1To address the important question of complexity within the world of science—and more modestly within the field of molecular physical—it is first necessary to define the contours of the question and to propose a consistent and reasonable definition of complexity, before moving on to discuss the paths that lead to it.
2Let us first set aside the notion that a complex world is necessarily “elusive” and “difficult” to apprehend—a notion that occasionally collapses these two epithets into one another by means of a dangerous shorthand. Let us adopt instead an understanding of a structure whose design is not reducible to the simple superposition of its constitutive elements. Out of this preliminary attempt at a definition, liberated at least in part from the notion of “difficulty,” emerges a vision originally proposed by Aristotle and according to which, “the whole is greater than the sum of its parts.” Within this historical perspective, it is important to point out that no hierarchization in the resultant structure, between the characteristics of the whole and those of its constituent parts, is a priori determined. For the scientist, quantities are typically algebraic. In other words, the forces of synergy and antagonism operate in the background and yield the result of their rivalries. A phenomenon may be attenuated, or on the contrary amplified, according to the important notion of interference. To conclude this introduction, and support our argument, let us briefly recall the famous experiment known as Young’s slits. A light wave propagating through a plate with two finely cut apertures produces a rather unexpected image on a screen placed behind the openings. Although light rays pass through both openings, the interference between light waves passing through the slits produces alternating zones of darkness (where light seems to have been absorbed) and zones of brightness (where light seems to have passed through). We can thus appreciate that, in the whole formed from two sources of secondary light (i.e. the slits), a seemingly rudimentary machinery manages to perform complex additive and subtractive operations. This surprising observation invites us to wonder not only about the nature of light waves but also about the conditions necessary for such complex “bookkeeping” to take place. Indeed, the various openings that perforate our own habitats are intended to create sources of light, not places of darkness! Without a correlation between the size of the openings and the spatial quality of light radiation, no darkness is perceptible and complexity is not expressed. On its own, light cannot produce such an unexpected result, but the presence of an appropriately proportioned partner positioned along its path becomes a source of troubling darkness. The phenomenon of chiaroscuro is simultaneous an oxymoron—a device from the visual arts practiced since the time of Greek antiquity— and an expression of the complexity of the material world.
3Cutting across disciplinary boundaries, the objective here is to understand complexity, its origins, and its manifestations in the molecular world. While the physical sciences generally aim to rationalize their observations, chemistry preserves in its forms something of the imagination and willpower of the sculptor. Molecular structures are aesthetic feats as much as they are opportunities to test the robustness of our theories. In the search for complexity, and for the emergence of new properties, prediction represents a constant quest while remaining a challenging endeavor. In the living world, the environment plays a decisive role in selecting the structures that are most suited to Nature’s “plan.” Likewise, in the molecular world, and more specifically in the world of matter, the complexity of a given system must be understood in terms of the relationships that it maintains with an external environment. In what follows, we will interrogate the concept of complexity independently of the various disciplines considered, since its origins appear to be shared; this will involve a subtle balance between specificity and generalization.
4In order to better grasp the question of complexity in the molecular world, let us first adopt a reductionist approach, by accepting a hierarchy of sizes of which we will specify the elementary unit. Once the question of a model becomes necessary to better explain our environment and to attempt to understand its structure, organization, and internal dynamics, it is useful to also define that model’s parameters and properties. Let us therefore limit ourselves to the molecular object, which is governed by a set of confusing social rules—some conceived as intuitive, others deduced from the mathematical formalism of quantum mechanics. How can we expect indistinguishable particles such as electrons, characterized by their aversion for one another, to form the pairs that hold together a cluster of atoms? The characteristics of the object can in no way be reduced to its individual properties, and we are forced to comprehend it in its entirety according to a kind of holism. Such complexity is all the more at stake when we consider that radically different expressions can emerge depending on the intensity of the relationship that each electron maintains with its neighbors. The explanation offered by quantum mechanics can be summarized as a rivalry between the individualism of constituent parts (the electrons) and the demands of a collective (the molecular structure). Our environment abounds with small molecules, quite similar to one another in structure but notoriously different in their properties. The molecule of dioxygen (O2) that we breathe—the source of life—is structurally similar to the small magnet dinitrogen (N2) which, however, lacks the properties of the former. We can speak here of paramagnetism and diamagnetism, respectively. To conclude, albeit rather elliptically, we can argue that the living structure is intimately linked to a singular organization of apparently very similar compounds. In no way could the chemist confuse dioxygen and dinitrogen, molecules that form the majority of our atmosphere. And yet the writer and chemist Primo Levi warned us of this very possibility in The Periodic Table  at a time when scarcity reigned everywhere, even in laboratories. The author chose to use the sodium’s near twin, potassium, to conduct his experiment. After the explosion that followed this unfortunate substitution, Primo Levi drew important lessons from the apparent similarities that govern the very different behaviors of two elements. Levi’s experience in the concentration camps naturally invited the author-chemist to question the nature of the universe and the bonds that exist between not simply between atoms but also between individuals. The latter may be amorous, amical or even hostile, whereas in appearance—and only in appearance—chemical bonds present far less nuance. History teaches us that social laws and the rights of the individual are fragile structures that a context of human folly can easily destroy. The search for peace and freedom can, unfortunately, turn into chaos and horror.
5Once again, considering the constitutive elements independently stands to lead us down a deceptive path, inviting us to perceive the characteristics of two elements (sodium and potassium) or two molecules (dioxygen and dinitrogen) as being similar. But the object must be considered as a whole, thus returning us to a holistic reading of even the most basic elemental structures of chemistry. The notion of liaison is as much an aspect of the verbal arts as it is a characteristic of the molecular world.  It is, moreover, a wonderful scientific model originally proposed by Gilbert Lewis at the beginning of the twentieth century. Lewis’s model is a rather democratic one: it is based on the idea of reciprocal comingling and allows for little variability. A more modern vision enables the electron cloud to move over the entire structure while still assuring molecular cohesion. As the origins of the link to the Other, the bonds of the physical world considerably stimulate our imagination. With respect to the molecular world, several clarifications appear necessary insofar as the specificity of this perspective seems not only reductive but also insufficient. In the traditional model developed by Gilbert Lewis, a marked rigidity prevents us from accounting for experimental evidence, the arbiter in any theoretical development. The model nonetheless remains a wonderful starting point—both simple and theoretically attractive—for trying to understand the zoology of chemical bonds. Prior to the advent of quantum mechanics, this description had the virtue of helping us to interpret, understand and make predictions about the behavior of molecular structures. Later theories, which were the fruits of formidable imagination and human capacity to harness the power of ideas, should not be conceived as competitors. The requirement is that a theory aligns with observable phenomena. The robustness of any theory or thought experiment depends, in particular, on this criterion. Furthermore, having an idea of the certitudes of a theory allows one to measure its control.
6In the two cases, the metaplasm  of language or that of the attractive forces between atoms gives rise to a surprising complexity and intricate choreography. In the context of music, the violinist’s legato, a privilege they are granted over other orchestral instruments, carries meaning. It unites sounds in order to mutually reinforce them. A particular nuance can emerge, so as to better communicate an ephemeral feeling. For the chemist, the outcome is the unification of parts into a cohesive whole—individual atoms into a single molecular structure—and the emergence of new meaning. The same is true of the combination of letters into words; we highlight the strong bonds that exist between letters by means of the compact and connected form used to distinguish one word from another in writing. Consider, for instance, the introduction of the Carolingian miniscule under Charlemagne, which resulted in a new typographic hierarchy. This form of writing aimed not only at a simplification—removing, at least in, the ligatures between letters—but also at the establishment of orthographic norms. In the same way that words are organized into sentences, molecules combine and exchange in order to form flexible and meaningful complex structures. Bonding (la liaison) is a multifaceted concept, both in language and in scientific discourse. The more general term of interaction is occasionally privileged in the context of molecular structures.
7As soon as the molecular structure expands due to introduction of a protocol that seems, on the surface, innocent but is in reality critical and generative of specific properties, modulations are observed and the boundary between two regimes becomes surprisingly permeable. Like the filaments of the chironex fleckeri jellyfish, this cohort remotely regulates and controls the characteristics of the assembly. The malleability of the whole also invites us to think in more dynamic terms, abandoning the static image of complexity with which we began. Molecular systems reveal a plasticity so pronounced that their “filaments” their connection to an external environment, are likely to modify the overall properties of the molecular structure. In this way, the peripheries function like funnels, channeling certain chemical compounds. The opening and closing of these hatches sets up the necessary gradient for the transmission of information. In dynamic behavior, the notion of reversibility occupies an important place. Constructing and deconstructing without exhaustion is not guaranteed, even on the molecular level. Any slow change, the result of superficial modifications, is potentially reversible without, however, this ever being certain. Complexity is also inscribed within a particular temporality. Previously, we evoked the musical illumination represented by legato, which connects several notes in mutually-reinforcing slur. The case of a recording notwithstanding, every musical performance is ostensibly unique, even if there persists an irresistible hope to think that replication is possible. The specific stroke of the bow, this connection (liaison) that is suggestive of various different meanings, is difficult to reproduce exactly. It is a source of complexity inscribed within the ephemeral and the singular.
8The time that passes is also a source of complexity, as much in terms of the dynamic observed in specific structures as in a form of irreversibility that each may be tempted to refute.
Complexity of assemblies
9The physical characteristic carried by the core and its potential modulation is sometimes expressed exclusively in the presence of these relays, which are themselves devoid of any property. To the existing dynamic of complexity is thus added the possible presence of a trigger. So-called “spin-crossover” molecules tend to alternate between two geometrically distinguishable forms, like the breathing of the subject under influence of an external stimulus . Within the same complex, the swelling of the core’s shell makes it possible to switch from a diamagnetic form (“of”) to a paramagnetic expression (“on”). Complexity arises from this combination in which links—chemical bonds—are established between two components— the core and the shell—to establish a form of logical information. The multiplication of bonds and the variability of their strength confers upon the same molecular structure a ductility that is comparable to that which we observed in the conversion of dioxygen into dinitrogen. Of course, the potential resides in the ingenuity exhibited by the chemist, who imbues complexity through the network of chemical bonds. His work resides in linking elementary units, or synthons, to then go on to extract and purify the result of the synthesis. In the living world, such combinations are frequent and govern a series of events, the design of which evolution alone determines. The active center of a protein may be of a relatively modest size in comparison to a seemingly incidental peripheral branching. Yet the former, when stripped of this inert cohort would not be able to provide the functions necessary for life.
10This brief detour into the living world reminds us of the importance of the environment, or what is more broadly termed the “external medium”. The latter exerts stress on the complexes that it simultaneously houses and controls. An arrangement or hierarchy naturally imposes itself given the relative abundance of actors. The physiological medium—mostly composed of water—structures, orders and transmits information. However, some interaction is necessary between the host and its guests, following the rules of sociability with which we are familiar. A contact much more labile than the previously mentioned bond is established and this recognition modifies the properties of the whole, once again giving rise to complexity. Not only does the medium condition the properties of each individual, but it also modifies their interactions and encounters. The behavior of a group of individuals plunged into darkness is profoundly different from the behavior of those same actors in daylight. Darkness first modifies the behavior of each individual by generating fear, apprehension, perhaps even a kind of panic. Then, each instinctively develops a form of prehension, adjusting to their own displacement through contact with that which they define as the outside. As soon as interpersonal contact is made impossible by the distance between individuals, each has recourse to sound perception—thus scanning the space by voice recognition. The surrounding environment, then, becomes significant, since it carries both message and structure simultaneously. The ringing of a bell becomes imperceptible when it is placed under a cloche in which a vacuum is gradually produced. The properties of the fluid condition the exchanges, regulating their intensity, range (a spatial concept) and repetition (a temporal concept). The living world essentially moves about in an aquatic environment. The salinity may vary, but water is the natural substrate for the majority. Its rarefaction leads to perturbations, such as the changes in color that are observable. Like Mercury, water functions as the messenger between chemical species, the guardian of information pathways, and the conveyer of knowledge. There is a plasticity in the kinds of contact that take shape. Chemical bonds, which are rigid and durable, are distinguished from interactions that are more ephemeral and sensitive to the environment.
11The ductility presented by molecular material—an assembly of more or less sophisticated units, with or without specific properties—is then inscribed within a temporality and dynamic of the whole. Molecular matter, like individuals, carries within it a memory of its past. Shape-memory refers to the ability of a molecular assembly to recover its original structure after a deformation. The past permeates these objects, systematically reminding them of an origin. As a result, we cannot study them with any degree of certainty without information about what they have been. Certain spin-crossover materials—three-dimensional assemblies of units that tend to switch between two states—exhibit a behavior that we will subsequently explore. In these constructions, we find the ingredients of complexity. While each unit consists of a core that forms chemical bonds with inert branches, the units communicate with one another by means of a sophisticated network of interactions. Properties on the level of the individual therefore arise from a competition between intramolecular bonds and inter-molecular interactions. Returning briefly to physical origins, the celibacy of electrons is in constant conflict with their penchant for coupling. Branching directly controls this organization at the center of the core, giving it a paramagnetic (lone) or diamagnetic property (paired). The state of an individual is not without consequence for its congeners, at least for its closest neighbors. The network of interactions is subsequently set into motion and gradually triggers neighboring units, signaling cooperativity within the structure. The rapidity with which this domino-effect unfolds represents a problem that is tricky to anticipate and describe. It is common to see a modification in the properties of the whole, and this change involves variable features. Specifically, the system’s response to a stress exerted by the environment, such as a change in temperature, can be mild or, on the contrary, dramatic. By analogy with the behavioral forms of living beings, we often evoke susceptibility as a gauge of this capacity for a more or less vigorous response. Susceptibility is regulated by an intertwining of chemical bonds, the expression and intensity of which is difficult to predict. From their elementary units to their three-dimensional structures, these materials are representative of a complexity of the molecular world. Their intimate workings as well as their potential for practical applications continue to raise questions for the scientific world.
12The unsuitability of a unit to a changing environment is certainly reminiscent of the notion of the hysteresis of habitus, as developed in the field of sociology by Pierre Bourdieu. The persistence of certain social dispositions can delay immersion in a modified environment. During the domino-effect mentioned above, adaptation can be gradual, but also conditioned by the system’s original state. In practice, the phenomena observed during an increase of temperature are not necessarily superimposable onto those recorded during a decrease in temperature. The composite’s history thus informs future behavior in response to environmental stress. This memory effect can be put to use when the physical origin of a compound is not entirely known. Recently, research has highlighted a profound reorganization of the bond accompanying the oscillation between the diamagnetic and paramagnetic states of an isolated unit . Bonding is not an immutable component of the molecular structure. Variations can be quantified, and these perturbations once again highlight the plasticity of chemical bonds. For the studies in question, the way in which the electron cloud is arranged between the core and its shell is especially sensitive to the magnetic state.
13This localized change within a unit obviously affects interactions with nearby units within the same molecular material. The conflict that is introduced depends directly on the initial state of an isolated unit and on the opportunity for competition offered by the environment. The environment can be understood as a juxtaposition of paramagnetic and diamagnetic units, the relative proportion of which is altered by an external stressor, namely temperature. Socialization differs in the assembly’s warming and cooling phases. In this framework, the phenomenon of hysteresis is indeed the expression of a complex cooperativity resulting from the modulations of intramolecular chemical bonds and intermolecular interactions.
14We would be remiss to conclude our overview of this striking physical behavior without mentioning the concept of supramolecular chemistry elaborated by Jean-Marie Lehn . By forming interactions between units, sophisticated assemblies can be produced without completely altering the integrity of the constituent units. The functional imitation of enzymes has led to the development of synthetic molecules capable of selective recognition. Complementarity of the geometric features of the host and guest is necessary for the assembly to form. The bonds that are established “beyond” the molecular level are, in fact, at the origins of self-assembly, recognition and mobility control at the molecular level , and the formation of molecular switches. For the latter, the idea of reversibility is essential, given that unidirectional change prevents the molecule from returning to its initial state. Once again, we can recognize the importance of having sufficiently labile links to ensure that oscillation between several forms with different properties can occur. The directional character imprinted on a structure, on a cluster of objects, can present an issue in the elaboration of molecular machines that combine a rotating component, rotor, and a stationary component, stator . In the absence of any control, the change is entirely random. A rectification is necessary, a part without which the structure cannot have the characteristics of a machine.
15Nature abounds with examples where recognition processes are manifest. For instance, a substrate recognizes the active site of an enzyme and forms an entity known as an enzyme-substrate complex, which catalyzes the conversion of substrates into products. Several stages, some reversible, follow one another in sequence, in accordance with the notions developed above. Note that the terminology used in biology borrows the epithet “complex.” However, we must attach this label to the phenomena that the enzyme assembly is likely to produce with its substrate.
16The world of complexity cannot, of course, be reduced to molecular structures. Even if we identify the elements that give rise to complexity, the very nature of the relations between those elements opens up the prospect of new expressions. Our senses are themselves vectors of complexity on the level of the individual. These sense perceptions allow us to survive; we mobilize them in function of their acuteness, and often combine them. Can the perception of a sound, for instance, cloud the vision of an imminent danger? The answer is definitively negative. The connections that we can establish allow us to better apprehend our environment, and to perceive its risks. With the increase in means of communication, we can observe a change in the modes of exchange—often driven by a desire for concision. In the face of misunderstanding, we have observed, for some years now, an interesting return to privileging physical exchanges, which bears witness to the insufficiency of email, even when supplemented telephonically. The superiority of the meeting seems indisputable. This mode of exchange, which mobilizes the senses of the different interlocutors guarantees clear communication. The other’s degree of comprehension appears to be tightly linked to this sensorial complexity, this synesthesia, which conditions human interaction. By analogy, mathematicians combine two so-called “real” numbers by means of an imaginary number in order to generate a world of “complex” numbers. This combination magnifies the world of real numbers, giving it the power to render otherwise unsolvable problems solvable.
17In an ever-changing environment whose codes are constantly being redefined, complexity as we have come to understand it finds new sources. It is found to be dynamic, subservient not only to the environment in a Darwinian sense, but also to a fragile codification that orders the relationships between individuals .
In French, liaison (from lier “to bind”) refers both to the phonological process and to a chemical bond.
In rhetoric, metaplasm refers to any phonetic or morphological change that modifies the internal structure of a word by processes of addition, deletion, substitution or permutation.
Vincent Robert wishes to thank the Excellence Network Chemistry of Complex Systems (LabEx CSC, ANR-10-LABX-0026CSC).