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An introduction to random rule-based chemical networks

Jeremie Unterberger

TL;DR

This work introduces a random rule-based framework for chemical networks in which a primitive rule can generate an infinite family of context-derived rules, serving as a toy model for reactivity. Reactions are probabilistically accepted according to a complexity index, yielding two main models: Model I with fixed acceptance and Model II with complexity-dependent acceptance, which produce distinct phase diagrams. A central methodological advance is mapping anabolic composition trees to an inhomogeneous Galton-Watson process, enabling analytic predictions for maximum levels, extinction times, and primitive-rule growth, corroborated by simulations. The results provide a foundation for extending to more realistic bond formation/breaking rules and offer a quantitative framework for examining large chemical networks and potential prebiotic evolution.

Abstract

We introduce in this article a random model of reactivity in which a primitive rule, if accepted, generates an infinite number of rules by context derivation. The model may be thought of as a toy model of chemical reactivity, where reactions are accepted if their randomly distributed activation energy is below a certain threshold. It may be simulated by induction on the level (length of the word). We describe some statistical features of the model, regarding the number and complexity of the rules, and the shape of the reaction network. The complexity index of a rule is defined as the number of covalent bonds involved in the rearrangement. The Bernoulli parameter (acceptation probability) of the rules is chosen as fixed in a first model (Model I), and exponentially decreasing in the complexity index in a second one (Model II). The two models have very different behaviors, Model II exhibiting a non-trivial phase diagram. The main tool for mathematical analysis is an approximate mapping to a Galton-Watson tree with generation-dependent progeny distributions. Detailed simulations, demonstrating a general agreement with theoretical estimates, are provided at the end. Extensions to realistic bond formation/breaking rules for molecules will be presented elsewhere.

An introduction to random rule-based chemical networks

TL;DR

This work introduces a random rule-based framework for chemical networks in which a primitive rule can generate an infinite family of context-derived rules, serving as a toy model for reactivity. Reactions are probabilistically accepted according to a complexity index, yielding two main models: Model I with fixed acceptance and Model II with complexity-dependent acceptance, which produce distinct phase diagrams. A central methodological advance is mapping anabolic composition trees to an inhomogeneous Galton-Watson process, enabling analytic predictions for maximum levels, extinction times, and primitive-rule growth, corroborated by simulations. The results provide a foundation for extending to more realistic bond formation/breaking rules and offer a quantitative framework for examining large chemical networks and potential prebiotic evolution.

Abstract

We introduce in this article a random model of reactivity in which a primitive rule, if accepted, generates an infinite number of rules by context derivation. The model may be thought of as a toy model of chemical reactivity, where reactions are accepted if their randomly distributed activation energy is below a certain threshold. It may be simulated by induction on the level (length of the word). We describe some statistical features of the model, regarding the number and complexity of the rules, and the shape of the reaction network. The complexity index of a rule is defined as the number of covalent bonds involved in the rearrangement. The Bernoulli parameter (acceptation probability) of the rules is chosen as fixed in a first model (Model I), and exponentially decreasing in the complexity index in a second one (Model II). The two models have very different behaviors, Model II exhibiting a non-trivial phase diagram. The main tool for mathematical analysis is an approximate mapping to a Galton-Watson tree with generation-dependent progeny distributions. Detailed simulations, demonstrating a general agreement with theoretical estimates, are provided at the end. Extensions to realistic bond formation/breaking rules for molecules will be presented elsewhere.

Paper Structure

This paper contains 30 sections, 3 theorems, 81 equations.

Table of Contents

  1. General motivation
  2. From reactions to rules
  3. Anabolic and catabolic rules, complexity indices
  4. Acceptation/rejection schemes, Models I and II
  5. Organization of the article
  6. Definition of Models I and II
  7. Anabolic and catabolic reactions
  8. Two families of random reaction networks
  9. Composition trees : the case of anabolism
  10. Anabolic composition trees : theoretical results for Model I
  11. Mapping to Galton-Watson tree
  12. $(k=0)$ and $(k=1)$ corrections
  13. Other quantities
  14. Anabolic composition trees : theoretical results for Model II
  15. Catabolic fragmentation trees: results for Models I and II
  16. ...and 15 more sections

Key Result

Proposition 3.1

Let $F\in {\cal F}$. Then the $F$-composition tree$({\mathbb{T}},V({\mathbb{T}}))$ and the set of primitive vertices Prim$= \uplus_{n\ge 0}$ Prim$_n$ are formed inductively as follows: • $(n=0)$ If $\omega_{F+a\to F\triangleright a}=1$, then $a\in \mathrm{Prim}_0$ is blue, otherwise $a\in V_0({\mat

Theorems & Definitions (5)

  • Definition 2.1
  • Proposition 3.1: composition tree, primitive vertices
  • Definition 4.1: inhomogeneous Galton-Watson process
  • Lemma 10.1: anabolic phase function
  • Lemma 10.2: catabolic phase function