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Self-organization in computation and chemistry: Return to AlChemy

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Chaos
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Cole Mathis
Cole Mathis
Cole Mathis
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Devansh Patel
Devansh Patel
Devansh Patel
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Westley Weimer
Westley Weimer
Westley Weimer
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Stephanie Forrest
Stephanie Forrest
Stephanie Forrest
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Abstract and Figures

How do complex adaptive systems, such as life, emerge from simple constituent parts? In the 1990s, Walter Fontana and Leo Buss proposed a novel modeling approach to this question, based on a formal model of computation known as the λ calculus. The model demonstrated how simple rules, embedded in a combinatorially large space of possibilities, could yield complex, dynamically stable organizations, reminiscent of biochemical reaction networks. Here, we revisit this classic model, called AlChemy, which has been understudied over the past 30 years. We reproduce the original results and study the robustness of those results using the greater computing resources available today. Our analysis reveals several unanticipated features of the system, demonstrating a surprising mix of dynamical robustness and fragility. Specifically, we find that complex, stable organizations emerge more frequently than previously expected, that these organizations are robust against collapse into trivial fixed points, but that these stable organizations cannot be easily combined into higher order entities. We also study the role played by the random generators used in the model, characterizing the initial distribution of objects produced by two random expression generators, and their consequences on the results. Finally, we provide a constructive proof that shows how an extension of the model, based on the typed λ calculus, could simulate transitions between arbitrary states in any possible chemical reaction network, thus indicating a concrete connection between AlChemy and chemical reaction networks. We conclude with a discussion of possible applications of AlChemy to self-organization in modern programming languages and quantitative approaches to the origin of life.
Example AlChemy simulation and the organization it produced. (a) The four λ expressions remaining at the end of a single simulation run, which constitute the L1 organization. (b) Time series of the simulated run where the vertical axis represents the fraction of the population occupied by any given expression. Each expression is assigned a gray-scale value, except the four winners shown in (a). After a few hundred collisions all four expressions in the final organization had emerged, and by ≈35k collisions they were the only expressions remaining in the simulation. (c) The reaction rules that correspond to the final organization, showing that it is closed under interaction between any two expressions. (d) Network representation of the reaction rules from (c).
Example AlChemy simulation and the organization it produced. (a) The four λ expressions remaining at the end of a single simulation run, which constitute the L1 organization. (b) Time series of the simulated run where the vertical axis represents the fraction of the population occupied by any given expression. Each expression is assigned a gray-scale value, except the four winners shown in (a). After a few hundred collisions all four expressions in the final organization had emerged, and by ≈35k collisions they were the only expressions remaining in the simulation. (c) The reaction rules that correspond to the final organization, showing that it is closed under interaction between any two expressions. (d) Network representation of the reaction rules from (c).
… 
L0 simulations. (a) Time series from two different representative simulation runs (red and blue), showing the number of unique expressions through the runs. In both cases, the maximum number of objects was set to 1000 and a total of 6 × 10 6 collisions were performed. Every 10 6 collisions (gray dashed lines), we introduced a perturbation by adding 100 random expressions to the simulation. (b) The distribution of unique expressions across 1000 different simulations after the fifth perturbation (median across the previous 10 6 time steps of the simulation); the inset shows the same data on log–log scales. The number of expressions at steady state seems to vary across orders of magnitude and is heterogeneously distributed. (c) The organizations stabilize with fixed macroscopic parameters; even though the expressions themselves are continually changing, the average expression length and population entropy (a measure of diversity) are shown for different simulations. Each color is a different simulation with the identical run parameters, and each point represents a snapshot of the simulation at different time points. (d) Survival rate over time of different organizations when they are perturbed by replacing p% of the expressions with the identity function. Different colors correspond to different values of p.
L0 simulations. (a) Time series from two different representative simulation runs (red and blue), showing the number of unique expressions through the runs. In both cases, the maximum number of objects was set to 1000 and a total of 6 × 10 6 collisions were performed. Every 10 6 collisions (gray dashed lines), we introduced a perturbation by adding 100 random expressions to the simulation. (b) The distribution of unique expressions across 1000 different simulations after the fifth perturbation (median across the previous 10 6 time steps of the simulation); the inset shows the same data on log–log scales. The number of expressions at steady state seems to vary across orders of magnitude and is heterogeneously distributed. (c) The organizations stabilize with fixed macroscopic parameters; even though the expressions themselves are continually changing, the average expression length and population entropy (a measure of diversity) are shown for different simulations. Each color is a different simulation with the identical run parameters, and each point represents a snapshot of the simulation at different time points. (d) Survival rate over time of different organizations when they are perturbed by replacing p% of the expressions with the identity function. Different colors correspond to different values of p.
… 
L1 simulations and similarity over time. (a) Abundance distributions illustrating pairs of similar organizations. Each plot shows two different pairs (blue and red); each bar corresponds to one expression that is a component of each member of the pair; the ordering of the expressions is determined by the sum of their abundance in both members of the pair. The vertical axis indicates the expression copy number with the red values inverted to facilitate comparison. High symmetry between blue and red indicates the similar copy number, and low symmetry indicates the different copy numbers between the organizations. (b) Stability analysis of four L1 organizations that emerged in simulation. Each organization was initialized with a modification, either “perturb” (ten random expressions added) or “seed” (different random number seed only). The similarity between the modified and original simulations is shown in solid orange lines. The similarity of the original simulation to itself at a previous time step (500 collisions earlier) is used as a control, and it is shown in the purple dashed lines. Each panel shows a different input organization, selected to highlight the diversity of possible outcomes.
L1 simulations and similarity over time. (a) Abundance distributions illustrating pairs of similar organizations. Each plot shows two different pairs (blue and red); each bar corresponds to one expression that is a component of each member of the pair; the ordering of the expressions is determined by the sum of their abundance in both members of the pair. The vertical axis indicates the expression copy number with the red values inverted to facilitate comparison. High symmetry between blue and red indicates the similar copy number, and low symmetry indicates the different copy numbers between the organizations. (b) Stability analysis of four L1 organizations that emerged in simulation. Each organization was initialized with a modification, either “perturb” (ten random expressions added) or “seed” (different random number seed only). The similarity between the modified and original simulations is shown in solid orange lines. The similarity of the original simulation to itself at a previous time step (500 collisions earlier) is used as a control, and it is shown in the purple dashed lines. Each panel shows a different input organization, selected to highlight the diversity of possible outcomes.
… 
L2 simulations. Composite organizations were formed by combining two different L1 organizations ( L and R) into the same simulation and running the combination for 10 6 collisions. (a) Time series of the similarity of the composite to the inputs; color corresponds to different choices of L and R. (b) Three possible outcomes: (i) mutual destruction (the composite retains almost no similarity to either input); (ii) dominance (one input eventually dominates the other, and the final organization reverts to the dominant input); (iii) coexistence (the final organization retains significant similarity to both inputs). The relative frequency of these outcomes is summarized in the table.
L2 simulations. Composite organizations were formed by combining two different L1 organizations ( L and R) into the same simulation and running the combination for 10 6 collisions. (a) Time series of the similarity of the composite to the inputs; color corresponds to different choices of L and R. (b) Three possible outcomes: (i) mutual destruction (the composite retains almost no similarity to either input); (ii) dominance (one input eventually dominates the other, and the final organization reverts to the dominant input); (iii) coexistence (the final organization retains significant similarity to both inputs). The relative frequency of these outcomes is summarized in the table.
… 
(a) Representative syntax trees for λ expressions generated by the two generators. Left (red): Two example trees generated by the permutation method. These trees are uniform in the ratio of abstractions to applications, and they are parameterized only by the size of the tree. Right (blue): Five trees generated using AlChemy original generator. These trees tend to have long chains of abstractions, seen here as long branches. Thus, their corresponding expressions vary greatly in complexity, and simple expressions (such as one or two-abstraction expressions) are common. (b) Statistical properties of the binary tree representation of the λ expressions. (c) The dynamical consequences of these different generators are dramatic; in the permutation method (left), the system collapses to an inert, trivial fixed point, while the original generator (right) produces a diversity of complex organizations.
+1
(a) Representative syntax trees for λ expressions generated by the two generators. Left (red): Two example trees generated by the permutation method. These trees are uniform in the ratio of abstractions to applications, and they are parameterized only by the size of the tree. Right (blue): Five trees generated using AlChemy original generator. These trees tend to have long chains of abstractions, seen here as long branches. Thus, their corresponding expressions vary greatly in complexity, and simple expressions (such as one or two-abstraction expressions) are common. (b) Statistical properties of the binary tree representation of the λ expressions. (c) The dynamical consequences of these different generators are dramatic; in the permutation method (left), the system collapses to an inert, trivial fixed point, while the original generator (right) produces a diversity of complex organizations.
… 
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Chaos ARTICLE pubs.aip.org/aip/cha
Self-organization in computation and chemistry:
Return to AlChemy
Cite as: Chaos 34, 093142 (2024); doi: 10.1063/5.0207358
Submitted: 7 March 2024 ·Accepted: 19 August 2024 ·
Published Online: 30 September 2024
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Cole Mathis,1,2,a)Devansh Patel,1,3 Westley Weimer,4and Stephanie Forrest1,2,3,5
AFFILIATIONS
1Biodesign Institute, Arizona State University, Tempe, Arizona 85281, USA
2School of Complex Adaptive Systems, Arizona State University, Tempe, Arizona 85281, USA
3School of Computing and Augmented Intelligence, Arizona State University, Tempe, Arizona 85281, USA
4Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, Michigan 48109, USA
5Santa Fe Institute, Santa Fe, New Mexico 87501, USA
Note: This paper is part of the Focus Issue on Topics in Nonlinear Science: Dedicated to David K. Campbell’s 80th Birthday.
a)Author to whom correspondence should be addressed: cole.mathis@asu.edu
ABSTRACT
How do complex adaptive systems, such as life, emerge from simple constituent parts? In the 1990s, Walter Fontana and Leo Buss proposed
a novel modeling approach to this question, based on a formal model of computation known as the λcalculus. The model demonstrated how
simple rules, embedded in a combinatorially large space of possibilities, could yield complex, dynamically stable organizations, reminiscent of
biochemical reaction networks. Here, we revisit this classic model, called AlChemy, which has been understudied over the past 30 years. We
reproduce the original results and study the robustness of those results using the greater computing resources available today. Our analysis
reveals several unanticipated features of the system, demonstrating a surprising mix of dynamical robustness and fragility. Specifically, we
find that complex, stable organizations emerge more frequently than previously expected, that these organizations are robust against collapse
into trivial fixed points, but that these stable organizations cannot be easily combined into higher order entities. We also study the role played
by the random generators used in the model, characterizing the initial distribution of objects produced by two random expression generators,
and their consequences on the results. Finally, we provide a constructive proof that shows how an extension of the model, based on the typed λ
calculus, could simulate transitions between arbitrary states in any possible chemical reaction network, thus indicating a concrete connection
between AlChemy and chemical reaction networks. We conclude with a discussion of possible applications of AlChemy to self-organization
in modern programming languages and quantitative approaches to the origin of life.
Published under an exclusive license by AIP Publishing. https://doi.org/10.1063/5.0207358
How life emerged from simple molecular constituents is a long-
standing question that has not been fully resolved. This paper
resurrects a computational model called AlChemy that was first
proposed over 30 years ago to study how a set of simple compu-
tational rules could produce complex, dynamically stable organi-
zations reminiscent of biochemical reaction networks. Using the
vastly increased computing resources available today, this paper
reports on new computational experiments and analysis, which
show that in AlChemy, stable and complex organizations emerge
more frequently than previously expected and that these organi-
zations are often robust against collapse. The paper also probes
a key component of AlChemy, its method for generating random
initial conditions, and finds that it is crucial to the model’s suc-
cess. Finally, the paper gives a constructive proof showing that
a slightly modified version of AlChemy can simulate state tran-
sitions in any chemical reaction network (CRN), thereby estab-
lishing a closer connection between the model and the chemical
reactions believed to have produced life.
I. INTRODUCTION
The origin(s) of life remains an unresolved mystery in science,
that is, how unconstrained reactive compounds were selected to
form the organized biochemical networks that form the basis of Dar-
winian lineages. In the early 1990s, Fontana and Buss proposed a
novel approach to this problem based on a formal model of compu-
tation known as the λcalculus.1–3In a departure from the prevailing
Chaos 34, 093142 (2024); doi: 10.1063/5.0207358 34, 093142-1
Published under an exclusive license by AIP Publishing
ResearchGate has not been able to resolve any citations for this publication.
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