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. 2025 Apr;122(13):e2422968122.
doi: 10.1073/pnas.2422968122. Epub 2025 Mar 27.

The emergence of eukaryotes as an evolutionary algorithmic phase transition

Enrique M Muro  1 , Fernando J Ballesteros  2 , Bartolo Luque  3 , Jordi Bascompte  4
Affiliations

The emergence of eukaryotes as an evolutionary algorithmic phase transition

Enrique M Muro et al. Proc Natl Acad Sci U S A. 2025 Apr.

Abstract

The origin of eukaryotes represents one of the most significant events in evolution since it allowed the posterior emergence of multicellular organisms. Yet, it remains unclear how existing regulatory mechanisms of gene activity were transformed to allow this increase in complexity. Here, we address this question by analyzing the length distribution of proteins and their corresponding genes for 6,519 species across the tree of life. We find a scale-invariant relationship between gene mean length and variance maintained across the entire evolutionary history. Using a simple model, we show that this scale-invariant relationship naturally originates through a simple multiplicative process of gene growth. During the first phase of this process, corresponding to prokaryotes, protein length follows gene growth. At the onset of the eukaryotic cell, however, mean protein length stabilizes around 500 amino acids. While genes continued growing at the same rate as before, this growth primarily involved noncoding sequences that complemented proteins in regulating gene activity. Our analysis indicates that this shift at the origin of the eukaryotic cell was due to an algorithmic phase transition equivalent to that of certain search algorithms triggered by the constraints in finding increasingly larger proteins.

Keywords: eukaryotic cell; gene length; protein length; scaling law.

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Conflict of interest statement

Competing interests statement:The authors declare no competing interest.

Figures

Fig. 1.
Fig. 1.
Gene and protein length distributions are lognormal. Length distributions for genes (blue, measured in number of base pairs) and their corresponding proteins (orange, measured in number of amino acids) for Danio rerio (zebrafish). Note that lengths are represented on a logarithmic scale. The red curves are fits to lognormal distributions. Mean lengths (and SDs) of genes and proteins are 31,084 (51,637) base pairs and 538 (562) amino acids, respectively. Total number of genes and proteins plotted is 25,432 and 25,706, respectively. These lognormal distributions are typically found in almost all species across the tree of life (Materials and Methods).
Fig. 2.
Fig. 2.
Scale-invariant relationship between mean gene length and variance. Each dot in this log–log plot represents the genome of a single species (n= 33,627), with H. sapiens highlighted by the circle. Gene length is measured in number of base pairs. Different colors identify major phyletic groups. As a guide to the eye, we have added the labels fishes, Aves, and Primates, marking the region where these groups cluster. The line represents the best fit to a power law (Eq. 4; σ2=0.0159(±0.0004)⟨L⟩2.511(±0.004), R2=0.92). Inset: equivalent representation for protein length (measured in number of amino acids) for 9,913 species. The line represents the best fit to an equivalent power law (SI Appendix; σp2=0.0168(±0.0015)⟨Lp⟩2.605(±0.016), R2=0.73). Major phyletic groups appear sequentially across the scaling law for genes, but not so clearly in the equivalent for proteins.
Fig. 3.
Fig. 3.
Threshold in the relationship between mean protein and gene lengths. Each dot represents a single species (n= 6,519) (see Merging Ensembl and Uniprot Annotations for details on data selection), with H. sapiens highlighted by the circle. Gene and protein lengths are measured in number of base pairs and amino acids, respectively. Different colors identify major phyletic groups. The black line (Eq. 6) describes the trend observed in the data separating two distinct phases. In the first phase, mean protein length grows proportionally to mean gene length. Beyond a critical mean gene length around 1,500 base pairs, mean protein size stabilizes and is not any more a function of mean gene length (SI Appendix, Fig. S9, Bottom).
Fig. 4.
Fig. 4.
Second-order phase transition in the fraction of gene noncoding sequences. Each dot represents a single species (n= 6,519), with H. sapiens highlighted by the circle. Gene length is measured in number of base pairs. Different colors identify major phyletic groups. The black line represents Eq. 7 assuming a critical mean gene length of 1,500 base pairs separating the coding sequence (CDS) and noncoding sequence (nCDS) phases. Inset: White squares describe the scatter of values for the fraction of gene noncoding sequences (i.e., fraction of states) at a given mean gene length, as described in Materials and Methods; colored points represent the experimental algorithmic complexity τexp for the groups of organisms shown in SI Appendix, Fig. S3. This is estimated as the quotient of their divergence time with H. sapiens, tdiv (in million years, obtained from TimeTree.org), measured from LUCA (3,600 My ago) (5), divided by the average mean gene length of the group, Lgr, in base pairs: τexp=(3,600−tdiv)/Lgr. Both measures become maximal around the threshold 1,500 base pairs.
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