The thermodynamic state map of biomolecular condensates

Abstract

Biomolecular condensates spatiotemporally regulate cellular biochemistry through liquid-liquid phase separation. In this Perspective, we refine the Condensate Code as a testable thermodynamic state map that integrates continuous cellular inputs, intrinsic molecular grammar and boundary conditions to govern measurable material states. Building on stickers-and-spacers and related grammar frameworks, we highlight a continuum from electrostatic coacervates to viscoelastic networks. We define active anti-grammar as an operational umbrella for localized repulsive or hydration-promoting effects that can counterbalance excessive cohesion when embedded in, modified within or recruited to dense phases. This view predicts measurable changes in partitioning, FRAP recovery, viscosity, aging and fibril-nucleation kinetics. Disruption of the code can drive pathogenic liquid-to-solid transitions and spatial uncoupling with organelle mechanotoxicity. We therefore conceptualize physical therapeutics as programmable macromolecular modulators that test whether localized anti-cohesive effects can remodel pathological material states while monitoring targeting, partitioning, oligomeric intermediates and safety.

Introduction

For decades, the foundation of cellular biology was firmly anchored in the sequence-structure-function paradigm. This classical framework operated on the deterministic premise that a rigid, precisely folded, energy-minimized three-dimensional architecture is the absolute prerequisite for dictating a macromolecule’s biological activity. A major conceptual leap occurred at the turn of the century when the histone code hypothesis elegantly revealed that biological information is not strictly linear; rather, it exists as a spatial, highly context-dependent language capable of dynamically rewriting local chromatin accessibility¹. Today, molecular and cellular biophysics is undergoing a parallel and equally revolutionary transformation: the intracellular space, once conventionally modeled as a dilute, well-mixed aqueous solvent containing discrete stoichiometric machines, is now unequivocally recognized as a highly compartmentalized, non-equilibrium multiphase viscoelastic fluid².

The historical foundation for viewing the cell through this material-state lens emerged independently from two pioneering physical models. The first originated from the biophysical interrogation of the nuclear pore complex (NPC). For years, the mechanism by which the NPC achieved extremely rapid, bidirectional nucleocytoplasmic transport while maintaining exquisite size-exclusion selectivity remained a paradox. The breakthrough came with the realization that intrinsically disordered nucleoporins containing repetitive phenylalanine-glycine (FG) motifs spontaneously self-assemble into a “selective phase.” Rather than functioning as a rigid mechanical gate, these FG-repeats form a dense, highly percolated hydrogel meshwork. This specific molecular architecture entropically excludes large, inert macromolecules while functioning as a highly conductive dynamic solvent for nuclear transport receptors, which locally melt the network to facilitate millisecond-scale molecular transit³. The second foundational model arose from the study of RNA-binding proteins (RBPs), revealing that the assembly of membraneless ribonucleoprotein (RNP) granules is frequently not amorphous, but is structurally scaffolded by “reversible fibers.” Unlike pathogenic, irreversible amyloids, these functional low-complexity domains (LCDs) adopt labile cross-β architectures that exhibit designed instability, allowing for the rapid spatiotemporal regulation of RNA metabolism⁴.

Importantly, this organizational principle—utilizing structural plasticity to assemble transient functional compartments—extends universally across biological kingdoms and diverse physiological scales. For instance, in the plant model Arabidopsis thaliana, the structural node protein FLL2 acts as a specific multivalent hub that directly co-phase separates with the RNA-binding protein FCA. This mesoscale condensation efficiently compartmentalizes the polyadenylation machinery, functionally rewiring the transcriptome to regulate transcriptional termination⁵. Rather than treating the NPC hydrogel, RNP reversible fibers, and plant nuclear bodies as disparate phenomenological observations, we propose that they represent diverse manifestations of a unified underlying physical principle: the Condensate Code. Here, we organize the Condensate Code as an integrative thermodynamic state map that links continuous cellular inputs and sequence-encoded molecular grammar to dynamic-liquid, viscoelastic-network, and arrested-solid states (Fig. 1). Conceptually, this Perspective is distinct from our recent IJMS review on condensatopathies and precision modulators⁶. Whereas that review surveyed disease categories and therapeutic opportunities, the present manuscript develops a more focused framework for the predictive logic linking molecular grammar, thermodynamic observables, active anti-grammar, and physical-therapeutic design. Importantly, the Condensate Code is not proposed as a replacement for established stickers-and-spacers, charge-patterning, phosphorylation-dependent regulation, or negative-design models. Instead, its added value lies in organizing these principles into an integrative state map, in which defined perturbations can be evaluated through measurable outputs such as Csat, partition coefficient, FRAP recovery, fusion kinetics, viscosity, aging rate, and fibril-nucleation kinetics. Accordingly, the framework is most useful for multivalent, tunable condensates, whereas it is less applicable to stable stoichiometric complexes or fully mature inert amyloid cores (Box 1).

Fig. 1. Decoding the condensate code: a state-map framework from cellular inputs and molecular grammar to condensatopathies and testable physical therapeutics.

The thermodynamic state map (center) links continuous cellular inputs to dynamic-liquid, viscoelastic-network and arrested-solid states. a Cellular inputs. Environmental factors, genetic mutations, epigenetic marks, and metabolite concentrations shape phase behavior; the upward arrow indicates increased ATP concentration. b Primary sequence grammar and active anti-grammar. Favorable sticker contacts are represented by ΔH < 0, whereas active anti-grammar denotes an operational class of localized anti-cohesive effects that can arise from charge patterning or regulatory modification and may delay pathological aging. c Condensatopathies. Path 1 represents an aberrant liquid-to-solid transition. In Path 2, loss of native tethers is followed by an ectopic condensate, viscoelastic aging, and a rigid solid; organelle mechanotoxicity is a possible downstream outcome. d Physical therapeutics. A programmable macromolecular modulator targets accessible regions and produces a local electrostatic effect that may introduce an entropy penalty. Material-state remodeling and clearance are presented as testable outcomes, with oligomeric intermediates monitored directly.

Box 1 Terminology in condensate biophysics.

• Intrinsically disordered region (IDR): protein sequences lacking a stable, unique three-dimensional equilibrium structure under physiological conditions. Instead of folding rigidly, they sample a vast ensemble of conformations, providing the extreme structural plasticity required for multivalent, transient interactions.

• Liquid-liquid phase separation (LLPS): a macroscopic thermodynamic demixing process where a supersaturated, homogeneous solution spontaneously separates into a dense, macromolecule-rich condensed phase and a dilute, highly solvated continuous phase, minimizing the overall free energy of the system.

• Complex coacervation: a highly specific, dynamic class of associative phase separation driven primarily by long-range electrostatic and entropic interactions between oppositely charged polyelectrolytes (e.g., polycationic IDRs interacting with polyanionic nucleic acids).

• Percolation: the physical topological transition where locally interacting multivalent molecules cross-link extensively to form a continuous, system-spanning network. A percolated network fluid possesses macroscopic viscoelastic, solid-like properties while rigorously maintaining localized microscopic liquid dynamics.

• Liquid-to-Solid Transition (LST): A time-dependent, stress-induced, or mutation-driven material maturation wherein transient, dynamic non-covalent interactions within a fluid condensate progressively become kinetically arrested, culminating in the formation of hyper-stable, irreversible structures such as amyloid fibrils.

A thermodynamic state-map framework

Deconstructing the primary grammar

To quantitatively decipher the Condensate Code and move the field toward predictive engineering, it is imperative to expand our physical models beyond the foundational parameters of associative polymer physics, notably the classical “stickers and spacers” framework⁷. It is a critical scientific misconception to overgeneralize the structural nature of condensates; they are neither exclusively ordered crystalline polymers nor generic, unstructured viscous droplets. Instead, biomolecular condensates exist and operate upon a continuous, highly tunable structural spectrum.

Electrostatic coacervation and rheological scaling

At one extreme of this structural continuum, many vital functional cellular compartments operate as pure electrostatic complex coacervates. Lacking detectable ordered secondary structures, these assemblies rely almost entirely on transient electrostatic and entropic interactions. In this regime, the Condensate Code is exquisitely sensitive to subtle chemical identities and microscopic stereochemistry. A prime illustration is the differential behavior of basic amino acids. Despite both side chains carrying a net positive charge at physiological pH, arginine and lysine exhibit profoundly different phase separation propensities. Explicit-solvent atomistic molecular dynamics simulations reveal the underlying cause: the planar guanidinium group of arginine is uniquely privileged. It forms extensive, multivalent hydrogen-bonding networks with RNA phosphate backbones while simultaneously engaging in robust, highly directional cation-π and π-π stacking interactions with nucleobases. This establishes an intricate, kinetically enduring cross-linking network that lysine’s simple primary amine fundamentally cannot replicate⁸.

Furthermore, the macroscopic material properties of these electrostatically driven condensates can be rigorously and quantitatively predicted from their primary sequence linear charge distribution. The normalized sequence charge decoration (nSCD) parameter acts as a highly predictive physical scaling law. Enhancing the segregation of oppositely charged residues into distinct “blocks” along a disordered polypeptide chain significantly amplifies interchain electrostatic attractions. Coarse-grained simulations confirm that this topological sequence feature directly dictates dilute-phase chain dimensions—drastically altering the radius of gyration (R_g)—which translates into an exponential scale-up of both the macroscopic zero-shear viscosity and the interfacial surface tension of the resulting condensed droplets⁹.

Reversible fibers and the active anti-grammar

Advancing along the structural continuum, we observe that functional condensates are frequently scaffolded by specific, albeit transient, ordered architectures. Atomic-resolution solid-state NMR studies have demonstrated that LCDs from proteins such as FUS form reversible cross-β polymers. Crucially, these functional structures utilize defined interaction motifs, such as “polar zippers” and kinked β-sheets. These features are evolutionarily selected to encode structural frustration into the polymer core, effectively preventing the dense, dehydrated steric packing that is characteristic of irreversible, pathogenic amyloid formation¹⁰.

However, relying exclusively on cohesive sticker interactions and passive flexible linkers (spacers) poses a severe and often overlooked thermodynamic hazard. Recent metadynamics simulations illustrate that when the specific interaction energy of cohesive stickers significantly exceeds the weak, non-specific entropic interactions of the spacer regions, intra-condensate stickers rapidly become fully saturated. This specific energetic trapping completely neutralizes the entropic driving force required for continued chain mixing and dynamic molecular exchange. This leads to a state of profound kinetic arrest, trapping the condensate in a metastable state that strongly resists normal liquid coalescence¹¹.

If a biological system were equipped exclusively with these assembly elements, it may become prone to coarsening, kinetic arrest, and progressive viscoelastic aging within the highly crowded cellular milieu. To actively evade this catastrophic physical fate and maintain a dynamic, functional phase, we define active anti-grammar as a testable operational umbrella rather than a single new physicochemical mechanism (Fig. 1b). Specific sequence-encoded regions defined by highly localized, dense electrostatic charge blocks—such as prominent poly-anionic tracts naturally present in certain IDRs or dynamically introduced via targeted, kinase-mediated multi-phosphorylation¹²—may function as context-dependent anti-cohesive modules. When embedded within or recruited to highly cohesive multivalent networks, these dynamic repulsive elements can counteract cohesive sticker forces; if untethered, they may remain in the dilute phase and have little effect. Thermodynamically, favorable sticker contacts lower the enthalpic contribution of assembly, whereas conformational confinement, dehydration, and like-charge repulsion can raise the free energy of dense or solid states. In this sense, active anti-grammar is best understood as a boundary-condition-dependent modulator of the free-energy landscape that can preserve exchange and delay pathological aging.

A direct experimental hypothesis follows: introducing or removing a tethered charge block within an aging-prone IDR scaffold should measurably shift dense-phase partitioning, FRAP recovery, fusion kinetics, microrheology, viscosity, and fibril-nucleation lag time. Matched constructs with the same composition but different charge-block placement, together with untethered controls, would test whether anti-grammar requires dense-phase enrichment.

Tunable metastability and chromatin organization

The precise, continuous interplay between the cohesive molecular grammar and the repulsive anti-grammar establishes a fundamental biophysical state: tunable metastability. A persistent physical paradox often arises in the field of IDP biophysics: if the highly ordered amyloid fibril represents the absolute thermodynamic ground state for many aggregation-prone disordered proteins (meaning the fibril saturation concentration is theoretically much lower than that of the liquid state), how can liquid condensates ever be considered functionally protective?

The resolution lies in the kinetics of classical nucleation theory. While the fibril is the ultimate energy minimum, forming a mature, stable solid fibril requires crossing a massive activation energy barrier associated with primary heterogeneous nucleation. By rapidly undergoing liquid-liquid phase separation, intrinsically disordered proteins effectively sequester themselves into a highly concentrated liquid phase that fundamentally acts as a “thermodynamic sink.” This spatial sequestration drastically buffers the bulk concentration of freely circulating soluble monomers in the dilute continuous phase, keeping it rigidly pinned strictly below the critical supersaturation threshold required to spontaneously nucleate fibrils. Consequently, the dynamic liquid droplet acts as a highly protective kinetic trap, reconciling the dual roles of condensates in cellular physiology¹³.

Epigenetic stickers and architectural modulators

The dynamic execution of this tunable metastability is perhaps most vividly illustrated in the high-order spatial organization of nuclear architecture. In this context, covalent epigenetic modifications on histone tails do not merely recruit isolated enzymes; they function effectively as specifically inducible, spatiotemporally regulated physical stickers. We previously demonstrated via size-exclusion chromatography coupled with multi-angle light scattering (SEC-MALS) that architectural scaffold complexes systematically engage H3K9me3 marks on nucleosomal arrays. By matching the specific chromodomain valency of the TRIM28-HP1 complex (e.g., reaching a valency threshold of six) with heavily methylated arrays, we observed robust LLPS. This process physically segregates the genome into dense, immiscible liquid compartments that actively exclude active transcriptional machinery like mTFIIB—elegantly mirroring the biophysical properties and functional silencing of constitutive heterochromatin observed in vivo¹⁴.

This condensation-driven organizational logic acts as a universal mechanism for genome architecture across remarkably diverse developmental contexts. In somatic cells, the linker histone H1 utilizes its highly disordered, charge-enriched C-terminal domain to drive pervasive heterochromatin condensation via phase separation¹⁵. Conversely, in the highly specialized context of flowering plant sperm, the evolutionary histone variant H2B.8 employs a uniquely evolved, profoundly negatively charged N-terminal domain to drive extreme, species-specific nuclear compaction, achieving massive volume reduction while preserving basal transcriptional capacity for fertilization¹⁶. The profound biological importance of reading the Condensate Code correctly is starkly highlighted in disease states. For example, severe Rett syndrome-causing mutations specifically abolish the capacity of the methyl-CpG binding protein MeCP2 to properly mediate chromatin LLPS, leading to a catastrophic, genome-wide structural collapse of three-dimensional architecture and subsequent severe neurodevelopmental deficits¹⁷.

Pioneer factors and physical expansion

If epigenetic stickers effectively condense the genome into repressed liquid droplets, how is this silenced phase dynamically accessed by the transcriptional machinery to initiate cell fate transitions? The pioneer transcription factor FOXA1 provides a compelling physiological application of biological anti-cohesive elements. We recently revealed that FOXA1 utilizes its intrinsic IDR-driven LLPS capability not to further condense the genome, but to selectively infiltrate and physically unpack pre-condensed heterochromatin arrays. By generating phase separation-mediated physical expansion forces that actively counteract the surface tension of the repressed chromatin droplets, FOXA1 functions as a natural mechanical wedge. It actively dismantles the rigid heterochromatic phase from the inside out to pioneer accessible, transcriptionally active euchromatin¹⁸.

Condensatopathies and mechanotoxicity

The Condensate Code meticulously maintains cellular homeostasis through tightly controlled, highly sensitive thermodynamic equilibria. When this biophysical code is irrevocably breached—whether by inherited genetic mutations, sustained oxidative environmental stressors, or the age-dependent decline of metabolic energy availability—cellular organization undergoes a catastrophic thermodynamic collapse. This systemic failure is formalized under the unifying mechanistic framework of Condensatopathies¹⁹. This framework dictates that sustained macroscopic deviation from the native physical phase state is the true root driver of cellular toxicity.

The liquid-to-solid transition

Pathological mutations in neurodegenerative diseases frequently act as insidious separation-of-function variants that severely disrupt the protein’s structural continuum. In functional polymers, specific structural “breaker” residues (such as proline, which lacks a backbone amide proton and introduces a structural kink) limit extensive main-chain hydrogen bonding. Disease-causing mutations frequently substitute these critical prolines, introducing new, unshielded hydrogen-bond donors. This seemingly minor sequence alteration drastically hyper-stabilizes the solid fibril phase while simultaneously destabilizing the functional liquid state²⁰. This tips the delicate metastable balance, serving as the direct atomic trigger for an aberrant, accelerated liquid-to-solid transition (LST)²¹ (Fig. 1c, Path 1). This time-dependent material hardening initiates a devastating dual-pathology mechanism in diseases like amyotrophic lateral sclerosis (ALS). In the cytoplasm, the aberrant seeding and progressive solidification of essential RNA-binding proteins like TDP-43 into rigid inclusions presents a Toxic Gain-of-Function (TGOF). Simultaneously, this irreversible cytoplasmic trapping severely depletes the functional, highly dynamic soluble pool of TDP-43 from the nucleus. This creates a catastrophic “double-hit” scenario, leading to a profound Loss-of-Function (LOF) characterized by massive transcriptomic collapse and widespread cryptic splicing failures in crucial neuronal survival targets²². Furthermore, the Condensate Code can be actively hijacked by oncogenesis. Abnormal chromosomal translocations generate fusion oncoproteins, such as the NUP98-HOXA9 oncoprotein in pediatric leukemia. By utilizing acquired low-complexity domains, they form dense, hyper-stable nuclear puncta that selectively miscompartmentalize super-enhancers and key transcriptional machinery, physically providing the driving force for relentless leukemic transformation²³.

Spatial uncoupling and ectopic mechanotoxicity

Moving beyond primary sequence mutations that alter intrinsic assembly rules, we must highlight a major, yet frequently overlooked, spatial dimension of pathogenesis: spatial uncoupling. In healthy physiological homeostasis, the local concentration of highly multivalent disordered proteins is tightly constrained by their physical tethering to precise architectural scaffolds, such as cytoskeletal microtubule tracks, specific organelle membranes, or distinct chromatin loci.

When acute cellular stress, traumatic injury, or specific genetic truncations abruptly sever these native structural tethers, the newly released multivalent proteins flood the unrestricted bulk cytosol. Unbound by local spatial limits, they rapidly exceed their global saturation concentration (C_sat), triggering an explosive wave of uncontrolled, ectopic phase separation. As these massive, unanchored condensates undergo viscoelastic aging and mature into rigid, glass-like solids, they transform into severe physical hazards. The hardened aggregates exert tremendous, localized mechanical pressure against highly compliant, delicate intracellular structures. This relentless mechanotoxicity is sufficient to physically crush the adjacent mitochondrial networks, violently deforming the outer mitochondrial double membrane and causing cristae collapse. This triggers the catastrophic efflux of double-stranded mitochondrial DNA (mtDNA) into the cytosol. Once engulfed by lysosomes via autophagy, the acidic environment denatures the mtDNA into single-stranded fragments. The subsequent high-density aggregation of these fragments generates a highly robust aggregation-induced emission (AIE) of fluorescence—providing researchers with a powerful, novel, label-free biophysical pan-marker capable of tracing mechanotoxicity-induced cellular senescence and cell death in real-time²⁴ (Fig. 1c, Path 2).

Outlook: future horizons and physical therapeutics

Recognizing that many of the most devastating, intractable human diseases are fundamentally driven by macroscopic physical phase transitions and mechanical stresses reveals a critical, systemic gap in modern pharmacology. Conventional small-molecule inhibitors, meticulously designed via lock-and-key principles to target static, rigid 3D enzymatic pockets, are often poorly suited to the highly dynamic, flat interfaces of disordered proteins. More importantly, they may not efficiently access the dense, dehydrated matrix of hardened solid aggregates²⁵.

Furthermore, therapeutic strategies that attempt to hijack the cell’s endogenous protein clearance machinery—such as chaperone-mediated refolding networks or targeted selective autophagy—may be insufficient in late-stage disease. These natural pathways rely heavily on robust, millimolar pools of cellular ATP, which acts both as a crucial energy source and as a biological hydrotrope²⁶, and require the target cargo to maintain highly dynamic, fluid liquid-liquid interfaces to form proper autophagic initiation hubs²⁷. Both of these prerequisites can be compromised when confronting hyper-stable, rigid pathological solids.

AI-driven decoding of the condensatome

To rationally design robust, next-generation interventions, we must first accurately and comprehensively decode the global “condensatome.” We have recently demonstrated that sophisticated machine-learning frameworks, which integrate multimodal molecular features—including post-translational modification frequencies, amino acid co-occurrence patterns, and AlphaFold2-derived structural parameters—can highly accurately screen and predict the active participants of complex membraneless organelles. This effectively overcomes the severe limitations of early predictive algorithms that relied almost exclusively on simple linear structural disorder scores, which biased discovery toward purely self-assembling proteins²⁸. This powerful AI-driven generative sequence design, paired synergistically with emerging, cutting-edge analytical tools like quantum-enhanced picoliter NMR—which enables the chemical resolution of specific chemical shifts and backbone proton dynamics directly within heterogeneous, crowded droplets of sub-nanoliter volumes²⁹—provides the requisite high-resolution empirical and theoretical foundation for highly targeted structural interventions.

Physical therapeutics

To overcome the physical barriers of neurodegeneration and cancer, we propose a complementary pharmacological frontier: Physical Therapeutics. Rather than relying solely on host chaperone networks, ubiquitination machineries, or vulnerable cellular ATP pools, we conceptualize the rational, bottom-up engineering of programmable macromolecular modulators.

Nature already provides the blueprint. For instance, the targeted, kinase-driven phosphorylation of the oncogenic transcription factor FOXM1 introduces repulsive anionic charges that disrupt its phase separation, suppressing breast tumor growth in vivo³⁰. By deliberately harnessing this repulsive anti-grammar, engineered agents could link a highly specific targeting module (such as a high-affinity intracellular nanobody or designed binder) to a densely charged or hydration-promoting effector through a structural spacer (Fig. 1d). This architecture would be directed to accessible surfaces or dynamic regions and used to test localized material-state remodeling.

If a targeting module recruits such a modulator to accessible surfaces or dynamic regions of a pathological assembly, the local concentration of the repulsive effector could increase near the assembly. According to foundational thermodynamics (ΔG = ΔH - TΔS), such targeted localization could introduce electrostatic repulsion, hydration, and conformational constraints that reduce the thermodynamic advantage of persistent contacts and thereby alter exchange, viscosity, aging, or clearance accessibility. Whether such remodeling produces fluidized species, avoids toxic oligomeric intermediates, or improves clearance is an empirical question that must be measured directly rather than assumed. If fluidized species are generated, they may become more accessible to clearance by targeted protein degrader technologies, such as emerging PROTACs utilizing TRIM21 or SPYTACs³¹.

Conclusions

The fundamental paradigm of biological organization has evolved dramatically beyond deterministic, static stoichiometry. By organizing these observations within a thermodynamic state map, we aim to move from a descriptive parts list toward a testable framework linking molecular grammar, cellular inputs, and measurable condensate material states.

The critical recognition that functional, dynamic metastability is an active state meticulously maintained by a precise molecular grammar—and that its disruption can contribute to Condensatopathies via pathways like the liquid-to-solid transition and spatial uncoupling—opens an entirely new biomedical frontier. By boldly moving beyond traditional lock-and-key pharmacology and embracing the concept of physical therapeutics to rationally and thermodynamically modulate the macroscopic material phase state of pathological assemblies, we gain a testable route toward interventions that measurably remodel pathological assemblies and may improve treatment of currently intractable protein-aggregation diseases.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Author contributions

Liang Wang conceived and supervised the work and provided the overall direction for the Perspective. Yaoyao Hou, Xin Li, Lingyun Jing and Kan Sun wrote the initial draft. Liang Wang, Jinrong Min and Xin Li developed the outline and structure. Liang Wang performed the major revisions and finalized the conceptual framework. Jingyue Zhao and Yufeng Chang discussed the content and provided critical feedback and edits. All authors reviewed and approved the final manuscript.

Peer review

Peer review information

Communications Biology thanks all reviewer(s) for their contribution to the peer review of this work. Primary Handling Editors: Nilanjan Banerjee and George Inglis. A peer review file is available.

Funding

This work was supported by the National Key R&D Program of China (Grant No. 2025YFA1309800), the National Natural Science Foundation of China (Grant No. 32470739), the Key Laboratory of Pesticide & Chemical Biology, Ministry of Education (Grant No. KL202506), the Beijing Life Science Academy (Grant No. 2025100CC0010), and the Beijing Natural Science Foundation (Grant No. L246003).

Data availability

No datasets were generated or analyzed during the current study.

Code availability

No custom code was generated or used in this study.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s42003-026-11056-4.