Heterogeneous Popularity of Metabolic Reactions from Evolution

Mi Jin Lee and Deok-Sun Lee

Phys. Rev. Lett. 132, 018401 – Published 4 January 2024

Abstract

The composition of cellular metabolism is different across species. Empirical data reveal that bacterial species contain similar numbers of metabolic reactions but that the cross-species popularity of reactions is so heterogenous that some reactions are found in all the species while others are in just few species, characterized by a power-law distribution with the exponent one. Introducing an evolutionary model concretizing the stochastic recruitment of chemical reactions into the metabolism of different species at different times and their inheritance to descendants, we demonstrate that the exponential growth of the number of species containing a reaction and the saturated recruitment rate of brand-new reactions lead to the empirically identified power-law popularity distribution. Furthermore, the structural characteristics of metabolic networks and the species’ phylogeny in our simulations agree well with empirical observations.

Received 10 December 2021
Revised 15 August 2023
Accepted 14 December 2023

DOI:https://doi.org/10.1103/PhysRevLett.132.018401

Physics Subject Headings (PhySH)

Biochemistry Biomolecular processes Network evolution Scaling laws of complex systems

Biological networks

Interdisciplinary PhysicsStatistical Physics & ThermodynamicsNetworks

Authors & Affiliations

Mi Jin Lee¹ and Deok-Sun Lee ^2,*

¹Department of Applied Physics, Hanyang University, Ansan 15588, Korea
²School of Computational Sciences and Center for AI and Natural Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea

^*deoksunlee@kias.re.kr

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Vol. 132, Iss. 1 — 5 January 2024

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Images

Figure 1
Statistics of the species-reaction association in empirical data and the network evolution model with $μ = 0.02$ . (a) Standardized distributions of the number of reactions $R^{s}$ in a species $s$ . The standardized variable $z^{s} = [(R^{s} - m_{R^{s}}) / (σ_{R^{s}})]$ is used with the mean $m_{R^{s}}$ and standard deviation $σ_{R^{s}}$ from empirical data (triangle) and from the model (circle). The solid line shows the Gaussian distribution $(1 / \sqrt{2 π}) \exp [- (z^{2} / 2)]$ . (b) Distributions of the reaction popularity. The solid line fits the empirical data for $f > 10^{- 3}$ , and the dashed-dotted and dashed lines fit the simulation results for $10^{- 3} < f < 0.4$ and $f > 0.4$ , respectively.
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Figure 2
Network evolution model. (a) A bipartite network of reactions (rectangles) and compounds (circles) represents each species. At every time step, each species may do nothing (rest) or evolve by either gaining a new reaction (expansion) or giving birth to a daughter species inheriting active components formed by a new reaction replacing an old one (speciation). (b) The species tree from a simulation with $μ = 1$ is shown, where nodes represent 5660 species and links represent the parent-daughter relationship. Node size and color vary with the birth time of the corresponding species. The metabolic network of the oldest species is shown in the left box.
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Figure 3
Growth of the species tree. (a) Simulation time $t_{\max}$ to generate $S \geq 5470$ species, the mean number of reactions in a species $m_{R^{s} (t_{\max})}$ , and the crossover scale of popularity $f_{*}$ versus the speciation rate $μ$ . Dashed lines fit the data, respectively. The empirical value $m_{R^{s}}^{(empirical)} = 1353.5$ is expected at $μ^{(empirical)} ≃ 0.0134$ . (b) The number of species $S$ versus the normalized time $\tilde{t} = (t / t_{\max})$ for $μ = 0.02$ . Inset: the probability $α$ that a potential reaction finds a similar reaction in a species versus $\tilde{t}$ . The solid and dashed lines fit the data for $\tilde{t} \leq {\tilde{t}}_{*} ≃ 0.4$ and $\tilde{t} > {\tilde{t}}_{*}$ , respectively. (c) Distributions of the degree of the compounds from the empirical data and simulations with $μ = 0.02$ . (d) Distributions of the metabolic distance of two species from the empirical data and simulations with different $μ$ ’s. Inset: Kullback-Leibler divergence and 1-Wasserstein distance versus $μ$ .
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Figure 4
First-recruitment and popularity of reactions in the network evolution model. (a) Distribution of the normalized first-recruitment time ${\tilde{τ}}_{r} = (τ_{r} / t_{\max})$ of a reaction $r$ with different $μ$ ’s. Inset: plot of the probability $β (τ)$ that a reaction recruited at time $τ$ is brand-new versus the total number of species $S (τ)$ with $μ = 0.02$ . (b) Time evolution of the fraction of distinct recruited reactions in the largest component $m^{(gc)}$ in the universal reaction-compound network. Also shown are the largest components at $\tilde{t} ≃ 0.18$ and $\tilde{t} ≃ 0.69$ , respectively. (c) The popularity $f_{r}$ versus ${\tilde{τ}}_{r}$ with $μ = 0.02$ . Inset: plot of $| (d f_{r} / d τ_{r}) |^{- 1}$ versus $f_{r}$ . Dashed and solid lines fit the data for $f_{r} < f_{*}$ and $f_{r} > f_{*}$ , respectively. (d) First-recruitment time distributions for $μ = 1$ and different probabilities of nonmetabolic speciation $p_{nm}$ .
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