Life sets off a cascade of machines

Significance

This paper follows an idea by Leibniz that life can be seen as an infinite cascade of machine-making machines, down to atomic machines. It proposes an oversimplified language of life, highlighting certain scaling aspects and the key step of self-reproduction, with a singular point at one micron and a thousand seconds.

Abstract

Life is invasive, occupying all physically accessible scales, stretching between almost nothing (protons, electrons, and photons) and almost everything (the whole biosphere). Motivated by seventeenth-century insights into this infinity, this paper proposes a language to discuss life as an infinite double cascade of machines making machines. Using this simplified language, we first discuss the micro-cascade proposed by Leibniz, which describes how the self-reproducing machine of the cell is built of smaller submachines down to the atomic scale. In the other direction, we propose that a macro-cascade builds from cells larger, organizational machines, up to the scale of the biosphere. The two cascades meet at the critical point of 10³ s in time and 1 micron in length, the scales of a microbial cell. We speculate on how this double cascade evolved once a self-replicating machine emerged in the salty water of prebiotic earth.

Multiscale organization, dense with interconnected cycles and loops, is one of living matter’s most striking features, and this paper proposes a view that it is also the most fundamental one. Concretely, we speculate that life can be presented as a hierarchical cascade of machines making machines. This simplified language of machines is false about the reality of living matter, but we hope it is still a useful vehicle to talk about the extreme varieties of life. Thus, the paper is more about accenting certain scaling aspects of life rather than about the infinite intricacy of life itself. And we hope there should be mathematical languages that capture life more fully than a purely realistic view. Such languages should fit their subject, living matter, like the formal language Lavoisier devised fit his elementary theory of chemistry (1, 2), where he followed Condillac’s logic “that the art of reasoning reduces to a well-formed language” (3). This is the subject of this paper.

To begin the discussion, let us gaze for a moment at a rudimentary bacterium and envisage how, from this minute creature, two infinite cascades emerge (Fig. 1). Let us explore first the micro-cascade, proceeding from the bacterium down to its tiniest parts: the membrane that envelopes the cell, the molecular machines inside the membrane, ribosomes, and cytoskeleton, the submachines that make the machines, proteins, RNA, and DNA, the parts that make the submachines, amino acids, nucleotides, and lipids, and down to atoms, electrons, protons, and photons. And then let us turn our gaze to the second, macro-cascade, as it assembles bacteria into larger, more intricate organizational machines: communities of bacteria, eukaryotic cells formed by the symbiosis of archaea and bacteria, tissues made of eukaryotic cells, organs constructed from tissues, organisms composed of organs, populations of organisms, ecosystems made of populations, including human society, up to the whole biosphere.

Fig. 1.

Schematic space-time scaling of the double cascade of machines, with its critical point (red) at the scale of the microbial cell, the smallest self-reproducing machine (1 μm, 10³ s). The micro-cascade describes how the cell is made from smaller machines, down to the atomic scale (green), and the macro-cascade depicts the construction of larger machines from cells, up to the whole biosphere ($1\mathrm{Y}\sim 3.{10}^7\mathrm{s}$). The spore (gray) can leave and return to the cascade (see text).

In exploring this cascade, our starting point is the general notion of a machine and the specific question of what makes living machines unique (Section 1). The cascade of machines appears as a solution to the problem of survival through self-reproduction realized in the world of salty water. We then ask how the machines self-organize into a cascade and elaborate on its scaling (drawn in Fig. 1), especially on what determines its fundamental space and time scales, 1 micron and ${10}^3$ s (Section 2). This is followed by a discussion of salty water, the world in which the cascade emerges and evolves (4). We examine what this physical boundary condition implies in realizing von Neumann’s self-reproducing machine, particularly the role of electrodynamic forces (Section 3). We conclude with tentative thoughts on the divergence of the macro-cascade (Section 4).

1. Machines and Their Self-Reproduction

The first to look at life in the language of cascading machines was most probably Gottfried Leibniz, more than three hundred years ago, in his famous Monadology (5). The idea of the cascade was so far ahead of his time that we must join Gottlob Frege’s admiration that “…in his writings, Leibniz threw out such a profusion of seeds of ideas that in this respect he is virtually in a class of his own” (6). Indeed, after building a computing machine and envisioning universal logic (7, 8), a seventeenth-century forerunner of the Turing machine (9), it was only natural for Leibniz to ask whether one could build a machine that imitates life. He answers: “…each organic body of a living thing is a kind of divine machine, or natural automaton, which infinitely surpasses all artificial automata, because a machine which is made by the art of man is not a machine in each of its parts…” (5), and lucidly states the essence of the cascade: “but the machines of nature, that is living bodies, are still machines in their smallest parts, to infinity.” Peering into the abyss of tiny molecular machines lit by modern technology, his insight is striking. As Leibniz anticipated, machines are present at any scale.

To follow Leibniz’s inquiry into living machines, we must first explain what “machine” means. Broadly speaking, a machine is an object that transforms a physical system nonrandomly to perform a definite function. Of these features of the machine, two require further clarification. First, regarding nonrandomness. Machines might be very noisy, especially molecular ones. Yet, even the noisiest machine must have a nonrandom bias, or else it cannot function: Molecular motors move erratically but in a definite average direction. Ion pumps transport charges intermittently but consistently across the same polarity of the membrane potential. And the division machinery splits cells and never merges them back. We see how machines inject local order into a universe drifting toward global disorder. To swim against this current of rising entropy, the machine demands an endless supply of energy. Hence, all machines are dissipative.

Second, we need to clarify the notion of functionality. Machines transform the system in a predictable manner, allowing them to perform tasks. By the scale of the task, the scale of the machine. Nanometric pumps convey ions, microtubules assemble into a micron-size machine that organizes the chromosomes in the cell, and cells form tissues, and this applies up to the scale of the biosphere. The existence of a function assumes no “purpose” or “goal;” these external human notions are beside the point. All we know is that the machine pushes the system along specific nonrandom paths in spacetime, and these paths are functional.

This is where thinking about life in the language of machines, as Leibniz did, leads us, away from the realm of pure physics and chemistry, where function is a meaningless word, and into the realm of engineering and technology, where function is everything. When molecules collide and exchange electrons and atoms, this chemical reaction serves no particular function by itself. Yet if this collision occurs in a living machine, say at the catalytic site of an enzyme, then it acquires a function through its context.

1.1. Survival.

What is then the function of life? The answer is as universal as simple: survival. This global task unifies all living machines. Thus, we can propose looking at a living organism as a survival machine that is part of a well-coordinated double cascade. Survival is the physical notion of continuity in space and time. Systems with continuous spacetime trajectories survive, and the living machine excels in selecting these continuous trajectories, as compared to random physical processes. Other systems with different organizing principles might have emerged and disappeared, but we observe only those that persisted over time, those that survived. Only two extreme solutions to the problem of survival are known: The standard solution is physical robustness, as ancient rocks survive through geological eons or rivers flow along slowly changing paths. The other extreme is the infinite cascade of life driven by self-replicating machines.

For the living machines, survival has a subtler meaning, with a different notion of continuity. First, the self-replicating machine must always remain continuously functional, implying that all levels of the cascade of machines must always be in synchrony. All fluxes of matter, energy, and information flowing through the levels and exchanged with the environment must be balanced, requiring numerous feedback loops (as discussed in Section 2). Second, the machine must reproduce new copies of itself, and the lineage of machines ensures continuity over long time scales—not continuity of matter but rather continuity of functional organization through the repeated cycle of self-reproduction. These cycles punctuate the continuous spacetime trajectories of life, adding a double-edged subtlety: For the sake of continuity, the new machine should be very similar to its ancestor machine, and this requires stringent error-correcting machinery. But, essentially, the correction cannot and must not be perfect, to allow new kinds of living machines to emerge through evolution.

Survival by self-reproduction is a singular exotic solution with no counterpart in technology.^* Constantly fed with the required resources, these machines will keep doubling themselves, and their number will explode exponentially with the number of generations $N$ as $2^N$. The steady-state solution of physical robustness is stable, whereas this exponential solution displays infinite plasticity in adapting to exploit the available resources for rapid doubling. Only one remarkable object can hop back and forth between these two extremes: the spore (Fig. 1). To endure adverse conditions, certain bacteria and plants metamorphose into spores (10, 11) that freeze their internal dynamics, thereby adopting the physical stability solution (12, 13). Once the stress is relieved, the spores germinate and return to the exponential solution of life.

1.2. Von Neumann’s Self-Reproducing Machine.

By its booming nature, the exponential solution is bound to be unstable and dissipative. Self-reproduction must therefore be tightly regulated and perpetually fed by a multiscale cascade. This endless dissipation inevitably ends up in ferocious competition over the limited resources of the physical world. Before we discuss this rough reality, let us look at the world of logic, where the state of affairs is much calmer. In this abstract world, self-reproduction was realized already in the 1940s by John von Neumann (15, 16).

The concept of the machine he envisioned was simple (Fig. 2A): It is a chimera of two conjoined submachines immersed in a sea of elementary building blocks. The first is a constructor that collects blocks from the sea and assembles them into an offspring machine, following the directions encoded along a blueprint tape. The second submachine is a copier that replicates the tape and passes it on to the offspring machine, which is ready to produce the next generation. And this loop of doubling reiterates to infinity as long as it is fed with building blocks. While von Neumann’s machine is an abstract logical construct, it captures some essence of living machines and their algorithmic nature by defining a precise language of self-reproduction. Thus, much of this paper is about constructors and copiers, only that those are realized in a physical world of carbons and salty water.

Fig. 2.

Self-reproduction in the worlds of logic and water. (A) von Neumann’s conceptual design of a logical self-reproducing machine (see text). (B) Superresolution image of a doubling and dividing E. coli (by permission from ref. 14), showing the DNA tape (blue) phospholipid membrane (red), and RNA polymerase (RNAP, yellow).

Von Neumann’s design prefigured the discovery of DNA (17, 18), the blueprint tape of the cell. In both biology and logic, the tape is a coded image of the machine, its “self,” which one can denote using quotation marks as “machine” (Fig. 2A). What is being replicated is always the object coupled to its image: machine + “machine” (where machine $=$ constructor + copier). In living machines, the tape is the DNA genotype that encodes the phenotype of the self-reproducing machine, the cell (Fig. 2B). This self-description is referenced—by the copier replicating the tape and by the constructor producing a new machine. These two submachines are two main archetypes of molecular machines: those that transform matter and energy, such as ion pumps, and those that process information, such as the enzymes that cut and paste DNA strings.

The two archetypes are never entirely distinct. Information processors consume matter and dissipate energy, and accurate processing requires even more energy to fuel error-correcting machines. The minimal cost of information is known from Maxwell’s demon (19) and Szilard’s engine (20) to be an energy of $\mathrm{kT}ln2$ for each bit of information. Conversely, circuits that shuffle matter and energy also process information, as the concentrations of molecules in the cell carry information about its physiological state. Information and matter machines are entangled at all scales of living matter and are built of the same molecular units. Thus, the letters of the DNA alphabet, A, T, C, and G, are nucleoside molecules that also shuffle energy throughout the cell in the form of phosphate groups, most notably in ATP (adenosine triphosphate), where the letter A (adenosine) is bonded to three phosphates.

1.3. Complexity and Self-Reproduction.

Thinking about life, it is hard to avoid the notion of complexity and the recognition that this notion remains defiantly undefinable. Life is said to be complex, but what does that imply? Returning to von Neumann’s machine, one can appreciate one aspect of this complexity even without a clear definition. A typical factory is more “complex” than the machines it makes, in the relative sense that the factory must contain all the facilities and knowledge needed to produce the machines. In contrast, von Neumann’s constructor is a factory whose product is an identical factory, equally complex. Hence, unlike typical production, which reduces complexity, it is conserved in self-reproduction. Yet, life is even more demanding, with more complex organisms occasionally emerging from simpler ones (15), again in the relative sense that these organisms have more parts that interact more elaborately. Von Neumann realized that increasing complexity becomes feasible only through evolution in the world of water, where the exponentially multiplying machines compete over finite resources, as discussed in Section 3.

Complex systems display features that are not apparent in their parts; as Anderson noted, “more is different” (21), but what these emergent features are remains elusive. Many measures of complexity have been proposed (22), and here, we will mention only two that are relevant to our discussion. The first is the algorithmic complexity of a string (23–25), defined as the length of the shortest computer program that can generate the string and, thus, its most compressed description. In a von Neumann machine, this measure is of particular significance: Since the string (the tape) encodes the machine, its algorithmic complexity measures the complexity of the machine itself.

Once we know the shortest computer program that describes the tape (or a close enough approximation), we can further estimate its intricacy by counting how many cycles it traverses while running. This geometric measure, called cyclomatic complexity, equals the number of independent cycles in the program’s flow chart (26). More complex programs with more decision points (“if” and “while” commands) have charts with more loops. Computer scientists use cyclomatic complexity to detect parts of the code that are excessively nested and dense with loops (27). Such programs are harder to test and debug and are therefore more prone to errors.

1.4. Loops and Innovation.

In the cascade of machines, loops and cycles are pervasive (and hence, the cyclomatic complexity is very high). First of all, the machines that make up the cascade are finite objects taking part in an infinite process and, therefore, must work in cycles, just like Carnot’s heat engine (28). Thus, through cyclic treadmilling, microtubules grow and shrink, molecular motors cycle through allosteric transitions to move along the microtubules, and the Krebs cycle produces the fuel needed to sustain this motion.

Second, all these cycles are combined to feed each other in synchrony, such that the fluxes of energy, matter, and information exchanged among the various levels of the cascade and with the external world are balanced. At all scales, production lines must work in concert to avoid surpluses and shortages in machine parts (for example, the number of ribosomes must be proportional to the number of RNA polymerases, Fig. 2B). All this requires numerous loops of control and regulation, especially error correction loops.

The self-reproducing machine carries its own coded image (Fig. 2A), allowing the machine to traverse along the two fundamental loops: self-reference and self-reproduction. These loops are the engines of innovation and discovery. To see why, one may look at the structure of mathematical theories. A theory is built like a tree: The roots are the axioms, and they are sequentially combined by the rules of logic into new theorems, which are the branches and leaves of the tree. One can always roll back the film of the growing theory tree to recursively reduce the theorems into the original axioms. Something essentially new cannot branch from the tree, or else it could be traced back to the old axiomatic roots. And if this new thing stems from other things, they should also be new and should all emerge simultaneously, as a connected loop, just like new concepts appear in a dictionary (29). A loop has no roots.

Thus, von Neumann profoundly understood that innovation is possible only through a self-reference loop, through random trial-and-error, because innovation is beyond what is written in the old records. In life, a multitude of such self-reproducing loops randomly walk through the enormous space of possible configurations by repeatedly tweaking the software (“machine”) and hardware (machine), and the fitter survive. This is how open-ended Darwinian evolution with its mutation–selection cycle works. To survive in a noisy physical environment, a realization of a von Neumann machine must contain many other loops. In the language of the present discussion, we say that the loops span and intertwine over all accessible space and time scales, forming a dynamic multiscale structure, a cascade of machines. The cascade is a dynamical system that can generate an infinite spectrum of adaptive responses, enabling survival in a high-dimensional changing environment.

2. The Cascade of Machines

Von Neumann’s machine is universal: It can read and perform the directions encoded on any blueprint tape, as long as it is written in the alphabet used by the machine. This follows Turing’s idea of a universal machine that can compute anything another machine can (30, 31). But the universality comes at the cost of excessive complication. Conceptually, von Neumann’s design (15), machine + “machine,” is simple (Fig. 2A), but its realization, even in the abstract world of logic, is notoriously elaborate. To achieve the capacity of universal construction, his logic machine had to be built of thousands of interacting blocks (a type of machine called “cellular automaton”). Each block dynamically switches among no fewer than 29 possible states, all necessary to pull through a cumbersome procedure of self-reproduction (16). While one can successfully simulate such complicated logic machines on a computer, they are much less feasible in reality (32, 33).

In the physical world, only life has achieved universal self-reproduction that is autotrophic—the capacity of a machine to make a new copy of itself from raw materials without external help (Fig. 2B).^† We propose that the cascade of machines can be seen as a realization of von Neumann’s machine in salty water, achieved by distributing its complexity over multiple scales. Interfacing with the physics of water at the atomic scale requires a gamut of specialized nonuniversal machines. While most universal self-reproduction is of protein (and RNA) machines, much of the nonuniversal production is invested in making nonprotein machines, especially a lipid membrane, as Section 3 explains.

2.1. Scaling of the Double Cascade.

The cascade of machine-making machines exhibits universal scaling. All these scales must work in concert, feeding and regulating each other, linking them through simple power laws that do not depend on microscopic details. Before delving into details, let us examine a rough sketch of the scaling observed in living systems, showing the typical size and period of the machines (Fig. 1). The range of the graph is astounding. It explodes to almost thirty orders of magnitude in time and eighteen in space. Notable is a kink in the graph at a scale of bacteria, about 1 micron in size and ${10}^3$ s in time.

At this critical point, located about the midpoint of the time axis, the micro-cascade of cellular machinery fuses with the macro-cascade of organizational machines driven by development, evolution, and ecology. The first cascade, envisioned by Leibniz, tells us how to build cells from smaller machines, and the second tells us what larger machines can be built from cells. The distinct nature of the two cascades shows in their scaling: The micro-cascade roughly scales with an exponent $\alpha \sim 4$, as $T\sim L^4$, while the macro-cascade scales as $T\sim L$, forming a sharply bent knee at the crossover. The critical point is not accidental; it is the minimal object that can self-replicate autonomously, the smallest von Neumann constructor.

Using the language of the cascade, we can observe in finer detail the processes that give rise to this multiscale organization (Fig. 1). Ascending the micro-cascade starts by covalently bonding atoms into small organic molecules, such as nucleic bases (50 to 60 atoms) and amino acids (10 to 27 atoms). These building blocks are then polymerized into large macromolecular machines, primarily DNA, RNA, and proteins. An average protein is a chain made of $\sim$300 amino acids, while the RNA pieces that, together with protein pieces, make the ribosome machine, are chains of 150 to 3000 nucleic bases. Collective noncovalent interactions among the monomers and the salty water around them, such as hydrogen bonds and hydrophobic forces (34), fold the one-dimensional polymers into intricate three-dimensional shapes. Similar hydrophobic–hydrophilic interactions make lipid molecules assemble into two-dimensional sheets of fluid membrane (as discussed in Section 3).

These first stages of the micro-cascade are marked by the emergence of slow, collective degrees of freedom from fast quantum mechanical processes at the atomic level. In this many-body dynamics, motions and forces propagate subdiffusively, with timescales that grow steeply with size. For example, the viscous relaxation time of a polymer scales with its size as $T\sim L^4$ or $T\sim L^3$ (35), and internal mixing of chromosomes scales as $T\sim L^5$ (36). The machines produced at these stages present elaborate surface patterns, allowing them to specifically recognize each other. As in a jigsaw puzzle, only building blocks whose shape and surface chemistry match will stick together. This specificity facilitates the combinatorial assembly of large supramolecular complexes, which marks the upper stages of the micro-cascade. The assembly is time-consuming as it involves numerous encounters of the pieces until they all fall into place. The larger complexes, such as the ribosomes, take minutes to assemble (as detailed below), and these combinatorial timescales eventually set the generation time of the whole self-reproducing machine, the microbial cell, at $T\sim {10}^3\mathrm{s}$ (37, 38).

Beyond this critical point of self-reproduction, the nature of the cascade entirely changes. While the lower micro-cascade is based on physical forces, covalent and noncovalent bonds, the essence of the upper macro-cascade is of multiagent networks in which cells communicate and process signals (39). These ramified networks can be viewed as organizational or social machines. Bacteria assemble into colonies and communities, such as the biofilm van Leeuwenhoek discovered on his teeth, already at the time of Leibniz (40, 41). Multicellular organisms grow their tissues and organs from stem cells following a precise algorithm encoded in their genome. This developmental program is a tightly synchronized sequence of events of cell doubling, differentiation, and death. But given the right cocktail of regulatory proteins, the sequence can be reprogrammed to restart from the initial state of stem cells (42).

Larger organisms take longer times to mature and reproduce, as can be seen from a naïve geometric argument: Since a growing organism produces a mass that increases with size as $M\sim L^3$, but is fed through a surface that increases only as $S\sim L^2$, the generation time should scale linearly with size (and a power of $1/3$ with mass) as $T\sim M/S\sim L\sim M^{1/3}$ (43).^‡ Considering the fractal nature of multicellular bodies, the scaling slightly alters to be sublinear with size, $T\sim M^{1/4}\sim L^{3/4}$ (44). The origins and validity of such power laws remain open questions (45, 46). But for us, it is only essential to note that the scaling of the macro-cascade is roughly linear and, at any rate, much less steep than that of the micro-cascade ($T\sim L^4$).

The interwoven networks of population dynamics, ecology, and social interactions that govern the slowest and largest scales of the macro-cascade (47) are beyond the scope of the present discussion, which focuses on the micro-cascade and its physical origins in the world of water. We only note one general feature: The upper cascade gradually departs from the dissipative, low-dimensional space of molecular machines into the abstract, high-dimensional information spaces. At these scales, data are a chief resource exchanged between consumers and producers and stored in long-term memories (48), most notably the DNA genome, which records the outcome of billions of years of Darwinian competition (49).

2.2. Protein Machines.

Close to the atomic scale, one can already see the most miniature machines at work. Prevalent among them are proteins (50). These nanometric machines are made of amino acids joined by the ribosome to form long chains. (The ribosome machine, which will be discussed later, is the living version of the von Neumann universal constructor.) The amino acid building blocks are twenty species of small organic molecules (0.4 to 1 nm) whose chemical diversity brings about a rich world of many-body electrodynamic interactions with the surrounding water and ions (51–53). This electrodynamics makes the amino acid chains collapse and fold into biochemical machines with intricate patterns of charges, polarity, hydrogen bonds, and hydrophobicity, allowing them to recognize and recombine numerous molecular species.

With their almost-universal capacity to process and produce, proteins open the gates to a world of chemical combinatorics of machine-making machines. Taking apart the self-replicating machine of the cell, one finds that most submachines are proteins. One then further sees how protein machines are joined into more complicated machines—often combined with RNA modules, as in the ribosome, or integrated into phospholipid membranes so they can harness the electric potential across the membrane as an energy source. Proteins also organize into chemical factories that produce smaller, nonprotein machines and submachines needed to interface with the physics of salty water, such as the fatty acids and polysaccharides used to build the bacterial cell wall (54). As part of the cascade, the elaborate 3D structure and motions of proteins are intimately linked to their large-scale cellular functions, such as ion channels that evolve to facilitate the whole-cell membrane potential (55).

To give an example, let us climb up the micro-cascade of cellular machinery, starting from the protein tubulin—a globular protein, about 4 nm in diameter, made of about 450 amino acids, each containing 20 atoms on average. Tubulin comes in two variants, $\alpha$ and $\beta$. Heterodimers are then built of $\alpha -\beta$ pairs, protofilaments are constructed by polymerizing dimers, and a 25 nm wide tube called microtubule is typically assembled of thirteen protofilaments joined side by side. Through dynamic instability of expansion and shrinkage, driven by GTP hydrolysis, the microtubule becomes a machine that operates at the scale of the whole cell, a few microns (56). But the microtubule extends only along one dimension. Coupling to the three-dimensional machine of the cell requires further steps: First, a supertube called a centriole is built by clamping together nine triplets of microtubules (57), then a centrosome is cast of a pair of centrioles encased in a dense protein material, and finally, additional microtubules are anchored to the centrosome (58) to form the spindle apparatus that divides the cell (59). Thus, in nine steps, atoms are bonded into nanometric protein machines that are then assembled into the micrometric division module of the self-replicating machine.

2.3. Information Machines, Codes, and Memories.

In the language of machine-making machines, there is no life without coded information and the machinery to process it. First, as we learn from von Neumann’s machine (Fig. 2A), to achieve universal production, each organism carries an algorithmic self-description in the form of a DNA tape. Second, to survive in the harsh, competitive world, this organism must continually acquire information about the state of the outside environment and its own internal state, store this information, process it, and use it to compute favorable responses. These two reasons are interrelated. Self-description allows universality, thereby enabling open-ended evolution in which organisms adapt to the environment by changing their self-description. Through these endless feedback cycles of evolution and adaptation, the self-description mirrors the environment and its history.

The cell is teeming with active processes that generate information by creating local order, such as concentration gradients and directed motion. In this broad sense of statistical mechanics, self-reproduction is an enormous generator of order and information as it synthesizes a whole cascade of machines out of tiny building blocks. However, we will narrow the discussion of information machines to processes in which the generation and transformation of information are central functions, not merely a byproduct of construction.

The creation of information machines follows the general scheme described above when we discussed the scaling of the micro- and macro-cascades. This cascading sequence is marked by the stage-by-stage emergence of slow collective degrees of freedom. The most elementary stage in the information micro-cascade is the polymerization of information bits into chains. By positioning a set of $N$ bits along a tape, polymerization increases their information content immensely, from $logN$, the entropy of an unordered set, to $N$.^§ Storing information in polymers also allows small, compact machines to sequentially process large amounts of data, just like the reader head of Turing’s machine (30, 31). Perhaps the most crucial polymer is the DNA tape, the template for the two core processes of the von Neumann machine (15, 16): copying by the DNA polymerase and construction by the ribosome.

Copiers, constructors, and other information-processing machines must be highly selective about their interaction partners and the building blocks they use. Otherwise, self-reproduction would be too inaccurate and depart from the approximate continuity essential for survival. Such precise molecular recognition is achieved in the second stage of the information micro-cascade, where polymers fold into complex three-dimensional machines, mostly proteins but also RNA machines. Their elaborate shapes and surface patterns allow these machines to solve the basic computational problem of classification: discerning the correct target among numerous lookalikes.

The folded polymers that make the information machines are held together by networks of noncovalent bonds, such as hydrophobic and electrostatic forces and hydrogen bonds. These relatively weak interactions generate a high-dimensional, rugged energy landscape with multiple metastable states. The landscape thus defines a finite-state computation machine (60) that can easily switch between energy minima by altering its molecular conformation in response to physical input received at the machine’s interface. Perhaps the simplest example is a two-state switch called “induced fit,” where a protein switches conformation while binding to its target (61, 62).

In the third stage of the information micro-cascade, the molecular finite-state machines are combined into large-scale circuits. These circuits rely on precise molecular recognition. Thus, proteins called transcription factors can accurately bind to specific DNA sequences that encode the production of other proteins by the ribosome constructor (63). These loops of protein machines referring to DNA images of other protein machines are interwoven into the genetic regulatory network that controls the synthesis of proteins in the cell. By similar rules of combinatorial assembly, the whole information circuitry of the cell is constructed.

From all this, we see that coded information is essential for building the hierarchy of the cascade. The logical calculus of survival must be encoded, and the cascade of machines provides a practical way to encode—by mapping fast levels to slower ones. The slower machines then become images or memories to be later retrieved, and these memory machines use even slower memories, up to the longest-term memories (48). These simple principles of symbolic representation make the cascade combinatorial and generative and provide it with the infinite plasticity required for survival. In the cascade, energy, matter, and information are entangled at all scales. In particular, information machines are necessary for regulating the distribution of energy and matter throughout the cascade.

2.4. Cells—Life’s Quanta.

A remarkable feature of living systems is the kind of quantization into cells—“all cells come from cells” (omnis cellula e cellula), wrote Virchow in 1858 (64). The essence of these cellular quanta lies in being the ideal self-reproduction machines for a world of water. The notion of a cell immediately brings to mind continuity and separability. The cellular self takes up a region that is continuous in space and, by self-replication, continuous in time. This continuum is sharply separated from the outside world—geometrically by a membrane, and logically through self-reproduction. For geometrical separation, the self-replicating machine necessitates topological submachines that cut and paste, as happens during cell division. Logical separation, on the other hand, involves immune-like mechanisms such as CRISPR (65), where the memory of the nonself is stored and retrieved by information submachines that cut and paste the genome. From this logic-geometry perspective, which defines the cell, the “self” is simply everything that is doubled.

The critical point of the cascade is positioned at $L_{\mathrm{c}}\sim 1\mathrm{\mu }\mathrm{m}$ and $T_{\mathrm{c}}\sim {10}^3\mathrm{s}$. This biological “Bohr model” is $\sim {10}^4$ larger than an atom and $\sim {10}^{16}$ slower. The enormity of these scales compared to atoms or molecular machines provokes a natural question: Why do the quanta of life need to be so many orders of magnitude larger and slower? (39) Many physical systems exhibit emergent collective excitations that behave as effective particles. Similarly, one can see the cell as an enormous compound particle—about ${10}^{11}$ atoms, which make ${10}^6$ protein machines, ${10}^4$ ribosomes, and ${10}^7$ DNA bases, all wrapped in a membrane made of ${10}^7$ lipids (46, 66). This emergent phenomenon is defined by its capacity to self-reproduce, and the reason for its gargantuan scale lies in the logic and geometry of self-reproduction.

2.5. Cell Scales.

The logic of doubling sets the timescale $T_{\mathrm{c}}$ (Fig. 3): The machines making machines for the new cell must also reproduce themselves, and this loop takes some time and resources. The most costly machine is the ribosome—a massive complex assembled of dozens of protein submachines and a handful of larger RNA submachines [55 proteins and 3 RNA pieces in E. coli (67)]. The ribosome is the engine of the living universal constructor: It reads messenger-RNA tapes and fabricates accordingly all protein machinery of the cell. To double the cell, each ribosome must fabricate, at least, all the protein submachines that make a new ribosome and another protein machine, called RNA polymerase, that produces the ribosome’s RNA modules. The protein submachines need to be synthesized from $\sim$10⁴ amino acids. Advancing at a rate of $\sim$10 amino acids/s (68), the fastest bacterial ribosomes accomplish this task in $T_{\mathrm{c}}\sim {10}^3\mathrm{s}$.^¶ Thus, the internal clock of the autocatalytic cycle of ribosome machines sets a strict minimum on bacterial self-reproduction time.

Fig. 3.

Critical scales. Doubling time as a function of cell size. At the critical size, $L_{\mathrm{c}}\sim 1\mathrm{\mu }\mathrm{m}$, the microbial cell achieves the fastest doubling time, $T_{\mathrm{c}}\sim {10}^3\mathrm{s}$, which is the lower bound set by the self-reproduction time of the ribosome. Larger cells ($L>L_{\mathrm{c}}$) have a lower surface-to-volume ratio and therefore self-reproduce more slowly, $T/T_{\mathrm{c}}\sim L/L_{\mathrm{c}}$. In smaller cells ($L<L_{\mathrm{c}}$), the cost of producing the membrane (of width $\mathrm{\Delta }\sim 10\mathrm{nm}$) slows down the doubling, as $T/T_{\mathrm{c}}\sim 1+18·\mathrm{\Delta }/L$.

The geometry of doubling sets the length scale $L_{\mathrm{c}}$ (Fig. 3). To self-sustain, the autocatalytic cycles must be confined within a boundary, a membrane; otherwise, newly produced machines would diffuse away, and the cycles die down. This boundary sets apart the “inside” where the self lies from the outside universe. The size $L_{\mathrm{c}}$ originates from the interplay between the need for a membrane large enough to feed the doubling process and the cost of producing this membrane.

To estimate the cost, consider a cell of size $L$ enveloped by a lipid membrane of width $\mathrm{\Delta }=$ 5 to 10 $\mathrm{nm}$ (54) that occupies a fraction $\varphi \sim 6\mathrm{\Delta }/L$ of the cell volume.^# Water molecules fill two-thirds of this volume, and the other third is taken by organic molecules (“dry mass”), mostly protein and RNA machines (66). Considering this factor of $1/3$, the membrane takes a fraction of $3\varphi \sim 18\mathrm{\Delta }/L$ of the net mass of all molecular machines. This is also the relative cost of generating the new membrane, which slows down self-reproduction by a similar factor (Fig. 3, Left). Setting an upper limit on the slowdown at $3\varphi \sim 20$% requires a minimal size of $L_{\mathrm{c}}\sim 6\mathrm{\Delta }/\varphi \sim 100\mathrm{\Delta }\sim$ 0.5 to 1 $\mathrm{\mu }\mathrm{m}$. In smaller cells, this cost further increases as $\sim L^{-1}$, becoming a severe drawback in a world governed by competitive exponential growth.

On the other hand, larger cells do not have enough membrane area to accommodate all the pumps needed to feed the doubling (Fig. 3, Right) because the surface-to-volume ratio, which is the ratio of flux to mass, decreases as $\sim L^{-1}$ as the cell grows. In Fig. 4, we give a rough order-of-magnitude estimate of the necessary energy flux, remembering that much of the organic chemistry in the cell is about reshuffling carbon atoms. The density of carbons in the cell is $n\sim 10{\mathrm{nm}}^{-3}$, and the synthesis of each carbon costs about the energy equivalent of one ATP ($\sim$20 kT).^‖ Hence, to double during a period $T_{\mathrm{c}}$, a bacterium that contains $L_{\mathrm{c}}^{3}n$ carbons, consumes the same number of ATPs, requiring an energy flux of $J_{\mathrm{ATP}}=L_{\mathrm{c}}n/T_{\mathrm{c}}\sim 10$ ATP/nm²/s (Fig. 4, Left).

Fig. 4.

Feeding the cascade. Left: Order-of-magnitude estimate on the energy flux needed to double the cell by estimating the cost of synthesis (by counting carbons at a cost of $\sim$1 ATP/carbon). Right: The proton flux and proton pump density required to provide the energy for cell doubling (see text).

The energy for synthesizing the ATP fuel is provided by pumping protons across the membrane potential (73). Each ATP costs $\sim$3 protons (69, 74), so the required proton flux is $J_{\mathrm{p}}\sim 3J_{\mathrm{ATP}}\sim 30$ proton/nm²/s (Fig. 4, Right). Assuming fast pumps, injecting currents of $i_{\mathrm{p}}\sim$ 300 protons/s (75, 76), one finds that the membrane must be packed with about one pump per $10{\mathrm{nm}}^2$.^** The bacterial membrane is therefore as much protein as lipid, and much more mosaic than fluid (78). A higher pump density might solidify the membrane into a protein shell that could not maintain the electric potential. Since the pumps approach their maximal packing density, larger cells with a lower surface/volume $\sim L^{-1}$ self-reproduce with a generation time of at least $T\sim T_{\mathrm{c}}(L/L_{\mathrm{c}})$ (Fig. 3).

Ultimately, we recognize the significance of the bacterial scales $L_{\mathrm{c}}\sim 1\mathrm{\mu }\mathrm{m}$ and $T_{\mathrm{c}}\sim {10}^3\mathrm{s}$: This is the most voracious consumer of resources, dissipating $L_{\mathrm{c}}^{3}n/T_{\mathrm{c}}\sim {10}^7$ ATP/s [power of 1 pW (46, 79, 80)] and the most industrious self-reproducer, using this power to synthesize $\sim$10⁷ carbons/s (81).^†† The physics of water sets both scales through geometric and logical constraints. Geometric separation of the self-reproducing machine from the surrounding water requires a lipid membrane, whose cost imposes a minimal size that is a hundred-fold larger than the membrane thickness, which is also the typical size of a protein machine, $L_{\mathrm{c}}\sim 100·\mathrm{\Delta }\sim$ 0.5 to 1 $\mathrm{\mu }\mathrm{m}$ (Fig. 3). The logic of self-reproduction in water sets the time scale. To double, millions of submachines must be produced and assembled in an elaborate combinatorial dance of molecules bumping and recognizing each other. Thus, each time the ribosome adds an amino acid to a newly built ribosomal protein, it has to try several tRNAs until it recognizes one carrying the correct amino acid. Moreover, to achieve higher accuracy, the ribosome proofreads each recognition step (62, 84, 85). All these molecular motions in salty water take time, adding up to a lower bound on the self-reproduction rate of $T_{\mathrm{c}}\sim {10}^3\mathrm{s}$.

3. Salty Water: Origin and Boundary of the Cascade

So far, we looked at the solution of life to the problem of survival through a synchronized self-reproducing cascade of machines. Nothing about this exotic solution makes sense unless one considers its origin and physical boundary: the world of salty water. Many believe the cascade was born in the hydrothermal vents buried deep in the ocean (4). There, the abundant currents of matter and energy suffice to feed an emergent cascade of machines. The richness of this bustling water world—with its multiple scales of time, space, information, and energy—is mirrored in the cascade (48).

Made of three light atoms, ${\mathrm{H}}_2\mathrm{O}$, water is a common molecule in the universe, but its properties are anything but common. This V-shaped molecule is a two-angstrom arm bent at a sharp 104.5^° elbow, endowing water with a strong electric dipole, rendering it highly polar (a dielectric constant $\sim$81) and prone to forming hydrogen bonds. All this gives water exceptional surface tension ($\sim$18 kT/nm²) and the capacity to induce strong hydrophobic and hydrophilic forces (86).

This rich electrodynamics makes water a superb solvent, one that can dissolve practically any polar or charged molecules. Thus, water becomes a perfect aether, a matrix upon which life can emerge (87). All organisms are mostly water, and all living machines are hydrated—with boundaries, singularities, and defects whose dynamics is the physical basis of life. First, there are ions: calcium, magnesium, sodium, potassium, chloride, sulfate, and others. These charged defects are so dense—ions are only 1 to 2 nm away from each other—that one should redefine the physical field of life to be that of salty water: water doped with ions.

Thus, living matter is the dynamic boundary of water, made mostly of carbon chains. For each carbon atom in the cell, there are, on average, two ${\mathrm{H}}_2\mathrm{O}$ (88). So each protein is surrounded by $\sim$10⁴ water molecules and $\sim$10² ions. In this logic of salty-water machines, the elementary components are ionic double layers. These thin layers form by swift rearrangement of water and ions, screening all electrodynamic forces and confining them within a Debye length of less than 1 nm (34, 89). Hence, the basic forces of life are all short-ranged, and long-ranged action can be achieved only effectively, as elastic and hydrodynamic forces propagate.

The thin hydration layers that envelop all molecular machines are essential for the cascade. The layers prevent undesired aggregation of machines in the crowded milieu of the cell that would disrupt the cascade (90), ending in various disorders, such as Alzheimer’s and Parkinson’s diseases (91). Instead of unspecific binding, the strong yet localized forces in the hydration layers equip the machines with the capacity to specifically recognize each other, opening the doors for accurate information-processing machines, and for the combinatorial explosion of the cascade by the assembly of more complex machines.

3.1. Universal and Nonuniversal Machine Making.

To a good approximation, living machines are combinations of carbon atoms immersed in water. Carbons readily polymerize and—aided by few other light atoms, mainly hydrogen, oxygen, nitrogen, phosphate, and sulfur—can span a broad spectrum of electrodynamic interactions with water and ions, opening infinite topological and chemical possibilities that life uses to build its machines. Much of this construction is universal: The cell, the materialization of von Neumann’s universal constructor, reads its self-description on the DNA tape, and uses it as a blueprint for synthesizing proteins by ribosome machines. Some of these proteins, such as enzymes, then function as nonuniversal constructors. These proteins form factories that perform organic chemistry reactions to produce all the machinery required for interfacing with the world of water. Notably, these nanoscale chemical factories produce the fatty acids that make the membrane machine, which forms the self-replicating machine’s boundary with the salty water outside. Thus, the cascade spreads from the universal scale of the whole self-replicating machine, the cell, down to the nonuniversal atomic scale of water physics.

A remarkable example of such nonuniversal construction factories is that of polyketide synthetases. These proteins form assembly lines for producing polyketides, chains of carbons alternately bound to oxygen (a ketone) or two hydrogens (a methyl) (92, 93). Examining the factory, using X-rays or cryo-EM, one finds that it contains several production units operating sequentially: The first module is loaded with the raw substrate, extends by two carbons, and channels it to the next module, which in turn extends it by two carbons and channels it further, down to the last module which completes the chain elongation. Looking at even smaller scales, one sees that each module comprises tinier submachines, protein domains that perform the substeps of binding, reshuffling chemical bonds and electrons, translocation, and release. The assembly line then sends its product to a battery of “posttailoring” enzymes, which further tweak the polyketide scaffolding according to the specification of its function in the cascade, often by closing the chain on itself to form aromatic rings or by sticking additional functional groups. As a result, the products of polyketide factories are highly diverse and, in particular, include many species of valuable antibiotics.

3.2. The Membrane Machine.

The same design principles of multimodular sequential assembly underlie the factories that make nonribosomal peptides and, most importantly, fatty acids, the building blocks of the membrane (92). Thus, we see that nonuniversal production is an essential part of the cascade of machine-making machines envisioned by Leibniz. The interaction of fatty acids with water gives the self-replicating machine its geometric identity. Water strongly attracts polar molecules as it fiercely repels symmetric carbon–carbon bonds. Thus, when phospholipids are immersed in water, their phosphate heads are attracted by the water while their lipid tails are repelled, making them stick together to form a membrane. This 5-nm thick fluid membrane is practically a two-dimensional singularity: Everything inside is the self and outside the nonself. This sharply defined compartment allows the self-replicating machine to amplify and regulate the concentrations of the interacting molecules and become an efficient chemical reactor.

But the membrane must not be perfect, completely isolating the self-replicating machine from the universe. Replication must be fed with large fluxes of matter, energy, and information. To allow such vigorous traffic, the fluid membrane must be semipermeable and densely packed with numerous pumps and pores. These gatekeeping machines regulate the traffic, but they cannot be perfect. Certain external agents will always sneak through the membrane, especially viruses that enslave the reproduction machinery for their own self-replication. Once the membranal line of defense is breached, the machine must develop means to tell the self from the intruding nonself and destroy intruders. Thus, the immune system is born.

3.3. Electrodynamic Life, Beyond Debye.

Dividing space into inside and outside, the membrane also defines the plus and minus sides of a battery or a capacitor. The battery is charged by ion pumps, protein machines that vigorously push ions in and out of the cell. By forming steep gradients of ion concentration, the pumps build up a voltage drop of $V_{\mathrm{m}}\sim 150\text{mV}$ across the $\mathrm{\Delta }\sim$ 5 to 10 nm membrane, thereby inducing a strong electric field of $V_{\mathrm{m}}/\mathrm{\Delta }\sim$ 15 to 30 $\text{kV}/\text{mm}$, the typical breakdown field of oil-filled transformers (94). But the phospholipid membrane is a superb ultrathin insulator that can sustain such fields without breakdown. It is also a superb boundary that keeps the battery from discharging by leaking ions. In particular, the membrane sustains a gradient of protons that powers the synthesis of ATP, the universal energy currency in the cell. Much like in a hydroelectric plant, the protons flow down their gradient through protein turbines called ATP synthase, and this proton-motive force (73) generates new ATP by adding phosphates to ADP molecules.

This steady-state macroscopic analogy to a battery, capacitor, or power plant captures some basic features of the membrane but misses its vibrant nature. This is because electrodynamics is generally assumed to be tamed in salty water: Whenever charges bounce around or are reshuffled during chemical reactions, the fields they emit are screened by swift rearrangement of water and ions around them. This physical picture of almost instantaneous screening, as introduced by Debye and Hückel (89), applies to large systems of dilute ionic solutions where water molecules are free to reorient in picoseconds. In the cell, these approximations may break down. In the membrane labyrinth crowded by molecular machines, water may be orders of magnitude slower, and hence, electrodynamics may be much less restrained.

In the catalytic cavities of protein machines, water molecules are slowed down by geometric confinement and the bonds they form with charged and polar amino acids (51). Ultraslow exchange of water between ions and protein pores dominates the dynamics of ion channels (95). Thus, the enormous fields that pierce through the membrane and its protein machines (96) violently pulsate as ions scuttle across the potential barrier. The dynamic fields shake the salty water over more extended periods and longer ranges than in the Debye–Hückel scenario. Pushing this speculation further, we can imagine how neighboring protein machines induce long-lived excitations that interact with each other, giving rise to extended dynamic modes that propagate around the membrane and the cell. In spores that are crammed with molecular machines and tenfold less water, such collective interactions may be even more potent (13).

What makes the breakdown of the screening picture even more likely is the high concentration of ions. As the ionic density $n_{\mathrm{e}}$ increases, Debye’s screening length decays faster (as $\sqrt{n_{\mathrm{e}}}$) than the distance between neighboring ions (which decays as $\sqrt[3]{n_{\mathrm{e}}}$). At physiological ionic densities of $\sim$150 mM, where the two length scales are comparable, the continuum picture breaks down. A sphere whose radius is Debye’s length is so minute and contains too few ions to screen electrodynamics, as suggested by recent surface force measurements of dense ionic solutions showing “underscreening” (97).

The ions that surround protein machines and actively participate in their function may dramatically restrain the screening of electric fields. In particular, magnesium ions take part in hundreds of enzymatic reactions, especially those driving the turnover of ATP. In the cell, what we call ATP molecules are, in fact, Mg–ATP complexes. The binding and unbinding rate of water to the Mg⁺² ion is prolonged by six orders of magnitude—microseconds compared to the picosecond rate of water rotation in bulk (34). The ion and its water shells can coordinate polar amino acids and become a bottleneck that hinders the release of product molecules.

All this hints that catalytic events induce slow, extended relaxation modes around the active sites. Minding such long-lived excitations, we may need to abandon the standard view of catalysis in the cell as a simple sum of localized random events and adopt a more dynamic view of interacting electrochemical machines (51). To find evidence in experiment, one should search for evidence for long-lived modes that spread over nanometers, for example, by measuring infrared scattering (98) from the vicinity of the active site and probing how it varies during the catalytic cycle, for example, using the Stark effect (96). Another path is looking for long-range correlations between macromolecules, indicating long-range recognition forces that steer binding surfaces toward each other (99).

In theory, one needs to explore beyond the simple two-body interactions of classical molecular dynamics. Two aspects of the interactions among water, ions, and proteins must be considered: the many-body nature of the forces, especially of network-forming hydrogen bonds, and their quantum-mechanical character, for example, in long-range van der Waals forces (100). Cooperativity and delocalized wave functions are two hallmarks of solid-state physics, suggesting that despite the lack of periodic symmetry, the worlds of water and crystals are closer than they seem.

3.4. Emergence and Expansion of the Cascade in Water.

So far, we have described the sophisticated machinery of the well-developed cascade, which evolves through genetic variation and selection. But we also argued that the organizing principles of the cascade stem from the physical boundary condition of salty water and its interactions with small molecules, which preexisted Darwinian evolution. What physical and chemical processes underlie life’s origin(s) is a question far beyond the present discussion. Here, we only mention certain general aspects in the language of the cascade:

The energy injected by the sun and the earth’s volcanic activity generated physical and chemical gradients that could drive cyclic currents of matter, energy, and information. The cycles could be harnessed to the chain reactions of self-reproduction and thus become the first machines; for example, simple thermal convection can drive the exponential replication of DNA polymers (101). The cycles provide the directionality needed for the emergence of the cascade from a pool of small molecules, for assembling the first nonrandom polymers, and for propelling the first motors along these polymers.

The motors become the reading heads of the first information machines, the copiers and constructors discussed in Section 2. Cascading brings forth molecular codes used by these machines. The first codes are noisy mappings between species of molecules. The mapping of nucleic bases to amino bases evolves into the genetic code, whose emergence can be seen as a transition in a noisy information channel (102). Mappings of fast recycling molecules become the first memories (48). With its generative power, the cascade combinatorially composes these information modules into a whole language.

Coding is essential for the emergence of the first selves, logically defined by the loops of self-reference (29) and self-reproduction of von Neumann’s machine. As for the geometrical definition, simple physical mechanisms that do not require biological machinery could have allowed primitive cells to form and compete with each other. Prebiotic amphiphile molecules can form membrane-like boundaries that may define a primitive “self,” even if not as sharply as our membranes (103, 104).

Already in the 1930s, Oparin proposed that macromolecules, such as protoproteins, could phase separate into viscous liquid droplets (105). Such liquid–liquid phase separation may generate dense ensembles of reactive molecules where concentrations are high enough to bootstrap autocatalytic circuits. These fuzzy protocells can then self-reproduce, growing by autocatalysis and then breaking, for example, by physical instability. Even if such doubling is highly inaccurate, we hypothesize that this is enough for life to begin with, and propose that the cascade will drive the following development of molecular technology, which set the period and size of the smallest self-reproducing machines at the present critical point ($L_{\mathrm{c}},T_{\mathrm{c}}$).

3.5. Summary.

As a summary, we write a provisional definition of life.

Life is a cascade of machine-making machines that solves the problem of survival by realizing in the world of water an autotrophic self-reproducing machine. The main features of the cascade are as follows:

• 1. Scaling down from the cell to atomic-size machines that interface with salty water.

• 2. Scaling up from the cell, filling all physically accessible scales, up to the whole biosphere.

• 3. Entanglement of matter and information: All machines have coded images.

• 4. “Self” defined by singularities in space, time, energy, and information:

  • – a semipermeable 2D boundary separates the self-reproducing machine from the water world.
  • – to self-reproduce: (i) high fluxes of matter and information are pumped through the boundary. (ii) the fluxes are used to double the machines and their images. (iii) the doubled matter and information are then cut into two new machines.

• 5. Open-ended evolution: Exponentially doubling self-reproducing machines compete over finite resources, leading to the appearance of new machines and new ways to organize them in a cascade.

Of these features, some are inherited from the abstract world of logic machines. The cascade inherits from von Neumann’s design coupling of machines to their coded images, the internal representation, and the procedure of self-reproduction by doubling and cutting. Other features originate from the physical boundary conditions of salty water, in particular, scaling down to atoms, electrons, and protons. This heuristic definition focuses on the cascade and does not mention explicitly other central aspects of life, such as homeostasis, which we consider an essential consequence of the interactions among different levels of the cascade and the environment. The cascade is an inherently open, collective system made of cells and their molecular machines. While the cell sets the scales of the critical point, it is never an isolated entity. Cells emerge, evolve, and operate as parts of this cascade and their functions emerge from the hierarchical organization of the cascade.

4. Conclusion

The cascade of machines introduces a universal language into a realm filled with very particular, intricate phenomena. Biological systems are traditionally fragmented into genetic and nongenetic, single-cell and multicellular, and so on. The cascade proposes an oversimplified language to look at all of them through one perspective, just at different scales. None of the phenomena described here is new, only the simple idea that they can be understood as an infinite hierarchy of machine-making machines, and that this hierarchy has a critical point at the scales of a self-reproducing machine, 1 micron, and ${10}^3$ s.

For example, tracing the history of regulation in biology, from genes to epigenetics (DNA methylation and histone modifications) to small RNAs, the notion of cascade suggests that this hierarchy expands to even smaller scales. It suggests that other layers of regulation loops are yet to be discovered, most speculatively, at the subnanometer scale of ionic charges and water. There, the ingrained view that electrodynamic interactions are screened might break down at physiological ion concentrations. This would open the possibility of rich collective interactions among charged surfaces of biological machines. Circuits of machine-making machines, codes, memories, and loops should be sought at all levels, following Leibniz’s insight about the micro-cascade that living machines are “still machines in their smallest parts” (5), and also in the other direction, up the macro-cascade of large organizational networks.

Figuratively speaking, the cascade is born when a von Neumann logic machine is thrown into a salty sea of Maxwell’s and Schrödinger’s equations. This injection of engineering into the physical world occurs at the critical scales of the cell, $1\mathrm{\mu }\mathrm{m}$ and ${10}^3$ s, and the cascade rapidly expands into the digital information world of Darwin (49), Mendel (106), and Crick and Watson (18), but remains tied, all the way down to atoms, to the dissipative world of water. At the high end of the cascade rests our civilization machine. Homo sapiens emerged only 300,000 Y ($\sim$10¹³ s) ago (107, 108), a blink of an eye compared to almost 4 billion years of evolution since the origin of life on earth ($\sim$10¹⁷ s). And in no more than 20,000 Y ($\sim$6 $·$ 10¹¹ s) of civilization and agriculture (109, 110), humans enslaved the whole planet ($L\sim {10}^7$ m). The astonishing speed of this takeover was facilitated by our brains.

Thus, at the end, we see that thinking about life in this language of the cascade is an act of self-reference. Humans have the idea of a self, self-consciousness (not discussed here at all), which leads to the idea of a self-reproducing machine. This reflection returns us to the seventeenth century, to Leibniz. His exploration of the cascade was motivated by the Pensées of Pascal (111, 112). There, questing for man’s place in the universe, Pascal finally finds him suspended “between those two abysses of the Infinite and Nothing” and concludes: “For in the end what is man in nature? nothingness against the infinite, everything against the nothingness, middle of nothing and everything …” (113). We can understand Pascal’s two abysses as the cascade, stretching from almost nothing to almost everything—with “almost” signifying the far-apart physical limits of quantum mechanics on extreme smallness and cosmology on extreme hugeness. Some 350 y later, this luminous prose still lacks a fitting mathematical language.

Data, Materials, and Software Availability

All study data are included in the main text. Previously published data were used for this work (see references).