# A dynamical framework for biological reproduction

Romain Brette

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# A dynamical framework for biological reproduction

Romain Brette¹

¹ Sorbonne Université, CNRS, Institute of Intelligent Systems and Robotics (ISIR), 75005 Paris, France

Corresponding author: Romain Brette, [romain.brette@inserm.fr](mailto:romain.brette@inserm.fr)

## Abstract

Classically, biological reproduction is depicted as the building of a new organism from replicated genomic instructions. The corresponding theoretical model is von Neumann's self-reproducing machine, which relies on an invariant universal constructor that can build any machine from instructions. However, the reproductive incompatibility of species and the diversity of developing processes speak against the existence of a universal constructor. Without a universal constructor, the genome as representation of the organism is circularly defined: what the genome represents is specified by the developmental processes represented by the genome.

I propose to take invariant reproduction not as a premise, but as an emergent dynamical property. Reproduction is seen as the iteration of a transform that maps one generation to the next, a transform shaped by the genome. Invariant reproduction then occurs when a reproductive sequence converges to a fixed point. A reproductive sequence may also diverge, converge to a cycle (multigenerational life cycle), or to one of several fixed points (non-genomic inheritance). When it does converge, it is necessarily to a stable point, implying that development is robust to perturbations. Finally, speciation can be understood as a process by which reproductive transforms become mutually incompatible, that is, the basins of attractions of the fixed points do not overlap any more.

In this view, the genome is a transmissible constraint on development, not a representation of the organism. I suggest that this dynamical framework is a more coherent model of biological reproduction than von Neumann's computational framework.
Introduction

Reproduction is an extraordinary property of living organisms and, consequently, a cornerstone of many biological theories. In *Chance and necessity*, the molecular biologist Jacques Monod argued that living organisms have a “teleonomic project”, which is invariant reproduction (Monod, 1970) – the concept of teleonomy was previously introduced by Pittendrigh (1958) and developed by Mayr (1961). To him, as to many contemporary biologists (Nicholson, 2014), an organism is a particular kind of machine programmed to reproduce itself. Evolutionary theory, whether in Darwinian or Lamarckian form, starts from the premise that living organisms can reproduce themselves, so that changes can propagate to future generations.

How do living organisms manage to reproduce themselves? The modern view of biological reproduction is that the living organism stores building instructions in the genome, which are copied over generations, ensuring faithful reconstruction. In the words of Sydney Brenner: *“The concept of the gene as a symbolic representation of the organism — a code script — is a fundamental feature of the living world and must form the kernel of biological theory.”* (Brenner, 2012).

Although very popular, the idea of the genome as representation of the organism suffers from known empirical and conceptual issues, in particular regarding the influence of the environment on development (Jablonka and Lamb, 2014; Oyama, 2000; Witzany and Baluška, 2012). Many previous critiques have also targeted genetic reductionism (one gene for one phenotype or instruction) and offered alternatives to the “program” and “blueprint” metaphors (e.g. “symphony” (Noble, 2008), “recipe” (Mitchell, 2018) or “generative model” (Mitchell and Cheney, 2025)). In this paper, I will focus on a more fundamental conceptual issue with the general idea of genome as representation of the organism. What organism does the genome represent? It represents the organism that grows from the egg that holds that genome. But not all eggs would do: it must come from the same species, or from a very close one (Beyhan et al., 2007). For example, if the nucleus of a chicken cell is transferred to a rabbit’s oocyte, the embryo develops to the blastocyst stage, a stage that exists in rabbits, not in chicken (Liu et al., 2004). Thus, the genome represents the adult organism, but only when it is expressed in the organism that it is supposed to represent. In other words, what the genome represents is defined circularly.

This paper offers a simple solution to this conundrum, based on dynamical concepts (Figure 1). Most discussions about the genomic code focus on animals, using the developmental program metaphor. I will start from the more fundamental problem of unicellular reproduction, which has been the dominant form of reproduction for most of life's history on Earth. I will later discuss how the arguments apply to multicellular reproduction. Consider for example the asexual reproduction of the budding yeast, where a bud grows and then detaches from the parent. How is it that this process produces a daughter that looks exactly as its parent? In Brenner's view, the adult organism is specified by its genome *g*, which is why the daughter is identical to its parent (Fig. 1a). But an organism always originates from another organism by a development process, never from a naked genome. Thus, an organism is mapped to the next organism, by a transformation that I shall call the *reproductive transform* *R* (Fig. 1b); this corresponds to the developmental and division process. The genome constrains this map, and is transmitted to the next generation. We do not presuppose that this process necessarily produces a copy of the parent. When we iterate the reproductive transform, we obtain a sequence of organisms with the same genome $O_{n+1} = R(O_n)$, where $O_n$ is the organism at generation *n* (Fig. 1c). When and if this sequence converges, the organism becomes a fixed point of *R*, that is, there is invariant reproduction and we can then speak of a “life cycle”: the process *R* transforms the organism into the same organism. In this view, invariant reproduction is an emergent dynamical property, rather than the making of a copy from a blueprint.
<figure><img src="image_2.png" /><figcaption>**Figure 1.** The dynamical view of reproduction, illustrated with the budding yeast. a, The genome as a representation or "blueprint" of the organism. b, The reproductive transform. The genome is a transmissible constraint on development. c, The reproductive sequence, converging to a fixed point, where reproduction is invariant. The blue disc is the nucleus, and the green shape is the vacuole.</figcaption></figure>

This dynamical framework has a number of implications. First, the organism cannot be directly deduced from the genome because the observable organism is recursively defined: it is that organism that is left unchanged by the reproductive transform constrained by the genome. In fact, in case the genome specifies a reproductive transform with several fixed points, several viable organisms are consistent with the same genome, as occurs in ciliates, as we will see. Second, the reproductive sequence may converge to a fixed point, but it may also converge to a cycle, as in species where the life cycle is multigenerational (Fusco and Minelli, 2019). Third, the reproductive sequence may diverge if the initial organism is not in the basin of attraction of the fixed point associated with the genome. This is what happens generally when a cell's nucleus is transferred to the oocyte of another species (Beyhan et al., 2007). Fourth, developmental robustness follows from the organism being at a fixed point of the reproductive transform. Finally, speciation can be understood as a process by which the basins of attractions of the fixed points do not overlap any more, which can occur in particular by through bifurcations.

Before I develop these themes, I will start by substantiating the concept of organism as a fixed point of the reproductive transform.

## The reproductive transform

### The organism as reproducing machine

In his famous book, *What is life?* Schrödinger (1944) argued that the nucleus must contain some stable molecules, "aperiodic crystals", that store the information to make an organism: "*these chromosomes [...] contain in some kind of code-script the entire pattern of the individual's future development*". Shortly after, John von Neumann explained how a machine could reproduce itself: it was an (abstract) automaton able to build any automaton from a description, and to copy the description (Neumann and Burks, 1966) (he presented it in lectures in 1949). This engineering perspective had considerable influence on molecular biology and more generally on biological theory (Nicholson, 2025).
Von Neumann's idea was popularized in *Scientific American* in an article entitled "*Man viewed as a machine*" (Kemeny, 1955). His machine was an (abstract) automaton with several parts (Fig. 2). Part A was an automaton able to build another automaton from a symbolic description. Part B was an automaton able to replicate a description. Part C was an automaton controlling the first two automata. Then, given any automaton D, a self-reproducing machine was built as the combined automaton A+B+C+D together with its description. The sub-automaton A would build the complete machine (A+B+C+D) from its description, and the sub-automaton B would copy the description. The modern biologist will identify the synthesis of proteins from DNA with automaton A and the replication of DNA with automaton B.

<figure><img src="image_3.png" /><figcaption>Figure 2. Von Neumann's self-reproducing machine.</figcaption></figure>

Von Neumann also considered the possibility of mutations. He noted that changes to A, B or C would lead to a sterile individual, because those parts are responsible for building the new machine from the description, including A, B and C themselves. In contrast, changes to D would propagate to the new machine, and would therefore support evolution.

I will now explain why von Neumann's self-reproducing machine, despite being very influential, is a poor model of biological reproduction.

### The limitations of von Neumann's reproducing machine

In this model, invariant reproduction is ensured by an invariant core (A+B+C) (Fig. 2), in charge of building any machine, which includes a replicator and a "universal constructor". Only changes to the automaton D are allowed, and any change to D leads to a viable machine (one that can self-reproduce).

If this were a good model of biological reproduction and evolution, then all biological organisms should share a set of universal mechanisms to build an organism from DNA. We know this is not true. For example, ciliates have a different genetic code from other eukaryotes, that is, nucleotide triplets are mapped to different amino acids (Lozupone et al., 2001). In chicken, development depends on mechanisms that produce the egg, but those mechanisms do not exist in humans, where development depends on the placenta. More generally, development depends on many factors that are in fact species-specific (Fusco and Minelli, 2019). As already mentioned, when a cell's nucleus is transferred
to the oocyte of another species, development general fails, unless the two species are closely related (Beyhan et al., 2007). Thus, there is no such thing as a universal constructor across life.

It appears that von Neumann's model fits the obsolete theory of evolution that Darwin was arguing against in *The Origin of Species* (Darwin, 1859), where only closely related species such as wolves and dogs would have common ancestors, each of those lineages independently created. Indeed, Darwin's contemporaries were perfectly aware that species could evolve, since this is what humans had been doing for ages with the artificial selection of domesticated species, Darwin's main source of inspiration for his theory of natural selection. What was at stake was the more radical idea that all species, including humans and earthworms, were in fact cousins (hence the famous caricature of Darwin as an ape).

This common ascendance is not disputed anymore, and this implies that evolutionarily significant mutations must also apply to the core of the “reproducing machine”.

How large is this core? Here we touch on another issue in von Neumann's model, namely that the reproducing machine builds a new machine outside of itself. Thus, the machine's core is not only invariant through generations, but also through the reproductive process. This is never the case in biological reproduction. It may superficially appear so because the chicken grows in an egg, spatially separated from the parent, but in reality, all biological reproduction relies fundamentally on cell division. A cell (e.g. an oocyte) does not reproduce by building a new cell, but by dividing. In this process, what we call the parent cell is the cell before splitting; it is in material continuity with the two daughter cells. The parent does not build the children; rather, children are pieces of the parent. This is particularly striking in ciliates, which are unicellular eukaryotes with complex structures. For example, during the asexual reproduction of *Tetrahymena*, a new oral apparatus forms near the center of the cell before the cell divides (Fig. 3). The anterior half conserves the oral apparatus of the parent, while the posterior half has a freshly built apparatus, but inherits the anal pore of the parent (tiny segment at the bottom). Thus, the formation of new structure depends on the cytoplasm and existing structures of the parent, which the two daughters also inherit.

<figure><img src="image_4.png" /><figcaption>Figure 3. Binary fission of *Tetrahymena* (Cole and Gaertig, 2022).</figcaption></figure>

This fact of cellular division applies to the reproduction of all multicellular organisms. Besides, many multicellular organisms also divide macroscopically by splitting. For example, many plants can reproduce from fragments, a mode called *vegetative reproduction*. In this case, the plant does not make a copy of itself, but rather splits and then further develops. Similarly, planarians (flatworms) can reproduce by fission: the animal tears itself into two pieces that then grow into two recognizable planarians (Malinowski et al., 2017).
Thus, there is a continuity between parent and child, because biological reproduction is primarily a
process of growth, not a process of assembling a new machine from external pieces. It follows that, in
biological organisms, we cannot distinguish an invariant universal building apparatus from a variable
part isolated from the reproductive process.

We conclude from these evolutionary and developmental considerations that biological reproduction
does not follow von Neumann's model. We now examine what happens when we reject the idea of an
invariant universal constructor within biological organisms.

Reproduction without a universal constructor

In contrast with von Neumann's machine, a chicken does not build another chicken. It lays an egg, and
the embryo then develops into an animal that might resemble the parent. Thus, to assess whether
reproduction is invariant, we must choose a particular time point in the life cycle, which could be the
time of reproductive competence (i.e., the adult chicken). We shall call *reproductive transform R* the
mapping from the parent organism *O* to the child *O*\* at a comparable stage of development (Fig. 1b).
In von Neumann's framework, *R* maps the original machine to the new machine, but here we do not
assume invariant reproduction (i.e., *O*\* = *O*).

An important subtlety is that a cell does not produce a new cell: it splits. Thus, it would be more
accurate to consider that the parent is mapped to two daughters. I will ignore it for now (e.g., picking
the anterior daughter), and return to this point later.

What is the role of the genome in the reproductive transform? Focusing for now on asexual
reproduction, the genome has a special role because it does not grow: it is replicated. Therefore, it is
(essentially) invariant through the reproductive process. On the other hand, it influences the
development. This means that the genome should be conceptualized as a *constraint*: something that
has causal effect on a phenomenon but is not affected by it (Brette, 2026; Juarrero, 2023; Kauffman et
al., 2008; Montévil and Mossio, 2015) – a classic example from biochemistry is the enzyme, which
channels certain reactions. Thus, we conceptually separate the genome *g* from the rest of the organism
*O*, and consider that *R* is the transformation from *O* to *O*\*, constrained by the genome *g* (Fig. 1b). This
conceptual distinction is similar to von Neumann's distinction between the machine and its
description, except the genome is not a description but a constraint. Leaving aside the influence of the
environment for now, *g* specifies the reproductive transform *R*, rather than the outcome *O*\* of the
process.

In this framework, invariant reproduction means $R(O) = O$, that is, the organism *O* is a fixed point of the reproductive transform *R*. Different genomes specify different transforms, and therefore different reproductive fixed points. This is illustrated very schematically on Figure 4, where the reproductive transform is shown as a one-dimensional nonlinear mapping.
<figure><img src="image_5.png" /><figcaption>Figure 4. The reproductive transform maps the parent *O* to the child *O*\* (curves). The drawings represent the ciliate species *Stylonychia mytilus* and *Vorticella* (from Jennings (1906)). The fixed point of the transform constrained by the *Vorticella* genome *g*₁ (resp. *Stylonychia* genome *g*₂) is the typical *Vorticella* organism (resp. *Stylonychia* organism).</figcaption></figure>

This formalism relies on a separation between the object of development and the constraints of development, i.e., what changes through development and what does not, at the relevant time scale. Constraints are associated with the transform $R$, while the objects of development $O$ form the domain of the transform. As a first approximation, we considered that the genome $g$ is the main constraint, and $O$ is the rest of the organism. This could be refined to take into account various factors influencing development. For example, organisms develop in a particular environmental niche, which can affect the developmental process in various ways (Gilbert and Epel, 2009). The niche then becomes a constraint alongside the genome. In contrast, when maternal behavior or physiology (e.g. placenta) influences development (Fusco and Minelli, 2019), these factors must be included in the domain of the reproductive transform, since they are both a parental condition and an outcome of development.

Let us see how this dynamical perspective accounts for various aspects of biological reproduction.

## Dynamical features of reproduction

*Convergence to invariant reproduction*
<figure><img src="image_6.png" /><figcaption>Figure 5. Convergence of a reproductive sequence to a fixed point, where reproduction is invariant (drawings of paramecia from Jennings (1906)).</figcaption></figure>

Figure 5 shows a schematic illustration of a reproductive sequence that converges to a fixed point. As I have previously argued, invariant reproduction cannot be a universal feature of reproduction. Therefore, in general, starting from a reproducing organism, the next organism in the reproductive sequence would be a different organism, even if the genome is unchanged. However, if the sequence converges to a fixed point, then parent and child become essentially identical: there is invariant reproduction. Thus, only the beginning of the reproductive sequence shows different organisms, and then we always observe invariant reproduction.

Cases of reproductive convergence have been well documented in ciliates. Jennings (1908) occasionally observed morphological peculiarities in *Paramecium caudatum*, and studied their descendance. Figure 6a shows an example of an individual with a small posterior spine. One of the two daughters inherits the spine. After two more divisions, the spine has disappeared and all descendants appear similar. Other examples are provided by Calkins (1911), who cut paramecia in two. In some cases, the cut cell can divide, and he observed their descendance. Usually, the first division resulted in one normal-looking cell and one abnormal cell. He then reported that *"in a few cases [the abnormal cell] continues to divide with a gradually decreasing abnormality until normal forms are regained"*, as shown in Figure 6b.

<figure><img src="image_7.png" /><figcaption>Figure 6. Reproductive convergence in *Paramecium caudatum*. a, Descendance of an individual with a spine (Jennings, 1908). Only the descendance of abnormal-looking cells is shown. b, Descendance of an individual cut below the oral groove (Calkins, 1911).</figcaption></figure>

An important consequence of this dynamical view is that the typical organism, seen as a fixed point of the reproductive transform, has a certain basin of attraction, meaning that it is stable with respect to various alterations. In other words, development is robust because we observe the organism near a fixed point. There is no robust development into a spiny paramecium (Fig. 6a), or into a truncated paramecium (Fig. 6b), but only into the organism that is a reproductive fixed point.
Waddington aptly explained the robustness of development in dynamical terms (Waddington, 1942), namely that the organism is an attractor of the developmental process. He suggested that “canalization” processes must have been selected for. I suggest that the primary reason why development is robust is that the observable organism must lie at a stable fixed point of the reproductive transform. Secondarily, it seems likely that natural selection would then enlarge the basin of attraction of the fixed point, as I will discuss later.

I will give a specific example in the context of size regulation. In *Paramecium aurelia*, transcription occurs in a large polyploid nucleus, called the macronucleus, which holds hundreds of copies of each gene. During the cell cycle, the macronucleus grows by replication then it divides amitotically. This division results in slight random differences in DNA content between daughters, of about 3% (Berger, 1988; Berger and Schmidt, 1978). Yet, we observe that, from one generation to the next, both the mean number of gene copies and its variance are stable. The constancy of size across generations is usually presented as evidence of regulating mechanisms. For example, Jorgensen and Tyers (2004) remark that “*when the normal size distribution is disrupted by some insult, it is usually restored quickly following removal of the insult, further suggesting that cell size is under homeostatic control*, [which] requires a coordination of growth with division”.

There is no doubt that cells have sophisticated regulatory mechanisms, but the mere fact that size is stable does not necessarily require coordination or control. In the case *Paramecium*’s macronucleus, Berger and Schmidt (1978) carefully analyzed the variance of DNA content across cell divisions, and found that a simple model accounting for their observations was an additive change in DNA content during replication, followed by division. Leaving stochasticity aside for now, this means that DNA amount *x* would change over successive cell cycles according to the following formula:

$$x_{n+1} = (x_n + \delta x)/2$$

where $\delta x$ is the amount of replicated DNA. This sequence converges to $x = \delta x$, where replication then leads to an exact doubling of DNA content. Moreover, the mapping from $x_n$ to $x_{n+1}$ is a contraction mapping, meaning that it maps any pair of values to two closer values (because $1/2<1$). The fixed point is therefore a global attractor of the dynamics, and it follows that it is also stable when noise is introduced. DNA content is stable over generations and exactly doubles in the growth phase, and yet, this model does not include any coordination between growth and division. This is a case of a robust invariant property of the organism that emerges from the existence of a fixed point of the reproductive transform.

## Diverging reproducing sequences

In this dynamical view, invariant reproduction is not a fundamental property. A reproductive sequence may converge, but it may also diverge. Jennings (1908) gives a number of examples of diverging reproductive sequences in *Paramecium*. Figure 7a shows the descendance of an irregular individual. One of the two daughters appears normal, but the other one develops into a “monster” with two mouths and digestive systems. This monster divides another time; the smaller daughter lives and divides, the other one dies after two days. Similar examples are shown by Calkins (1911) in cut cells. Mottram (1941) observed that transiently exposing paramecia to various blastogenic agents would produce monsters that would then repeatedly divide into cells with heterogeneous abnormalities, even when the blastogenic agent was withdrawn. Figure 7b shows the descendance of an abnormal cell over two days. Jennings observed such diverging sequences in normal cultures, but only in a minority of cases. This is expected since, by definition, converging sequences are stable while diverging sequences are not, and may ultimately die.
<figure><img src="image_8.png" /><figcaption>Figure 7. Diverging reproducing sequences. a, Descendance of an irregular paramecium (Jennings, 1908). b, Descendance of an abnormal paramecium exposed to heat shock, after one and two days. c, Schematic representation of the reproductive transform of two species (“frog” and “chicken”). Each transform has a stable fixed point corresponding to the typical organism (filled circle), and two unstable fixed points. When the frog’s genome is transferred to a chicken’s embryo, the reproductive sequence diverges.</figcaption></figure>

Another important case is when the nucleus of one species is transferred to the oocyte of another species (Beyhan et al., 2007). In general, this does not yield a viable individual, although the oocyte may divide a number of times. This is illustrated in Figure 7c. Here, the frog's reproductive transform has a stable fixed point, but with a finite basin of attraction delimited by two unstable fixed points. When the frog's genome is inserted into the chicken's egg, a reproducing sequence starts outside the basin of attraction and diverges. Closely related species may allow for viable nucleus transfer because one's fixed point lies in the basin of attraction of the other's fixed point. This allows us to propose a reformulation of the species concept in dynamical terms, as a group of individuals whose reproductive fixed points lie in each other's basin of attractions. We will return to this theme later when we discuss speciation and evolution.

Finally, another example of diverging reproductive sequence is clonal aging. In *Paramecium*, cells dividing by binary fission look very similar to their parents, but over tens of divisions, differences appear. For example, *Paramecium tetraurelia* can undergo a form of self-fertilization called autogamy (Wichterman, 1986). After autogamy, the cell will divide by binary fission, and the descendants will go through several phases. Until about 20 divisions, paramecia are not able to undergo autogamy. Then they can reproduce by autogamy, when conditions are stressful (e.g., lack of food). Otherwise, they continue dividing by binary fission, then start aging and ultimately die after about 200 divisions. Autogamy “rejuvenates” the clone and allows reproduction to continue indefinitely. Thus, if we focus on vegetative reproduction, the reproductive sequence slowly diverges. If we extend our analysis to the entire reproduction process, then we see that the reproductive sequence follows a stable cycle, rather than a stable fixed point.

Stable cycles and multigenerational life cycles

Indeed, iterations of a map may converge to a fixed point, diverge, or converge to a cycle (other possibilities exist, such as chaos). This is illustrated for a two-generation cycle in Figure 8. Many
species have multigenerational life cycles. Medusozoa, for example, alternate between larva, polyp and medusa (Fusco and Minelli, 2019) – in practice, such life cycles usually mix sexual and asexual reproduction, and therefore are more complex than our formalism.

<figure><img src="image_9.png" /><figcaption>Figure 8. A multigenerational life cycle resulting from a stable cycle of the reproductive transform.</figcaption></figure>

### Multiple fixed points and heredity beyond genes

A reproductive transform may have several stable fixed points. In this case, the same genome (or more generally developmental constraint) may support the development of very different organisms. Again, this case has been well documented in ciliates. In *Paramecium*, cells exchange genetic material by a sexual process known as conjugation, where two cells transiently join their cytoplasms, before detaching. Sometimes, the cells fail to detach and form a doublet, with two mouths and two digestive systems (Fig. 9a) (Sonneborn, 1964, 1963). In some cases, the doublet may then divide again for many generations (Fig. 9b).

<figure><img src="image_10.png" /><figcaption>Figure 9. *Paramecium doublets*. a, A cell with two mouths and two anuses (Sonneborn, 1964). b, Fission of singlet (left) and doublet (right), with the fission and growth of the macronucleus (Sonneborn, 1963).</figcaption></figure>

This phenomenon has been traced to the developmental processes of *Paramecium* (Beisson, 2008; Beisson and Sonneborn, 1965; Jerka-Dziadosz and Beisson, 1990). When *Paramecium* grows, it must generate new cilia, which grow from basal bodies. The basal bodies are regularly organized along antero-posterior meridians (called rows), except for the oral region, which has a more complicated
structure. New basal bodies are produced by budding: a new basal body grows on the side of an existing one, then detaches and inserts itself in the existing structure. The orientation of the basal body is determined by the local orientation of the neighbors. In this way, the existing structure drives the formation of the new structures. In some accidents of detachment after conjugation, one of the cells acquires a piece of its partner, where basal bodies can be oriented in the opposite way, resulting in an unusual swimming behavior (“twisty”) (Beisson and Sonneborn, 1965). When the cell grows, the structural pattern propagates over several generations until an entire row has the inverted pattern, and then the cell reproduces faithfully for many generations, with the twisty swimming phenotype. This is a case of convergence to an atypical fixed point of the same reproductive transform.

Sonneborn remarked that doublets converge to singlets (normal cells) after a few generations if the two oral meridians are less than 90° apart (Sonneborn, 1963). Otherwise, they may reproduce indefinitely. Thus, singlets and doublets are two stable fixed points, and the singlet has a basin of attraction corresponding to a 90° angle between the two oral meridians.

A direct consequence of the existence of multiple fixed points is that the genome (or any constraint shaping the reproductive transform) is not the only source of long-term hereditary transmission. The case discussed here was called "structural inheritance". There are other known epigenetic modes of transmission (Jablonka and Lamb, 2008), for example cytoplasmic inheritance. The genetic transcription network has many stable modes, which support gene regulation. When the cell divides, since the two daughters inherit the cytoplasm of the parent cell, these modes may be conserved through the division.

It is rare to observe *Paramecium* doublets, and therefore we consider the singlet form as the "normal" *Paramecium*. Sonneborn pointed out that, since doublets appear viable and perfectly functional, the reason for their scarcity is likely natural selection: if the growth rate of doublets is even slightly lower than that of singlets, then singlets would quickly dominate. However, such effects are contextual; they depend on the particular environmental niche where the organisms happen to live. It does not seem completely implausible that, under different circumstances, doublets might thrive. In fact, Sonneborn (1963) pointed out that certain protists appear morphologically to be multiples, like *Giardia* or *Theutophrys*, and suggests that their evolutionary origin might be a merging event.

Even when a single fixed point dominates, the existence of multiple fixed points of the reproductive transform implies that it is impossible to infer the organism from its genome, not only because the relation is very indirect, but also because which fixed point comes to dominate depends on the ecosystem, and not just on the reproductive transform. For example, it is not possible to infer from the genome of *Paramecium* whether the dominant form is a singlet or a doublet.

We now examine the implications for evolutionary theory.

## Dynamics and evolution

### Dynamical selection

Classically, evolutionary theory takes invariant reproduction as the starting point, and aims at explaining change. Thus, evolution occurs when some alteration is invariantly reproduced. The debate then revolves around whether alterations are random (mutations) or not (e.g. epigenetic changes induced by the environment). Here we inverted this perspective, and tried to explain how an organism that changes continuously through the life cycle can reproduce identically. In this dynamical perspective, change is the default: the reproductive transform maps an organism to another organism.
Thus, there is an endogenous evolution in the absence of alterations (whether mutations or environmentally-induced). Is this evolution adaptive? In a sense, it is: when the organism is in the basin of attraction of a fixed point, it converges to a form that reproduces faithfully and robustly.

What happens when there is an alteration? In some cases, the organism will exit the basin of attraction and will converge, over a number of generations, to another fixed point. As is well-known in stochastic optimization, the result of such random hopping is that the organism is more likely to end up in the largest basin of attraction, i.e., the most robust form. This leads to a principle of dynamical selection of the most stable forms, a principle that is not competitive and does not require differential reproduction rates. This might explain why, in *Paramecium*, the dominant form is singlets rather than doublets: if spontaneous merging events are less frequent than spontaneous reversions (from doublets to singlets), then singlets will dominate – all else being equal, of course.

## Interaction with natural selection

This principle of dynamical selection interacts with the principle of natural selection. For example, an alteration resulting in a diverging reproductive sequence would, by definition, be selected against. This reinforces the dynamical selection of stable forms.

Let us now consider specifically the classical case of a genomic alteration, or more broadly the alteration of the reproductive transform (which could include constraints other than genomic). The most straightforward case is when this alteration results in a slight shift of the fixed point. In this case, the reproductive sequence quickly converges to the new fixed point (Fig. 10a). This altered fixed point will dominate the population if its net reproduction rate is higher. This is essentially the classical case of natural selection, with the qualification that the effect is contextual: the same genomic alteration may have different effects on the different fixed points of the reproductive transform. For example, it could be beneficial for one fixed point and detrimental for the other. What gets selected is the alteration that is beneficial for the fixed point that has been dynamically selected (i.e., the present form of the organism). It follows that natural selection should tend to reinforce the currently selected fixed point.

Another case is when the genomic alteration leaves the fixed point unchanged, but changes its basin of attraction (Fig. 10b). An alteration leading to an increase in the stability of the fixed point is likely to be beneficial, first because it avoids diverging accidents, and second because, as argued above, natural selection would tend to increase fitness in the vicinity of the fixed point, and therefore an alteration leading to a quicker return to the fixed point is likely to be beneficial. Thus, the interaction of dynamical and natural selection should lead to strengthening invariant reproduction. This would explain, for example, the emergence of the sophisticated mechanisms that regulate cell division.

a

<figure><img src="image_11.png" /></figure>

b

<figure><img src="image_12.png" /></figure>

c

<figure><img src="image_13.png" /></figure>
Figure 10. Alterations of the reproductive transform (e.g. mutations). a, Shift of the fixed point. b, Change in fixed point stability. c, Bifurcation: a fixed point disappears.

An alteration may also be silent, leaving the vicinity of the fixed point unchanged, while changing the reproductive transform far from the current fixed point. This should result in a hidden drift of the reproductive transform, far from the fixed point.

Finally, an alteration might cause a fixed point to disappear or become unstable (Fig. 10c). In dynamical systems theory, this is called a bifurcation. In this case, the reproducing organism may converge to another stable fixed point. Thus, a gradual alteration of the reproductive transform can lead to a continuous change in the fixed point, until a bifurcation is reached, where the invariant reproducing organism suddenly becomes very different. This leads us to the question of speciation.

### Dynamical principle of speciation

Classically, the concept of species revolves around the reproductive isolation of a group. In this dynamical framework, a species can be seen as a group of individuals whose reproductive fixed points lie in each other's basin of attractions. If we consider two similar groups of individuals, speciation then corresponds to an event when the fixed point typical of one group moves out of the basin of attraction of the other group's reproductive transform.

<figure><img src="image_14.png" /><figcaption>Figure 11. Dynamics of speciation. a, Two reproductive transforms (solid black and blue curves) evolve separately (thin curves), such that fixed points exit the basin of attraction of the other reproductive transform. b, A reproductive transform with two fixed points evolves in two different ways (green), leading to two reproductive transforms with fixed points outside the other's basin of attraction. c, A reproductive transform (1) gradually shifts, creating a new fixed point (2), then deleting the initial fixed point (3), and finally shrinking the basin of attraction (4).</figcaption></figure>

One way this can happen is by an accumulation of alterations, as illustrated on Figure 11a. First, the two groups may have slightly different reproductive transforms, perhaps because of adaptation to a slightly different environment (thick black and blue curves). These differences could be very subtle. Then each group undergoes alterations of its reproductive transform. As argued above, in a constant environment, natural selection could favor silent alterations, but an alteration that is silent to one group is not necessarily silent to the other group. Thus, the expectation is that the fixed points evolve slowly, but the rest of the reproductive transform may drift considerably. At some point, one group's fixed point may leave the basin of attraction of the other group's fixed point (Fig. 11a, thin curves). When this happens, transferring the genome from one group to the other would start a diverging reproductive sequence. This is essentially the concept of genetic drift, but the subtlety is that the drift occurs not on the typical organisms (fixed points) but on the deviations from typical organisms. In
Waddington's terminology, the epigenetic landscapes of the two groups change so as to become mutually incompatible, without substantially affecting the developmental outcomes.

Another route to speciation starts with organisms corresponding to two different fixed points of the reproductive transform (Fig. 11b). This could be, for example, *Paramecium* singlets and doublets. Again, if these two groups are isolated, natural selection could then progressively alter the reproductive transforms far from the fixed points (Fig. 11b, green curves), so that one group's fixed point lies outside the basin of attraction of the other fixed point.

Finally, another possibility is a bifurcation of the reproductive transform (Fig. 11c). Here, a drift of the reproductive transform makes a second fixed point appear (1, 2), then makes the first fixed point disappear (3). At this point, the organism becomes very different from the initial organism, but still compatible. Then alterations away from the fixed point make the initial and final reproductive transforms incompatible (4).

# Discussion

## Summary

This analysis started by pointing out the logical incoherence of the conventional view of biological reproduction, according to which the adult organism is encoded in the genome. The logical issue comes from the fact that development is a process of growth from an existing organism, and therefore what the genome "codes for" depends on the developmental processes of the organism, those very processes that the genome is supposed to encode. This is not a logically coherent concept. Recognizing that the environment influences development does not fix this logical issue. This view inherits the conceptual legacy of von Neumann's reproducing machine (Neumann and Burks, 1966), which relies on an invariant core, the universal constructor, which builds an identical machine from replicated instructions. The reproductive incompatibility of species means that living organisms do not have a universal constructor, and that invariant reproduction must then be an emergent property, appearing in certain contexts.

When expressed in a dynamical framework, where successive generations correspond to iterations of a reproductive transform, invariant reproduction correspond to a fixed point of the reproductive transform. Thus, invariant reproduction may emerge as a result of convergence of a reproductive sequence to a fixed point. Convergence is not obligatory, and when it occurs, it can be to one of several fixed points, or to a cycle. All cases are instantiated in known biological situations. Cycles correspond to multigenerational life cycles; divergence can occur in response to various accidents; multiple fixed points correspond to certain cases of non-genomic inheritance, such as structural or cytoplasmic inheritance.

This alternative perspective offers a new angle on the relationship between genome and organism, which goes beyond criticisms of genetic reductionism (one gene encodes one phenotype) and the emphasis of interactions between genes and the environment. The revised view that the whole genome encodes the organism jointly with the environment still conceives reproduction as a mapping from a source of instructions (genome plus environment) to the final organism. The dynamical perspective emphasizes that genes and environment specify the developmental mapping, not its product. In Waddington's terminology, genes shape the epigenetic landscape, not the endpoints of development. The same genome and the same environment can be consistent with different reproducing organisms,
as illustrated by the case of *Paramecium* doublets. It follows in particular that the organism cannot be inferred from its genome.

If invariant reproduction is emergent, then it does not require a specific mechanism. For example, a ball rolling on a hilly landscape will eventually stop at the bottom of a hill (Fig. 12a). If the ball is then slightly pushed, it will return to its equilibrium position (Fig. 12b). This “homeostatic” behavior does not rely on any particular mechanism encoded in the ball’s structure, or in the landscape. In the same way, robust invariant reproduction is a consequence of reproductive convergence, rather than a cause.

<figure><img src="image_15.png" /><figcaption>Figure 12. Landscape analogy of convergence to robust invariant reproduction. a, The reproductive sequence converges to a fixed point of the reproductive transform. b, Invariant reproduction is robust because it is a stable fixed point.</figcaption></figure>

In the dynamical perspective, evolution occurs by default: iterations of the reproductive transform produce a sequence of distinct organisms, which is biased towards convergence to stable fixed points or cycles, in the same way as the rolling ball will spend more time at the bottom than at the top of hills. This dynamical principle of evolution is distinct from natural selection, which interacts with it. First, natural selection eliminates diverging sequences. Second, natural selection increases the fitness of the selected fixed point, thereby making convergence to that fixed point more desirable, which results in increasing the stability of the fixed point. In this way, invariant reproduction processes may emerge secondarily as a result of natural selection. Third, mutations silently alter the dynamical landscape away from the selected fixed point.

Finally, in the dynamical perspective, speciation is a process by which reproductive transforms become mutually incompatible, that is, the basins of attractions of the fixed points do not overlap any more. This can occur by different dynamical mechanisms, namely bifurcation or alteration of the dynamical landscape away from the fixed points. In particular, bifurcations open the possibility of abrupt changes with gradual alterations.

I will now discuss the formalism of the reproductive transform.

## The dynamical formalism

The formalism relies on a separation between what changes and what is conserved through reproduction. As a first approximation, I considered that the genome is conserved, as it is replicated, and what changes is the rest of the organism. In this view, the genome is unchanged but influences development. This is the concept of a causal *constraint*: something that has causal effect on a phenomenon but is not affected by it (Brette, 2026; Juarrero, 2023; Kauffman et al., 2008; Montévil
and Mossio, 2015). For example, an enzyme represents a constraint on chemical reactions: the enzyme does not change but channels certain reactions; the reactions that actually take place and their products depend on initial conditions. In the same way, the genome is unchanged and channels development, but what actually develops depends on the initial organism. Furthermore, since the reproductive transform is to be iterated, the formalism relies on the genome being a *transmissible constraint*, that is, invariant through development and through repeated iterations, at least at the time scale of interest – Darwinian evolution constituting a longer time scale.

Thus, the relevant distinction is between a transmissible constraint and the space in which change occurs. For the sake of simplification, I have identified the genome with the transmissible constraint and the rest of the organism with the object of change, focusing mainly on the asexual division of cells. To take into account the influence of the environment in this formalism, we must ask whether the environment is inherently stable through reproduction, or may change as a result of development, at the time scale of interest. For example, the typical environmental niche of a species could be considered as an invariant constraint, and therefore would shape the reproductive transform, along with the genome. A change of environment could then have the same kind of effect as a mutation (shift of fixed point, divergence, bifurcation, etc.). However, if the environment of the growing organism results from either the structure or the behavior of the parent organism, as in egg laying, nest making or development in the womb, then this environment is both the condition and the outcome of development, and should therefore be included in the domain of the reproductive transform.

We have left aside the question of sexual reproduction. By definition, if there are two different parents, then there cannot be invariant reproduction. Therefore, the question must be framed somewhat differently. If the genome changes at each generation (e.g. by recombination), then so does the reproductive transform. We must then consider a family of reproductive transforms, and each generation applies a transform sampled from this family. A broader notion of invariant reproduction can then be understood as the convergence to a set of fixed points with overlapping basins of attractions.

Another subtlety that we left aside is the fact that cells reproduce by splitting. This means that, in reality, the reproductive transform should have two products, rather than just one. This can be included in the formalism by either following a particular reproductive sequence (e.g. the anterior daughter in ciliates, or the bud in the budding yeast), or by taking a statistical approach, that is, the reproductive transform is sampled from two alternatives on each iteration. It is possible, for example, that invariant reproduction holds only in a statistical sense.

Finally, most of the examples I presented were in unicellular rather than multicellular organisms. In multicellular organisms, the formalism may be applied at the cellular level or at the multicellular level. The cellular level is of course the phenomenon of development. Here, the generic case is precisely not invariant reproduction: cellular differentiation illustrates the fact that the same genome is compatible with very different kinds of cells, which in most cases stop reproducing after a number of divisions. Thus, Monod's idea of the "teleonomic project" of invariant reproduction does not apply to most cells of multicellular organisms. We may instead apply the formalism at the level of the multicellular organism. One possibility is to consider the reproductive transform that maps a reproductive cell of the parent into a reproductive cell of the child. This may work for the spores of fungus, for example. It would generally not work, however, for the gametes of animals. The reason is that embryological development depends on conditions created by the parent, e.g. the mother's egg or the mother's uterus. These conditions are then both an influence on and an outcome of development, and must then be included in the domain of the reproductive transform. Therefore, the formalism should apply to the reproductively mature organism, rather than the reproductive cells. Thus, the fact that epigenetic
marks on the genome are not transmitted to the descendants in mammals (Heard and Martienssen, 2014) does not prevent the possibility of non-genomic inheritance. Parthenogenesis is another case where the reproductive transform must be applied at the multicellular level, not at the cellular level.

## Cancer

The theory of cancer deserves a brief discussion, as it is a pathology of cellular division. Cancer is characterized by abnormal tissue growth, with cells displaying various structural or functional abnormalities. The dominant theory of cancer, somatic mutation theory, derives logically from the idea of the genome as a representation of the organism: an abnormality that is transmitted through cell division must correspond to an abnormal genome. While in many cases cancer has been traced back to specific mutations, it is also known that cancer can develop in the absence of mutation, or conversely that an identified carcinogenic mutation does not necessarily develop into cancer; in other words, mutations are neither necessary nor sufficient (Huang et al., 2025). This has motivated the development of alternative theories of cancer, such as tissue organization field theory (Soto and Sonnenschein, 2011), the atavistic theory (Bussey et al., 2017) and the systemic evolutionary theory (Mazzocca et al., 2025).

I have briefly mentioned experiments in which *Paramecium* was exposed to blastogenic agents (Mottram, 1941). An interesting observation in that report is that the daughters of abnormal cells would also tend to be abnormal, but the specific abnormality could change. In the dynamical perspective, this means that the transient perturbation due to the agent can trigger a diverging reproductive sequence. This perturbation could be a mutation, in which case the dynamical phenomenon is a bifurcation. But it could also be a non-genomic perturbation causing an exit from the basin of attraction, possibly converging to another fixed point of the same reproductive transform. In this view, cancer is not so much a failure of a well-tuned machine as a switch to a latent dynamical mode of the reproducing cell, possibly a degraded evolutionary relic. This is in line with the idea of cancer as a pathological attractor of gene network dynamics (Huang et al., 2009).

## Philosophical notes

As Nicholson (2014) puts it, the view of the genome as a representation of the organism that unfolds in development can be characterized as *neo-preformationism*: in preformationism, the embryo contains a miniature version of the adult organism; in neo-preformationism, it holds a symbolic representation of the organism (see also (Griffiths and Neumann-Held, 1999)), where the miniature version of the adult organism is replaced by a symbolic representation of the organism. It is an instance of the fallacy that structure must somehow already preexist “somewhere” (Oyama, 2000), when one should explain how it emerges.

More broadly, neo-preformationism is an example of a view grounded in substance metaphysics, which consists in explaining phenomena by the presence of some substance with a special quality. For example, burning used to be explained by the presence of a special fire-like substance, phlogiston, which moves to the air when wood burns and is later captured by growing trees. The modern theory sees combustion as a self-sustaining chemical process, which does not involve the transport of some special matter with burning properties. As Bickhard (2009) has pointed out, science has progressively replaced substance-based by process-based theories. Substance metaphysics remains common in biological theory (Brette, 2026; Nicholson and Dupre, 2018), yet living organisms are out-of-
equilibrium organizations that actively renew themselves. Most stable cellular structures are in fact materially renewed at a fast pace, and must therefore be conceptualized as attractors of biological processes, rather than as devices assembled according to a plan.

In this respect, the machine analogy is a poor model of living systems (Nicholson, 2019). The machine is inherently stable, and therefore the focus of explanation is on how it can change. In contrast, a living organism is inherently fluctuating, and one must explain how it can be stable. Thus, von Neumann's reproducing machine explains invariant reproduction by the existence of an invariant universal constructor. As I have argued, this cannot apply to living organisms, and the computational framework should be replaced by a dynamical framework, in which invariant reproduction can emerge. This emergence may be favored by the fact that living systems are dissipative systems, which tend to constrain trajectories to lower-dimensional spaces (Prigogine and Nicolis, 1971), but it is not obligatory. The fact that the lack of invariant reproduction or the existence of multiple stable reproductive forms is not often acknowledged, despite well-documented evidence in ciliates, may be due in part to the self-reinforcing nature of theories. As Minelli and Fusco (2019) point out: "*the hitherto studied model species in developmental biology are scarcely representative of most biological diversity, precisely because they have been chosen by virtue of their ability to develop in the laboratory in a highly repeatable way*". I suggest that the dynamical perspective on reproduction better accounts for the processual nature and diversity of living phenomena than the computational analogy.

## Acknowledgments

This work was supported by Agence Nationale de la Recherche (ANR-20-CE30-0025-01 and ANR-21-CE16-0013-02). I thank Bruno Delord for his suggestions.

## Competing interests

The authors have no competing interests to declare.

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