Mycobacterium tuberculosis (Mtb), most commonly manifests in patients with respiratory systems. However, it can also colonize other tissues including skeletal. In our case, a 77-year-old Caucasian male presented to the emergency department following a rollover motor vehicle collision with chief complaints of neck and lower back pain. After clinical improvement and a preliminary negative workup, the patient was deemed stable for discharge. Four months later, the patient was subsequently admitted for worsening back pain with workup suspicious for T9 and T10 discitis osteomyelitis and abscess formation on computed tomography (CT). During this admission, spinal Mtb was confirmed by acid-fast stain and real-time polymerase chain reaction of a CT-guided disc space aspirate of a left paraspinal cystic collection at approximately T9-T10. Given these findings, the patient was subsequently put on standard four-drug therapy for Mtb. Our case demonstrates the importance of considering Pott’s disease in the diagnosis of lumbar spinal pain, especially in patients living in areas with high international migration and travel.Tuberculosis infection, which is caused by the bacterium  Mycobacterium tuberculosis (Mtb). These are non-motile, weakly-gram positive, acid-fast bacilli part of the order Actinomycetales [Tuberculosis is caused by a bacterium called s . Mtb moss . Spinal s -6. The ms . Chronics ,8.Here, we report a case of a patient initially admitted with lower thoracic back pain following a motor vehicle collision found to have Pott's disease upon subsequent admission.A 77-year-old Caucasian male initially presented to the emergency department (ED) following a rollover motor vehicle collision with chief complaints of neck and lower back pain. The patient was driving while wearing his seatbelt and was under sternal precautions for a coronary artery bypass graft (CABG) five months prior for which he was taking clopidogrel and aspirin. Physical examination was noteworthy only for midline spinal tenderness to palpation throughout the cervical, thoracic, and lumbar spine without deformities or neurological deficits. A computed tomography (CT) scan of the entire spine revealed fracture of the second cervical vertebrae C2), multiple fractures of the lumbar transverse processes, and a lucent spinal lesion of the eighth cervical vertebrae (C8) concerning for malignancy. A nuclear bone scan revealed increased uptake at the eighth and ninth thoracic (T8-T9) vertebrae that correlated with the lesions seen on the CT scans of the spine Figures , 2., multiplThe patient’s past medical history was extensive and included sick sinus syndrome with placement of permanent cardiac pacemaker, diabetes mellitus with associated nephropathy and neuropathy, hypertension, coronary artery disease, hypothyroidism, hyperlipidemia, and peripheral vascular disease with left third-toe amputation. The patient also reported distant early-stage bladder cancer that was initially treated with intravesical therapy followed by annual cystoscopy. He further reported a 120 pack-year smoking history and that he quit smoking 10 years ago. He noted a recent unintentional weight loss of 60 pounds over the prior 10 months without loss of appetite and chronic lumbar pain from a work injury about 33 years prior.The patient was admitted for a trauma evaluation and his neck was immobilized and placed in a cervical collar. He was seen by the neurosurgeon who recommended no acute surgical intervention. CT-guided needle biopsy of the T9 paravertebral soft-tissue abnormality and core biopsy of the T9 vertebra were performed, revealing necrotizing granulomas with rare giant cells and no evidence of carcinoma or acid-fast bacteria; no mycobacteria were identified by Fite's stain. Kappa and lambda light-chain tests, serum levels of immunoglobulinIg G (IgG), IgA, and IgM, protein electrophoresis, and immunoelectrophoresis for the assessment of multiple myeloma were all within normal limits.Four months later, the patient was seen at another ED due to worsening dull back pain in the lower thoracic area radiating to his left chest and left flank. He reported the pain as continuous since his motor vehicle collision without aggravating or alleviating factors, and that since his prior discharge, his pain was unsuccessfully controlled by his primary care and pain management physicians. Furthermore, he was not a surgical candidate for the spinal fractures he sustained from the collision. He denied saddle anesthesia and associated bowel or bladder symptoms. A pleural effusion suggestive of empyema in the right chest was found incidentally on chest CT without contrast. CT of the abdomen and pelvis without contrast revealed an abnormal disc space at T9-T10 with destruction of adjacent endplates suggesting acute discitis with osteomyelitis Figure . A subseHe was admitted for further management, and pulmonology, hematology-oncology, neurosurgery, and infectious disease departments were consulted. When interviewed, the patient endorsed working for the U.S. Navy as a young adult, living in multiple regions of the USA and Germany throughout his life, and recreationally hunting wildlife.Purified protein derivative (PPD), QuantiFERON, and Brucella serology were negative. Ultrasound-guided aspiration of the right-sided pleural effusion was positive for exudative fluid; cytology was negative for malignancy and insignificant for bacterial stains and culture. Because the patient was a poor surgical candidate for open biopsy of the lesion, a CT-guided disc space aspiration at approximately T9-T10 (which was the area of concern for discitis/osteomyelitis) was performed with drainage catheter placement. Acid-fast staining of the specimen was positive and the patient was started on a four-drug antituberculous treatment regimen of oral rifampin 600 mg/day, isoniazid 300 mg/day, pyrazinamide 500 mg thrice/day, and ethambutol 1200 mg/day (RIPE).Real-time polymerase chain reaction (PCR) of the aspirate resulted positive for Mtb. The patient was diagnosed with clinically stable Pott’s disease. He tolerated the antituberculous treatment regimen well and was discharged with instructions to receive an ophthalmological evaluation within two weeks of discharge to monitor for optic neuropathy, weekly complete blood count (CBC) and complete metabolic panel (CMP) for four weeks and then monthly until completing drug regimen, and to continue the RIPE drug regimen for two months followed by the rifampin/isoniazid drug regimen for six months.Pott’s disease is relatively uncommon in high-income countries such as the United States . Thus, iSeveral case reports have documented Pott’s disease in patients where other diagnoses were initially pursued -15. The Diagnosis of spinal tuberculosis is multifaceted and includes components of clinical history, laboratory results, and imaging findings . As mostThe initial laboratory workup for Mtb infection should include a PPD skin test and CBC; if a PPD cannot be done or is inconclusive, a QuantiFERON Mtb gold test (a form of enzyme-linked immunospot assay) should be obtained. Bodily fluids and pus from abscesses should undergo testing for acid-fast bacteria. Biopsies of solid lesions should undergo PCR testing and evaluation for granulomas and caseating necrosis. The gold standard of laboratory testing for Mtb is Mycobacterium cultures from collected fluids or samples. Magnetic resonance imaging is the image of choice for spinal Mtb infection and typically shows lesions that originate from the vertebral endplate, vertebral body collapse or destruction, paraspinal abscess formation, or abscess calcification .Delayed diagnosis of Pott’s disease can lead to detrimental outcomes including degenerative spinal stenosis and paralysis . DespiteBecause of the relatively high proportion of adult patients with Mtb infection caused by drug-resistant Mtb, four drugs are necessary in the initial phase for the six-month regimen to be maximally effective . In mostMycobacterium tuberculosis. However, it should be considered in the diagnosis of spinal pain in patients with significant risk factors and accompanying history such as travel and unexplained weight loss. Initial smearing for acid-fast bacilli may be negative, but it is important to confirm the diagnosis with biopsy results. Early management of Pott's disease is crucial to preventing long-term complications of the disease and improving prognosis.Pott’s disease is a relatively unusual complication of infection with


Here, we show that by breaking σz-symmetry of the PhCS, asymmetry in upward and downward directions and arbitrarily polarized BIC can be realized with a bilayer-twisted PhCS. It exhibits elliptical polarization states with constant ellipticity angle at every point in momentum space within the vicinity of BIC. The topological nature of BIC reflects on the orientation angle of polarization state, with a topological charge of 1 for any value of ellipticity angle. Full coverage of Poincaré sphere  have been demonstrated to be associated with topological charge and vortex polarization singularities in momentum space. For conventional symmetric photonic crystal slabs (PhCSs), BIC is enclosed by linearly polarized far fields with winding angle of 2 Bilayer twisted photonic crystal slabs enable arbitrarily polarized bound states in continuum (BICs) covering full Poincare sphere. Destructive interference or symmetry mismatch can make these modes become nonradiative with infinite lifetime, that is, bound states in the continuum (BICs)7. The mode at BIC remains localized and cannot couple with free-space radiations even in a continuous spectrum carrying energy away. Since there is no outgoing power and thus undefined polarization vector, BIC manifests as a momentum space singular point representing vortex center (V point) in the far-filed polarization state10. Conventionally, when the PhCS supporting BIC maintains σz-symmetry (up-down mirror symmetry) and in-plane inversion symmetry, the far-field is linearly polarized and the polarization angle winds around the V point, generating topological radiation pattern quantized by topological charges12. These winding topologies of resonances in momentum space can inherently act like spatially winding configurations in real space for generating optical vortex beams16. The topological property guarantees the robustness of BIC against change of system’s parameters. Recently, modulating polarization of light and exploring polarization singularities using PhCS has been of greet interest due to abundant polarization features emerging in momentum space20. To overcome the limit of only linearly polarized resonance around BIC, the in-plane inversion symmetry of square PhCS is broken into trapezoid and triangle, which causes the V point at BIC decomposed into two circular polarized states (C points) and equator lines on the Poincaré sphere (L lines)21. This broken symmetry enriches the features of polarization modulation using BIC topology and achieves full coverage of Poincaré sphere. Later, with a honeycomb lattice, it has been demonstrated that C points can occur near BICs and Dirac points with preserved in-plane inversion symmetry, which remarkably contributes to diversity of polarization around BIC22. While for PhCS with inversion symmetry but broken σz-symmetry, two C points emerge without splitting the V point, and the evolution demonstrates merging processes governed by the global charge conservation23. These findings promote the applications of PhCS supporting BIC for polarization modulation and manipulating various polarization singularities in momentum space.Photonic crystal slab (PhCS) is dielectric structure with periodically modulated refractive index and finite thickness, supporting Bloch modes that are mostly leaky with finite lifetimes23. The relevant circular and elliptical polarization states originate from destroyed radiative modes that are almost linearly polarized enclosing BIC. This is due to unperturbed σz-symmetry and unidirectional scattering of eigenmode of the PhCS26. Although broken σz-symmetry has been analyzed in ref. 23 in the one-dimensional PhCS, the lack of rotation asymmetry still ends up with unidirectional guided resonances with global charge of zero. Most recently, the concept of twistronics has been extended from two-dimensional heterostructures to optics like twist-stacked metamaterials and chiral PhCS, which provide new platforms for tailoring chiral-optical effects and elliptical polarizations with new degree of freedom32.However, in previous studies, the circularly polarized states spawning from BIC can only occur in pairs at off-Γ point due to the conservation of an integer topological charge induced by the V point, and pure elliptical/circular polarization states within a large span of momentum space cannot be achievedχ of polarization states, we define the polarization singularity induced by the Γ-BIC from the twisted PhCS as χ point, which can be encircled with constant ellipticity angle χ. BIC in the twisted PhCS has infinite quality factor robust against rotation angle of the hole and is encircled by far-field polarization state carrying the same ellipticity angle (χ). Taking advantage of BIC-induced topological distribution of orientation angle (ψ), polarization states around BIC encircles a full latitude of the Poincaré sphere for each BIC. The ellipticity angle of χ point can be arbitrarily controlled by changing the rotation angle of the hole, resulting in a full coverage of Poincaré sphere of the radiating polarization states.In this letter, based on the ellipticity angle σz-symmetry and in-plane mirror symmetry standing in free space. The PhCS has a thickness of 600 nm and a lattice constant of 580 nm. The hole has a side length of 330 nm. BIC and topological far-field polarization distribution are expected to be observed in momentum space. This PhCS supports a BIC encompassed by linear polarization symmetrically radiating in upward and downward, as proved by temporal coupled-mode theory (TCMT)34. Breaking in-plane inversion symmetry leads to an undermined BIC and two C points spawning from the original V point at BIC , the realization of χ of the BIC can be fully covered by controlling both It is shown that the ellipticity angle αtop = 18°, αbot = 0° and the quality factor of the eigenmode is color-mapped to the iso-frequency surface. The quality factor is not undermined by rotation of the hole, and the twisted PhCS can still maintain the Γ−BIC  while forming the χ point. The bounded resonance at BIC is shown in the field distribution of the twisted PhCS at Γ point . It should be noted that even though the relative rotation angles of the two PhCS layers in both cases of Fig. To verify the generation of circularly polarized BIC, the rotation angle is fine-tuned to αtop on far-field polarization states and demonstrate full coverage of Poincaré sphere assisted by BIC topology, we illustrate the evolution of upward far-field radiation with varying αtop in Fig. αtop = 45°, equivalent to the conventional PhCS  to circular states (C points). Continuing to increase αtop from 18° to 45°, the far-field polarization gradually changes from C points to L lines. A similar trend holds true for negative αtop that covers the polarization states on the lower hemisphere. The Poincaré sphere can be covered twice with αtop ranging within a span of 90°. This trend can be understood in view of the square’s four-fold rotation symmetry and the difference of polarization angles induced by the two equivalent scatters.To better understand the effect of rotation angle hCS Fig. . Decreasχ can be defined as the winding number of the polarization vortex22L is a closed trajectory in momentum space around the BIC of different χ. Figure χ. Arbitrarily polarized BIC can be uniquely defined by the ellipticity angle of polarization states enclosing it. When χ = 0°, it is a conventional linearly polarized BIC occurring in the in-plane and σz- symmetric PhCS. When χ has a value between −45° and 45°, it is the elliptically polarized BIC. At χ = ±45°, there emerges circularly polarized BIC. Although with circularly polarized BIC the orientation angle ψ still reveals topological feature of q = 1, the far-filed polarization states are the same since the C point on Poincaré sphere is actually a singular point different from L or elliptical line. This also demonstrates the robustness of BIC topology against an ellipticity angle. Each χ point at BIC and its topology correspond to a point on the higher-order Poincaré sphere as proposed in ref. 35. By employing the new degrees of freedom in rotation angles of both holes, full coverage of higher-order Poincaré sphere can be achieved with BIC in the twisted PhCS .The topological charge carried by BIC with different 42, followed by direct wafer bonding of two PhCS layers43. Furthermore, other techniques using spin-on glass for the fabrication of bilayer silicon structures have also been demonstrated in refs. 45.Regarding the fabrication processes, the single-layer PhCS can be fabricated by using conventional e-beam lithography and etchingσz-symmetry of a PhCS, arbitrarily polarized BIC can be realized by controlling the twist angle between two layers of the PhCS. Unlike BIC in unperturbed PhCS or broken in-plane symmetric PhCS where only linear and at most two circular polarization states can occur, the twisted PhCS can exhibit circular polarization states at every point in momentum space within the vicinity of BIC. At different twist angles, other elliptical polarizations can be approached surrounding BIC singular point with the same ellipticity angle. Based on the topological nature of BIC, polarization states on the Poincaré sphere can be fully covered twice with a rotation angle of 90 degrees. Full coverage of the higher-order Poincaré sphere is also attained by tuning the rotation angles of both holes. More diverse polarization control is demonstrated by introducing new degrees of freedom like opposite rotation angle for two layers, breaking both in-plane and σz- symmetry, and changing the surrounding index. The demonstrated work may contribute to applications of BIC topology in chiral-optical effects, polarization control, structure light like more exotic vector beam, and twistronics.In conclusion, we reported that by breaking Supplementary Information


Our study revealed the persistence of terrestrially-sourced carbon in severely energy limited deep continental groundwaters supporting deep microbial life.The deep terrestrial biosphere encompasses the life below the photosynthesis-fueled surface that perseveres in typically nutrient and energy depleted anoxic groundwaters. The composition and cycling of this vast dissolved organic matter (DOM) reservoir relevant to the global carbon cycle remains to be deciphered. Here we show that recent Baltic Sea-influenced to ancient pre-Holocene saline Fennoscandian Shield deep bedrock fracture waters carried DOM with a strong terrigenous signature and varying contributions from abiotic and biotic processes. Removal of easily degraded carbon at the surface-to-groundwater transition and corresponding microbial community assembly processes likely resulted in the highly similar DOM signatures across the notably different water types that selected for a core microbiome. In combination with the aliphatic character, depleted δ Dissolved organic matter in the Fennoscandian Shield deep continental bedrock fracture waters of varying characteristics and ages carries a strong terrigenous signature, and only a small proportion of this potential energy source links to the deep biosphere microbial community. As the water moves into deeper layers, nutrients and available organic carbon are removed, severely limiting the energy supply to the deep biosphere life estimated to contain 2–6 × 1029 cells3 that is approximately one quarter of the global microbial biomass4. Key processes in deep subsurface carbon cycling depend on the recalcitrance of the organic matter to microbial utilization5, input of metabolic oxidants and reductants, and aquifer permeability6. While acting as a nutrient and energy source for heterotrophic life7, the composition and concentration of the DOM also impacts trace element and contaminant transport9, especially in anthropogenically impacted groundwater systems. Climate change and increasing urbanization exert pressures on the deep groundwater systems affecting DOM input, storage, and turnover10. Yet to date, few studies have addressed DOM composition and its associated bioavailability in deep terrestrial groundwaters.Groundwater is the earth’s largest active source of freshwater11. Plant-derived molecular constituents and a high overlap with riverine fulvic acids, carrying a large number of preferentially biodegraded nitrogen- and sulfur-containing compounds, were attributed to groundwater DOM at depths <3 m13. Sulfur-containing organic matter can be a dominant component of rainwater14, but can also be released by microbes or abiotically produced in anoxic settings, enhancing the recalcitrance of the organic matter17. Permeating through soils and sediments, selective interactions of the organic matter occur with microbes, minerals, and water that integrated in the “regional chromatography” effect shape a homogenized DOM signature in the saturated zone19. Nevertheless, the source and history of groundwater DOM remain imprinted in its composition, with bulk organic carbon and isotopic compositions similar to surface soils indicating a surface origin and vertical transport into aquifer systems20 that varies seasonally22. The proximity and connectivity to organic-rich layers drive microbial abundance and activity as well as microbial community composition24 and DOM from chemolithoautotrophic communities in deep basaltic or granitic intrusions may be important as the long timescales increase the proportion of microbial-derived and transformed DOM over allochthonous, terrigenous material in the oligotrophic systems27.Plant litter and soil comprise the main sources of DOM in shallow groundwaters30. The present groundwaters broadly fall into four main types according to water chemistry signatures: marine, meteoric, old saline, and glacial30. While the brackish Baltic Sea water is classified as a marine source, its DOM is of predominantly terrestrial origin, i.e., comprising about 75% in the Baltic Proper31. During the 30 years since the Äspö HRL tunnels were constructed, complex mixing has occurred where some fracture zones facilitate downward percolation of Baltic Sea water whereas other fractures promote upward flux of deep saline waters. The Fennoscandian Shield microbial community has a core microbiome between the different groundwater types often comprised of small cells with streamlined genome sizes that likely represent adaptations to an oligotrophic lifestyle34. While isotopically light carbon indicates slow, but persistent microbial transformation of surface-derived DOM35, the link between the DOM reservoir and the microbial community is largely unknown.The Äspö Hard Rock Laboratory (HRL) on the Swedish Baltic Sea coast facilitates access to deep continental groundwaters in the Fennoscandian Shield. Its 3.6 km long tunnel partially extends below the Baltic Sea at depths down to 460 meters below sea level (mbsl). The groundwater components consist of old saline water, brackish water from the overlying Baltic Sea and its predecessors such as the Littorina Sea, and fresh waters from temperate and cold climate  of the Pleistocene36. We hypothesized that terrestrial surface DOM fuels microbial life in the deep continental bedrock fractures. In addition, we suggest that abiotic transformations occurring on long timescales and microbial reworking of the aged DOM is likely in the low carbon and energy groundwaters with cycling of sulfur compounds playing an important role.In this study, DOM concentration and molecular composition in conjunction with stable and radiogenic carbon and water isotopic investigations, water chemistry, and microbial community structure in Äspö HRL fracture waters Fig.  of diffe18O values, and Cl/Mg plus Br/Cl ratios to distinguish water origin in the aquifer at Äspö HRL30 grouped the samples  ‘BrMarine1’ recent brackish-marine waters intruded into the boreholes; (iii) ‘BrMarine2’ Holocene marine waters older than present day Baltic Sea; (iv) ‘Transition’ waters that carry signatures from both BrMarine and old saline waters; and (v) brine-type old ‘Saline’ water of unknown origin with residence times in the bedrock of ≥100,000 years38 mixed to various degrees with old meteoric water of cold climate origin, likely glacial and pre-Holocene. The origin of the brine component is complex and may involve leaching of marine sediments  later influenced by water rock interaction through time38. The Baltic Sea samples clustered with BrMarine1 borehole SA1229A that accessed a large section with high connectivity to the Baltic Sea. All other BrMarine1 and BrMarine2 groundwaters grouped within the area of mixed water origin, with one borehole sample  moved toward meteoric characteristics. KA2865A01_1 and KA3385A_1 waters associated closest with KA3510A_2 classified as Transition-type waters of intermediate Cl/Mg ratios.Using chloride concentrations, δes Table  into:  and near Äspö Island Baltic Sea values framed the BrMarine samples  mostly as ammonium of an unknown source, dissolved C/N ratios were not significantly different between water types at an average of 19.4 ± 6.1.DOC concentrations decreased with increasing conductivity and were below 1.15 mg L39, including few O-poor aromatics and mostly highly unsaturated O-poor or unsaturated MFs, some containing N  were widely spread across van Krevelen space besides the area of O-rich aromatics and highly unsaturated MFs.Solid-phase carbon extraction efficiencies were 81 ± 4% and 66 ± 4% across all deep fracture waters and the two Baltic Sea samples, respectively Table  in the 918O, δ2H, SO42−, D, and Cl−) similar to those used previously for water type classification at Äspö HRL29 and DOM composition were significantly related , 9999 permutations, n = 15, F = 3.6, p < 0.01). Overall, hydrochemistry explained about 83% of DOM variability with some differences in classifications remaining. BrMarine1 samples KA2511A_5 and SA1229A_1 grouped closest with the Baltic Sea waters, while KA2051A01_5 associated with BrMarine2 and Transition boreholes KA3105A_3 and KA2865A01_1. The Transition DOM formed a tight group in the hydrochemical classification, but their mixed origin stood out in DOM molecular space. The Transition-type waters KA3510A_2 and KA3385A_1 grouped closest to Saline KA1755A_3 and HA2780A_1, respectively, while KA2865A01_1 more closely affiliated with BrMarine2 borehole KA3105A_3.A gradient from Baltic Sea, through Transition, to Saline groundwater origins was partially reflected by the distribution of DOM compositions along the first axis of NMDS ordination Fig. . HydrochMF and P/CMF ratios (weighted means from MF assignments) that were low compared to older groundwater samples  were detected in this dataset . The AbioS peaks were not present in Baltic Sea water and barely present in KA2051A01_5 (BrMarine1) and KA3105A_3 (BrMarine2). In addition, the S/CMF ratio was highly correlated to the number of S-containing MFs , implying the production of new S-containing MFs rather than addition to already existing ones. The presumed lability of the DOM based on the lability index MLBw was highest in the oldest, Saline- and some Transition-type waters  but the organic fraction showed a higher pMC value than the inorganic fraction, except for the Baltic Sea waters, that was mainly attributed to dissolution of calcite within the overburden or in the uppermost fractures. The relatively young age of the inorganic and organic carbon in the Saline-type fracture waters was surprising regarding their implicit origin. The saline component endmember  has a long residence time manifested in Mg/Cl, Br/Cl ratios, He content, and δ18O/δH ratio38. However, the Cl− concentrations were below 15,000 mg L−1 with the dilute components being of glacial origin or possibly an older meteoric water. Introduction of modern water portions or contamination through e.g., biofilm formation in the sampling system is unlikely as sampling section and tubing were flushed with several volumes of fracture water before each sample was taken, similar carbon extraction efficiencies were found over all section and tubing lengths, and FT-ICR-MS spectra showed no obvious contaminants. The detected trends between water types thus likely hold true but nevertheless, these potential caveats need to be considered especially for the low-DOC Saline-type water.The radiocarbon age of the bulk inorganic and organic carbon fractions was assessed as percent modern carbon  and batch 2 (δ13C = −23.70‰). As all other samples were consistent between the years and the second sampling of SA2600A_1 was not otherwise conspicuous, this value was disregarded during further interpretations. δ13C values of organic and inorganic carbon were positively related  with Δ13CDIC−DOC values of −16.6 ± 4.1‰ for all samples and −14.1 ± 3.3‰ for old saline waters. The solid-phase extracted DOC stable carbon isotope signature (δ13C-SPE-DOC) did not mirror the full extent of δ13C-DOC and ranged from −26.1 to −28.4‰ . The agreement between δ13C-SPE-DOC and δ13C-DOC values was good, but deviation was highest for Baltic Sea water with the lowest extraction efficiencies  that comprised between 4 and 9% of the relative intensity.δely Fig.  except fF = 1.40, n = 10) of the microbial community composition. However, this relationship was not statistically significant, likely due to the low number of samples used in this study . Comparison of the DOM composition and the Äspö HRL groundwater microbial community compositions via Procrustes analysis resulted in an intermediate m2 of 0.65  that indicated the datasets were not significantly related along main gradients of variability. Therefore, individual associations between microbial ASVs and DOM MFs were analyzed via network analysis based on proportionality. The MFs retained in the network were centered in van Krevelen space  that aligned within the Proteobacteria (28% mean relative abundance) including classes Gamma- and Alphaproteobacteria, Patescibacteria (24%), Campylobacterota (13%), and Desulfobacterota (8%)  and relatively low variability in the Äspö HRL fracture system where lignin-like MFs contributed >50% of the relative intensities in all water types. In addition, that the sixfold change of bulk DOC concentrations between Baltic Sea and the deepest groundwaters plus the increased DON in the Baltic Sea were not mirrored in the narrow variation in DOM molecular characteristics can partially be attributed to the analytical window inherent to the applied methods43. However, the higher deep groundwater solid-phase extraction efficiencies suggested they contained a lower proportion of poorly captured high molecular weight material, colloidal organic matter, or small monomeric compounds compared to the Baltic Sea water. Despite a considerable terrestrial fingerprint31, the Baltic Sea DOM carries freshly produced matter comprising e.g., proteins, amino acids, and carbohydrates44. These compounds are preferentially respired in the subsurface water column and may adsorb to sediment grains leading to the strongly terrigenous DOM signature entering the deep fractures. In line with the changes in DOM composition with depth and groundwater type from the Baltic Sea downwards, Äspö HRL groundwater microbial communities supported by the available DOM also considerably differ from the overlying Baltic Sea water, suggesting species sorting occurred prior to the infiltrated waters reaching the Äspö HRL boreholes45. Thus, the existence of a Fennoscandian Shield core microbial community adapted to the oligotrophic subsurface32 may be at least partially due to the low variability and recalcitrant nature of the DOM in the different Äspö HRL groundwaters.Selective interactions of organic substances, especially N-rich materials, with water, microbes, and mineral surfaces during groundwater transport was reported by Aiken−1 organic carbon are commonly located close to organic-rich sediments or receive organic-rich recharge46. The shallow BrMarine1 and BrMarine2 groundwaters with high surface connectivity exhibited large contributions of lignin- and tannin-like MFs of high aromaticity, similar to the high humic and fulvic acid concentrations from soil carbon of aromatic and (highly) unsaturated molecular classes detected in shallow soil-derived DOM21 and the Baltic Sea31. Toward the old Saline-type waters, DOC concentration, molecular mass and aromaticity, as well as O/CMF and P/CMF ratios consistently followed the dominant mixing process. The intrusion of terrigenous DOM into deep aquifers may confound previous interpretations of a largely chemolithoautotrophic origin of DOC in other systems47.Groundwaters carrying >1 mg L49. Adsorption on surfaces or formation of insoluble complexes with hydrolyzing metals, especially Fe and Al, preferentially removes aromatic and lignin-like DOM moieties from solution51. The relatively high aromaticity of the BrMarine and partially the Transition-type DOM together with the high DOC concentrations indicated saturated surfaces and/or sufficient recharge29. Overall, that more than two thirds of the variability in DOM composition was explained by water chemistry parameters confirmed mixing of different source waters as the main driver of water compositions in the deep bedrock. The remaining variability points toward additional abiotic or biotic processes including sulfurization, sorption, condensation, or microbial uptake and reworking6 shaping DOM composition in Fennoscandian Shield groundwaters.The Äspö HRL is built in Paleoproterozoic granitoids with fracture surfaces carrying calcite, chlorite, pyrite, and clay minerals53. Microbial community data indicate the presence . The S-content of the SPE-DOM (count of S-containing MFs or S/CMF ratio) correlated to the relative contribution of the AbioS index indicative of abiotic sulfurization40, while the double bond equivalent (DBE) of S-containing MFs of the groundwaters was on average one unit lower than that of CHO-only MFs. Together with a negative relationship of S-containing and CHO-only contributions to sample compositions, this could indicate the formation of new DOC compounds through bisulfide addition16. However, the deviation in DBE was even higher at around two DBE for the oxic Baltic Sea samples, where a different mechanism must take place.The high sulfate concentration in Saline waters and two Transition-type boreholes originated from Baltic Sea input and gypsum dissolution and were consistent with previous investigations in a nearby fracture systemMF ratios, of organic matter is thought to enhance preservation17. It may therefore seem counterintuitive that a higher S/CMF ratio characterized DOM in the old Saline-type DOM that overall also comprised DOM of higher lability as indicated by the MLBw. Notably, on average, MFs in the dataset carrying at least one S (N) atom contributed approximately 13% (17%) of the total sample intensities. Separating relative contributions included in the MLBw (H/CMF > 1.5), CHO MFs contributed 65 ± 5% followed by S- (29 ± 4%) and N-containing (5 ± 2%) MFs. S-containing MFs thus contributed disproportionally highly to the labile DOM pool, especially compared to the often bioavailable N-containing compounds. The abiotic sulfurization process is not selective in regard to saturation, aromaticity, degree of oxidation, or heteroelement content of the precursor compound, and bisulfide nucleophilic addition shifts the MF toward higher H/CMF ratios16. Furthermore, sulfur-containing metabolites produced though bacterial assimilatory S-reduction likely contributed to the DOS pool57. Hence, abiotic sulfurization of the subsurface DOM was an important process in the Äspö HRL fracture waters, but sulfur integration in DOM not only acted as a stabilizer but also increased the assumed lability of the DOM (% MLBw), an ambiguous role that needs further investigation in this system.Sulfurization, manifested in increased S/C13C-DIC carbon isotopic values of −0.7 to −18.5‰ decreased with conductivity and represented recharging groundwaters with BrMarine influence, terrigenous organic matter oxidation and dissolution of calcite from different generations with only minor contributions of extremely depleted DIC derived from anaerobic oxidation of thermogenic (δ13C −30 to −50‰) or biogenic methane (δ13C −60 to −90‰) present at low concentrations in the fracture waters53. δ13C-DOC correlated with δ13C-DIC, but isotopic compositions were in a narrow range from −23.7 to −28.7‰ that were close the values reported for boreal forest soil (−24 to −29‰58), the dominant river DOM source in the region. In addition, the Δ13C(DIC-DOC) values were mostly higher than the range of measured fractionation factors for the reverse tricarboxylic acid cycle (−2 to −12‰), emphasizing that CO2 fixation was not the main driver of DOC production as suggested for the hot Costa Rican backarc system47. Degradation by microbes typically lowers the δ13C of the remaining DOC pool59 and the most depleted δ13C-DOC value of −28.6‰ were found for the old saline DOM, indicating a stronger imprint in these waters to which a minor contribution of higher-plant derived material was previously attributed60. Usually, higher H/CMF ratios or lower aromaticity of DOM are associated with younger DOM that is potentially more accessible to microbial utilization63, opposite to what was observed at the Äspö HRL. Here, in conjunction with the low AImod, low molecular mass, and high H/CMF ratios, permanent reworking and release of an aged but apparently labile organic matter fraction through the resident microbes was likely in the Saline-type waters10. In addition, the MFs unique to the deep Saline DOM were enriched in low-O/CMF and high-H/CMF MFs  compared to the average molecular weight of all samples (397 ± 5 Da). Methane in the deep crystalline bedrock can originate from abiotic  or biotic processes via methanogenesis65, but methanotrophic contribution to the DOM pool was low at Äspö according to the stable isotopic composition of DOC.Microbial remineralization, transformation, and production affected the deep groundwater DOM compositions in addition to the dominant mixing signature. The microbial imprint was not only reflected in the old Saline DOM fingerprints, but also in the carbon isotopic composition. δ66. However, these studies deal with DOM originating at least partially from recent primary production, whereas the DOM examined in the anoxic Fennoscandian Shield fractures had a terrigenous signature and had cycled through the subsurface bedrock fractures for long timescales. The fundamentally different source DOM quality and environment can entail that the opposite trend was found for the P/CMF ratio with increasing age in the Äspö HRL fracture waters, and no consistent change in N/CMF ratio was observed. Although terrestrial organic matter is generally depleted in nitrogen compared to marine organics, both the DON and DOC concentrations were lower in Transition- and Saline-type waters compared to the more recent BrMarine with no significant differences in bulk C/N ratios. The nitrogen-containing DOM may still be rapidly cycled in the environment limited in soluble nitrogen and phosphorus67, keeping the concentrations measured at discrete time points low. The increase in P/CMF ratio was not easily explained and literature including DOP on MF level analysis is scarce. Phosphate was depleted in BrMarine2, Transition, and Saline waters, such that DOP comprises a potential P-source for microbial life rather than being released to the DOM pool. Yet, with the increase in P/CMF ratio from Baltic Sea to Saline waters, also the number of P-containing MFs increased from around 50 to 150, pointing toward new production and no P-limitation of the subsurface microbes.With ongoing microbial degradation of the DOM, N- and P-containing molecules are often preferentially utilized by heterotrophic bacteria68. Reworking of DOM due to microbial reuse of necromass adds to the labile characteristics of Saline-type DOM. In addition, carbon dioxide and hydrogen geogas-driven microbial metabolism may leave little imprint in the DOM and the fast uptake of products of this metabolism by other microbes further diminishes the detectable imprint34. However, all DOM fingerprints included in this study shared more than 85% of the MFs relative intensity, so that the proportion of the DOM responsible for the labile signature in the deep Saline waters was small in comparison to the recalcitrant, terrigenous signature common to all groundwaters. This potentially explains the slow growth rates and “stop-and-start” cell replication processes that are suggested to be activated or deactivated as and when carbon and energy sources are intermittently available to the deep fracture water microbial community32.The possible reasons why the microbial imprint was primarily observed in the deep Saline-type DOM were threefold: (i) the DOC concentration was the lowest such that minor removal or dilution of terrigenous organics resulted in a visible microbial imprint in the relative abundance spectra; (ii) microbial adaptation to the refractory C sources takes time and thus only occurs in the oldest waters; or (iii) more accessible energy sources were available in other fracture waters. Previous studies show that the deep biosphere microbial community is viable and active, rapidly recycling dead cells as substratew) of the network MFs and in line with previous studies reporting major proportions of refractory DOM in aquatic systems18. Most ASV nodes were affiliated with the phylum Patescibacteria that also constituted more than one-third of the module hubs that were highly connected and proposed to be keystone taxa69. These module hub Patescibacteria belonged to the classes Paceibacteria, ABY1, and Gracilibacteria, which were also present in a shallow groundwater system70 and are suggested to be prevalent in deep groundwaters by their ease of mobilization from soil coupled to their adaptation to low energy conditions71. Patescibacteria were not linked with specific DOM MFs, but with MFs of varying characteristics and all compound classes. In addition, they mostly co-occurred with either other Patescibacteria ASVs or ASVs affiliated with Chloroflexota, Dusulfobacterota, and ‘Unclassified’ phyla in the same module. This indicated coupling of the production and consumption of DOM compounds via a symbiotic host-associated lifestyle72. The coupled breakdown of complex DOM compounds was also supported by connector MFs, representing nodes that connect different modules  and therefore might reflect key intermediate compounds during microbial DOM degradation69. These connector MFs were positively and negatively linked to various microbial taxa, which might represent potential producers and consumers, respectively. Members of the phylum Desulfobacterota were also abundant in the network with module hub ASVs belonging to Desulfocapsa, Desulfatiglans, Desulfobacula, and unclassified genera of the order Syntrophales. Characterized members of the Desulfatiglans and Desulfobacula are heterotrophs and able to degrade aromatic compounds like phenol73, thus potentially being capable of degrading complex aromatic compounds like tannins and lignins in line with links between these two genera and mostly lignin-like MFs in the network  at the Äspö HRL on Äspö Island in the Misterhult archipelago in Kalmar County, Sweden. Since the tunnels were constructed between 1990 and 1995, contamination from excavation has been minimized although the boreholes have been instrumented for varying times that has introduced materials into the deep groundwaters. Samples from boreholes SA1229A_1, KA3105A_3, SA1730A_1, KA3600F_2, and SA2600A_1 were taken in November 2018 (batch 1) followed by all other samples in May 2019  with an analytical uncertainty of 6% (Swedish Standard 028136 edition 1), and SO42− concentrations were determined using ion chromatography (DIN EN ISO 10 304-1: 2009) with an analytical uncertainty of 12% (4.5–70 mg L−1) and 31% (0.5–4.5 mg L−1). Dissolved Fe2+ and total iron  concentrations were determined immediately after sampling using a spectrophotometer  and a modified ferrozine method77 with an analytical uncertainly of ±0.005% (0.02–0.05 mg L−1), 8% (0.05–1 mg L−1), and 13% (1–3 mg L−1) depending on concentration range. Oxygen concentrations could not be reliably measured due to the setup installed to obtain water from the boreholes, but Fe2+ concentrations equal to the Fetotal indicated anoxic conditions in all samples except those from the Baltic Sea. HS− concentrations were determined with a spectrophotometer (as above) with an analytical uncertainty of 32% (Swedish Standard SIS 02 81 15). Concentrations of Ca, K, Mg, and Na cations were determined with inductively coupled plasma atomic emission spectroscopy at ALS Scandinavia AB in Luleå (Sweden) with an analytical uncertainty of 12%. Analysis of the water 18O/16O and 2H/1H ratio  were determined by laser spectrometry  with an analytical uncertainly of ±0.25 and ±1.5 unit, respectively. δ13C- and pMC-DIC and DOC were determined on whole water samples at the Ångström Laboratory in Uppsala (Sweden) using Accelerator Mass Spectrometry, with an analytical uncertainty of 0.3‰ PDB for δ13C and 0.2–0.5 pMC.Hydrochemical variables were determined in the ISO 17025 rated Äspö HRL chemical laboratory as follows: pH was determined potentiometrically (±0.10 pH units), Cl13C-SPE-DOC) was analyzed to better characterize the analytical window of the SPE method. Aliquots containing ~10 µmol C were dried in doubled Sn caps  and analyzed using an Elemental Analyzer (Thermo Scientific Flash) coupled to a continuous‐flow isotope ratio mass spectrometer (Thermo Scientific Delta V) via an Advantage Conflo IV interface. Values were corrected with procedural blanks, using the dried extracts from ultrapure water processing in doubled Sn caps. Technical replicates deviated <0.3‰, accuracy was better than 0.2‰.All glassware used for sampling was pre-combusted  or thoroughly soaked in ultrapure water at pH 2 and rinsed with sample before use. Samples for DOM quantification and molecular characterization were filtered (Durapore PVDF 0.22 µm) and acidified to pH 2 with HCl before shipping to the University of Oldenburg (Germany). DOC and TDN were quantified from duplicate subsamples using a Shimadzu TOC-L with an ASI-L autosampler with concentrations determined via an L-arginine standard curve. Then, ~1 L of each sample was extracted onto 1 g solid-phase extraction columns  and eluted with 6 mL methanol (ULC grade). DON was calculated from TDN − (nitrate + nitrite + ammonium). Extract DOC concentrations were determined by drying an aliquot of the extract and re-dissolving it in ultrapure water at pH 2. The extracts were then stored in the dark at −20 °C until further analysis. Stable carbon isotope composition of solid-phase extracted DOM  were processed together. The method detection limit was set to 3. Junction of mass lists along m/z was performed via fast join at a tolerance of 0.5 ppm while standard smooth and additional isotope tolerance was 10‰. Singlet peaks occurring only once in the dataset were removed, then MF attribution was done with a tolerance of 0.5 ppm in the range m/z 0–1000 within the limits C1–100H1–100O0–70N0–4S0–2P0–1. For isotopic verification of MF attributions, up to one 13C, 15N, and 34S isotopologue was allowed. Additional tolerance using quantile-based isotope ratio tolerances was set to 1000‰ and isotope ratio mismatches for peaks below the method detection limit <5 were permitted. MF suggestions with >3 heteroelements (except N4) and four common contaminant MFs were removed  were removed from all samples. Before statistical analysis, relative intensities were normalized to the sum of intensities per sample and the highest minimum between all samples was set as detection limit and all peaks with lower normalized relative abundance were set to 0 to improve comparability between spectra. Furthermore, MFs were only retained in the dataset if they occurred at least three times across all samples. Molecular indices indicating lability through relative contribution of MFs above the molecular lability boundary of H/CMF = 1.5, MLBw79, and aromaticity as modified aromaticity index, AImod80, were calculated. MFs were assigned to molecular categories within the boundaries described by ref. MF).Fourier-transform ion cyclotron resonance mass spectrometry (FT-ICR-MS) analysis was performed on a 15 T Solarix . Samples were diluted to yield a concentration of ~5 ppm in ultrapure water and methanol 50:50 (vol/vol). This dilution was filtered through pre-cleaned 0.2 µm polycarbonate syringe filters before analysis performed in random order. Electrospray ionization in negative mode (Bruker Apollo II) was done at 200 °C and the capillary voltage was set to 4.5 kV. The sample was injected at a flow rate of 120 µL h55. Microbial communities and the corresponding water chemistry data were sampled for boreholes SA1229A_1, KA2051A01_5, KA2511A_5, KA3105A_3, KA3600F_2, KA3385A_1, SA2600A_1, KA1755A_3, KA2862A_1, and SA1730A_1, whereas no data was available for the boreholes KA2865A01_1, KA3510A_2, and HA2780A_1. For a detailed description on sampling and sample processing refer to ref. 82. High-throughput sequencing was done on the Illumina MiSeq platform at the Science for Life Laboratory (Sweden) according to ref. 83, 1.14.1) in R . After the filter and trim step, remaining primer sequences were removed with Cutadapt . (2) Statistical analyses including DOM and water chemistry were based on a dataset containing mean spectra of DOM from the same borehole regardless of sampling date (n = 15). (3) For statistical analyses including DOM composition, hydrochemistry, and microbial community compositions, the number of samples was further reduced due to the availability of the sequencing data (n = 10). Relative intensities of spectra, relative abundances of microbial ASVs, and environmental parameters were averaged per borehole for the analyses including more than one of these datasets.Three different datasets were used throughout the manuscript: (1) DOM composition was analyzed based on all available samples including technical and analytical replicates . MFs and ASVs were only considered for the analysis if they were present in at least five samples; low-abundant ASVs were removed (relative abundance < 0.1%); zeros were replaced by their respective minimum value/10 to account for the fact that the range of relative intensities and abundances covered several orders of magnitude; and each dataset was center log-ratio (clr) transformed separately. Proportionality measure ρ was then calculated for the combined ASV and MF data matrix (propr package). The selection of a cutoff for ρ was based on permutation of the false-discovery rate (FDR) for different ρ88. To control for a FDR < 5%, |ρ | ≥ 0.85 was chosen as cutoff for network construction. The resulting network retained a relationship between the relative abundances of 229 ASVs and relative intensities of 1388 MFs. Network visualization was done in Gephi (version 0.9.289). Network modularity, module separation and roles of nodes were determined via the interdomain ecological network analysis pipeline  with the fast greedy modularity optimization91. The potential ecological role of each node was assigned based on within-module connectivity (z) and among-module connectivity (P)92 and nodes were classified into four categories: peripheral nodes , connectors , module hubs , and network hubs 69.Bray Curtis dissimilarities were used to assess the overall dissimilarity of the DOM fingerprints and microbial community composition and major gradients of variability were visualized with non-metric multidimensional scaling (NMDS). The congruence of DOM molecular and microbial community composition was analyzed via Procrustes rotation. The impact of mixing on DOM and microbial community composition was assessed via distance-based redundancy analysis. These statistical analyses were performed using R (Version 4.0.3) and the vegan packageSupplementary InformationDescription of Additional Supplementary FilesSupplementary Data 1


The cell division cycle is regulated by a complex network of interacting genes and proteins. The control system has been modeled in many ways, from qualitative Boolean switching-networks to quantitative differential equations and highly detailed stochastic simulations. Here we develop a continuous-time stochastic model using seven Boolean variables to represent the activities of major regulators of the budding yeast cell cycle plus one continuous variable representing cell growth. The Boolean variables are updated asynchronously by logical rules based on known biochemistry of the cell-cycle control system using Gillespie’s stochastic simulation algorithm. Time and cell size are updated continuously. By simulating a population of yeast cells, we calculate statistical properties of cell cycle progression that can be compared directly to experimental measurements. Perturbations of the normal sequence of events indicate that the cell cycle is 91% robust to random ‘flips’ of the Boolean variables, but 9% of the perturbations induce lethal mistakes in cell cycle progression. This simple, hybrid Boolean model gives a good account of the growth and division of budding yeast cells, suggesting that this modeling approach may be as accurate as detailed reaction-kinetic modeling with considerably less demands on estimating rate constants. Saccharomyces cerevisiae, is a model organism for studying regulation of the eukaryotic cell cycle. The complex gene-protein interaction network controlling the yeast cell division cycle has been modeled by several different mathematical techniques, including discrete Boolean networks and nonlinear ordinary differential equations (ODE). ODE models track variables with continuous values (concentrations or activities of proteins) in continuous time. Boolean models, on the other hand, assign variables to discrete states (0 or 1) in discrete time .Budding yeast, 2 developed a highly detailed ODE model to describe the synthesis, degradation, and modification of proteins in the cell-cycle pathway. The model captured essential features of the budding yeast cell-cycle control system observed in wild-type and > 100 mutants. Later models incorporated probabilistic components to simulate the stochastic nature of cell-cycle progression4. These models, when properly parametrized, match experimental results with exquisite quantitative detail, but the models employ a large number of kinetic parameters that must be measured from experiments or estimated with care.Chen et al.5 proposed a Boolean model tracking the discrete states (0 = off or 1 = on) of eleven proteins that are major regulators in the budding yeast cell cycle. The rules for updating the Boolean variables, 11 = 2048 initial states, 1764 progress to a single state of the eleven proteins corresponding to a ‘stationary G1’ phase of the cell cycle.  Furthermore, the large basin of attraction drains quickly onto a sequence of thirteen states  corresponding to the known sequence of molecular transitions as a budding yeast cell progresses from late G1 to S phase, mitosis and cell division into stationary (early) G1 phase. Li et al. interpreted their model as evidence that the yeast cell-cycle network, from a qualitative point-of-view, is ‘robustly designed.’ However, this conclusion is dependent on updating the Boolean model synchronously. If the model is updated asynchronously , then the thirteen-state sequence (the ‘cell-cycle highway’) is no longer a robust attractor of the dynamics, as shown in Supplementary Text At the other extreme, Li et al.entclass1pt{minimaentclass1pt{minima9, fission yeast11, and mammalian cells15 (Supplementary Table 12 to ‘hybrid’ models (Boolean + ODEs)14 to ‘stochastic’ models (random asynchronous updating)15 to models with subcellular compartments9 and a ‘comprehensive’ model tracking 357 mRNAs and proteins participating 1238 elemental states 8. Our intention is not to improve on these studies but to consider the consequences of random asynchronous updating on a Boolean model of the budding yeast cell cycle. We update the Boolean variables asynchronously and stochastically in continuous time, using the ‘Boolean Kinetic Monte-Carlo’ (BKMC) approach developed by Stoll et al.15 As an example of their approach, Stoll et al. applied BKMC to the mammalian cell cycle model of Faure et al.12 but did not explore the model in any detail or compare simulations to experimental data.Since the pioneering study of Li et al., many other authors have proposed Boolean models of cell cycle progression in budding yeast32.In this paper, we apply BKMC to a modified version of the Li et al. model. First, we reduced the model to seven discrete variables by lumping together some proteins that perform analogous functions in Li’s network, in order to focus on the consequences of asynchronous updating and stochastic progression. Second, we introduced a continuous variable for cell size, because growth of yeast cells to a ‘critical size’ is known to play a crucial role in progression through the cell cycleUsing this model, we perform thousands of simulations  of a population of budding yeast cells undergoing unsynchronized growth and division. From this collection we calculate statistical properties of cell-cycle progression that can be compared directly to experimental observations. The Boolean rules are crafted to produce a repetitive sequence of transitions (a cell-cycle highway) that corresponds to the normal progression of molecular events observed in the budding yeast cell cycle. Our model includes the effects of cell growth on cell cycle progression, so we can compute the distributions of cell size at birth and division and the correlations of cell size with cell-cycle phase durations. By comparing model simulations to experimental observations, we show that the BKMC approach (random asynchronous updating of a Boolean model) can account in quantitative detail for many characteristics of cell cycle progression in wild-type budding yeast cells. Finally, we consider all single-state ‘flips’ of the Boolean variables away from the cell-cycle highway, in order to address the robustness of the model to ‘small’ perturbations of the control system. About 9% these flips induce lethal mistakes, in circumstances where lethality would be expected from the underlying molecular biology of the control system.1 phase);Cdh1 = combined activities of Cdh1 and Sic1 ;Cln2 = combined activities of Cln1 and Cln2;Clb5 = combined activities of Clb5 and Clb6;G = combined activities of Clb1 and Clb2 in prophase;Clb2M = higher activities of Clb1 and Clb2 in metaphase;Clb2Cdc20 = combined activities of Cdc20 and Cdc14 .In all eukaryotic cells, events of the cell division cycle are coordinated by cyclin-dependent protein kinases, whose activities initiate DNA synthesis (S-phase cyclins) and mitosis (M-phase cyclins). As cells exit mitosis, these cyclins are ubiquitinated and degraded by ubiquitin ligase enzymes and proteasomes, respectively. In budding yeast, the S-phase cyclins are Clb5 and Clb6, the M-phase cyclins are Clb1 and Clb2, and another pair of cyclins (Cln1 and Cln2) initiate bud emergence. Our model employs seven Boolean variables to track the activities of these key regulators of the budding yeast cell cycle. The Boolean variables, which can be either 0  or 1 , are:G, Clb2M, Cdc20); for example, a cell in early G1 phase would be in state (1000000). The Boolean variables are updated asynchronously, i.e., only a single variable changes its activity (from 0 to 1 or 1 to 0) in each time step.The ‘state’ of the dynamical system is given by a septuplet of Boolean variables: , ‘geometric size’ is presumably a proxy for some still-unknown ‘physiological size.’ Despite lingering uncertainties about how yeast cells measure their ‘size,’ we introduce a continuous variable, called ‘Size(t),’ which increases exponentially in time with a ‘mass doubling time’ of approx. 100 min (appropriate for well-nourished yeast cells in a laboratory culture) and is apportioned unequally between mother and daughter cells at division.Our model also includes a variable representing the ‘size’ of a budding yeast cell, in order to account for the considerable body of evidence for size-control of the yeast cell cycle33. The general framework is:Bj, to determine the potentially new activity of variable The possible new activity of each variable is determined by a logical functionpi is a constant .We calculate the propensity that each variable will change its activityj that satisfies r1 is a random number drawn from a uniform distribution between 0 and 1.Of all the variables with a potential to change, only one is selected with a probability proportional to its propensity calculated in step 2. Technically, letr2 is another random number drawn from a uniform distribution between 0 and 1, independently of r1.The time interval at which the activity change chosen in step 3 occurs is calculated asj is selected, and Once the index 34Lastly, we update Size using an exponential growth function We update the variables in each simulation step by a Boolean version of Gillespie’s stochastic simulation algorithmThe steps 1–6 are repeated iteratively until the total time of the simulation reaches a specified value.The seven Boolean variables regulate one another according to logical functions that are derived from known interactions of the budding yeast cell-cycle control system Fig. a, as folG, or Clb2M in our model)35. During mitotic exit, Cdh1 is dephosphorylated and activated by Cdc14 (which is included as part of the ‘Cdc20’ variable in our model)36. We assume that Cdc20  activates Cdh1 even in the presence of Clb2G. These logical relations are implemented by Eq.  and then with a probability proportional to (Size − S0)2. The critical size for a cell is assigned randomly at birth from a lognormal distribution with meanlog = log(S0_mean) and sdlog = S0_CV, where S0_mean and S0_CV are (approximately) the mean and coefficient of variation of size at birth. Once turned on, SBF maintains its own activity until it is inactivated by Clb2 (Clb2G or Clb2M)37. These logical relations are implemented by Eq.  is activated when the cell grows to a sufficiently large sized by Eq. :8\docume40. Cln2 is an unstable protein, so its activity drops rapidly after SBF is turned off. On the other hand, Clb5 maintains its production via a positive feedback loop39, and Clb5 degradation is promoted by Cdh1 or Cdc2041. These relations are implemented by Eqs. (SBF promotes the production of Cln2 and Clb5 by Eqs. :9\documeG and Clb2M)42. Clb2 also promotes its own production by activating its transcription factor Mcm143, which is not included in the model. A high level of Clb2 (Clb2M) can be degraded by both Cdh1 and Cdc2044, but a residual level of Clb2 (Clb2G) can only be degraded by Cdh144. In addition, we assume that Cln2 suppresses the activation of Clb2M for reasons to be explained shortly. These relations are implemented in Eqs. (Clb5 enhances the production of Clb2 (Clb2 in Eqs.  and 12)G and Clb45. In our model, we assume that Cdc20 is activated by the high level of Clb2 (Clb2M). Once Cdc20 is activated, its activity can be maintained by the activity of a residual level of Clb2 (Clb2G), as provided in Eq. . Therefore, instead of using Eq.  distribution with meanlog = log(fmean) and sdlog = fCV. The remaining fraction (1−f) belongs to the daughter cell. The rationale behind this rule is explained on p. 375 of Chen et al.2When Clb2In addition, we introduce three extra rules:−1, because we assume that the Boolean switches are fast and we do not want to bias one switch over another during asynchronous updating. The exception, pCln2, was set much faster (10 min−1) because Cln1/2 are very unstable proteins that quickly switch on and off as SBF switches on and off. The parameters tM_mean and tM_CV were chosen to match the mean and CV of the budded period Tbud. The parameters S0_mean, S0_CV, fmean and fCV were chosen to match experimental data on TG1 and its correlation to cell size at birth.All simulations are carried out using the parameter values in Table 7 = 128 possible states eventually approach a sequence of 14 recurring states (the cell-cycle ‘highway’), which is the global attractor of the network , SBF is activated with a probability proportional to (Size − S0)2 and the state transitions to (1100000). The rise of SBF promotes the production of Cln2 (1110000), which subsequently turns off Cdh1 (state = 0110000), allowing SBF to promote Clb5 (0111000). Next, Clb5 initiates a two-step activation of Clb2. First, Clb5 activates Clb2G (0111100). Then Clb2G turns off SBF (0011100), and subsequently Cln2 turns off (0001100). Clb2G then activates Clb2M (0001110). It is crucially important that Clb2G turns off Cln2 before it turns on Clb2M; otherwise, the sequence of cell cycle events during asynchronous updating can go awry  is synthesized in response to SBF activity, but (unlike Cln1 and Cln2) Swe1 remains active during G2 phase until a bud is properly formed. Hence, Clb2M cannot be activated until ‘Cln2’  is inactivated by Clb2G.The order of states on the global attractor corresponds to the known sequence of molecular transitions in the budding yeast cell cycle Table . A newboM activates Cdc20 (0001111), which in turn inactivates Clb5 (0000111) and Clb2M (0000101). The inactivation of Clb2M allows Cdh1 to be re-activated (1000101) and to shut down Clb2G (1000001). Finally, Cdc20 spontaneously inactivates, and the cell returns to G1 (1000000).The inactivation of Clb2 occurs in two steps during exit from mitosis. First, Clb2G + Clb2M), and Cdc20 when the simulation tracks a mother cell, starting in G1 phase (1000000) with a cell size of 0.65, through four cycles over the course of ~ 300 min. The cell-cycle transitions follow a precise order of 14 protein states as discussed above and shown in Table 34, in our simulations the ‘Size’ of a mother cell can get smaller from one division to the next, as a consequence of our rule  with initial Size = 0.65 (the average size of mother cells after division). We then averaged the value of each variable over all simulated cells to represent activity measurements from a population of cells  newborn cells. Cell synchrony was assessed by plotting the % budded cells as a function of time. The population of daughter cells  loses synchrony rapidly. The cohorts of mother cells with one-, two- or three birth scars show synchronous waves of budding, as observed34.Supplementary Fig. Tc = time from birth to division).the cell cycle period (TG1 from mitotic exit (state 1000001) to SBF activation (state 1100000), which corresponds experimentally to the interval between cell separation and bud emergence17.the period Tbud from bud emergence (state 1100000) until the end of the cycle (1000001).the period cell size at birth.Next, we simulated an asynchronous population of cells by following lineages of mother and daughter cells over many generations and collecting random samples of cells from these simulations. From these samples, we calculated several statistics and compared them to experimental observations. The data include, for both mother and daughter cells,17, except CV of the G1 period (TG1). Clearly, the model overestimates the variability in progression through G1, despite the fact that size control in the Boolean model is implemented analogous to size control in our stochastic ODE model50, which provides a good fit to the mean and CV of TG1 in both mother and daughter cells . (We haven’t yet tracked down the cause of this discrepancy.) Finally, in Fig. TG1 for mother cells and daughter cells and compare them to experiments in Di Talia et al.17 In Fig. 1 would correspond to a correlation coefficient = − 1 for µTG1 in dependence on ln(Size-at-birth). Using the same ‘binning’ method proposed by Di Talia et al., we compute correlation coefficients of − 0.33 for mother cells and − 0.79 (− 0.33) for small (large) daughter cells, which are in good agreement with experimental slopes of − 0.1, − 0.7 and − 0.3, respectively17. For a thorough statistical analysis of the experimental data, see Supplementary Text 17.As Fig. 51. To model these effects, we simulated yeast cells growing with mass-doubling time of 150 min . From many such simulations, we computed statistics for mother and daughter cells , Lord & Wheals51 observed average cycle times of 118 min and 188 min for mother and daughter cells, respectively. For cells growing in raffinose (mdt = 155 min), Ball et al.52 observed a lesser spread between cycle times of mother (132 min) and daughter (160 min) cells.When budding yeast cells are grown on a poor carbon source , they are generally smaller than cells grown on glucose, and they exhibit a longer interdivision time, primarily because of a lengthened unbudded intervalG has just turned on) by flipping Cdc20 (component 7) from 0 to 1. We then follow all possible sequences of states until the system reaches the stable sink G1 = 1000000, and we check to see if the perturbed sequence of events is ‘normal’ or ‘aberrant’. In this case, in 77% of the sequences, the simulated cell divides before activating Clb2M, i.e., before aligning replicated chromosomes on the mitotic spindle, which would produce aneuploid (inviable) progeny.We analyzed all non-trivial, single-state perturbations of the Boolean variables during normal progression through the 14 states of the cell-cycle highway. Each of the 14 states has seven components, one of which was updated from the previous state and another one is to be updated in the next state; so five of the components are subject to perturbation away from the highway, by flipping from 0 to 1 or vice versa. Hence, there are 14 × 5 = 70 single-state perturbations to consider. For example, suppose we perturb State 6 = 0111100 (Clb2G turns on),DNA replication (either Clb5 or Clb2G turns on),Prophase entry (Clb2M turns on),Proper spindle fiber attachment (Clb2on), and.Sister chromatid segregation (Cdc20 turns G turns off).Mitotic Exit  before mitotic entry. Aberrant progressions are classified according to the problem encountered.Supplementary Table G or Clb2M is pre-maturely activated during two early states (1000000 and 1100000). In these cases, either Clb2 is immediately inactivated by Cdh1 or Cdh1 is inactivated by Clb2. In the former case , the cells resume normal cell-cycle progression. In the latter case (67%), however, the cells proceed to activate Cdc20 and exit mitosis without budding (Cln2 and Clb5 never turn on or they turn on after cells enter prophase).First, Clb2M. These cells exit mitosis without proper metaphase alignment of chromosomes and produce progeny with DNA copy-number abnormalities.Second, Cdc20 is activated pre-maturely during four states corresponding to S phase and prophase . These simulations resemble a defective anaphase checkpoint . From 20,000 simulations of those four states, 52% proceed to mitotic exit without activating Clb2Third, Cdh1 is activated prematurely during four states corresponding to prophase to metaphase . In 41% of 20,000 simulations, the cells divide without activating Cdc20 (no segregation of replicated chromosomes) and produce progeny with DNA copy-number abnormalities.53 is their ability to simulate accurately the phenotypes of hundreds of mutant strains. Of course, simple Boolean models, like Li et al.’s and ours, cannot possibly compete because they do not account for all the gene products that are perturbed in the mutant strains, nor for the redundancies among the regulatory proteins. For example, knocking out any component (setting Bi = 0) is lethal in these models but not in reality.A hallmark of detailed ODE simulations of the budding yeast cell cyclecln3 mutant cells, which are about 70% larger than wild-type cells54, by increasing S0_mean, the minimum size requirement for executing Start. The results are displayed in Supplementary Fig. We can easily simulate Δ55.S-M strain : viable; bud emergence normal, DNA synthesis delayed.Clns-Clb2MET3prCln2-Clb2G1-S-M strain : inviable in methionine, no bud emergence, DNA replication and mitosis in mother cell, no cell division.A more challenging test is to simulate two interesting mutant strains recently studied by Pirincci Ercan et al.To simulate these mutants with our BKMC model, we made the following changes demanded by the mutant genotypes:S-M strain: Clb5* = 0, Clb2G* = Clb2M | ( ( Clb5 | Clb2G ) & ~ Cdh1 ) | SBF.Clns-Clb2MET3prCln2-Clb2G1-S-M strain in methionine: additional changes Cln2* = 0, S0_mean = 0.8 (100% increase) because Δcln3.G is turned on. In the majority of trajectories (431/500), Cdh1 turns off (1 → 0 in the first digit) with Clb2G on (1 in the fifth digit); consequently, Clb2M turns on, then Cdc20 turns on, and the cell exits mitosis and returns to G1. So, the majority of cells go through a viable sequence of cell-cycle states. DNA synthesis may be delayed if Clb2 is not as effective as Clb5 in initiating S phase. In a minority of trajectories (69/500), Cdh1 stays on (1 in the first digit) and turns Clb2G off, so the cell returns to G1 (1000000) and tries again. In either case, the cells are viable, with some delay in S phase. In the second case , there are two options: (1) Cdh1 kills Clb2G (135/500) and the cell returns to G1 and tries again, or (2) Clb2G kills Cdh1 (365/500) and activates Clb2M, followed by Cdc20 activation and exit from mitosis, a more-or-less normal sequence of cell cycle events. However, because the cell never made a bud, it cannot divide, which is the observed phenotype.In each case we simulated 500 cells starting at state (1000000) until they returned there by a variety of trajectories Fig. . In the ctories 41/500, Cd15. Inspired by Gillespie’s stochastic simulation algorithm33, the Boolean state (a binary seven-digit number) and the simulation time  are updated stochastically. The logical rules governing updates of the Boolean variables reveal that all 128 possible initial states are attracted to a 14-state ‘cell-cycle highway’ that corresponds to normal progression through major cell cycle phases (late G1 → S → G2 → M → CD → early G1). Adding a simple rule for a growth-controlled Start transition , we are able to simulate the proliferation of mother and daughter cells in an exponentially expanding population of budding yeast cells. From these simulated populations we computed several statistical properties of cell cycle progression, and they are in good agreement with experimental observations  approachBi to ~ Bi), we find that 91% of the perturbations return to the normal cell-cycle sequence of events without difficulties, but 9% induce abnormal sequences, such as aberrant partitioning of chromosomes at mitosis, or division without bud formation  activation of exit proteins  in S or G2 phase, before the mitotic spindle can be assembled; and (3) activation of Cdh1 and Sic1 (cell division) in prophase or metaphase, before Cdc20 can induce anaphase separation of replicated chromosomes. All these perturbations can be expected to have lethal consequences.We can also use the model to explore the ‘robustness’ of cell-cycle progression with respect to perturbations away from the cell-cycle highway. Because the highway is globally attracting Fig. b, any peon Table . The abn50), whose roles are necessary to understand the phenotypic properties of mutant strains of budding yeast. Success in this endeavor will require a careful consideration of the logical rules describing protein interactions and a disciplined effort to optimize parameters over a range of experimental data types. It will be interesting to see if a more complete BKMC model of the budding yeast cell cycle can be as successful as comprehensive nonlinear ODE models in giving a quantitative account of mutant phenotypes. If so, then BKMC modeling may prove competitive with ODE modeling, which is more computationally demanding in terms of simulation time and parameter estimation.Despite the fact that the model, with only seven Boolean variables + cell size (a continuous variable), is greatly oversimplified, it conforms in quantitative detail to many experimental observations of the growth and division of wild-type cells. The next logical step would be to extend the model by separating the ‘lumped’ variables  and to include other crucial regulatory proteins (e.g., Whi5, Mcm1, Swi5, Cdc5; see Fig. 8 in Laomettachit et al.Supplementary Information.