AXIOM-Bio: A Foundational Model for Biological Logic and Disease

1 | Page Opinion Title: AXIOM-Bio: A Foundational Model for Biological Logic and Disease Running title: If Life Computes with Logic—So Should Medicine Authors Saptarshi Sinha1, 3* and Pradipta Ghosh1-3* Departments of 1Cellular and Molecular Medicine and 2Medicine, University of California, San Diego, CA, 92093, USA. 3UC San Diego Institute for Network Medicine, University of California, San Diego, CA, 92093, USA. *Correspondence to [email protected] (SS) or [email protected] (P.G). Keywords Invariants · Systems Biology · StepMiner · Dose–Response Alignment · Cellular Intelligence · Function-Agnostic Modeling 2 | Page Abstract Biology and medicine often mistake complexity for understanding. We build black-box AI models and amass terabytes of omics, yet the logic of life remains hidden in plain sight. Cells, the smallest decision-makers, compute not probabilistically, but logically: “on” or “off,” “commit” or “retract.” Boolean mathematics decodes this digital behavior, distilling molecular noise into invariant “if–then” rules that persist across tissues, species, and diseases. By quantifying how populations shift between binary states, Boolean logic maps disease as an analog continuum—dynamic, graded, reversible and measurable. By formalizing a framework that has long existed but unnamed, we introduce AXIOM-Bio (Abstract eXpression Inference and Ontology-free Modeling for Biological Logic and Disease): a platform that replaces static biomarkers with mechanistic rules, advancing a new premise, if life computes in logic, medicine should too. 3 | Page When Mathematics Defies Biology 1 The laws of nature, from Newton’s gravitation to Einstein’s relativity, astonish us by their 2 simplicity, invariance, and predictive power. These mathematical regularities, i.e., conditional 3 statements describing what must follow from what is, form the scaffolds of physics. Biology (and 4 Medicine), however, refuses such precision. For centuries, the living world has resisted the kind of 5 mathematical formalism that so elegantly explains the cosmos. The cell—life’s smallest decision-6 making unit—is a system of staggering complexity, yet it operates on remarkably simple, universal 7 principles. Within it, thousands of molecules interact through circuits that sense, decide, and act. These 8 decisions—whether to divide, differentiate, migrate, or die—trace the continuum of transition states 9 between health and disease. Despite breathtaking advances in molecular profiling and AI, our ability to 10 extract the invariant rules that govern these transitions remains limited. Some biologists argue this is 11 not due to ignorance, but due to impossibility [1]: states of life—evolution, emergence, adaptation—12 defies closure within fixed mathematical systems. Concepts such as “genes”, “species”, or “fitness” 13 remain context dependent and fluid. Biology transcends computation: it is a science of thresholds, 14 feedback, and exceptions. 15 And yet, if mathematics cannot define life, perhaps it can describe its logic. Boolean frameworks, 16 through sophisticated thresholding [2], embrace rather than erase biological complexity, translating 17 noise into rules, randomness into predictability. Instead of exact equations, they seek invariant rules—18 binary “if–then” relationships that persist across tissues, species, and diseases. These Boolean 19 implications, first formalized by Sahoo et al. [3,4], expose the simple decision-making architecture 20 within the cellular storm. At its core lies StepMiner[5], an thresholding algorithm that converts noisy 21 gene-expression data into binary outcomes—“low (0)” or “high (1)”—mirroring how cells themselves 22 decide to act. In an era of black-box AI, the Boolean framework offers something rare in modern biology: 23 a mathematical abstraction that captures life’s decision logic rather than its descriptive chaos. In this 24 Opinion, we argue that these Boolean rules, instantiated in AXIOM-Bio (Abstract eXpression Inference 25 4 | Page and Ontology-free Modeling for Biological Logic and Disease), can serve as a foundational model for 26 biological logic and disease. In the sections that follow, we distill them into five simple rules, examine 27 their strengths and limitations, and outline how they could underpin AI-ready virtual cells (Figure 1; 28 Box 1). 29 30 Rule 1 | Life Computes in Thresholds 31 Every living cell is a decision engine. Molecules fluctuate continuously, but at critical junctures the 32 system commits — to divide or arrest, differentiate or retain stemness, activate immunity or remain 33 quiescent. These commitments are neither probabilistic nor gradual; they are threshold crossings. A 34 cell does not partly divide, somewhat differentiate, or tentatively die. It flips. 35 Thresholds are evolution’s answer to noise: regulatory circuits convert variable molecular inputs 36 into binary outcomes, i.e., the molecular equivalents of yes/no, go/stop, live/die. These switches mark 37 the points where possibility becomes physiological action. 38 The Boolean framework captures this architecture mathematically. At its core lies StepMiner[5], 39 an adaptive thresholding algorithm that detects inflection points in gene-expression profiles (Figure 40 1A-B). StepMiner fits a step function to each gene’s sorted expression values to determine the most 41 significant transition separating “off (0)” from “on (1)” states. This converts noisy continuous data into 42 clear low (0) or high (1) states, with a noise buffer removing uncertainty around the threshold. 43 44 This digital abstraction is not an oversimplification; it mirrors how cells compute: 45 • Fate choice: mutually inhibitory transcription factors enforce lineage commitment. 46 • Apoptosis: feedback-driven switches distinguish survival vs. death. 47 • Immune activation: T cells fully engage only beyond receptor–signal thresholds. 48 5 | Page By imposing such thresholds, the Boolean framework transforms high-dimensional, stochastic omics 49 data into interpretable decision landscapes, and converts continuous molecular inputs into binary 50 outcomes through signal transduction, gene regulation, and metabolic feedback. 51 Crucially, digital abstraction at this step 52 does not discard analog information that is vital to 53 track cellular processes in multicellular life that 54 continues to evolve through complex cell-cell and 55 cell-environment crosstalk. The StepMiner-56 normalized continuous score is retained and 57 becomes foundational in Rule 4 (placing samples 58 along the analog health → disease continuum) 59 and in Rule 5 (quantifying population-level drift 60 and therapeutic reversibility). 61 Although the Boolean framework can, in 62 principle, operate on any omics layer, we 63 prioritized the transcriptome for its optimal 64 information density and signal-to-noise ratio. 65 Transcriptomic data offer the breadth of genomics 66 (>20,000 features) with measurable dynamic 67 range across cellular states, while proteomic[6,7] 68 and metabolomic datasets remain limited by 69 coverage (<60%), batch variability, and missing 70 values due to low-abundance analytes and 71 incomplete annotation of post-translational or 72 allosteric modifications. By contrast, RNA 73 Box 1. The Five Rules of Biological Logic 1. Life Computes in Thresholds – Thresholding algorithms such as StepMiner convert noisy expression into decisive 0/1 states, revealing where cells commit to differentiation, survival, activation, or death — the binary logic underlying biological decisions. 2. Noise Hides Invariants – Boolean implication reveals relationships that persist despite biological or experimental noise, exposing rules that hold across tissues, species, and disease. 3. Feedback Is Fidelity – Boolean edges trace push–pull signaling architectures that maintain Dose–Response Alignment (DoRA), identifying the checkpoints where proportionality is preserved when biology maintains order and where dysregulation begins. 4. Disease Is a Reversible Continuum – Boolean implications define the directionality of health-to-disease transitions, while analog composite scores (StepMiner-normalized) position each sample along a reversible, quantifiable continuum. 5. Function Follows Logic, Not Annotation – Composite scores from data-derived Boolean networks retain analog behavior while exposing causal constraints and enable function-agnostic benchmarking, defining biological meaning through empirical invariants rather than evolving ontologies, thereby improving target discovery and model fidelity. 6 | Page expression captures integrated outputs of genetic and epigenetic regulation, as well as the feedback 74 loops that maintain homeostasis (Rule 3: Feedback Is Fidelity). Empirically, transcript levels correlate 75 more strongly with clinical phenotypes than protein abundance [8], and transcriptome–proteome 76 concordance increases under to stress [9-11], when cells engage conserved adaptive programs. Thus, 77 the transcriptome offers the most comprehensive and quantitative window into cellular decision logic 78 currently available. 79 80 From Analog Chaos to Digital Clarity: StepMiner and the Birth of BINs 81 Traditional network methods, e.g., co-expression, mutual information, Bayesian inference, are 82 symmetric and probabilistic. They capture correlation but not direction, and falter in real-world 83 heterogeneity and noise. Boolean Implication Networks (BINs) [2] shifted the field by treating logic as 84 biology’s native language: simple “if–then” rules linking high/low gene-expression states. The resulting 85 network of Boolean implication relationships (BIRs), such as “A high ⇒ B high” or “A high ⇒ B low” 86 (Figure 1C), forms a digital fingerprint of cellular logic. Each gene becomes a logical variable; each 87 implication, a constraint biology consistently obeys. The network reveals direction, dependency, and 88 hierarchy as statistical facts, not assumptions. 89 In biology and in math, robustness emerges from thresholds. Rule 1 defines these digital 90 decisions. But thresholds alone cannot tell us which relationships stay true as noise and context shift. 91 For that, we turn to Rule 2: Noise hides invariants. 92 93 Rule 2 | Noise Hides Invariants 94 If thresholding exposes the moment of decision, Boolean implication uncovers the rules that endure. 95 Biology is inherently noisy—gene expression fluctuates, proteins misfold or misfire, pathways 96 crosstalk—yet within this apparent chaos lie invariants: relationships that remain true across tissues, 97 7 | Page species, and perturbations. These are nature’s constraints, the hidden grammar of gene-regulatory 98 logic that evolution preserves. 99 The Boolean framework [5,12] detects these invariants not just through symmetric correlation 100 but also through asymmetric logic (Figure 1C). For every gene pair (A, B) across thousands of samples, 101 their joint expression pattern, after thresholding, divides the samples into four groups (or four quadrants 102 in a two-dimensional plot)—A low/high × B low/high. When one quadrant is significantly 103 underpopulated, indicating that a particular combination of states almost never occurs, an if–then 104 Boolean implication is established. This asymmetric exclusion reveals the directional constraints that 105 biology consistently enforces. 106 107 The Six Boolean Relationships as Natural Invariants (Figure 1C) 108 1. Four asymmetric: “A high ⇒ B high,” “A high ⇒ B low,” “A low ⇒ B high,” “A low ⇒ B low” 109 2. Two symmetric: “Equivalent” (A ⇔ B), or “Opposite” (A high ⇒ B low AND B high ⇒ A low) 110 Each relationship represents an empirical invariant, a directional, data-driven relationship that holds 111 across diverse biological contexts. These are not hypotheses; they are statistical facts, distilled directly 112 from huge real-world expression data. They define which molecular states co-exist, which exclude one 113 another, and which form nested hierarchies of regulation. They represent constraints enforced at the 114 cellular level and sustained through heterotypic feedback and crosstalk among cells within tissues—115 emergent rules that hold at the population scale. 116 117 Across cancers, developmental programs, and cross-species comparisons, such invariants repeatedly 118 surface: 119 • B-cell maturation: Stem marker high ⇒ differentiation marker low pinpoints transitional genes 120 defining the midpoints of developmental hierarchies[13]. These Boolean intermediates helped 121 8 | Page identify previously unrecognized regulators of lineage progression—genes silent in stem states 122 but active as differentiation begins. 123 • Bladder cancer: KRT5 high ⇒ KRT20 low demarcates basal progenitors from luminal cells [14], 124 defining a binary architecture of epithelial plasticity. This rule persists across patient samples 125 and species, signifying a deeply conserved regulatory toggle between basal identity and terminal 126 differentiation. 127 • Colon tumors: CA1 high ⇒ KRT20 high captures a nested lineage constraint in differentiated 128 tumors, reflecting hierarchical fidelity within intestinal epithelium. Conversely, ALCAM or 129 CCDC88A high ⇒ CDX2 or PRKAB1 low marks a stemness axis—cells reverting toward an 130 undifferentiated, regenerative program, often predictive of therapeutic resistance and poor 131 prognosis. 132 • Lung injury and fibrosis: ACE2 high ⇔ IL15/IL15RA high signals a conserved epithelial–133 immune co-regulation circuit stable across human and mouse models [15-18]. 134 • Macrophage polarization: Conserved implication patterns trace immune trajectories from 135 reactivity to tolerance, revealing logical paths that remain fixed across tissues and species [19]. 136 • Inflammatory bowel disease (IBD): Genes preserving epithelial integrity and energy 137 homeostasis share a high ⇒ low relationship with those driving inflammation and fibrosis—138 computationally trackable across organoids, animal models, and patient biopsies [2,20]. 139 140 Together, these and numerous other examples [21-24] demonstrate how the Boolean framework 141 reproducibly extracts invariant logic beneath biological diversity. Whether in hematopoiesis or fibrosis, 142 immune tolerance or epithelial regeneration, the same mathematical grammar applies, i.e., directional 143 relationships, conserved across scales, that define what biology allows or forbids. Boolean relationships 144 thus emerge as constraints imposed by life itself; rules so fundamental that they endure through 145 evolution, adaptation, and disease.. 146 9 | Page 147 This reframes “mechanism” not as a static pathway map but as a set of logical constraints: the 148 boundaries of biological possibility that remain stable despite noise and context shifts. Yet invariants 149 alone do not explain how information is transmitted with precision. That principle resides in Rule 3: 150 Feedback is Fidelity, which details how push–pull architectures enforce dose–response alignment. 151 152 Rule 3 | Feedback Is Fidelity 153 Life maintains order not by silencing noise but by aligning feedback to preserve proportionality. This 154 is better known as Dose–Response Alignment [25] (DoRA), a principle that describes how cells 155 preserve faithful information transfer by aligning the input-to-output relationship across signaling 156 cascades[26]. When feedback loops achieve DoRA, downstream responses scale predictably with 157 receptor activation, converting molecular chaos into coherent behavior. Mechanistically, DoRA is 158 achieved not by fine-tuning every parameter but by architectural motifs [24,27-29]—push-pull regulatory 159 networks in which the active form of a signaling species drives output (“push”) while the inactive form 160 counter-acts it (“pull”) [30]. 161 The Boolean framework captures these equilibria at the transcriptomic level in complex 162 biological circuits[24,28,31]. Each Boolean edge, A high ⇒ B high or A high ⇒ B low, represents a 163 feedback-stabilized dependency, the digital trace of push–pull motifs that maintain biological fidelity. 164 By elevating DoRA from molecular cascades to transcriptomic logic, the Boolean framework reveals 165 the transistor-like decision points in cellular regulation—where biology imposes invariant logic rather 166 than variable curves. These are the nodes where “noise” is filtered, fidelity is enforced, and decisions 167 become digital. In this way, the DoRA concept serves as the bridge from analog chaos of molecular 168 networks to the digital clarity of Boolean implication. 169 Remarkably, Boolean relationships often align with protein-level behaviors (with surprising 170 fidelity), as confirmed by cytochemistry-based validations [2,32-34]. This suggests that many 171 16 | Page analog trajectories (Rules 4–5), AXIOM-Bio bridges computation and experiment, converting 316 molecular noise into the structured logic biology already uses. The result is a framework able to 317 forecast biological behavior, therapeutic directionality, and the reversible nature of disease 318 transitions, with a clear set of strengths and limitations. 319 320 Strengths: 321 • Digital direction from noisy data: Unlike correlation-based models, BINs extract “if–then” 322 causal constraints (A high ⇒ B low), capturing directional dependencies missed by correlation 323 networks. Human–mouse meta-analysis has revealed 3.2 million conserved BIRs, far beyond 324 equivalences alone [4]. 325 • Robust to noise: StepMiner thresholds and noise margins (plus a noise-margin: e.g., ±0.5 326 log2) ensure that only decisive “low/high” states are retained, converting variability into 327 invariance[3]. 328 • Analog fidelity: Composite signature scores quantify where each sample lies along disease 329 trajectories. 330 • Scalable and generalizable across species and tissues, revealing conserved rules of 331 regulation. 332 Numerous studies [2,15,16,18,19,22,23,32] have now shown that BIRs survive across species: 333 human–mouse, and human–mouse–fly conserved edges with very low false-discovery rates 334 [85] (e.g., <2.4 ×10⁻⁵ for human/mouse/fly [4]), implying BIRs represent evolutionarily hard-335 wired logic. 336 • Aids rapid formulation of testable, mechanistic hypothesis and objective validation: 337 Directional and invariant edges uncover hidden hierarchies and feedbacks, predicting 338 intermediate states later validated experimentally (reviewed in[3]) and several other use case 339 17 | Page since[19,32,48,86]. These success stories suggest regulatory or hierarchical relationships 340 rather than mere correlation. 341 • Regulatory aligned: Logic and quantification accelerate Phase-0 / NAM suitability and 342 benchmarking. 343 344 Caution and Limitations: 345 • Sample size matters: rare quadrants require statistical power; still, refined BINs from mid-size 346 cohorts (n = ~125-200) have proven robust, when subjected to further refinement and rigorous 347 validation using independent datasets [2,15,16,19,21]. 348 • Context dependence: Implications discovered in one tissue or stage should not be assumed 349 as universal; care must be taken when extrapolating implications beyond the original sample 350 space. 351 • Not all biology is binary: BINs reduce continuous expression to “low/high”. Some regulatory 352 relationships may not necessarily have corresponding implications. Similarly, genes regulated 353 by combinatorial logic may escape capture. 354 355 Concluding remarks and future perspectives 356 AXIOM-Bio reframes biology as decision logic, not statistical correlation. It captures the four essentials 357 of cellular intelligence (Figure 2A): thresholds of commitment when cells decide; constraints of 358 possibilities and the endurance to act/react, enforced via feedback loop of mechanochemical signals; 359 and trajectories of progression/reversibility as cells learn and adapt over time to the evolving 360 environment. AXIOM-Bio instantiates these principles as an ontology-free implementation that fuses 361 Boolean implication networks with composite scores to model biological logic across tissues, species, 362 and experimental systems. Its future potential (see ‘Outstanding questions”) underscores a central 363 realization: Life computes in logic, and medicine should too. 364 18 | Page To build a virtual cell, we must first understand how real cells compute. AXIOM-Bio provides that 365 language: a compressive, interpretable representation of cellular behavior grounded in rules rather than 366 heuristics. Anchored to clinical endpoints, it transforms NAMs from experimental platforms into 367 reasoning systems, introducing the essential elements for causal ‘digital twins’ capable of forecasting 368 disease progression and therapeutic response. Robust cross-cohort generalization of ontology-free 369 gene signatures demonstrated by others[64,87-90] provide key precedent for the type of stable, 370 transferable logic that AXIOM-Bio seeks to extract directly from expression data. 371 Where other sciences gained transformative power only because they rest on immutable 372 governing laws (Figure 2B), e.g., Maxwell for communication, Newton for motion, the Thermodynamic 373 Hypothesis for protein folding, etc., biology and medicine now stand at an analogous threshold. To 374 make AI truly biological, biology must first become intelligible. Neural networks may emulate life, but 375 Boolean logic explains it. In doing so, AXIOM-Bio positions itself as a foundational model for cellular 376 intelligence, a transparent mathematical scaffold on which future virtual-cell architectures can be built. 377 AXIOM-Bio is an early prototype of this scaffolded system, already supporting NAM-compatible, 378 regulator-ready reasoning at the level of transcriptomic logic. 379 19 | Page Acknowledgments 380 This work is supported by NIH grants R01-AI141630 and R01-AI55696 and by the Leona M. and Harry 381 B. Helmsley Charitable Trust. S.S. was supported through The American Association of Immunologists 382 (AAI) Intersect Fellowship Program for Computational Scientists and Immunologists. 383 384 Declaration of generative AI and AI-assisted technologies 385 During the preparation of this work, the author(s) used Microsoft Co-Pilot, Gemini and ChatGPT to 386 generate icons in figures and edit text. After using this tool or service, the author(s) reviewed and edited 387 the content as needed and took full responsibility for the content of the publication. 388 389 Declaration of interests 390 The authors declare no conflicts of interest.391 20 | Page Figures and Legends: Figure 1. A Rules-Based Framework for Modeling Biological Logic, Cellular State Transitions, and Disease Reversibility 21 | Page Rule 1 – Life computes in thresholds. A. Gene expression datasets from healthy and diseased cohorts serve as input for digital threshold detection. B. StepMiner adaptively identifies threshold positions by fitting step-like functions to normalized gene expression values, merging noise regions and converting continuous data into binary (high vs. low) states. These thresholds enable the identification of Boolean relationships between genes. Rules 2 and 3 – Noise hides invariants; feedback is fidelity. C. Boolean implication analysis identifies asymmetric relationships (e.g., A high ⇒ B high, A high ⇒ B low) and symmetric relationships (e.g., Equivalent or Opposite), while excluding forbidden quadrants that represent biologically impossible states. These relationships reflect universal constraints and push–pull feedback motifs that maintain dose–response fidelity. Rule 4 – Disease is a reversible continuum, not an absolute category. D. Clustering Boolean implication relationships yields higher order “logic networks” of gene clusters, forming a stable scaffold for downstream interpretation and model construction. E. Summary of invariant Boolean relationship types (Equivalent, Opposite, and directional implications), which define the causal structure and allowed transitions between cellular states. F. A sample-agnostic Boolean Implication Network (BIN) is constructed by connecting the largest gene clusters through Boolean paths, generating a continuum model of cellular states rather than discrete categories. G. Boolean paths reveal hierarchical and sequential processes embedded in transcriptomes, enabling charting of regulatory transitions that underlie phenotypic change. H. ML-driven exploration of the BIN uncovers the global architecture of biological logic across datasets, tissues, and disease contexts. I. The derived Boolean paths map a reversible health–disease continuum, positioning individuals by directionality (Boolean implication) and degree of progression (analog composite scores). 22 | Page Rule 5 – Function follows logic, not annotation. J. Function-agnostic composite gene signatures are constructed from BIN clusters using StepMinernormalized analog scores, enabling quantitative assessment of cellular state, disease trajectory, and response to perturbation. K. These quantifiable signatures, serving as digital biomarkers, operationalized through COMPASS™[71] composite-score algorithms provide an objective and precise basis for multiple downstream applications, including validation in independent cohorts (bulk or single-cell), benchmarking the fidelity and “humanness” of models, defining clinically relevant endpoints that align with regulatory standards, iteratively refining NAMs and disease models, and identifying high-value therapeutic targets through causal, directional logic. Together, these five rules unify thresholding, invariants, feedback control, reversible state trajectories, and function-agnostic logic into a cohesive, ontology-free framework for modeling biological behavior and guiding translational decision-making. 23 | Page Figure 2: AXIOM-Bio unifies cellular intelligence with first principles of biological logic. (A) Essentials of Cellular Intelligence and the Five Rules of Biological Logic. Left, the four foundational capabilities of cellular intelligence—Sense, Decide, Act & React, and Learn & Adapt—are depicted as core behaviors that enable cells to interpret cues, commit to actions, execute mechanochemical responses, and refine future behavior through gene regulatory networks. Right, AXIOM-Bio formalizes these capabilities through five rules of biological logic. 24 | Page (B) AXIOM-Bio positions biology alongside other fields grounded in scientific rules and laws. Across disciplines, breakthroughs have emerged when systems were reframed through fundamental principles. From Left to Right: Shannon’s and Maxwell’s laws for wireless communication, Newton’s and Archimedes’ laws for robotics and transportation, thermodynamics for weather prediction, and biophysical principles for protein folding. 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