a sample of a random network following a given proba- bility distribution. This is also true for cell networks: in a assumption is necessary for the use of statistics in biol- tissue, cell to cell interactions or the production of proteins ogy: when performing an experiment on — say — rats, are largely a context- and history-dependent phenomenon. one supposes that all rats are independent realizations of For instance, the “normalization” of a cell transferred from a random variable taken from the same underlying distri- a cancer tissue to a healthy one can be understood as the bution and that this distribution is stemming from their effect of tissue context (and its history) controlling indi- most recent common ancestor. The most recent common vidual cellular activities (Soto et al., 2008; Sonnenschein ancestor plays the role of the funnel; and subsequent muta- & Soto, 2016). These examples show that the standard tions, effects of the environment, spontaneous variations, modeling strategies of a biological system struggle against etc., play the role of the pins. Variation can occur, but it the historicity and contextuality of biological organisms. will be merely quantitative and measured by the position We interpret the “big data” approach, that aims at on the horizontal axis in the bean box. taking into account a massive amount of data in a model, By contrast, the principle of variation posits that unex- as an attempt to address the consequences of the histori- pected (and unknown) qualitative variation permanently cal nature of biological objects while keeping the physical occurs. This means that different organisms are not differ- methodology of establishing intelligibility on the basis of ent realizations of a random variable taken from an under- generic objects. Such an approach, however, raises the lying single distribution, as this distribution cannot even question of the intelligibility of their object, because the be defined. In terms of the bean box, this means that the complicated mathematical structures of models based on pins unexpectedly open new dimensions (i.e. new relevant big data make only computer simulations possible. Other features arise), which would not be defined before the re- more physical approaches focus on generic features that alization, and would not be reproducible after either. This even these historical systems would meet. For example, is what we mean by divergence in a strong sense. Galton scaling laws in networks have been extensively investi- used his device to illustrate normal variability where vari- gated, but their validity is criticized (Fox Keller, 2005). ability would be quantitative, in a pre-defined space. By Globally speaking, however, the methodological emphasis contrast, the principle of variation implies that variation on generic features implies that the biological meaning of can be qualitative (i.e. symmetry changes) and that the specific variations, and their role in a given organism, is space of variation is not pre-defined. This, to reiterate, lost. The issue is that without stable generic features, the applies both at ontogenetic and phylogenetic scales18 . this experimental methodology, which aims at selecting biological 18 Inexperimental biology, organisms are often kept as historically objects in such a way as to reduce variability “symmetrization”. A close as possible, they may be siblings for example. The aim is more detailed analysis of biological experiments will be the object of then to keep the divergence in their organization limited. We call a specific paper. 12 question of the objectivity of these models is open, insofar 4. Bringing organization into the picture as their description and behavior will have a high degree of arbitrariness: the models will miss the consequences of Let us begin with a methodological remark on the ar- the principle of variation and, thereby, display invariants ticulation of the principles of variation and organization. which are not valid. The theoretical definition of a biological organization at Most mathematical models do not aim at capturing a given time is closely related to how it may change, and features of whole organisms but, rather, at singling out that for two related reasons. First, the organization of ev- some constraints, corresponding to specific parts of an or- ery current organism is the result of a cascade of changes ganism. Typically, they focus on the morphogenesis of an over ontogenetic and evolutionary time scales. Second, the organ or a tissue, for example the formation of leaf dis- appropriate theoretical definitions and representations of position (phylotaxis) (Douady & Couder, 1996), the or- scientific objects are, generally speaking, those that enable ganization of the cytoskeleton (Karsenti, 2008), the mor- us to understand the changes of these objects. For exam- phogenesis of vascular networks (Lorthois & Cassot, 2010), ple, positions, momenta, and the mass are both necessary etc. Even though this approach has obvious merits and has and sufficient to understand the changes of position of the provided remarkable insights, it does not take into account planets of the solar system in classical mechanics. This jus- that these organs or tissues are parts of the whole organ- tifies the theoretical representation of the planets on the ism and that the possible reorganizations of these parts are basis of these quantities. In this respect, an appropriate essential to variability, development, and evolution. From framework for organisms requires the articulation of orga- a mathematical viewpoint, one aspect of this weakness can nization with the changes that it may undergo. To some be expressed as the fact that models miss some degrees of extent, this question has been neglected in the past inso- freedom corresponding to the changes of phase space that far as biological organization has been mostly approached follow from changes of symmetry, in accordance with the as a mathematical fixed point, which leads to the concept principle of variation. of organizations as maintaining themselves identically over Although mathematical models are more and more used time. in biology, we think that the key challenges raised by bio- logical organisms, in particular variability, historicity and 4.1. Organization grounds constraints in specific objects contextuality, have not yet found a proper methodological Even though organisms should be understood as spe- and epistemological treatment. We hope that the princi- cific objects, as the principle of variation posits, we would ples discussed in this issue will contribute to better identify argue that some of their parts exhibit generic features in and address these challenges. a restricted sense. As mentioned in section 3.2 above, we refer to these parts as constraints. More precisely, con- 3.8. Conclusive remarks on the principle of variation straints are characterized as entities that control biologi- The principle of variation leads to a change of perspec- cal dynamics (processes, reactions, etc.) because of some tive with respect to physics. Historicity, contextuality, and symmetrical (conserved) aspect which they possess at the variability are fundamental every time an organism is un- relevant time scales. For example enzymes are not con- der scientific scrutiny. Rather than trying to avoid the sumed in a chemical reaction that they nevertheless change intrinsic difficulties in mathematizing these features, our completely. Similarly, the geometry of the vascular system theoretical frame aims at building on them. To be sure, the is conserved at the time scale of blood transport, and this randomness of symmetry changes limits the actual knowl- transport is constrained by the vascular system. edge we can obtain on a given organism. At the same time, The stability of constraints, however, has to be ex- however, this new kind of randomness can be studied as plained by a sound theory of biological organisms, espe- such, and opens up new avenues of investigation. cially in the long run. Indeed, beyond the time scale at Last, underpinning our discussion above is the fact that which a constraint operates, constraints undergo degrada- the principle of variation involves two kinds of changes: tion. A constraint may be further stabilized by a process changes of the biological object itself (philosophers would being under the control of another constraint, which is it- say this is an ontological change) and changes in the ques- self stabilized by another constrained process, and so on: tion asked about this object (philosophers would say this if the chain of dependencies folds up and the constraints is an epistemological change). For example, developmental can be said to be mutually dependent, the system of con- biology studies features that appeared with multi-cellularity: straints is organized. The constraints that constitute an the field is thus a result of biological variation. Recipro- organism are the organized ones, which (i) act on a process cally, growing cells in lab conditions comes with modifica- stabilizing a constraint of the organism and (ii) depend on tions of their behaviors which in turns affects the questions at least another constraint of the organism. The key as- at stake and possibly their future culture conditions. Thus, pect in this framework is that constraints are stable at a in our view, the instability of biological objects goes hand given time-scale, while being stabilized by processes tak- in hand with the instability of biological questions: they ing place at other time scales, so that constraints behave co-constitute each other. as local invariants with respect to the processes they con- strain. 13  However, while the time scales of constraints in the In a theoretical sense, the generality of biological anal- principle of organization are the intrinsic time scales of ysis is made possible by the permanent relevance of orga- the processes and constraints under study, they do not nization as closure, despite the continual symmetry and preclude a change at these or other time scales for reasons phase space changes. To a certain extent, the situation extrinsic to these objects. Changes of organization stem- of closure is similar to that of the energy of a physical ming from the principle of variation can alter a constraint system being conserved despite its permanent changes of at any time scale. In this event, the former constraint may state. In the case of a change of constraints, an organized lose its status of constraint or may operate differently. object goes from one closed regime to another, unless the The cohesion of organisms is one of their fundamental organism does not succeed in establishing a new regime features, and this cohesion has been the object of many and dies. theoretical investigations, for example as autopoiesis (Varela et al., 1974) or as work-constraints cycles (Kauffman, 2002). 4.3. Relevant variation with respect to closure Following this line, we argue that biological organisms real- The principle of organization understood as the closure ize closure of constraints (Mont´evil & Mossio, 2015; Mossio of constraints leads to the idea that the relevant changes et al., 2016): functional parts of organisms act as con- of the organism involve constraints subject to closure. The straints on each other, and they realize a mutual depen- changes of constraints that do not impact the constraints dence. Closure is basically the circularity in the relation subject to closure fall in two categories: those that af- of dependence between constraints. The principle of or- fect the environment and those that affect the organism ganization that we propose states that the constraints of (in other aspects than the constraints subject to closure). organisms realize closure. If a change of constraint affects the environment, it may We postulate that the stability of functional constraints be biologically relevant, for example if it affects other or- hinges on their mutual dependence (Mont´evil & Mossio, ganisms. If the change affect the organism, but not its 2015; Mossio et al., 2016), so that the overall stability organized constraints, then it is not significant for the or- of biological organisms is justified by the closure of con- ganism in the light of the principle of organization: these straints. When we consider the principle of organization constraints do not play a role in the biological system (al- and the principle of variation together, constraints are con- though they may be involved in an unpredictable organi- tingent in two complementary ways. They are contingent zational change). because of their historical nature and because their exis- As for the changes that may affect the organization, a tence depends on the circularity of closure instead of being general distinction can be made between irrelevant and grounded on other stable first principles. functional variations. On the one hand, processes and constraints may undergo irrelevant variations, for exam- 4.2. The epistemological status of closure under variation ple small quantitative variations, i.e. quantitative fluctua- By relying on the principle of organization, it is the- tions that neither undermine nor modify the overall orga- oretically meaningful to work on sets of constraints that nization. This is variation that, in a word, the organism verify closure. Following the principle of variation, how- does not need to control in order to ensure its stability, ever, constraints are not necessarily conserved over time by hypothesis. On the other hand, variation can be func- and may undergo changes which cannot be stated in ad- tional, in the sense of resulting in a change of one of more vance. As a result, the validity of closure must extend constraints, of their relations, and hence of the very organi- beyond a given configuration of constraints. The validity zation. Examples of quantitative variation are for example of the principle of organization should not be understood moderate differences in the weight of some organs like the as based on a given set of constraints (or invariants) which liver, or in enzyme concentrations; examples of functional would happen to realize closure. Accordingly, the princi- variations are the reshaping of bones and musculature to ple of organization is not deducible from a set of invariants perform a new function or to perform differently an old and symmetries (as in “physical laws”), rather, it is the function (West-Eberhard, 2003). Of course, the quantita- condition of possibility for the existence and persistence tive variation of a given constraint can also be potentially of biological constraints (i.e. local invariants and symme- functional, in the sense of enabling the possible further tries). For this reason, we suggest that closure constitutes emergence of functional variation, including pathological the principle of organization that, alongside variation and ones. other principles, frames the biological domain as a whole Another example of functional variation is random (see Mont´evil & Mossio, 2015; Mossio et al., 2016). gene expression, which has been studied both in unicel- In epistemological terms, stating that the principle of lular (Eldar & Elowitz, 2010) and multicellular organisms organization is a fundamental principle implies that it can- (Dueck et al., 2016). In this literature however, functional not be deduced from underlying stable symmetries and be- variation is mostly understood in an evolutionary sense, comes an (irreducible) theoretical principle for biological while closure provides a systemic interpretation of func- organisms. Closure becomes an a priori that replaces the tions (Mossio et al., 2009). As a result, closure enables a priori of space and time in physics, or, more precisely, us to conceive functional variation that is not necessar- of the phase spaces of physical theories. 14 ily inherited, provided that the constraint resulting from cells by albumin: after a cell division (which is a variation is still subject to closure. symmetry change19 ), the same albumin maintains its inhibitory effect on the new cells. 4.4. Closure remains closure under variation • Another situation corresponds to a change involv- As discussed in Mossio et al. (2016), closure contributes ing a modification of the relationship between pre- to making both internal and external variations possible. existing constraints as they come together to gener- The circularity of closure weakens the coupling between ate a new biologically structure or dynamics. Such what is going on inside a system and its boundary condi- a change is fundamentally non-local with respect to tions (Barandiaran & Moreno, 2008). Such a decoupling the graph of interacting constraints. In this kind of enables variation beyond what would be permissible if the situation, some constraints act on processes which system were completely determined by its boundary condi- they did not constrain before the change. This cor- tions (such as in physical self-organization). Reciprocally, responds typically to the notion of exaptation. In an organism can stand a relatively unstable ecosystem be- general, such a change implies the alteration or the cause of its autonomous stability due to the closure of con- appearance of specific constraints that establish the straints. new behavior: the important difference with respect Under our principles, functional variation cannot lead to the case described in the previous paragraph is to a violation of the organization principle — except in the that various other constraints are also mandatory to case of death. This means that any change affecting the enable the emergence of the new behavior. constitutive constraints are changes from one organized situation to another. In our frame, closure is always met, • Finally, a change in organization might result from even though the constraints relevant to closure may and the appearance of new constraints. In order for a new do change. The continuous alteration, loss or acquisition constraint to be included in the closed system, the of functions result in the realization of new organizational organization must be reshaped so that the new con- regimes; each regime, in turn, achieves a form of stability straint be integrated to the organization (Mont´evil determined by closure as a mutual stabilization of con- & Mossio, 2015; Mossio et al., 2016). There are two straints. Being subject to both the organization and vari- aspects to this: the new constraint must be main- ation principles, biological organisms realize a succession tained by other constraints (I) and maintain another of different instances of organized regimes over individual constraint (II). Whether (I) or (II) occurs first cor- and evolutionary times. Then, the stability achieved by responds to different scenarios. It is fairly easy to the organism is not conservative, but it is for a part cumu- picture a constraint being maintained (criterion I) lative, insofar as it sustains many functional innovations, starting to play a role in an organization after some and makes their preservation over time possible. time (criterion II). For instance, in mammalian de- Changes of the organization may correspond to several velopment, lungs are first formed and maintained (I) situations depending on the constraints involved. They and they acquire a functional role only after birth may be more or less local with respect to the rest of the (II). However, the opposite may also happen, for ex- organization. We propose a typology on this basis: ample, thanks to generic physiological responses dis- cussed above: a change of behavior leading to me- • A first situation consists of a local change of a con- chanical friction (II) leads to the strengthening of the straint, such that it does not induce a change in skin by keratinization (I). Lastly, the two aspects can the relationship between constraints. For example, a be coupled. For instance, some structures (such as supplementary branching event in a network or tree muscles, bones, etc.) which are not used (II) may structure (such as vascular networks or mammary atrophy (I), and reciprocally their use (II) may lead glands) does not correspond to a major reorganiza- to their further development and strengthening (I). tion of the constraints of a system. Let us remark, still, that this situation corresponds to a basic sym- The key issue about changes of organization is the in- metry breaking involving the appearance of new rele- scription of the change in a new organization. After a vant quantities of preexisting kinds (for example the change of constraints takes place at relatively short time angle between the new branches). Therefore, such a scales, the altered constraints involved may be stabilized change is generic (a branching among many possible by other constraints, at longer time scales. These sta- branchings). In section 3.1 and 3.3 above, we dis- bilizing constraints are then typically solicited differently cussed such examples of generic symmetry changes than before the change: they maintain, for example, the in the context of specific objects. In turn, the new constraints can be stabilized by generic constraints 19 Cell division corresponds to the disappearance of an object and (insofar as the new branch is stabilized in the same manner than the preexisting ones). In the context of the appearance of two new non-identical objects, see Sonnenschein & Soto (1999; Longo et al. (2015; Soto et al. (2016; Mont´ evil et al. closure, a simple example of a generic stabilization is (2016). the inhibition of the proliferation of estrogen-target 15 same tissues but in a different macroscopic shape or con- a “organismal specification” of the framing principle. The figuration. This can happen through generic physiological latter is an informal overarching principle that can be fur- responses (e.g. keratinization of oral mucosa subjected ther specified by the two principles of organization and to friction, resorption of bones under compressive stress, variation. etc.). These changes do not happen only in the interac- The framing principle applies to morphogenesis under- tion with the environment, they happen in essential devel- stood in a general sense, that is, both to organogenesis opmental, metabolic and regulatory processes (as in the and to proliferation with variation at the cellular level. developmental processes mentioned above). Another ex- In other words, both in organ generation (for example, ample is given in David et al. (2013): the authors show lungs, vascular systems, plants’ organs etc.) and in repro- that “jamming” the regulation of key metabolic genes of duction, a form is iteratively (and hereditarily) produced, yeast cells did not lead to their death but, instead, to new yet never identically. Let us now develop what it means dynamic behaviors which enabled them to thrive after a for biological phenomena to be “non-identical iterations of transition period. morphogenetic processes”. A change of constraint, or the appearance of a new con- By non-identical, we mean (as discussed above) not straint does not necessarily lead to a stabilization of the just quantitative changes but rather unpredictable changes new situation. In particular, organized constraints might of symmetry, thus unpredictable qualitative changes in the tend to restore the initial situation because constraints behavior of the object. In the context of the organism, the subject to closure are maintained by another constrained relevant changes are the ones impacting the organization, process. For example, a mutation in mrna is not going that is to say, the ones changing the constraints subject to to be sustained because the production of new mrna will closure. not carry the same variation. One might refer to such a The iterations are those of organized objects, subject tendency as a form of organizational “inertia”. In such a to closure. However, they should be understood in several case, the new constraint may vanish at a relatively short ways depending on the particular kind of objects they refer time scale. The diametrically opposite situation (among to. others) is also possible. It corresponds to an amplification First, closure is by definition about circular causal ar- of a change affecting a constraint, which in turn destabi- chitectures. For instance, consider a simple closed system, lizes other constraints in the longer run. It is typically the where C1 generates C2 (at time-scale τ1 ), C2 generates C3 case in carcinoma where, as stated by the Tissue Field Or- (at time-scale τ2 ), and C3 generates C1 (at time-scale τ3 , ganization Theory of carcinogenesis, the lack of sufficient say this is the fastest of the three). To discuss iterations, constraints on the epithelium can lead to a progressive let us consider a perturbation on C1 at t0 . This pertur- disorganization of the tissue and, sometimes, disrupt the bation impacts C2 significantly at time t0 + τ2 . Then, C2 organization of the whole organism leading to death (Son- impacts C3 at time t0 + τ2 + τ3 . Finally C3 impacts C1 at nenschein & Soto, 2016). time t0 +τ2 +τ3 +τ1 , and this closes the loop.20 Thus, with Overall, the principle of variation complements the prin- the flow time, the circularities of closure lead to iterations ciple of organization, which should not be conceived as a of closed patterns. More generally, in a loop described by “fixed point” that iterates itself always identically. Rather, closure, the duration of the loop as a whole corresponds organisms change while staying organized. Variation par- to the scale of the slowest process. At this time scale, the ticipates in the robustness of closure in changing envi- iterations are the whole set of constrained processes which ronments. Changes of organization actually enabled the stabilize and maintain the organization of the organism. maintenance of organisms over very long time scales (dur- Following the principle of variation, these iterations are ing evolution). Last, but not least, current organisms are associated with unpredictable changes of symmetry. the product of such variations. Current biological orga- Second, the organizations themselves are iterated. This nizations are determined by their (partially) cumulative adds to the principle of organization the notion of repro- variations, and this process enables organisms to explore duction. By reproduction we mean the process of go- more and more complex organization (Gould, 1997; Bailly ing from one organized object to two (or more) organized & Longo, 2009; Longo & Mont´evil, 2012; Soto et al., 2016). objects21 . Reproduction pertains to the notion that the default state of cells is proliferation (with variation and 5. Non-identical iteration of morphogenetic pro- motility) (Soto et al., 2016; Longo et al., 2015) which com- cesses plements the principles of organization and variation. Reproduction is also essential in that organizations which As a last step, we discuss in this section the connection undergo variations may undergo deleterious variations. As between the organization and variation principles and the “framing principle” proposed in Longo et al. (2015), ac- 20 Note that the iterations of these loops are not just about suc- cording to which biological phenomena should be under- cessive operations. Instead, all constraints are active simultaneously. Incidentally, this is why a perturbation approach is better suited to stood as “non-identical iterations of morphogenetic pro- show the iterative structure underlying closure. cesses”. As mentioned in Mossio et al. (2016), we submit 21 Note that some situations can be fairly complex. Indeed, some that organization and variation, taken together, constitute organizations include constraints which act across generations. 16 a thought experiment, a cell which would never prolifer- without stressing the importance of variation, its perva- ate but would undergo variation should have a finite life siveness and its conceptual consequences. This has led expectancy because at some point a deleterious variation modeling attempts to focus on generic objects, which are, would occur. As a result, varying organizations require we think, unable to adequately represent current biological reproduction to be sustained in an open-ended manner. objects. Reproduction enables a balance between the exploration In order to understand current biological objects, ar- of possibly morbid variations and the maintenance of a ticulating the principle of variation with the principle of strain of organized systems. organization is necessary. In our framework, organization Finally, the framing principle applies also to organ for- grounds the relative stability of a set of constraints by the mation. Iteration is a very common morphogenetic process circularity of closure. It controls and counters (a part of which takes place for example in branching morphogenesis the) variation that would be deleterious and would un- of glands such as the mammary glands, the lungs, etc22 . dermine the very existence of the organism. At the same Iteration processes explain the abundance of fractal-like time, organisms undergo quantitative and functional vari- structures in biology (Longo & Mont´evil, 2014b). Such ations, both of them being crucial requirements for their multi-scale structures play a particular role because they increase in complexity, their adaptability, and, in the end, link different scales, coupling macroscopic and microscopic the sustainability of organization itself as suggested in entities. As such they constitute spatio-temporal coher- Ruiz-Mirazo et al. (2004). One of the central challenges ence structures, which we propose elsewhere to interpret of a full-fledged theory of organisms consists in providing as biological levels of organization (Longo et al., 2012b). a coherent account of how they manage simultaneously to restrict and undergo variation. The epistemological structure of our framework is dis- 6. Conclusions: back to theoretical principles tinct from the one of physical theories. In physical the- Biological variation is relevant at all levels of organiza- ories, assumptions on the validity of stable mathematical tion, and, for example, it is manifested in the default state structures (symmetries) come first, and they may lead to of cells (proliferation with variation and motility). The randomness in a given mathematical space. In our frame- principle of variation that we propose states that biologi- work, variability comes first and closure justifies the valid- cal organisms are specific objects and, thereby, fundamen- ity of constraints. tally different from the objects defined in physical theories. The notion of constraint is central to our framework. The principle, which draws directly on Darwin’s insights Constraints are the building blocks of mathematical mod- on biological variation, embeds a specific notion of ran- eling in biology and are the main objects of experimental domness, which corresponds to unpredictable changes of investigation. The theoretical notion of constraints that the mathematical structure required to describe biological we propose should lead to a reinterpretation of mathemat- objects. In this framework, biological objects are inher- ical models that are based on them. In our framework ently variable, historical and contextual. A specific object constraints depend on the rest of the organism and the such as an organism is fundamentally defined by its his- rest of the organism depends on them (principle of organi- tory and context. Its constraints which may be described zation). Moreover, constraints may undergo unpredictable by mathematical structures are the result of a history and variations (principle of variation). may change over time. The principles of variation and organization do not aim Our approach to variation contrasts with a relevant at providing a complete framework to understand biologi- part of the theoretical literature on biological organization cal objects (the default state, for instance, is also required), which aims at investigating the origin of life by the means but they elaborate on both the Darwinian and organi- of minimal or physical models. The strong point of these cist traditions and lead to a significant departure from the models is that they lead to tractable mathematics (see for physical methodology, which opens the way to original re- example Luisi, 2003; Ruiz-Mirazo & Moreno, 2004). Here, search directions. we aim instead at combining organization and variation in a framework that focuses on current organisms, with the Acknowledgments massive amount of history that they carry. This difference between the two methodologies corresponds to distinct but Ma¨el Mont´evil’s work is supported by the ‘Who am complementary aims, and, crucially, to the fact that the I’ Labex grant (ANR-11-LABX-0071 WHOAMI). Arnaud concept of organization has been traditionally approached Pocheville has been partly supported by Ana Soto’s Blaise ´ Pascal Chair, Ecole Normale Sup´erieure, R´egion ˆIle-de- France. 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