The analysis of quantitative laws

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Table of contents

1 Introduction 
2 Family-specific scaling laws in bacterial genomes
2.1 Introduction
2.1.1 High-level functional categories of genes follows quantitative laws
2.1.2 The analysis of quantitative laws at the domain-family level may explain how the scaling of functional categories emerges from the evolutionary dynamics
2.2 Families have individual scaling exponents, reflected by family-specific scaling laws
2.2.1 Data analysis
2.2.2 Comparison with a null model supports the existence of scaling laws at the family level is not simply due to sampling e↵ects
2.2.3 Family exponents correlate with diversity of biochemical functions but not with contact order or evolutionary rate of domains
2.3 The heterogeneity in scaling exponents is function-specific
2.4 Determinants of the scaling exponent of a functional category
2.4.1 Super-linear scaling of transcription factors is determined by the behavior of a few specific highly populated families
2.5 Grouping families with similar scaling exponents shows known associations with biological function and reveals new ones
2.6 The main results of our analysis hold also for PFAM clans
2.7 Discussion
3 Dependency networks shape frequencies and abundances in component systems 
3.1 Introduction
3.1.1 The emergence of universal regularities in empirical component systems may be the e↵ect of underlying dependency structures of the components
3.2 Model: description of the dependency structure and the algorithm that defines a realization
3.3 Our positive model recovers the empirical regularities of component systems, namely the Zipf’s law and the Heaps’ law
3.3.1 The analytical derivation of the components abundance distribution matches simulations and satisfies the Zipf’s law
3.3.2 The power-law distribution of components occurrence is a “null” result of our model
3.4 The analytical mean-field expression of the Heaps’ law matches the results of numerical simulations of the model
3.4.1 The analytical expression of the Heaps’ law shows three di↵erent regimes
3.4.2 The stretched-exponential saturation is a remarkably good approximation of the simulated data
3.5 Conclusion
4 Signature of gene-family scaling laws in microbial ecosystems 
4.1 Introduction
4.2 Methods
4.3 The analytical implementation of family scaling laws results in the definition of a metagenome invariant
4.3.1 Analytical derivation of the abundance of a protein family in a metagenome
4.3.2 The metagenomic invariant gives access to the moment of the distribution of genomes size in the metagenome
4.4 The mean genome size and the number of genomes in a metagenome are estimated reliably in simulated metagenomes
4.4.1 The rescaled family abundance in simulated metagenomes shows clear scaling with family exponent
4.4.2 The total number of sampled genomes can be estimated reliably in simulated metagenomes
4.4.3 The average genome size can be estimated reliably in simulated metagenomes
4.4.4 The variance of the genome size distribution deviates from the predicted behavior
4.5 The mean genome size and the number of genomes are estimated reliably in real metagenomes
4.6 Conclusions
5 Conclusions and perspectives