# Formalising the Pathways to Life Using Assembly Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Results

#### 2.1. Graph Theoretical Prerequisites

**Definition**

**1.**

**quiver**Γ consists of

- 1.
- A set of vertices $V\left(\mathrm{\Gamma}\right)$;
- 2.
- A set of edges $E\left(\mathrm{\Gamma}\right)$;
- 3.
- A pair of maps ${s}_{\mathrm{\Gamma}},{t}_{\mathrm{\Gamma}}:E\left(\mathrm{\Gamma}\right)\to V\left(\mathrm{\Gamma}\right)$.

**Definition**

**2.**

**path**$\gamma ={a}_{n}\dots {a}_{1}$in $\mathrm{\Gamma}$of length$n\ge 1$is a sequence of edges, such that $t\left({a}_{i}\right)=s\left({a}_{i+1}\right)$for$1\le i\le n-1$. The functions $s$and $t$can be extended to paths as$s\left(\gamma \right)=s\left({a}_{1}\right)$and $t\left(\gamma \right)=t\left({a}_{n}\right)$. We write $\left|\gamma \right|$ to denote the length, or number of edges, in the path. Additionally, for each vertex $x\in \mathrm{\Gamma}$there is a

**zero path**, denoted ${e}_{x}$, with length 0 and$s\left({e}_{x}\right)=t\left({e}_{x}\right)=x$.

**Definition**

**3.**

**directed cycle**if $\left|\gamma \right|\ge 1$with $t\left(\gamma \right)=s\left(\gamma \right)$.

**Definition**

**4.**

**acyclic**if it has no directed cycles.

**Definition**

**5.**

**reachable**from $x$if there exists a path $\gamma $such that $s\left(\gamma \right)=x$and$t\left(\gamma \right)=y$, where $\left|\gamma \right|\ge 0$.

**Lemma**

**1.**

**reachability relation**on $\mathrm{\Gamma}$.

**Proof.**

**Definition**

**6.**

**upper quiver**of $x\in V\left(\mathrm{\Gamma}\right)$is $x\uparrow $with vertices$V\left(x\uparrow \right)=\{y\in V\left(\mathrm{\Gamma}\right)|x\le y\}$, edges$E\left(x\uparrow \right)=\left\{e\in E\left(\mathrm{\Gamma}\right)|{s}_{\mathrm{\Gamma}}\left(e\right),{t}_{\mathrm{\Gamma}}\left(e\right)\in V\left(x\uparrow \right)\right\}$,${s}_{x\uparrow}={s}_{\mathrm{\Gamma}}{|}_{E\left(x\uparrow \right)}$, and${t}_{x\uparrow}={t}_{\mathrm{\Gamma}}{|}_{E\left(x\uparrow \right)}$. The

**lower quiver**of$x\in V\left(\mathrm{\Gamma}\right)$is$x\downarrow $with vertices $V\left(x\downarrow \right)=\{y\in V\left(\mathrm{\Gamma}\right)|y\le x\}$, edges$E\left(x\downarrow \right)=\left\{e\in E\left(\mathrm{\Gamma}\right)|{s}_{\mathrm{\Gamma}}\left(e\right),{t}_{\mathrm{\Gamma}}\left(e\right)\in V\left(x\downarrow \right)\right\}$,${s}_{x\downarrow}={s}_{\mathrm{\Gamma}}{|}_{E\left(x\downarrow \right)}$, and ${t}_{x\downarrow}={t}_{\mathrm{\Gamma}}{|}_{E\left(x\downarrow \right)}$.

**Definition**

**7.**

**maximal**in $\mathrm{\Gamma}$ if, whenever$x\le y$in $\mathrm{\Gamma}$, we have $x=y$. Dually, $x$is

**maximal**in $\mathrm{\Gamma}$if, whenever $y\le x$in $\mathrm{\Gamma}$, we have $x=y$. The set of all maximal vertices of$\mathrm{\Gamma}$ is denoted $max\left(\mathrm{\Gamma}\right)$with $min\left(\mathrm{\Gamma}\right)$defined dually.

**Definition**

**8.**

**finite**if its vertex and edge sets are both finite. Similarly, a vertex $x$in a quiver $\mathrm{\Gamma}$is said to be finite if$x\downarrow $in $\mathrm{\Gamma}$is a finite quiver.

**Definition**

**9.**

**subquiver**of $\mathrm{\Gamma}$if$V\left(\mathrm{\Gamma}{}^{\prime}\right)\subseteq V\left(\mathrm{\Gamma}\right)$,$E\left(\mathrm{\Gamma}{}^{\prime}\right)\subseteq E\left(\mathrm{\Gamma}\right)$,${s}_{\mathrm{\Gamma}{}^{\prime}}={s}_{\mathrm{\Gamma}}{|}_{E\left(\mathrm{\Gamma}{}^{\prime}\right)}$ and${t}_{\mathrm{\Gamma}{}^{\prime}}={t}_{\mathrm{\Gamma}}{|}_{E\left(\mathrm{\Gamma}{}^{\prime}\right)}$. We will denote this relationship as $\mathrm{\Gamma}{}^{\prime}\subseteq \mathrm{\Gamma}$.

**Lemma**

**2.**

**Proof.**

**Definition**

**10.**

**quiver morphism**, denoted$m:\mathrm{\Gamma}\to \mathrm{\Gamma}{}^{\prime}$, consists of a pair $m=\left({m}_{v},{m}_{e}\right)$of functions${m}_{v}:V\left(\mathrm{\Gamma}\right)\to V\left(\mathrm{\Gamma}{}^{\prime}\right)$and ${m}_{e}:E\left(\mathrm{\Gamma}\right)\to E\left(\mathrm{\Gamma}{}^{\prime}\right)$such that ${m}_{v}\circ {s}_{\mathrm{\Gamma}}={s}_{\mathrm{\Gamma}{}^{\prime}}\circ {m}_{e}$and ${m}_{v}\circ {t}_{\mathrm{\Gamma}}={t}_{\mathrm{\Gamma}{}^{\prime}}\circ {m}_{e}$. That is, the following diagrams commute:

#### 2.2. Assembly Spaces

**Definition**

**11.**

**assembly space**is an acyclic quiver$\mathrm{\Gamma}$together with an edge-labelling map$\varphi :E\left(\mathrm{\Gamma}\right)\to V\left(\mathrm{\Gamma}\right)$which satisfies the following axioms:

- 1.
- $min\left(\mathrm{\Gamma}\right)$is finite and non-empty;
- 2.
- $\mathrm{\Gamma}=min\left(\mathrm{\Gamma}\right)\uparrow $;
- 3.
- $Ifa$is an edge from $x$to$z$in $\mathrm{\Gamma}$with $\varphi \left(a\right)=y$, then there exists an edge $b$from $y$to $z$with $\varphi \left(b\right)=x$.

**Definition**

**12.**

**basis**of $\mathrm{\Gamma}$and is denoted ${B}_{\mathrm{\Gamma}}$. Elements of the basis are referred to as basic objects, basic vertices, or basic elements.

**Lemma**

**3.**

**Proof.**

**Definition**

**13.**

**assembly pathway**of an assembly space$\mathrm{\Gamma}$is any topological ordering of the vertices of $\mathrm{\Gamma}$with respect to the reachability relation.

**Definition**

**14.**

**split-branched**if for all $x,y\in \mathrm{\Gamma}$,$x\le y$or$y\le x$whenever $V\left(x\downarrow \right)\cap V\left(y\downarrow \right)\ne \varnothing $.

#### 2.3. Assembly Subspaces and the Assembly Index

**Definition**

**15.**

**assembly**

**subspace**of$\left(\mathrm{\Gamma},\varphi \right)$if$\mathrm{\Gamma}{}^{\prime}$is a subquiver of$\mathrm{\Gamma}$and$\psi =\varphi {\left.\right|}_{E\left(\mathrm{\Gamma}{}^{\prime}\right)}$. This relationship is denoted as$\left(\mathrm{\Gamma}{}^{\prime},\psi \right)\subseteq \left(\mathrm{\Gamma},\varphi \right)$, or simply$\mathrm{\Gamma}{}^{\prime}\subseteq \mathrm{\Gamma}$, when there is no ambiguity.

**Definition**

**16.**

**rooted**in$\mathrm{\Gamma}$if${B}_{\mathrm{\Gamma}}$is non-empty, and${B}_{{\mathrm{\Gamma}}^{\prime}}\subseteq {B}_{\mathrm{\Gamma}}$as sets.

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Figure 3.**An assembly space comprised of objects formed by joining together white and blue blocks. Some of the arrows have been omitted for clarity. The dotted region is an assembly subspace, and the topological ordering of the objects in the subspace represents a minimal assembly pathway for any subspace containing the sequence of four blue boxes.

**Definition**

**17.**

**cardinality**of an assembly space$\left(\mathrm{\Gamma},\varphi \right)$is the cardinality of the underlying quiver’s vertex set,$\left|V\left(\mathrm{\Gamma}\right)\right|$. The augmented cardinality of an assembly space$\left(\mathrm{\Gamma},\varphi \right)$with basis${B}_{\mathrm{\Gamma}}$is$\left|V\left(\mathrm{\Gamma}\right)\backslash {B}_{\mathrm{\Gamma}}\left|=\left|V\left(\mathrm{\Gamma}\right)\right|-\right|{B}_{\mathrm{\Gamma}}\right|$.

**Definition**

**18.**

**assembly index**${c}_{\mathrm{\Gamma}}\left(x\right)$of a finite object$x\in \mathrm{\Gamma}$is the minimal augmented cardinality of all rooted assembly subspaces containing$x$. This can be written$c\left(x\right)$when the relevant assembly space$\mathrm{\Gamma}$is clear from the context.

#### 2.4. Assembly Maps

**Definition**

**19.**

**assembly map**is a quiver morphism$f:\mathrm{\Gamma}\to \mathsf{\Delta}$such that$\psi \circ {f}_{e}={f}_{v}\circ \varphi $. That is, the following diagram commutes:

**Theorem**

**1.**

**Proof.**

#### 2.5. Bounds on the Assembly Index

**Lemma**

**6.**

**Proof**.

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

#### 2.6. Computability

**Theorem**

**4.**

**Proof.**

## 3. Discussion

#### 3.1. Addition Chains

#### 3.2. Vectorial Addition Chains

#### 3.3. Strings

#### 3.4. Pixels and Voxels

#### 3.5. Graphs

#### 3.6. Other Applications

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Algorithms

Algorithms A1: Basic Assembly Index Algorithm |

Function Main (B, t) Global Variable A // the pathway assembly index Set A = upper bound of assembly index + |B| AssemblyIndex (B, t) Return A - |B| End Function Function AssemblyIndex (S, t) For each pair of objects ${\mathrm{s}}_{1},{\mathrm{s}}_{2}\in \mathrm{S}$ If there exists an edge $e~\left[t{s}_{1}\right]$ with $\varphi \left(e\right)={s}_{2}$ and $\mathrm{A}>\left|\mathrm{S}\cup \mathrm{t}\right|$ $\mathrm{A}=\left|\mathrm{S}\cup \mathrm{T}\right|$ Else if there exists an edge $e~\left[u{s}_{1}\right]$ with $\varphi \left(e\right)={s}_{2}$ for some $\mathrm{u}\in \mathrm{\Gamma}$ AssemblyIndex $\left(\mathrm{S}\cup \mathrm{u},\mathrm{t}\right)$ End If End For End Function |

Algorithms A2: Practical Assembly Index Algorithm |

Function Main (t) Global Variable A // the best assembly index Set A = upper bound of assembly index AssemblyIndex({t}) Return A End Function Function AssemblyIndex(P) remnant = last element of P For each substructure subLeft in remnant For each substructure subRight in remnant\subLeft If subRight = subLeft newP = append subLeft to P newRemnant = remnant\subLeft with subRight split out as a separate connected component CurrentPathwayIndex = CalculateIndex(newP) If CurrentPathwayIndex < A A = CurrentPathwayIndex newP = append newRemnant to newP AssemblyIndex(newP) |

Algorithms A3: Split-Branched Assembly Index Algorithm |

Function SplitBranchedAssemblyIndex($\mathrm{\Gamma}$, B, t, I) Set A = upper bound of assembly index for t For each partition of U into connected sub-objects ${\mathrm{\Gamma}}_{\mathrm{P}}=\left\{{\mathrm{\Gamma}}_{1}\dots {\mathrm{\Gamma}}_{\mathrm{n}}\right\}$ Set PartitionIndex = 0 Partition ${\mathrm{U}}_{\mathrm{P}}$ into $\mathrm{K}=\left\{\left\{{\mathrm{\Gamma}}_{11},\dots ,{\mathrm{\Gamma}}_{1\mathrm{i}}\right\},\left\{{\mathrm{\Gamma}}_{21},\dots ,{\mathrm{\Gamma}}_{2\mathrm{j}}\right\},\dots ,\left\{{\mathrm{\Gamma}}_{\mathrm{m}1}\dots {\mathrm{\Gamma}}_{\mathrm{mk}}\right\}\right\}$ Where for each ${\mathrm{K}}_{\mathrm{n}}$, the ${\mathrm{\Gamma}}_{\mathrm{nx}}$ are identical for all $\mathrm{x}$ For each ${\mathrm{K}}_{\mathrm{i}}\in \mathrm{K}$ If ${\mathrm{K}}_{\mathrm{i}1}\in \mathrm{B}$ $\mathrm{PartitionIndex}+=1$ Else $\mathrm{PartitionIndex}+=\mathrm{SplitBranchedAssemblyIndex}\left(\mathrm{\Gamma},\mathrm{B},{\mathrm{K}}_{\mathrm{i}1}\right)$ $+\left|{\mathrm{K}}_{\mathrm{i}}\right|-1$ End If End For A = $\mathrm{min}\left(\mathrm{PartitionIndex},\mathrm{PA}\right)$ End For Return A End Function |

## Appendix B. Proofs

**Proof of Theorem**

**1.**

**Axiom**

**1.**

**Axiom**

**2.**

**Axiom**

**3.**

**Proof of Theorem**

**3.**

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**Figure 1.**The assembly process (centre) [13] is compared to the implementations of Shannon entropy [14] (left) and Kolmogorov complexity [15] (right) for blue and white blocks. The Assembly process leads to a measure of structural complexity that accounts for the structure of the object and how it could have been constructed, which is in all cases computable and unambiguous.

**Figure 2.**The basic assembly concept is demonstrated here. Each of the final structures can be created from white and blue basic objects in four joining operations, giving an assembly index of 4. Pathway (

**a**) shows the creation of a structure that can only be formed in four steps by adding one basic object at a time, while pathway (

**c**) represents the maximum increase in size per step, by combining the largest object in the pathway with itself at each stage. Pathway (

**b**) is an intermediate case.

**Figure 4.**An assembly map that maps an assembly space of white and blue blocks onto integers representing the object size.

**Figure 6.**Examples of text assembly pathways for 16-character strings. The first example demonstrates the shortest possible assembly index of any such string. The second example has a nontrivial assembly pathway, while the third example is a string without any shorter pathway than adding one character at a time. This model assumes that text fragments cannot be reversed when concatenating.

**Figure 7.**Illustrative assembly pathway of a two-dimensional image. This does not necessarily represent the minimal assembly pathway for this shape. Here, images that are rotated or reflected are considered equivalent.

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**MDPI and ACS Style**

Marshall, S.M.; Moore, D.G.; Murray, A.R.G.; Walker, S.I.; Cronin, L. Formalising the Pathways to Life Using Assembly Spaces. *Entropy* **2022**, *24*, 884.
https://doi.org/10.3390/e24070884

**AMA Style**

Marshall SM, Moore DG, Murray ARG, Walker SI, Cronin L. Formalising the Pathways to Life Using Assembly Spaces. *Entropy*. 2022; 24(7):884.
https://doi.org/10.3390/e24070884

**Chicago/Turabian Style**

Marshall, Stuart M., Douglas G. Moore, Alastair R. G. Murray, Sara I. Walker, and Leroy Cronin. 2022. "Formalising the Pathways to Life Using Assembly Spaces" *Entropy* 24, no. 7: 884.
https://doi.org/10.3390/e24070884