Beyond Algorithmic Phase Transitions: Integrating Evolutionary Contingency, Biophysical Constraints, and System‐Level Organization in Eukaryogenesis

ABSTRACT

Understanding the origins and determinants of biological complexity requires integrating theoretical and empirical perspectives across the tree of life. The present work discusses a recent theoretical proposal suggesting that the emergence of eukaryotes can be understood as an algorithmic phase transition governed by universal scaling laws in gene structure. The model identifies broad regularities including protein length stabilization and increased regulatory complexity. The predicted stabilization of protein length around 500 amino acids (~1500 bp), coinciding with the prokaryote‐to‐eukaryote transition, represents a distinctive strength of the proposed framework. Drawing on evolutionary theory, this analysis highlights the importance of historical contingency, natural selection, and stochastic processes in shaping biological complexity. At the same time, biophysical, biochemical, and evolutionary mechanisms may contribute to constraining protein length, although these mechanisms do not currently explain the specific ~500‐amino‐acid threshold predicted by the phase‐transition model. More broadly, complexity emerges from the interplay between evolutionary dynamics, physical constraints, and system‐level organization. Factors such as epistasis, epigenetic regulation, population dynamics, gene–environment interactions, and ecological dynamics contribute to shaping evolutionary trajectories and constraining possible outcomes. This perspective therefore proposes an integrative framework in which phenomenological scaling relationships, mechanistic constraints, evolutionary dynamics, and historical contingency are viewed as complementary levels of explanation.

1. Background

The advent of next‐generation sequencing (NGS) technologies has profoundly transformed the life sciences, enabling the generation of biological data at a scale unimaginable only a few decades ago. In parallel, advances in computational power and methodology have made it possible to interrogate complex, high‐dimensional datasets that once lay beyond analytical reach. Together, these developments have reshaped the epistemological landscape of biology: a discipline historically grounded in hypothesis‐driven inquiry [1] now increasingly operates in tandem with data‐driven approaches [2].

This convergence has not merely expanded the scope of empirical observation; it has fundamentally altered the kinds of questions that can be meaningfully addressed. Among these, one of the most enduring and intellectually compelling challenges is to understand how biological complexity emerges. This problem lies at the heart of modern biology. It has, for decades, drawn the attention of a deeply interdisciplinary community spanning genetics, biochemistry, and physiology, as well as mathematics and physics, all united by the search for unifying principles underlying the organization of living systems.

Recently, the study by Muro et al. [3], which proposes an explanation for the emergence of eukaryotes as an evolutionary algorithmic phase transition, received the prestigious 2025 Cozzarelli Prize in the field of Biological Sciences, awarded by the Proceedings of the National Academy of Sciences under the auspices of the U.S. National Academy of Sciences (https://www.nasonline.org/news/pnas‐2025‐cozzarelli‐recipients/). This work investigates the evolutionary emergence of eukaryotes, a foundational event underlying multicellularity, by analyzing how gene regulatory mechanisms evolved to facilitate increased complexity. In a large‐scale bioinformatic effort, the authors examined gene and protein length distributions across more than 6500 species (> 30,000 genomes), revealing a scale‐invariant relationship between mean gene length and its variance across evolutionary history. A multiplicative model of gene growth, which accounts for the observed lognormal length distributions, highlights a two‐phase process. Initially, in prokaryotes, protein length scales directly with gene length (Figure 1). However, at the origin of the eukaryotic cell, mean protein length stabilizes around 500 amino acids (Figure 1), while genes continue to grow exponentially, primarily through the incorporation of noncoding regulatory sequences. This expansion includes the emergence and progressive expansion of introns, which are characteristic noncoding elements of eukaryotic gene architecture. This prokaryote‐to‐eukaryote transition in genomic organization is presented as an “algorithmic phase transition,” analogous to computational search algorithms encountering limits when optimizing increasingly larger proteins. From a dynamical systems perspective, this transition is naturally associated with hallmark signatures of phase transitions, including the emergence of critical slowing down (also discussed by Ref. [4], see below), whereby the system's return to equilibrium after perturbations becomes progressively slower as it approaches the critical point (Figure 1). In this regime, the reorganization of regulatory configurations is no longer rapid or locally constrained; instead, it occurs over progressively longer evolutionary timescales, reflecting reduced dynamical responsiveness and enhanced sensitivity to fluctuations in genomic and regulatory architecture. This transition is proposed to alleviate the computational constraints of protein‐based regulation, enabling the development of more sophisticated genetic regulatory mechanisms. The model predicts that eukaryotes emerged ~2600 million years ago, consistent with chronological data, and suggests that evolution combines conserved gene growth with novel regulatory strategies to overcome complexity barriers and pave the way for multicellular life (Figure 1). Importantly, the framework proposed by these authors should not be interpreted as an exhaustive or exclusive explanation of eukaryogenesis, but rather as a phenomenological description of a potentially general regularity in genomic organization.

FIGURE 1.

Relationship between gene and protein length and the associated shift in genomic organization across the proposed evolutionary transition according to Muro et al. [3]. The algorithmic phase transition reflects a profound reorganization in genetic and regulatory evolution within the tree of life, akin to phase transitions in physics, marked by shifts in computational complexity and the organizational architecture of biological systems. (A) Relationship between mean protein and gene lengths. (B) A second phase transition in the fraction of noncoding sequences, indicating a threshold at approximately 1500 base pairs in mean gene length, where a regime shift occurs: At the onset of the eukaryotic cell, the mean protein length stabilizes around 500 amino acids, while genes continue growing at the same rate as before, but this growth primarily involves noncoding sequences that complement proteins in regulating gene activity. The inset (also adapted from Muro et al. [3]) shows that the computational cost reaches a maximum of hardness (peak computational difficulty) around L _C and decreases as the system moves away from this point. The peak corresponds to the critical slowing down point. The colored contour lines (manually generated) provide a schematic simplification of the data points corresponding to the phyletic groups represented by [3] in their figures 3 and 4; they approximate the distribution of the 6519 species included in the original analysis. Aa, amino acids; bp, base pairs; L _C, critical mean gene length.

The Cozzarelli Prize highlights the potential significance of this work for theoretical biology, as it addresses a central question in the field [5] and has garnered substantial recognition. The proposal has received theoretically grounded comments, as noted by Ferrada [4], who highlights several significant challenges and nuances. These include the difficulty of accurately inferring past evolutionary events solely from the genomic features of extant species, given the incompleteness and often poor annotation quality of many genomes. Additionally, Ferrada points out that the influence of environmental factors on evolutionary patterns remains insufficiently understood, complicating interpretations of genomic data. Moreover, the interplay among various types of evolutionary constraints (such as physicochemical, genetic, selective, and developmental constraints) adds multiple layers of complexity that must be unraveled to fully grasp the mechanisms underlying major evolutionary transitions. Ferrada [4] suggests that advances in functional genomics, dynamic biological data, and comparative analyses of genotype–phenotype maps across species will be crucial for addressing these challenges and refining the theoretical framework.

The present perspective builds on the framework proposed by Muro et al. [3] by placing their algorithmic phase‐transition model within a broader conceptual framework of biological complexity. The aim is not to propose an alternative model of eukaryogenesis, but to distinguish between different explanatory levels: phenomenological regularities, mechanistic constraints, evolutionary dynamics, and system‐level organization. These levels are considered complementary, with the explanatory power of the phase‐transition model potentially strengthened when its quantitative predictions are considered together with the biophysical, evolutionary, and ecological constraints within which eukaryogenesis occurred.

2. Constraints, Contingency, and the Limits of Scaling Approaches

The proposal by Muro et al. [3] is insightful and intellectually compelling insofar as it identifies the existence of a deep regularity in biological evolution, a kind of scaling law that spans the entire tree of life. This approach recalls the notion that nature does not explore an unbounded space of possibilities, but is instead constrained by structural limitations, much as in geometry, where only a finite number of regular polyhedra can exist. This analogy resonates with early approaches, such as those presented in Patterns in Nature [6] by Peter S. Stevens, where many natural patterns are understood to emerge from general physical and mathematical principles. From this perspective, the algorithmic proposition has clear merit, as it seeks to identify a universal pattern in life, namely, the scale‐invariant relationship between mean gene length and its variance.

The existence of such regularities, however, does not imply that evolutionary history is deterministic. Biological evolution is shaped by both structural constraints and historical contingency: multiple trajectories and outcomes may be possible, but not all configurations are equally accessible or viable. Biological systems operate within constrained regions of phenotypic and genomic space, as illustrated by scaling relationships such as the well‐known Kleiber's law, which describes the approximate ¾‐power scaling of metabolic rate with body mass. The observation of such regularities does not eliminate historical contingency; rather, it suggests that contingent evolutionary trajectories unfold within boundaries imposed by physical, biochemical, informational, and organizational constraints.

Importantly, this work does not present this phenomenological approach as a comprehensive account of the evolutionary mechanisms underlying eukaryogenesis. Rather, its value lies in identifying a statistical regularity and a potential higher‐level transition in genomic organization that can be integrated with mechanistic explanations operating at other levels of biological organization. In a broader theoretical context, this perspective also recalls the twentieth‐century debate between the neo‐Darwinian emphasis on selection‐driven adaptation and Motoo Kimura's neutral theory of molecular evolution, highlighting the interplay between deterministic and stochastic forces in shaping genomic and phenotypic outcomes, a distinction that remains central to understanding the origins of biological complexity. This tension is particularly relevant because it frames the extent to which complex traits arise through adaptive optimization rather than the accumulation and subsequent co‐option of neutral or nearly neutral variation. Rather than requiring these evolutionary processes to be incorporated explicitly into the phase‐transition model, they provide complementary mechanistic and historical contexts within which the observed genomic regularities may emerge. In this regard, the algorithmic phase transition model could be further enriched by engaging more fully with the extensive theoretical framework of evolutionary and population genetics. For instance, it does not yet explicitly incorporate factors such as selective costs, epistasis (which involves pervasive and often non‐additive interactions among genetic variants that shape fitness landscapes and evolutionary trajectories), epigenetic regulation (e.g., chromatin dynamics, DNA methylation, and other heritable but reversible modifications affecting gene expression), or population dynamics, all of which are essential for understanding how particular genetic innovations become fixed, lost, or maintained within populations, and therefore how complexity is ultimately generated and constrained over evolutionary time.

Moreover, these evolutionary forces do not operate in isolation but interact in complex and context‐dependent ways, making their joint effects difficult to disentangle. In particular, epistatic interactions can generate rugged fitness landscapes with multiple local optima, thereby constraining accessible evolutionary pathways, while epigenetic mechanisms can modulate phenotypic expression without altering the underlying DNA sequence, adding layers of regulatory complexity that are both dynamic and context‐sensitive.

While these evolutionary processes (or, more broadly, causal determinants of evolutionary change) are often studied independently in simplified models for analytical tractability, this decomposition should not be taken too literally. As has been argued in the philosophical literature, even in physics, many so‐called forces, such as friction or elastic forces, are not fundamental but are context‐dependent and resist straightforward isolation [7]. In this sense, evolutionary processes may be more appropriately compared to such non‐ideal forces than to canonical examples like gravitation [7], which are often treated as fundamental and context‐independent.

3. Complexities in Eukaryogenesis

The Muro et al. [3] proposal that eukaryotic complexity does not arise from an indefinite increase in protein size, but rather from a shift toward enhanced regulatory architecture, is conceptually compelling and consistent with a broad body of empirical observations. For instance, large‐scale functional genomics efforts such as the ENCODE project [8, 9] have revealed that a substantial fraction of the human genome is engaged in regulatory activity, including transcription factor binding sites, chromatin accessibility regions, noncoding RNAs, and long‐range enhancer–promoter interactions. These findings highlight the density and hierarchical organization of gene regulatory networks, as well as the extensive noncoding regulatory landscape that contributes to cell‐type‐specific gene expression programs. Together, the intricate regulatory interactome uncovered by ENCODE [9, 10] illustrates the central role of regulatory complexity in shaping eukaryotic biology, beyond changes in protein‐coding sequence alone.

In addition, many scholars have reported that organismal complexity does not correlate straightforwardly with genome size [11], nor is there a clear correspondence between genome size and the number of protein‐coding genes, further indicating that gene count in eukaryotic genomes provides only limited insight into organismal complexity [12]. Historically, this disconnect was first recognized as the “C‐value paradox,” whereby genome size (C‐value) fails to reflect organismal complexity, largely because protein‐coding genes occupy only a small fraction of eukaryotic genomes. Subsequent genomic data extended this observation to the “G‐value paradox,” highlighting that gene number is likewise a poor predictor of complexity [13]. Proposed explanations have often focused on mechanisms that increase the functional and informational output of individual genes (such as protein–protein interactions, intrinsic disorder, post‐translational modifications, and alternative splicing), while comparatively less attention has been paid to processes that expand genomic content without proportionally increasing functional diversity, including gene duplication and domain shuffling [13]. The diversity of protein families, domains, and motifs has been suggested as a more reliable correlate of organismal complexity, as these features tend to scale with the number of cell types and functional specializations.

In this framework, it is particularly noteworthy that the mean protein length stabilizes around 500 amino acids following eukaryogenesis [3], reinforcing the idea of a structural limit beyond which complexity shifts from protein size to regulatory architecture. The notion of a “phase transition” at the origin of eukaryotes further adds an appealing interdisciplinary dimension by importing concepts from physics and algorithmic theory into evolutionary biology. Importantly, the relevance of the algorithmic phase‐transition model lies not simply in proposing yet another possible constraint on protein length, but in its ability to predict a transition around ~500 amino acids (~1500 bp of coding sequence) coinciding with the prokaryote‐to‐eukaryote transition. This quantitative prediction represents an important strength of the model that alternative biophysical or evolutionary constraints considered here do not, by themselves, currently explain.

The proposed limit on protein length may nevertheless have multiple contributing causes grounded in well‐established biochemical principles, including structural stability, domain modularity, folding efficiency, translational costs, and energetic constraints, which may therefore be viewed as complementary mechanistic explanations; see also Ferrada [4]. These mechanisms may help explain why protein length is constrained in general, but they do not, on their own, explain why a marked regime change should occur specifically around 500 amino acids and coincide with the prokaryote‐to‐eukaryote transition. For instance, longer polypeptide chains are inherently more prone to misfolding and aggregation, as increased folding complexity and slower folding kinetics prolong exposure to unstable intermediate states, thereby enhancing aggregation propensity and compromising cellular homeostasis and organismal fitness [14, 15]. In addition, proteins often evolve as modular assemblies of independently folding domains, favoring architectures that can be recombined or regulated without requiring excessive length [16]. Constraints related to co‐translational folding and the kinetics of protein synthesis may further limit the feasible size of functional proteins, as longer sequences increase the likelihood of translational errors and energetic costs [17]. Energetic constraints, error rates in transcription and translation, and limitations in intracellular transport may further contribute to bounding feasible biological complexity. Together, these considerations provide plausible mechanistic constraints on protein length and may contribute to the patterns captured by the phase‐transition model, while leaving unexplained the specific ~500‐amino‐acid threshold [3].

A schematic overview of the principal mechanisms and constraints that contribute to the emergence and limitation of biological complexity across evolutionary scales is provided in Figure 2; specifically:

1. Core evolutionary processes and mechanisms, including natural selection, genetic drift, mutation, recombination, gene flow, gene duplication, exaptation, and epistasis (i.e., non‐additive interactions among genes that influence phenotypic effects and fitness).

2. Systems‐level properties and regulatory complexity, considering gene regulatory networks, noncoding DNA, epigenetic regulation, and post‐transcriptional and post‐translational regulation (e.g., microRNAs, lncRNAs, transcription factor binding, alternative splicing, and post‐translational modifications).

3. Biophysical and biochemical constraints, including protein stability, protein‐folding, protein–protein interaction specificity, and energetic costs.

4. Information‐theoretic and algorithmic perspectives on biological organization, where complexity arises from how information is encoded, transmitted, and transformed, as well as from the algorithmic rules governing these processes. This includes: algorithmic phase transitions in regulatory systems (i.e., where small changes in regulatory parameters can lead to abrupt shifts in system behavior), information capacity of genomes (governing how much functional information a genome can store and reliably transmit, constrained by genome size, mutation rates, and the need for error tolerance), computational constraints in regulatory systems (which limit the ability of gene regulatory networks to process information, imposed by factors like network architecture, timing, energy costs, and biochemical precision), and signal‐to‐noise constraints in gene expression (i.e., the impact of molecular noise on the reliability of gene expression, which restricts how precisely cells can regulate and interpret biological signals).

5. Constraints imposed by genomic architecture (genome size, gene structure, intron‐exon organization, transposable elements, among others).

6. Ecological and environmental influences (including gene–environment interactions, environmental variability, symbiosis, and coevolution).

7. Major evolutionary transitions (e.g., eukaryogenesis and multicellularity) are highlighted as major reorganizations of the architecture and regulation of biological complexity.

8. Multiple constraints, such as energetic limitations (i.e., the energy/ATP budget, whereby cells have finite energy resources and the maintenance of large genomes, complex regulatory networks, and high expression levels is metabolically costly) [18], error catastrophe (i.e., a maximum mutation rate beyond which genetic information cannot be reliably maintained) [19], mutational load (where increased complexity raises the number of mutable sites, thereby elevating the burden of harmful mutations and reducing overall fitness) [20], and developmental restrictions [21], among others, act to bound biological complexity.

9. Emergent properties and self‐organization are presented as higher‐order phenomena arising from the interaction of these factors; including self‐organization in biological systems (where ordered structures or patterns emerge from local interactions without central control) [22], criticality and phase transitions (where biological systems may operate near critical points such that small perturbations produce large effects, for example, switching behavior in gene regulation) [23], trade‐offs between robustness (stability against perturbations) and evolvability (capacity to generate adaptive variation) [24], modularity [25], and other collective network properties.

FIGURE 2.

Multidimensional determinants of biological complexity. Created using elements from BioRender (www.biorender.com).

4. Toward an Integrative Framework for Biological Complexity

Understanding biological complexity requires accounting for the interplay between physical constraints, evolutionary dynamics, and system‐level organization. This perspective provides a conceptual framework in which scaling laws and phase transitions can be interpreted as emergent outcomes of interactions across levels, linking genomic patterns, evolutionary processes, and functional constraints. This integrative perspective does not imply that a phenomenological scaling or phase‐transition model must explicitly incorporate all underlying mechanisms. Rather, it provides a basis for relating different explanatory levels and understanding how regularities can emerge from their interaction. In doing so, it shifts the analysis from a descriptive enumeration of factors toward a more explicit and testable account of how complexity is generated and constrained across scales.

These components can be organized within a hierarchical framework in which they play distinct but interacting roles across levels of organization (Figure 2). Specifically, this framework distinguishes between conditions of possibility, evolutionary dynamics, and functional organization. At the most fundamental level, biophysical and biochemical principles (e.g., molecular stability, folding kinetics, energetic costs), together with informational constraints, define the space of realizable biological configurations. Within this space, evolutionary dynamics (driven by e.g., mutation, selection, drift, and epistasis) determine which configurations are generated, explored, and maintained over time. At a higher level, regulatory and systems‐level properties (e.g., gene regulatory networks and epigenetic mechanisms) determine how genomic information is deployed, coordinated, and stabilized. Within this framework, genomic scaling relationships, e.g., Muro et al. [3], can be interpreted as emergent regularities arising from interactions across these levels. In particular, the proposed algorithmic phase transition may reflect a shift at the interface between evolutionary dynamics and regulatory organization, where increases in genomic complexity necessitate more elaborate regulatory architectures.

This hierarchical organization integrates the diverse factors summarized in Figure 2 as interacting components rather than independent determinants. It also introduces a natural prioritization: physical and informational constraints define the space of the possible; evolutionary dynamics determine which configurations are realized; and regulatory systems shape how complexity is expressed and stabilized. Although necessarily schematic, this framework provides a structured basis for a more unified account of biological complexity.

A key challenge for future research is to operationalize this framework by linking physical constraints, evolutionary dynamics, and regulatory organization across scales. This requires multi‐scale models that generate quantitative predictions linking genomic features, evolutionary parameters, and regulatory complexity. These predictions can be tested using integrated datasets combining comparative genomics (e.g., gene length distributions, intron structure), functional data (e.g., gene expression profiles and regulatory network architectures), and ecological or life‐history variables. A key step is identifying measurable variables that bridge levels (e.g., energetic costs, expression noise, robustness, information capacity), enabling direct links between genomic patterns, evolutionary processes, and phenotypic outcomes. Controlled experimental systems (e.g., microbial evolution or synthetic regulatory circuits such as toggle switches or repressilators) provide a direct way to test how constraints on energy, noise, and regulatory architecture influence the emergence of complexity. Ultimately, progress will depend on integrating theoretical, computational, and experimental approaches to achieve a predictive and mechanistic understanding of biological complexity.

The main conceptual contribution of this perspective is therefore not to replace the algorithmic phase‐transition model, but to situate it within a multilevel framework in which quantitative scaling relationships, mechanistic constraints, evolutionary dynamics, and historical contingency are treated as complementary components of the same explanatory problem.

5. Final Considerations

The findings of Muro et al. [3] also invite further reflection. Their proposal elegantly foregrounds structural and formal (potentially algorithmic) constraints as key explanatory factors, suggesting that they may play a central role in shaping the emergence of biological complexity. At the same time, the phenomenological nature of their approach means that it need not specify, or exclude, the full range of mechanisms and historical contingencies that contribute to the observed transition. At this point, the classical critique articulated by paleontologist Stephen Jay Gould becomes particularly relevant. Gould emphasized that evolution cannot be understood solely in terms of general laws or constraints, but is instead profoundly shaped by historical contingency, natural selection, and processes such as genetic drift and exaptation, among others. It is not sufficient to describe the space of the possible; one must also explain why evolution follows particular trajectories rather than others. As emphasized by Gould [26], evolutionary history is fundamentally contingent; to use his well‐known analogy, if the “tape of life” were replayed, the outcomes would likely differ substantially [27].

The algorithmic phase transition proposal may thus be best understood as a highly valuable contribution to a tradition that seeks to identify general laws and constraints in nature. Rather than constituting a competing explanation for the mechanisms of evolution, it provides a phenomenological account of a potentially general transition in genomic organization that can be integrated with mechanistic and historical accounts. In this sense, it is closer to a “theory of conditions of possibility” than to a complete theory of evolution. The proposal helps illuminate aspects of the boundaries and statistical organization of the evolutionary space, while leaving open the diverse mechanisms and contingent historical processes through which that space is navigated. The tension between these two perspectives, general laws and constraints on the one hand, and historically contingent (non‐deterministic) processes on the other, is not a flaw of this approach, but rather a reflection of one of the deepest debates in biology: to what extent life is determined by universal rules, and to what extent it is the irreproducible outcome of its own history. Rather than being mutually exclusive, these perspectives can be seen as complementary, each capturing different but equally essential aspects of evolutionary dynamics. Future work may benefit from more explicitly integrating these complementary perspectives into a unified and predictive theoretical framework that links constraints, dynamics, and historical trajectories.

Author Contributions

A.S. conceived and designed the study, performed the analyses, interpreted the data, and wrote the manuscript.

Conflicts of Interest

The author declares no conflicts of interest.

Data Availability Statement

No new data were generated or analyzed in this Perspective, and no existing datasets or databases were used. Therefore, data sharing is not applicable.