Comparing biological models and theories of life with process-enablement graphs

For the remainder of this paper we use the term organisational closure to refer to the existence of a cycle in a relevant $pe$-graph. This definition of organisational closure is weaker than definitions put forward in other contemporary accounts. In particular, we have defined organisational closure in terms of a closure of processes which differs from \citeAmontevil2015biological’s characterisation of biological organisation as a closure of constraints. There, constraints are conceived of as “contingent causes, exerted by specific structures of dynamics, which reduce the degrees of freedom of the system on which they act.”

Processes and constraints differ from each other in a few important ways. Processes, in general, often contribute energy and matter to other downstream processes. Constraints, on the other hand, merely canalise processes at lower levels of causation, without directly contributing energy or matter Montévil \BBA Mossio (\APACyear2015). This allows constraints to do work, in the strict sense of physics, which we consider here as a constrained release of energy Kauffman (\APACyear2000). Constraints also remain unaltered as they act on systems, when viewed from an appropriate perspective. For example, our vasculature acts as a constraint on the flow of blood through our body since, without blood vessels, blood would flow in all directions instead of being neatly transported to our cells. Further, over short time periods our vasculature remains largely unaltered. But by virtue of existing in a world where everything flows, constraints are themselves processes. Using the present example, over longer timescales our vasculature will change its physical structure through processes like angiogenesis. However, processes are not necessarily constraints, as not all processes reduce the degrees of freedom of a particular system.

It is arguably important to make a distinction between the processes themselves and the constraints that impinge on them Mossio \BBA Bich (\APACyear2017). We have not made a distinction between processes and constraints in our current framework and will therefore not be able to identify where a system is doing work. Further, we also lose the powerful concept of self-constraint in our analysis which has become a fruitful framework to understanding biological organisation Juarrero (\APACyear2023); Moreno \BBA Mossio (\APACyear2015). We make this sacrifice for means of mathematical simplicity, since it is far easier to make mathematical comparisons when there is only one kind of vertex (processes) and one kind of arrow (enablements) as opposed to, for example, the multiple types of arrows used by \citeAmontevil2015biological.

Organisational closure is nevertheless biologically interesting since cycles in $pe$-graphs embody a more general form of self-organisation. To see why, consider the following two perspectives. First, whenever we have a closure of constraints, we can also see such closure as a corresponding cycle in a $pe$-graph. Cycles of processes are called self-organising by \citeAmossio2017makes, but are distinguished from self-determining systems which realise a closure of constraints El-Hani \BBA Nunes-Neto (\APACyear2020). We can therefore understand self-determination as a special case of self-organisation, so studying the latter can inform our understanding of the former (Juarrero, \APACyear2023, p. 101). Second, a cycle in a $pe$-graph is collectively impredicative, meaning that it can only exist if all of its constituent processes are already unfolding and generating one another Jaeger \BOthers. (\APACyear2024). It is unclear how such a system could have come to exist in the first place, since none of the processes in this network can begin unless all the other processes have already begun. This is what \citeAhofmeyr2021biochemically refers to as the starting problem, analogous to \citeAturing1936computable’s halting problem. So cycles of processes possess an interesting property which is not shared by noncyclical process-enablement structures.

Every $pe$-graph represents a particular perspective of a system. As a result, for any given system we could draw many distinct $pe$-graphs that each represent a different way of looking at the same system. For instance, if two scientists observe a bird in a forest, a biochemist would likely draw a very different $pe$-graph to describe it compared to an ecologist. This is a fundamental feature of the nature of scientific modelling. Every model represents a particular point of view and a particular way of thinking. Every model leaves things out and distorts reality in one way or another. It is only by comparing many models together that we can really begin to understand nature Massimi (\APACyear2022).

This doesn’t mean that every $pe$-graph is a valid representation of a system. Each time you draw an arrow in a $pe$-graph you are claiming the existence of a genuine lawlike dependency that could be investigated. Thus, whether the $pe$-graph is accurate or not is ultimately up to empirical investigation. If such an investigation shows that the enablement is wrong then the graph should be updated accordingly.

We now want to develop the mathematical tools required to conduct such a comparative analysis of $pe$-graphs. We want to enable the biochemist and ecologist, for example, to compare their points of view with each other and see where and how their models differ, but also to investigate where there are similarities. However, as we conduct these perspective shifts we want to ensure that the appropriate structural features of $pe$-graphs remain intact. This desire motivates our following notions of preservation and reflection of closure.

Note that standard graph homomorphisms always preserve closure, but this is not necessarily true for weak graph homomorphisms. Figure 2 illustrates the notion of preservation of closure.

The following result shows that weak graph homomorphisms that preserve closure are composable.

The next result establishes useful conditions for determining the preservation of closure based on structural properties of the induced images of cycles under the weak graph homomorphism.

When considering $pe$-graphs we want homomorphisms to preserve additional structure with respect to the processes.

We also refer to homomorphisms of $pe$-graphs as $pe$-graph homomorphisms, or simply homomorphisms if it is sufficiently clear that we are referring to $pe$-graphs. We will often abbreviate the notation when referring to a $pe$-graph homomorphism and say ‘$p$ is happening within $\phi(p)$’ when we really mean ‘$p_{{}_{X}}$ is happening within $q_{{}_{Y}}$’ where $q_{{}_{Y}}=\phi(p_{{}_{X}})$.

We now show that $pe$-graph homomorphisms are composable.

Homomorphisms of $pe$-graphs allow us to translate between two perspectives representing the same system, mapping all of the relevant structural properties of one $pe$-graph to another. However, it is possible for the target graph to contain more structure than the source graph. If every cycle in the target graph is reflected appropriately in the source graph then we say that a homomorphism reflects closure.

We now show that weak graph homomorphisms that reflect closure are composable.

The following result establishes a useful condition for determining the reflection of closure.

A homomorphism which reflects closure provides a correspondence between all of the key structural features of the two $pe$-graphs, namely the processes and the cycles. As such, these maps indicate a greater similarity between two $pe$-graphs. We therefore call these maps homorheisms to distinguish them from homomorphisms which do not necessarily reflect closure.

The notion of a homorheism is illustrated in Figure 3. The following result shows that homorheisms are composable.

Homorheisms allow us to identify two vantage points that convey roughly the same description of the system under investigation, though the source graph may be a more finely-grained perspective than the target graph. Note, however, that homorheisms are not symmetric so they are not an equivalence relation on $pe$-graphs. That is, just because there exists a homorheism $G\mathrel{\vphantom{\rightarrow}\mathchoice{\ooalign{$\displaystyle\circ\mkern 1.0mu$\cr$\displaystyle\longrightarrow$\cr}}{\ooalign{$\textstyle\circ\mkern 1.0mu$\cr$\textstyle\longrightarrow$\cr}}{\ooalign{$\scriptstyle\circ\mkern 1.0mu$\cr$\scriptstyle\longrightarrow$\cr}}{\ooalign{$\scriptscriptstyle\circ\mkern 1.0mu$\cr$\scriptscriptstyle\longrightarrow$\cr}}}H$ does not necesarily mean there is a homorheism $H\mathrel{\vphantom{\rightarrow}\mathchoice{\ooalign{$\displaystyle\circ\mkern 1.0mu$\cr$\displaystyle\longrightarrow$\cr}}{\ooalign{$\textstyle\circ\mkern 1.0mu$\cr$\textstyle\longrightarrow$\cr}}{\ooalign{$\scriptstyle\circ\mkern 1.0mu$\cr$\scriptstyle\longrightarrow$\cr}}{\ooalign{$\scriptscriptstyle\circ\mkern 1.0mu$\cr$\scriptscriptstyle\longrightarrow$\cr}}}G$.

But if there are no homorheisms between two $pe$-graphs that represent the same system, then there is at least one fundamentally different organisational feature between them. Namely one must contain a cycle that the other does not capture in any way. Graphs like these are therefore different vantage points that offer significantly different views of the same system, with respect to their organisational structure. It is possible, however, to use intermediate $pe$-graphs to translate between perspectives when there are no homorheisms directly between the graphs available. We will demonstrate this process in Section 4.3.

As a linguistic note to conclude the discussion here, the term homorheism translates to same flow. It is inspired by C. H. Waddington’s neologism homeorhesis to describe dynamical systems that return to the same dynamical trajectory, following a disturbance Waddington (\APACyear1957).

Our formalism allows for a $pe$-graph to have loops, but we have not yet described what this means conceptually. We wish to avoid paradoxes of self-enablement, so we do not want a loop in a $pe$-graph to indicate that a process is directly enabling itself. Instead, we use loops as a convenient shorthand. Consider a system $S$ of processes and direct enablements (the prokaryotic cell is a good example). If this system $S$ contains at least one cycle then instead of writing all of the processes and enablements within $S$ we can simply write $S\to S$. Therefore, a loop within a $pe$-graph should always indicate to the reader that the loop in question is really a finer-grained network of processes containing at least one cycle. There are, therefore, really two kinds of arrows in $pe$-graphs: direct enablements (not loops) and loops.

In contrast, many models often choose to ignore lower-level organisation and treat systems as if they were not self-organising. For instance, organisms within many evolutionary models are treated as objects that can be mostly reduced down to their genes Lewontin (\APACyear1983); Walsh (\APACyear2018); Chiu (\APACyear2022). For this reason, population genetics has been referred to pejoratively as ‘beanbag genetics’ Rao \BBA Nanjundiah (\APACyear2011). We can see this disappearance of the self-organising organism in, for example, the Price equation where the self-organising properties of individuals are ignored in favour of a simple statistical equation Price (\APACyear1970); Frank (\APACyear2012). Accordingly, if we were to treat such models as reality and then constructed the resulting $pe$-graphs, loops like ‘Organism $\to$ Organism’ would never appear.

With the use of loops as a shorthand we need to keep in mind that more detailed descriptions of a system may result in the loss of a homorheism. For instance, suppose $G$ is the graph with a single process ‘Hepatocyte’ and a single loop (since cells are self-organising), and $H$ is a similar graph with a single process ‘Liver’ and a single loop. Then there is a natural homorheism $\phi\colon G\mathrel{\vphantom{\rightarrow}\mathchoice{\ooalign{$\displaystyle\circ\mkern 1.0mu$\cr$\displaystyle\longrightarrow$\cr}}{\ooalign{$\textstyle\circ\mkern 1.0mu$\cr$\textstyle\longrightarrow$\cr}}{\ooalign{$\scriptstyle\circ\mkern 1.0mu$\cr$\scriptstyle\longrightarrow$\cr}}{\ooalign{$\scriptscriptstyle\circ\mkern 1.0mu$\cr$\scriptscriptstyle\longrightarrow$\cr}}}H$, but clearly $G$ and $H$ model quite different systems. If we unpack the loops in $G$ and $H$ and express these graphs as more detailed networks of the systems in question then we lose the homorheism: once we unpack the process ‘Liver’ we will see how intricate and complex the network here really is, with hundreds of processes and cycles between many different cell types. Indeed, there are far more process-enablement cycles within the liver compared to those just occurring within a single hepatocyte.

The concept of autocatalytic sets emerged during the 1980s Dyson (\APACyear1982); Kauffman (\APACyear1986). Dyson and Kauffman were both interested in the origins of life, and wanted to explore how random sets of chemical reactions could come together to form an organised network. Their ideas were then refined via a mathematical formalism known as RAF sets: reflexively autocatalytic systems generated by a food set Hordijk \BBA Steel (\APACyear2004); Hordijk \BOthers. (\APACyear2012). Here, reflexive autocatalysis means that every reaction in the system is catalysed by at least one molecule already in the system, and food-generated means that all the molecules in the system can be ultimately generated from the food set through chains of reactions within the system (Figure 4).

To study autocatalytic sets using the language of $pe$-graphs, we first need to consider the kinds of processes and enablement relationships going on within these chemical systems. The processes in each case are relatively clear: they are the chemical reactions of the autocatalytic set, together with the process of food diffusing freely into the system. Since at least one food molecule is required for each reaction displayed in Figure 4, food diffusion must directly enable each reaction. However, whether reactions directly enable each other depends on the nature of molecule degradation and the rate constants of the reactions. For instance, if molecule $B_{3}$ degrades rapidly within the system, then Reaction 2 will directly enable Reaction 3 because Reaction 2 will need to continuously produce $B_{3}$ in order to keep Reaction 3 going (Figure 4). On the other hand, if $B_{3}$ is more stable and remains abundant within the system then this direct enablement could be undermined. For example, if $B_{3}$ has been stockpiled by the system in large quantities, then the occurrence of Reaction 2 may be irrelevant to Reaction 3. For now, we will assume that this RAF set operates in a high degradation environment, so that whenever a reaction ceases then its respective products disappear almost instantaneously from the system¹¹1We could also allow for a low-degradation environment, but add the diffusion processes of each molecule into the $pe$-graph. For simplicity we instead assume a high-degradation environment, yielding direct enablements between reactions and ignoring the diffusion processes.. We can then draw two $pe$-graphs to describe the RAF sets (Figure 5).

Interestingly, the $pe$-graph $R_{1}$ does not have a cycle, whilst $R_{2}$ does (Figure 5). This demonstrates that organisational closure is partially perspective dependent. Unlike \citeAcusimano2020objectivity, we do not see this failure of objectivity as a disadvantage of the $pe$-graph framework. Rather, we see it is a significant advantage because it introduces a way to identify qualitatively different analyses of the system under investigation. A comparison of these perspectives can then inform future empirical investigations and help us to better understand the system. For instance, if we see Reactions 1 and 3 as really two parts of the broader process ‘Reactions $1+3$’, then the analysis provided in $R_{2}$ will make more sense. The system contains a cycle and this part of the system therefore achieves organisational closure. Thus, depending on where this system is located (e.g. whether it be in a Petri dish or a cell) we may even expect the system to evolve structures, or act in a particular way, to preserve this cycle (Moreno \BBA Mossio, \APACyear2015, p. 71).

In a similar manner, using the homomorphism $\phi_{1}$, the scientist can come to understand how a collectively impredicative system like $R_{2}$ may have evolved. This is because, in $R_{1}$, Reaction 3 does not enable Reactions 1 or 2. Thus, we could have had a reaction network with Reactions 1 and 2, but not Reaction 3. We therefore have a solution to \citeAhofmeyr2021biochemically’s starting problem for this case. From one point of view, we may see a cycle that cannot seem to start without all the processes already existing ($R_{2}$). But from another point of view, related via an appropriate homomorphism ($\phi_{1}$), we can see how the system may have actually come to exist, without any collective impredicativity ($R_{1}$). It is not clear to us if all collectively impredicative systems can be dissolved like this using a perspective shift, however our example here shows that it is certainly possible in particular situations.

Humberto Maturana and Francisco Varela together developed the concept of autopoiesis, which proved to be a hugely influential account of life Maturana \BBA Varela (\APACyear1980). The concept went through a number of iterations, but we will focus on a later definition given by \citeA[p. 34]varela2000fenomeno and discussed in relation to cognition in \citeA[pp. 99–103]thompson2007mind. Here, a system is defined as being autopoietic if the following criteria all hold:

1. 1.

   The system has a semipermeable boundary constructed from molecular components. This boundary allows us to discriminate between the inside and outside of the system.

2. 2.

   The components of the boundary are produced by a metabolic reaction network (MRN) located within the boundary.

3. 3.

   There is an interdependence between Criteria 1 and 2. That is, the MRN is regenerated and maintained because of the existence of the boundary which keeps all of the molecules in the system in close proximity to each other.

For an autopoietic system to be maintained, much like an autocatalytic set, there must also be a ready supply of external nutrients that can flow into the system. Although this final criterion is not made explicit by Maturana and Varela, it is implicitly assumed throughout their work Varela (\APACyear2000). The prototypical example of an autopoietic system is the prokaryotic cell. It is still debated whether multicellular organisms, eusocial insect colonies, or even human societies are also autopoietic. The discussion around these trickier cases typically centres on how one defines a semipermeable boundary Thompson (\APACyear2007).

From this basic outline we can see that a minimal autopoietic system has three key processes: the MRN, boundary maintenance, and the passive diffusion of food into the system. Food diffusion directly enables the MRN since it provides the basic reactants for the system, and boundary maintenance and the MRN directly enable each other because of Criteria 2 and 3 (Figure 6).

There is no direct requirement that the MRN itself be organisationally closed, but in practice life has incorporated many cycles into its metabolic reaction networks (for example, the Krebs cycle). For these more complex autopoietic systems, the MRN itself achieves organisational closure (Figure 6). Some autopoietic systems also play an active role in bringing nutrients and other molecules into the system. For instance, cells often use processes like facilitated diffusion, primary and secondary active transport, and endocytosis to control which molecules enter into their internal metabolic reaction networks Jeckelmann \BBA Erni (\APACyear2020). In these autopoietic systems the MRN directly enables transport by producing the relevant transport proteins (for example GLUTs, Na⁺/K⁺-ATPase, and Clathrin), and the maintenance of the boundary directly enables these transport processes since they would not occur without a cell membrane (Figure 6).

So using the language of $pe$-graphs we can see at least three distinct kinds of autopoietic systems, with differing levels of organisation (Figure 6). We can also connect this perspective of life to autocatalytic sets by identifying homomorphisms between the two accounts. To do this, we first need to consider a system that could be viewed either through the lens of autopoiesis or autocatalytic sets. For example, we can imagine an extended version of the RAF set in Figure 4 where molecule $B_{2}$ is a lipid which is produced in sufficient quantity to form a semipermeable boundary. This boundary then keeps all of the relevant molecules of the RAF set together, and is completely permeable to the food molecules $F_{1}$, $F_{2}$, $F_{3}$, and $F_{4}$. Depending on how we view the RAF set as discussed in the previous section, and depending on how much of the system we would like to study, this analysis yields four related $pe$-graphs (Figure 7).

Figure 7 illustrates quite well how loops in $pe$-graphs should be understood. At first glance, $\phi_{4}$ can be analysed in the same way as $\phi_{1}$, as discussed in the previous section. But in fact, from this zoomed out perspective, we really have no understanding as to how closure is being realised in $A_{2}$ compared to $A_{1}$: it is all hidden away inside the loop. Once we contextualise what we mean by the ‘metabolic reaction network’, via either $\phi_{2}$ or $\phi_{3}$, it then becomes clear what is actually happening at the more detailed level.

If there is a homomorphism between two $pe$-graphs, then the relationship between them is relatively easy to analyse. We can quickly check which processes and enablements match up, what gets abstracted away from the source graph, and what extra phenomena the target graph may be able to capture that are invisible to the source graph. However, homomorphisms often do not exist between $pe$-graphs, so it can be useful to have some more general tools for comparison. In particular, when there are no homomorphisms between two $pe$-graphs we can sometimes develop an intermediary graph which indirectly shows how the two models are related.

To illustrate this process we will compare autopoietic systems, as described in Section 4.2, to a biochemically realisable version of Robert Rosen’s metabolism-repair $(M,R)$-systems Rosen (\APACyear1991). Rosen’s theory of biological organisation and $(M,R)$-systems is highly abstract and has essentially no reference to biochemistry. As a result, Rosen’s framework has been largely ignored by biologists. However, his work has recently been refined by \citeAhofmeyr2019basic, hofmeyr2021biochemically to fit it onto real biochemical processes. The result is what Hofmeyr calls fabrication-assembly $(F,A)$-systems. As considered within the context of the cell, an $(F,A)$-system has three classes of processes: (1) covalent metabolic chemistry (fabrication); (2) supramolecular chemical processes (assembly) and; (3) maintenance of the intracellular milieu via the transporting of ions.

Covalent molecular chemistry or fabrication involves the transport of nutrients into the cell, the degradation of nutrients into the building blocks of metabolism (for example amino acids, lipids, and nucleotides) and the synthesis of macromolecules like polypeptides and nucleic acids from these building blocks. Supramolecular chemistry or assembly involves the folding of proteins, the self-assembly of ribosomes, and the maintenance of the cell membrane via noncovalent interactions. This kind of chemistry is facilitated by the regulated ionic composition, temperature, and pH of the intracellular milieu, and by chaperone proteins. Finally, the maintenance of the intracellular milieu is primarily driven by the transport of ions into and out of the cell via membrane-bound transporters.

The enablements between these three classes of processes within an $(F,A)$-system can be summarised via three cycles (Figure 8).

Fabrication enables assembly by providing the raw materials that need to be assembled. This enablement is direct because as macromolecules are being produced within the cell, they immediately begin to self-assemble as they interact with the intracellular milieu. Conversely, assembly enables fabrication by providing the enzymes needed to catalyse the covalent chemistry of the cell. This enablement is direct because enzymes do not have a single static conformation, they are highly dynamic entities that have an ensemble of equilibrium configurations Yang \BOthers. (\APACyear2003); Nicholson (\APACyear2019). So we have the first cycle: Fabrication $\rightleftarrows$ Assembly.

In turn, the stochastic fluctuations of proteins are modulated by the constantly shifting, local composition of the intracellular milieu. Thus, the maintenance of the intracellular milieu directly enables assembly. Conversely, the maintenance of the cell membrane directly enables ions to be transported across it via membrane transporters, since without a membrane there would be no ion channels. Thus, assembly directly enables the maintenance of the intracellular milieu. This enablement is direct because the cell membrane is constantly shifting and undergoing repair. This gives us a second cycle: Assembly $\rightleftarrows$ Maintenance of the intracellular milieu.

Finally, assembly has a loop because protein folding and cell membrane maintenance, both processes occuring within assembly, directly enable each other. Maintenance of the cell membrane keeps ions within the cell, directly enabling protein folding. Conversely, the folding of enzymes like flippase and scramblase is needed to keep the membrane regulated and intact.

Both the $(F,A)$-system and autopoiesis are models of the biological cell, and we can model each with the $pe$-graphs $F_{A}$ and $A_{3}$, respectively, as shown in Figure 8. But it is impossible to directly compare $F_{A}$ and $A_{3}$ via a $pe$-graph homomorphism. This is because the two perspectives partition the processes of the cell differently. Explicitly, assembly involves both the maintenance of the cell’s boundary and protein folding; the latter is a part of the MRN as viewed from the perspective of autopoiesis. So if we wanted a $pe$-graph homomorphism $\phi\colon F_{A}\to A_{3}$, we would need to map assembly to both boundary maintenance and the MRN. Conversely, the MRN also involves the synthesis of nucleic acids from building blocks, which is a part of fabrication in $F_{A}$. So if there were a $pe$-graph homomorphism $\psi\colon A_{3}\to F_{A}$, we would need to map the MRN onto both fabrication and assembly. Since a $pe$-graph homomorphism cannot map a single vertex to two vertices, there are no $pe$-graph homomorphisms between $F_{A}$ and $A_{3}$. Yet, both $F_{A}$ and $A_{3}$ model the same system, so we would expect some kind of connection between them. Indeed, we can find one, if we construct an intermediate perspective as follows.

First, we need to choose a finer partition $\mathcal{P}$ of the processes described in each of the models. There are many choices we could make here, and different sets of finer-grained processes will reveal different similarities between the models. As an illustrative example we will choose the set $\mathcal{P}:=$ {nutrient transport, membrane maintenance, protein folding, ion transport, covalent chemistry}. Here, covalent chemistry is equivalent to $(F,A)$-system fabrication, excluding nutrient transport, and is also equivalent to the autopoietic MRN, excluding protein folding. Membrane maintenance refers to the trafficking of lipids and proteins to the cell membrane, the lateral diffusion of membrane components, the flipping of lipids from one side of the membrane to the other, membrane repair processes, and the containment of cytosolic molecules within the cell as they bounce off the inner plasma membrane. Ion transport refers to the transport of ions into and out of the cell via facilitated diffusion and active transport.

Second, we match each of the processes in $\mathcal{P}$ to processes in the target graphs in question. We have illustrated this matching for our example in Figure 8 using two colourings. We must ensure that the choice of $\mathcal{P}$ is fine enough such that every process $p\in\mathcal{P}$ is only happening within a single process in each of the target graphs. This will ultimately allow us to construct well-defined homomorphisms.

Third, we construct the corresponding $pe$-graph, $I_{\mathcal{P}}$, by drawing in the direct enablements between processes in $\mathcal{P}$. If the arrows are drawn correctly then, using the correspondence from the previous step, we should have weak graph homomorphisms from $I_{\mathcal{P}}$ to the two target graphs. Finally, we need to check whether these weak graph homomorphisms preserve or reflect closure. If this is successful then we will have indeed constructed $pe$-graph homomorphisms, or possibly homorheisms, from the intermediate $pe$-graph to the two target graphs. We prove this for our case in Theorems A.1 and A.2 in the Appendix. If, however, the weak graph homomorphisms do not preserve closure then we can choose a different set $\mathcal{P}$ and repeat the steps above. Alternatively, if we cannot construct homomorphisms from $I_{\mathcal{P}}$ to the target graphs then it may be an indication that there is a misplaced or missing arrow in these target graphs. Accordingly, this process of constructing an intermediate $pe$-graph can facilitate model development, as well as improve existing models.

The existence of the graph $I_{\mathcal{P}}$ and corresponding homorheisms $\phi_{5}\colon I_{\mathcal{P}}\to F_{A}$ and $\phi_{6}\colon I_{\mathcal{P}}\to A_{3}$ in Figure 8 demonstrates that lower-level perspectives of the cell, with more molecular detail, can be organisationally compatible with more abstract models like autopoiesis. We have also shown that autopoietic and $(F,A)$-system models can both arise from the same set of biological processes.

Additionally we have shown that there are no $pe$-graph homomorphisms between $F_{A}$ and $A_{3}$, so they must have at least one different organisational feature: and indeed they do, as the graphs $F_{A}$ and $A_{3}$ have different cycle structures. Thus, the processes prioritised as being central to the self-organisation of the cell are different in each case, since each perspective views the self-organisation of the cell itself in different ways. From the $(F,A)$-system perspective, the key process in the model is assembly as it sits within all three cycles in $F_{A}$. The way in which the cell arranges its parts from disorganised building blocks into functional assemblies is the central process that keeps the cell alive. But from the autopoietic perspective given in $A_{3}$, the key process is the metabolic reaction network as it also sits within all three cycles.

If we analyse the constraints on the central processes in each model, we can also flesh out more of the differences between these two perspectives. The key constraint on assembly is the maintenance of the intracellular milieu, but the key constraint on the MRN is the maintenance of a boundary. These are similar processes, though they are ultimately different. The maintenance of the intracellular milieu involves keeping the ionic composition of the cytoplasm stable, whilst the maintenance of a boundary refers to keeping all of the molecules in the cell within spatial proximity, thereby enabling reactants to interact and form a reaction network. This highlights a subtle but important distinction between these two models.

Another important remark is that because $\phi_{5}$ and $\phi_{6}$ are homorheisms, not just homomorphisms, all of the cycles in $F_{A}$ (respectively $A_{3}$) can be found in $I_{\mathcal{P}}$ and are therefore embodied in $A_{3}$ (respectively $F_{A}$) via $\phi_{6}$ (respectively $\phi_{5}$). This means that we have not ‘lost’ any of the self-organisational features when we move from $F_{A}$ to $A_{3}$ or vice versa, as they are just rearticulated in a different way and into different cycles. In short, we could simply say that $F_{A}$ and $A_{3}$ model the same self-organising processes in the cell, but they each have a different focus as to what they consider important. In the Appendix we consider the $pe$-graph $I_{\mathcal{P}}$ in more detail, showing that each of the edges in $I_{\mathcal{P}}$ is a direct enablement. We also prove that $\phi_{5}$ and $\phi_{6}$ are homorheisms.

There are some natural problem cases that arise whenever philosophers of biology ask themselves the question ‘what is life?’. In this section we have no intention of resolving demarcation problems surrounding what counts as a living organism or not. We will, however, translate some of these interesting edge cases into the language of $pe$-graphs, allowing for them to be studied from a new angle. We will also focus less on homomorphisms between $pe$-graphs in this section and instead look at when cycles are present or absent from the graphs. This will facilitate our study of organisational closure in these lifelike systems.