Entropy

MotivationWith the rapid development of AI methods that predict protein structures from sequence, understanding the structure-function relation increasingly depends on quantitative structural descriptors that are both biologically meaningful and scalable to large datasets. Here, we introduce mathematical topology metrics that quantify the entanglement complexity of a tertiary protein structure while respecting uncrossability constraints. ResultsBy employing only three such metrics across all protein structures in the Protein Data Bank, we represent the proteome structural space in a continuous three-dimensional space. Distances within this space capture structural similarity and correlate with functional similarity. We find that the mathematical entanglement based landscape of protein structural space diversifies with the evolutionary expansion of protein function across species. Moreover, this continuous representation reproduces CATH classifications with high accuracy for major structural classes. These results indicate that these metrics efficiently encode structural features linked to protein function and provide a more informative description than conventional metrics. AvailabilityData used in this study are available in the Protein Data Bank. Details of the machine learning model used can be found in https://github.com/roshitac/CATH_Classification-. ContactBanu.Ozkan@asu.edu, Eleni.Panagiotou@asu.edu Supplementary informationSupplementary data are available at Journal Name online.

AO_SCPLOWBSTRACTC_SCPLOWCellular automata and graph reaction-diffusion systems encode local spatial interactions in different mathematical forms. We develop a cochain-operator calculus for these two settings. Over a finite field Fq, every local rule on a finite neighborhood has a unique reduced polynomial representative. On an oriented line, the coboundary and endpoint maps recover the left and right shifts. Our main theorem shows that these operators, together with linear operations, constant cochains, and the degree-zero cup product, generate every finite-radius polynomial cellular automaton. Explicit formulas for Rules 30, 110, and 22 show how reflection-invariant linear coupling, directed transport, and nonlinear neighbor interactions enter the calculus. On a general graph, d*d is the unweighted combinatorial Laplacian and enters a graph reaction- diffusion recurrence. Over [R], the term - Dd*d with D [≥] 0 admits the usual diffusion interpretation; over Fq, the corresponding expression defines modular coupling without an intrinsic order. In the morphogenetic examples, we therefore distinguish pattern-generating dynamics from finite-state observation and use the Betti numbers of active induced subcomplexes to summarize observed patterns. This yields a common algebraic representation without identifying real-valued diffusion with finite-field dynamics.

Direct reciprocity, a mechanism for evolution of cooperation, rests on a promise of the future: that the cost of cooperation will be returned in subsequent encounters. Whether reciprocity works is often asked as a question about evolutionary stability: can a population of cooperators resist invasion by mutants that defect? Here we answer this question exhaustively for a large, but finite strategy space projected onto an uncountable infinity of evolutionary games. We take the binary memory-two strategies, in which each of the sixteen possible two-round histories is answered by cooperation or defection up to a small error rate. We work out what every one of them achieves on its own and against every rival, in every symmetric two-player game. Alone, they realise 475 distinct patterns of play carrying 229 distinct cooperation rates. Against each other, a game supports between 299 and 22069 Nash equilibria, that is, resident strategies that no rare mutant can outperform. Efficient strategies, which are those that reach maximum payoff, are present as equilibria at every game, and for 3/8 of games efficient strategies constitute the only equilibria. Those results belong to the limit of a vanishingly small error rate. For any positive error rate the map changes: under 1/4 of all games support no equilibrium, over 3/8 support equilibria but none are efficient, over 1/8 support equilibria and all are efficient, and under 1/4 of games support equilibria of both kinds. Every share quoted here is an exact natural density over the plane of games. The map is the landscape over which any evolutionary dynamics in this strategy space moves.

How morphological complexity evolves is still enigmatic. While there is evidence in algae and plants as well as animals that diversification of the repertoire of transcription factors (TF) is causative for evolution of organismal complexity, there are many examples from lineages that follow their own way of complexity evolution, for example by expansion of particular families. For land plants, correlation of the size of the TF complement with number of cell types (as a proxy for morphological complexity) has been shown, and several families were identified as candidates to drive complexity evolution. Here, we expand a previously available dataset of cell type numbers from 12 to 82 proteomes and introduce a four class body plan scheme. We find that the total TF complement correlates with the number of cell types of Archaeplastida (primary plastid bearing plants and algae). We used TabPFN (Tabular Prior-data Fitted Network) for binary (uni- vs. multicellularity) as well as for four class Bauplan classification. TabPFN is able to predict the morphological complexity with high accuracy. This approach allows to determine organismal complexity based on the gene space of an organism. Based on our results, we can confirm that plant morphological evolution is driven by gain and expansion of TF families.

Ecological interactions play a crucial role in driving speciation and adaptive radiation. According to the adaptive dynamics theory of evolution in one-dimensional trait spaces, such evolutionary diversifications can occur when evolutionary branching points (convergence stable and evolutionarily unstable singular points) exist in the trait spaces. However, these evolutionary diversifications may occur across multiple traits at once, as exemplified by various within-species ecotypic diversifications involving multiple traits or genes. This paper attempt to extend this branching point condition to evolution in multi-dimensional trait spaces. Two criteria, termed branching possibility and branching inevitability, are developed to evaluate the likelihood of evolutionary branching in multi-dimensional trait spaces. Branching possibility guarantees the existence of evolutionary pathways that lead to branching (where deterministic parts of these pathways are described with the canonical equation of adaptive dynamics theory). Branching inevitability guarantees a steady progression of evolutionary branching (where the progression degree is described with a time-increasing function related to the local Lyapunov function, the potential function in an evolutionary potential game, and Fisher's fundamental theorem). Numerically simulated evolution in multi-dimensional trait spaces show that points having branching possibility or inevitability can induce evolutionary branching not only under asexual reproduction but also under sexual reproduction.

Network centrality is the workhorse of gene prioritisation, yet what a ranking omits is rarely audited. Scoring each selection against an annotation-count-matched maximum-entropy reference--asking whether a selected gene set covers the genomes functional space or collapses it-reveals that the criterion in standard use has a measurable blind spot in exactly the class it is meant to surface. Degree, the most widely used criterion, returns the cross-module bridges that are also locally dominant--connector hubs--and omits the non-hub connectors: where 26% of the genome occupies these coordinating roles, a degree-ranked list holds 18% and an EDVS-ranked list 55%, and degrees top-1% collapses functional coverage below the reference on all five networks tested. We repurpose EDVS (Entropy of Degree-Vector Sums), an information-theoretic diversity measure, as an annotation-free, partition-free centrality that recovers this omitted class. The coverage it preserves is carried by cross-module participation P, which cannot be computed without a community partition; EDVS matches P-level coverage on all five networks using none, and retains 0.84 of its selection under edge perturbation that leaves partition-based selections at 0.21-0.46. The deficit is general: the collapse holds in the same direction on the two networks built without functional annotation (0.5-1.1 bit; co-expression, physical interaction) as on the three supervised by it (1.6-3.3 bit; RiceNet, AraNet, STRING), so supervision amplifies it rather than creates it. The remedy is bounded: EDVS ceases to preserve coverage on the sparse physical-interaction network. And the class EDVS isolates is organizational, not an importance signal: pre-registered probes--essentiality, transcription-factor identity, tissue-specificity, date/party-hub character, phenotype co-localisation--return null or reversed throughout. The conclusive ones are equivalent to their degree-matched nulls within {+/-}5 percentage points (demonstrated, not merely undetected), and the classical coupling of centrality to importance itself holds only network-dependently. Author SummaryGenes rarely act alone: many diseases and agricultural traits are shaped by genes that coordinate several biological processes rather than specialising in one. The standard way to find such genes in a network of gene interactions is to count each genes connections--its "centrality"--and rank genes by that count. We show this standard approach has a blind spot: it favours genes that dominate one process over genes that quietly bridge several processes without dominating any, and this blind spot appears across rice, thale cress, and yeast gene networks. We repurpose a diversity measure from an unrelated field (originally used to compare citation patterns) as a new way to rank genes that finds these bridging genes from network structure alone, without needing gene-function annotations--which are themselves incomplete and biased toward well-studied genes--or a prior, unstable step of splitting the network into modules. We are careful to show where the new approach also falls short: on sparse, noisy networks it stops working, and the genes it recovers are not shown to be more biologically important than other genes, only differently positioned. What that position is for is a question this work leaves open.

Structural network representations of metastatic dissemination typically focus on static topology without resolving transport dynamics, relaxation timescales, or steady-state behaviour. Here, we formulate a discrete Markovian transport model on a directed higher-order network with transition rates derived from qualitative clinical affinity classes. By constructing a non-Hermitian row-stochastic transfer operator, we characterise the relaxation dynamics through its spectral decomposition. The system exhibits a fast-mixing regime characterised by a spectral gap of {gamma} {approx} 0.67, corresponding to a characteristic relaxation timescale of {tau} {approx} 1.49 discrete steps, with the influence of the primary tumour origin progressively attenuated during dissemination. Convergence towards a non-equilibrium steady state (NESS) is accompanied by a reduction in Shannon entropy, concentrating probability mass within specific topological sinks. This spectral relaxation delineates two distinct dynamical regimes: early transient dissemination (n < {tau}), dominated by local organ-specific transition probabilities (organotropism), and the asymptotic regime (n > {tau}), determined increasingly by the global transport architecture of the network. Comparison with independent clinical and autopsy observations across 21 primary tumours and 23 target organs indicates that the predicted stationary distribution is consistent with the observed hierarchy of metastatic organ involvement.

Artificial intelligence (AI) assistants can improve individual performance but may weaken unaided performance and limit interest diversity, raising questions about their long-term effects on collective intelligence (CI). Here, we consider an evolutionary model in which individuals predict, consult AI, and imitate better-performing peers. Classic reward structures that promote CI without AI can fail under Chatbot AI, whose predictions depend on users' interests, by reducing interest diversity or propagating bias. Directly rewarding or penalizing AI use does not reliably mitigate these effects. Instead, a balanced incentive that preserves minority expertise and rewards collective contributions can promote CI, but an additional incentive is needed to restore unaided collective accuracy. Our framework offers an explanation for emerging evidence of AI's societal impact and clarifies its long-term implications.

The folding thermodynamics of proteins are dominated by two opposing forces, the loss in backbone entropy and the packing of hydrophobic groups. The same forces are major contributors to the extension thermodynamics of elastic proteins with the distinction that both processes act in concert, favoring the higher chain and solvent entropy of a relaxed conformation. The relative entropic contributions specify the recoil mechanism; human elastin recoil is primarily driven by hydrophobic forces, whereas fly resilin has a rubber-like mechanism driven by backbone entropy. Despite the importance of elastic proteins to tissue biomechanics, few have been identified, let alone characterized to the same extent as elastin and resilin. We develop a thermodynamic framework that maps proteins by sequence-derived estimates of extension-induced backbone and solvent entropy changes. Putative elastic proteins are proposed and classified by recoil mechanism based on estimated thermodynamic features. Proteins that map to elastic regions are overrepresented by the skin proteome. The set of predicted elastic domains is further extended by incorporating sequence context embedded in protein language models. Protein domains with distinct thermodynamic recoil mechanisms cluster on the latent space manifold. Some of these domains are anticipated to have roles within molecular machines, expanding the scope of elastic protein function beyond mechanical materials like elastin and resilin.

A central goal of neuroscience is to establish how similar (or different) brain networks are across individuals, development, psychiatric conditions, and even between biological and artificial brains. Whenever one such comparison is made, it relies on the assumption that we have a measure that produces an accurate and plausible similarity score. Yet, the field lacks one such measure, and the wide variety of existing measures often produces conflicting results. To address this problem, we perform a systematic comparison of 16 established similarity measures. First, we show that different measures consistently disagree on which brain networks are most similar. Second, we rank all measures according to multiple criteria such as biological plausibility, computational efficiency, and sensitivity. We show that DeltaCon is the most accurate measure in terms of parameter recovery, and that it ranks among the fastest and most noise-tolerant. For this reason, we conclude that DeltaCon is the best general-purpose similarity measure. However, no measure ranks best on more than two of the five criteria we assessed, and different measures should be preferred depending on the exact research question. Measure selection can now be made on evidence and should be a crucial decision in any study that compares brain networks.

Contextualized protein-protein interaction networks provide crucial insight into diseases and other biological processes, but for a profound understanding of such processes and their distinct effects on individuals, the protein-protein interactions within individual samples must be investigated. A straightforward approach to estimate the PPI network of a sample is to restrict a general network of known PPIs to the proteins that are found in the sample. Although proteomics methods are becoming more accessible and precise, large-scale and single-cell studies still mainly target characterizing the transcriptomics profile of the samples, which is then often used as an approximation of the protein activities. The correlation of gene expression and protein abundance has been addressed in the past, but information about the deviations of the different omics-based estimates of the PPI networks is still lacking. In this study, we performed a comparative analysis of transcriptomic-based and proteomic-based sample-specific PPI network estimates to fill this gap. We created a framework for a comprehensive and transparent comparison of the two omics levels in two independent datasets, with a special focus on time-related network dynamics. We found that the size-adjusted characteristics of the different omics-based networks are very similar; the overall trend of how they change with time is also often the same, but the rate of the changes typically differs. The characteristics of the nodes present in both types of networks also show high similarity and often different time-related rates of change, but this varies among metrics. These results shed light on the properties of PPI network estimations and advise caution in interpreting them appropriately.

Extensive studies suggest the brain performs Bayesian inference to infer the latent world states. It is a fundamental neuroscience question that how canonical recurrent neural circuits in the brain implement Bayesian inference. Many existing theoretical studies focused on how the recurrent circuits compute the posterior, while largely overlooking how the circuits compute the model evidence (normalization constant in Bayes' theorem), a key quantity that measures how well a model explains the observed data.Thus, it remains largely unknown about how the recurrent circuits compute the model evidence. The present study performs rigorous theoretical analyses of the continuous attractor networks, a canonical recurrent circuit model, and reveals that the nonlinear circuit dynamics can simultaneously compute the posterior and model evidence in first two dominant subspaces within the circuit dynamics. Specifically, the circuit dynamics in the stimulus feature subspace implements the Langevin posterior sampling, and the circuit dynamics in the subspace of total neuronal activity computes the model evidence in a way analogous to the evidence lower bound in stochastic variational inference. We further extend the circuit model to compute the model evidence of multiple inputs, and simulations validate the computation in the network. Our work for the first time reveals the concurrent model evidence and posterior sampling in subspaces in continuous attractor networks, significantly deepen our understanding of the computational algorithms adopted by the neural circuits.

Functional protein sequence space is often described either by the sparsity of functional sequences or by the connectivity of neutral networks. These quantities characterize different properties: density measures the fraction of all sequences that are functional, whereas connectivity describes relationships among functional sequences. Neither alone determines how much of sequence space lies close to function. The coverage function measures the proportion of sequence space within a prescribed Hamming distance of functionality and therefore provides a direct geometric measure of local accessibility. Here we connect this geometric framework to characteristic discovery times using an explicitly heuristic model of stochastic exploration. The explored region is represented by an effective Hamming ball, and radial displacement is modelled as an outward-biased substitution process. A numerical illustration, calibrated to an empirical human germline mutation rate, shows how a coverage radius can be converted into a timescale. We then extend the framework to parallel search by defining an overlap-adjusted effective number of trajectories. The purpose is not to predict exact evolutionary waiting times, but to separate the geometric distribution of function from the dynamics by which sequence space is explored.

The genetic code is the rule that (apart from minor modifications) is used by all contemporary life on Earth to translate genes into proteins. It is summarized by a 4x4x4 table which maps the 64 possible triplet RNA codons to the 20 canonical amino acids ("AAs"). Understanding the origin of this code is a fundamental open problem in biology. We review the idea of a comma-free code, and put forward a general reason to suspect the primordial code was comma-free (even though the modern code is not). We explain why this primordial code should then persist as a comma-free subcode within the modern code, and identify the five optimal candidates for such a subcode. We then introduce the idea of "compatible comma-free codes" and explain how they allow multiple codes to evolve in parallel, circumventing a standard argument suggesting that code evolution should effectively freeze when the associated genome becomes sufficiently complex. In particular, we highlight the possibility that the original code was a comma-free doublet code, which was then overtaken by a compatible comma-free triplet code, which finally evolved (after the evolution of the start codon and ribosome) to the modern saturated triplet code. To assess this idea, we classify all inequivalent compatible comma-free doublet-triplet codes and, among these, identify the three most promising predecessors of the modern code. Consistency between these three best codes and the "wobble rules" in gene translation provide non-trivial evidence for this possibility.

Biochemical Systems Theory (BST) represents nonlinear biochemical rate laws by local power-law approximations in logarithmic concentration coordinates. First-order coefficients are elasticities, whereas higher-order derivatives describe local log-synergism and its variation. Ordinary higher derivatives, however, are not tensorial under nonlinear reparameterizations and can mix biochemical response structure with coordinate artifacts. We formulate a covariant hierarchy, cBST1-cBST3, on a positive operating-point space equipped with a declared reference connection. cBST1 recovers classical elasticities in a flat logarithmic chart, cBST2 is the covariant Hessian of the log-response, and cBST3 is the symmetrized covariant derivative of cBST2. The framework is quantitatively evaluated using three representative rate laws from the curated yeast glycolysis model BIOMD0000000064: glucose transport, glucose phosphorylation, and phosphofructokinase. For 10,000 finite log-concentration perturbations at each of five radii, cBST2 reduced the cBST1 log-rate root-mean-square error by 96.6-98.6% at the largest tested radius, and cBST3 provided a further 68.4-98.1% reduction. The contracted cBST2 and cBST3 terms strongly predicted the corresponding lower-order truncation errors. Under the nonlinear transformation qi = sinh(ui), covariant contractions agreed across coordinates to within a 95th-percentile relative error of 1.2 x 10-13, whereas ordinary higher derivatives showed order-unity coordinate mismatches. Supplementary analytic tests recovered the expected second-, third-, and fourth-order truncation-error scaling. These results show that cBST1-cBST3 are not only coordinate-consistent descriptors but also practical diagnostics of where local power-law approximations require higher-order correction.

Recent advances in large-scale electrophysiological recording technologies now allow simultaneous measurement of spike trains from ensembles of neurons. A central challenge in mathematical neuroscience is therefore to identify functional assemblies and their collective organisation directly from this data. The goal of this work is to devise a method to infer the collective organisation of the neurons from their spiking activity. We construct weighted functional networks from neural spike trains using the van Rossum distance to quantify pairwise similarity. The functional network captures similarities in neuron firing patterns and provides a representation of their collective organisation. We hypothesise that similar neuronal assemblies will appear as clustered communities in the network, and employ the Louvain algorithm to detect such assemblies. We validate our approach using synthetic spike train data and simulated data from a stochastic block model of Leaky Integrate and Fire neurons subjected to external Poisson drives, where the ground truth is known. Finally, we apply our approach to large-scale recordings from the Allen Institute Visual Coding: Neuropixels dataset. We find that our methodology works well as long as a sufficient amount of data is available and the temporal structure is strong relative to the noise.

The global potential of a chemical reaction network has many applications and is closely related to the stochastic detailed balance. However, many fundamental questions concerning stochastic detailed balance remain unresolved, such as whether it depends on the system volume and how to construct new systems that satisfy it. In this paper, we show that stochastic detailed balance may depend on the system volume. We therefore introduce four types of stochastic detailed balance according to their dependence on volume and rate constants, and systematically investigate the relationships among them. Our results distinguish detailed balance arising from particular choices of volume and parameters from that enforced by network structure, and identify conditions under which detailed balance at one volume extends to all volumes. We further obtain a class of networks satisfying stochastic detailed balance for every volume and every positive choice of rate constants, and construct new systems whose global potentials exhibit double-well structures.

Understanding the relationship between gene expression dynamics and cellular identity remains a central challenge in single cell biology. Here, we introduce a novel computational and mathematical framework that integrates information geometry, fuzzy topology, and UMAP analysis to model gene expression landscapes derived from single cell RNA sequencing data. We formalize gene expression data as a fuzzy topological space, where interactions between expression points are governed by probabilistic distributions inspired by manifold learning approaches such as UMAP. Within this framework, we define an information geometric structure through a Fisher metric induced by these distributions, enabling the computation of geodesic trajectories that capture cellular differentiation processes. A key contribution of this work is the derivation of analytical conditions, expressed as expression radius formulas, that characterize local neighborhoods in gene expression space. These conditions allow for the identification of genes associated with stem cell states and predictions in transitional cell types in future work. Application of the proposed framework to single cell datasets reveals biologically meaningful gene sets enriched in key regulatory pathways and transcription factors, demonstrating the capacity of our approach to uncover latent structure in complex gene expression data. Our results suggest that integrating differential geometry with statistical learning theory offers a powerful paradigm for modeling genotype and phenotype relationships and cellular state transitions, with potential implications for precision medicine and systems biology.

Inferring "whether a change in the expression of a given gene causally affects the disease state" from observational single-cell transcriptomic data is one of the central problems in single-cell biology. The difficulty lies in confounding: cell state, batch, cell cycle, and the co-expression of other genes may all simultaneously influence the target gene (treatment variable T) and the disease label (outcome variable Y), so that naive correlation analysis cannot distinguish causation from covariation. Double machine learning (DML), via orthogonal scores and cross-fitting, allows machine learning to estimate high-dimensional nuisance functions, thereby addressing the causal inference problem in high-dimensional data. Constrained by computational resources, this study takes a small-sample dataset with p{approx}n (2,120 cells, 1,999 background genes) as the experimental testbed and systematically compares three nuisance function construction strategies under this critical condition; strategies for the n>>p regime are then addressed by theoretical argument. Using systemic lupus erythematosus (SLE) peripheral blood memory B cells (GSE189050, 2,120 cells), we construct a three nuisance function construction strategies x (in-sample / cross-fitting) 2x3 factorial experiment and compare them in terms of resolution, biological plausibility, stability, and deconfounding ability in causal effect estimation. The three strategies are: (S1) direct linear nuisance regression on the high-dimensional background genes without dimensionality reduction; (S2) learning a shared low-dimensional representation with an unsupervised autoencoder, with the treatment and outcome residuals sharing that representation; (S3) fitting the treatment and outcome with two independent deep networks. The results yield a clear three-part pattern with mechanistic meaning: (1) both modes of S1 fail; (2) S2 achieves the optimum in-sample and requires no cross-fitting; (3) S3 is "rescued" under cross-fitting and becomes effective. We explain this pattern starting from the convergence rate condition of DML (Sec.5 Theoretical Foundations) and give the applicability boundaries of the two strategies: S2 (shared unsupervised deep learning) achieves estimation quality comparable to standard cross-fitted DML at a tiny fraction of the computational cost, making it a feasible scheme for large-scale screening; S3 (dual networks + cross-fitting), as the standard DML recipe, can serve as a broad-spectrum reference control for S2 results.

Biological populations have been evolving in multi-dimensional trait spaces. Ecological interactions can induce the evolutionary diversification of a biological population into distinct multiple populations, which is called evolutionary branching in adaptive dynamics theory. If possible mutations are restricted to certain directions due to developmental, physiological, or physical constraints, the resulting evolutionary dynamics may be restricted to subspaces with fewer dimensions than the original trait spaces. In general, such constraints are also evolving slowly, due to rare and slight mutational deviations from the constraint subspaces. In this sense, constraints are not strict. However, the method for describing non-strict constraints that is globally consistent has yet to be developed. This study proposes a general and flexible method for constructing a non-strict constraint subspace from the corresponding strict constraint subspace. This method enables efficient analysis of adaptive evolution including evolutionary branching under non-strict constraints in the framework of adaptive dynamics theory. Applying this method to a resource competition model shows that the shape of the carrying capacity distribution can be treated as a non-strict constraint curve, and that the likelihood of evolutionary branching is affected not only by the shape of the constraint curve but also by its slow evolution.